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Università degli Studi di Salerno

FACOLTÀ DI INGEGNERIA CORSO DI LAUREA IN INGEGNERIA CIVILE

Tesi di Laurea

in

TECNICA DELLE COSTRUZIONI II

“A PPLICATION OF SOME CLASSIC

CONSTITUTIVE THEORIES TO THE

NUMERICAL SIMULATION OF THE

BEHAVIOR OF PLAIN CONCRETE ”

RELATORE Ch.mo P rof. Ing. Ciro Faella CORRELATORI CANDIDATO

Ch.mo P rof. Ing. Antonio Caggiano Guillermo Etse Matr.: 06201000025 Dott. Ing. Enzo Martinelli Ing. Paula Folino

ANNO ACCADEMICO 2007/2008

Abstract

In past years, the methods of analysis and design for concrete structures

were mainly based on elasticity combined with various classical procedures

as well as on empirical formule developed on the basis of a large amount of

experimental data. Such approaches are still necessary and desirable and

continue to be the most convenient and effective methods for ordinary

design.

However, the rapid development of modern numerical analysis techniques

and high-speed digital computers has provided structural engineers with

powerful tools for complete nonlinear analysis of concrete structures.

Indeed, stress and strain response of concrete structures can be efficiently

reproduced by using the finite-element method and performing an

incremental inelastic analysis. The increasing use of fully three-

dimensional finite-element analysis in reinforced and prestressed concrete

structures motivates to development of sophisticated constitutive

formulations when the structural response is to be predicted beyond the

linear elastic limit.

The present Thesis deals with the description and validation of a concrete

material model based on non-associated plasticity models which can be

used for an easy and robust numerical implementation. All the analysis are

performed by means of the “Constitutive Driver Interactive Graphics”

program (namely Co.Dri.) and by Concrete Damage-Plasticity constitutive

model implemented in Abaqus (general-purpose nonlinear finite element

analysis program).

A comparison of the numerical predictions with experimental tests

available within the scientific literature is also presented. Actual limits and

further developments of the proposed models are finally outlined in this

job.

CONTENTS

I

1 INTRODUCTION…………………………………...….….….1

1.1 COSTITUENTS OF CONCRETE MATERIAL………….…….…….1

1.1.1 PORTLAND CEMENT……………………………………..….….…..3

1.1.2 AGGREGATES……………………………………………..……......3

1.1.3 WATER……………………………………………………..….…....4

1.2 BASIC FEAUTERES OF CONCRETE BEHAVIOR……………...4

1.2.1 NONLINEAR STRESS-STRAIN BEHAVIOR…………………………....5

1.2.2 DIFFERENT RESPONSES IN TENSION AND COMPRESSION…………..6

1.2.3 MULTIAXIAL COMPRESSIVE LOADING……………………………..7

1.2.4 VOLUME EXPANSION UNDER COMPRESSIVE LOADING……………..9

1.2.5 STRAIN SOFTENING……………………………………………….11

1.2.6 STIFFNESS DEGRADATION………………………………………...13

1.3 CONSTITUTIVE MODELING OF CONCRETE MATERIALS…...14

1.3.1 EMPIRICAL MODELS………………………………………………14

1.3.2 LINEAR ELASTIC MODEL………………………………………….15

1.3.3 NONLINEAR ELASTIC MODEL……………………………………..15

1.3.4 PLASTICITY BASED MODEL………………………………………..17

1.3.5 STRAIN SOFTENING AND STRAIN SPACE PLASTICITY……………..18

1.3.6 FRACTURING AND CONTINUUM DAMAGE MODELS……………….19

1.3.7 MESOMECHANIC ANALYSIS OF CONCRETE BEHAVIOR…………...20

1.3.8 MICROPLANE MODELS……………………………………………21

CONTENTS

II

1.4 OBJECT OF THE THESIS………………………………………….21

REFERENCES OF THE FIRST CHAPTER………………………22

2 BASIC EQUATIONS…………………………….…………30

2.1 STRESS AND STRESS TENSOR……………………….…………30

2.1.1 PRINCIPAL STRESSES AND INVARIANTS OF THE STRESS TENSOR…33

2.1.2 STRESS DEVIATION TENSOR AND ITS INVARIANTS………………..34

2.1.3 HAIGH-WESTERGAARD STRESS-SPACE …………………………...36

2.2 YIELD AND FAILURE CRITERIA………………….…………....40

2.2.1 YIELD CRITERIA INDEPENDENT OF HYDROSTATIC PRESSURE……40

2.2.1.1 The Tresca Yield Criterion ……………………….…….41

2.2.1.2 The von Mises Yield Criterion…………………………..43

2.2.2 FAILURE CRITERION FOR PRESSURE-DEPENDENT MATERIALS…..45

2.2.2.1 The Mohr-Coulomb Criterion…………………………..49

2.2.2.2 The Drucker-Prager Criterion………………………….53

2.3 LINEAR ELASTIC ISOSTROPIC STRESS-STRAIN RELATION………………………………………………………...56

2.4 STRESS-STRAIN RELATION FOR WORK-HARDENING MATERIALS………………………………………………..……..59

2.4.1 PLASTIC POTENTIAL AND FLOW RULE……………………………60

2.4.2 INCREMENTAL STRESS-STRAIN RELATIONSHIP……….……….…62

2.4.3 SOFTENING BEHAVIOR…………………………….….………….66

CONTENTS

III

2.5 INTEGRATION SCHEME FOR ELASTO-PLASTIC MODELS...........................................................................................66

2.5.1 GENERAL DESCRIPTION OF A GENERAL ELASTOPLASTIC

INTEGRATION………………………………………………..……67

REFERENCES OF THE SECOND CHAPTER……...……………72

3 CO.DRI. INTERACTIVE GRAPHICS……………...74

3.1 USER OPTIONS...............................................................................74

3.2 EXPERIMENTAL DATABASE.......................................................78

3.2.1 IDENTIFICATION OF EXPERIMENTS………….................................78

3.2.2 TEST APPARATUS…………………………….................................79

3.3 CONSTITUTIVE MODELS............................................................81

3.3.1 ASSOCIATED VON MISES PLASTICITY MODEL................................81 3.3.1.1 Quadratic hardening/softening function.........................81

3.3.1.2 Simo hardening/softening function..................................82

3.3.1.3 Simo Modified hardening/softening law..........................84

3.3.1.4 Calibration and validation of the von Mises model........85

3.3.2 NON-ASSOCIATED DRUCKER-PRAGER PLASTICITY MODEL (TWO PARAMETERS)……………………………………………….........90

3.3.2.1 Calibration and validation of Drucker-Prager model....91

3.3.3 NON-ASSOCIATED DRUCKER-PRAGER PLASTICITY MODEL (THREE PARAMETERS) ...............................................................................98

3.3.3.1 Calibration and validation of Drucker-Prager model.................................................................................103

3.3.4 NON-ASSOCIATED BRESLER-PISTER PLASTICITY MODEL.............108 3.3.4.1 Calibration and validation of Bresler-Pister

model.................................................................................110

CONTENTS

IV

REFERENCES OF THE THIRD CHAPTER……...………….…116

4 CONSTITUTIVE MODELS AVAILABLE IN ABAQUS…………………………………………………….....119

4.1 YIELD AND FAILURE SURFACE……………………………...123

4.2 HARDENING/SOFTENING LAWS…………………………….130

4.2.1 COMPRESSIVE BEHAVIOR……………………………………..…130

4.2.2 TENSILE BEHAVIOR…………………………………..………….131

4.3 NONASSOCIATED FLOW LAW………………………………..133

4.4 CALIBRATION AND VALIDATION OF DAMAGE PLASTICITY

MODEL………………………………………………………..….134

REFERENCES OF THE FOURTH CHAPTER……...…………..140

5 FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES..……………………………….....142

5.1 SOME CLASSICAL FAILURE CRITERIA..……………………143

5.1.1 LEON FAILURE CRITERION……………………………………....143

5.1.2 HOEK AND BROWN FAILURE CRITERION………………………...146

5.1.3 WILLAM AND WARNKE (THREE PARAMETERS) FAILURE CRITERION……………………………………………………....149

5.1.4 WILLAM AND WARNKE (FIVE PARAMETERS) FAILURE CRITERION……………………………………………………....153

5.1.5 OTTOSEN FOUR-PARAMETER MODEL…………………………....156

5.1.6 HSIEH – TING – CHEN FOUR-PARAMETER MODEL………….…....159

CONTENTS

V

5.1.7 EXTENDED LEON MODEL (ELM) PROPOSED BY ETSE……............162

5.2 APPLICATION OF PLASTICITY BASED MODELS TO PASSIVE CONFINEMENT ……...................................................................166

5.2.1 STEEL CONFINEMENT……...........................................................169

5.2.2 FRP CONFINEMENT……................................................................178

REFERENCES OF THE FIRST CHAPTER..................................181

6 SUMMARY AND CONCLUSION...............................184

7 APPENDIX : “CODRI.F”………….................................186

INTRODUCTION

1

1. INTRODUCTION

This chapter is subdivided into three sections:

- the first section deals with the princi pal components of concrete material;

- the second section contains the principal features of the mechanical behavior of

concrete under ordinary typica l solicitations in the field of the civil engineering.

The nonlinear behavior of concrete material and its being a composite material

in nature is treated.

- the las t part of the in tro duction pres ents var ious developmen ts in the field of the

constitutive modeling of concrete o n differ e nt approaches such as elasticity,

plasticity, continuum damage mechanics, plastic fracturing, microplane models,

etc.

1.1 COSTITUENTS OF CONCRETE MATERIAL

Concrete has been the most common building material for many years and

the same trend is expected for the coming decades. Reinforced concrete

structures and infrastructures are quite common throughout the developed

world and are more and more frequent in developing countries; the greater

number of buildings for various uses and purposes are made on concrete as

well as bridges, massive dams, nuclear power plants and so on.

In pre-historic times, some form of concrete using lime-based binder may

have been used (Stanley [100]), but modern concrete using Portland cement

dates back to mid-eighteenth century, with the patent by Joseph Aspdin in

1824.

Traditionally, concrete is basically a composite natural consisting of the

dispersed phase of aggregates (ranging from its maximum size coarse

aggregates down to the fine sand particles) embedded in the matrix of

cement paste. This is a “Portland cement concrete” with the four

constituents:

INTRODUCTION

2

- Portland cement;

- water;

- stone;

- and sand.

Fig. 1.1 – The basic components of concrete: aggregates (stone and sand), Portland

cement, and water (Chen and Liew [32]).

These basic components remain in current concrete but other constituents

are now often added to modify its fresh and hardened properties. The

quality of concrete in a structure is determined not only by the proper

selection of its constituents and their proportions, but also by appropriate

techniques of production, transportation, placing, compacting, finishing,

and curing of the concrete of the actual structure, often at the job site.

The constituents of modern concrete have increased from the basic four

(Portland cement, water, stone and sand) to include both chemical and

mineral admixtur e s . These admixtures have been in use for decades, first in

special circumstances, but have now been incorporated in more and more

general applications for improving technical and performance cost-

effectiveness.

INTRODUCTION

3

1.1.1 PORTLAND CEMENT

In the past, Portland cement was restricted to that used in ordinary concrete

and is often called or dinary Portland cement . There is a general trend

towards grouping all cement types as Portland cement, included those

blended with molten iron slag or pozzolan such as fly ash (also called

pulverized fuel ash ), and silica fume into cements of different sub-classes

rather than special cements. This approach has been adopted in Europe (EN

197–1 [41]) but the American practice subdivides them into two separate

groups (American Society for Testing and Materials provides rules for both

Portland cement within ASTM C150 [3] and blended cements in ASTM

C595 [4]).

Raw materials for manufacturing Portland cement basically consist

calcareous and siliceous (generally clay-based) materials. Mixture is heated

to a high temperature (1400°-1600° C) within a rotating kiln to produce a

complex group of chemicals, collectively called “cement clinker” .

Further details about manufacturing process, the formation of these

chemicals and their reactions with water are well beyond the scopes of this

Thesis and can be found in various specific textbooks (e.g., Hewlett [53]).

1.1.2 AGGREGATES

Aggregates in concrete are usually grouped according to their size in fine

and coarse aggregates. The separation is based on materials passing or

retained on the nominally 5 mm sieve (No. 4 sieve after ASTM D2487 [5]).

Fine aggregates basically consist in sand, while coarse aggregates are

represented by small stones. Traditionally, aggregates are derived from

natural sources in the form of river gravel or crushed rocks and river sand.

INTRODUCTION

4

Fine aggregates produced by crushing rocks to sand sizes are referred as

manufactured sands. Aggregates derived from special synthetic processes

or as a by-product of other processes are also available.

In most concrete mix, volume fraction of aggregates is about twice the

volume of cement paste matrix. Hence, the physical properties of concrete

are dependent on the corresponding properties of the aggregates.

1.1.3 WATER

Water is basically needed for the hydration of cement, but not all is used for

this purpose only. Part of the water is aimed to provide workability during

mixing. This latter usage can be reduced by the introduction of chemical

admixtures, e.g., plasticisers (Chen and Liew, [32]).

Where possible, potable water is used. Other sources may contain

impurities introducing undesirable effects on properties of fresh and

hardened concrete. A good list of concrete mixtures is given in the PCA

Manual (Kosmatka & Panarese, [64]). ASTM C94 [2] and BS 3148 [19]

both provide guidance on acceptance criteria for water of questionable

quality in terms of expected concrete strength and setting time.

Seawater should not be used as mixing water for reinforced concrete due to

the presence of chloride and its effect on corrosion of steel reinforcement

(Chen and Liew, [32]).

1.2 BASIC FEAUTERES OF CONCRETE BEHAVIOR

Mechanical behavior of concrete is very complex, being largely determined

by the structure of the component related issues, such as water-to-cement

ratio, cement-to-aggregate ratio, shape and size of aggregates, the kind of

cement used, and so on. The present dissertation deals with stress-strain

behavior of an average ordinary concrete. The physic-chemical structure of

INTRODUCTION

5

the material is ignored and the rules of material behavior are developed on

the basis of continuum mechanics. Under this standpoint the material is

basically assumed homogeneous and isotropic.

Concrete is a brittle material; its stress-strain behavior is affected by micro-

and macro-cracks developing within the material body during the loading

process. Furthermore, concrete is affected by a large number of micro-

cracks, especially at interfaces between aggregates and mortar, even before

the application of external load. These initial microcracks are caused by

segregation, contraction, or thermal expansion in the cement paste (Chen &

Han, [31]). Under applied loading, further development of micro-cracks

may occur at the aggregate-cement interfaces, which is the weakest link in

the composite system. The progression of these cracks, which are initially

invisible, become visible when the cracks occur with the application of

external loads and contribute to the overall nonlinear stress-strain behavior.

1.2.1 NONLINEAR STRESS-STRAIN BEHAVIOR

A typical stress-strain curve for concrete in uniaxial compression tests is

shown in Fig.1.2 and three fundamental deformation stages can be

observed even in this simple test (Kotsovos & Newman, [65]):

- the first stage corresponds to a stress in the region up to 30% of the

maximum compressive stress f ’ c. At this stage, cracks initially

existing in concrete remain nearly unchanged. Hence, the stress-

strain behavior is assumed linearly elastic. Therefore, 0.3f ’ c is

usually proposed as the limit for elastic constant range of concrete;

- beyond this limit, the stress-stain curve begins diverting from the

original straight line. Stress between 30% and about 75% of f ’ c

characterizes the second stage, in which bond cracks start to increase

INTRODUCTION

6

in length, width, and number; material comes out as micro-cracks

develop within nonlinearity;

- after further load increases, in the third stage, the progressive failure

of concrete is classically caused by cracks through the mortar (Chen

& Han, [31]). These cracks form a crack zone or “internal damage”;

at this load-like deformations may be localized in the damage zone

and the nonlinear behavior is very pronounced. Finally, the load

reaches the value of peak, loading the concrete specimen to failure.

Fig. 1.2 – Typical uniaxial compressive stress-strain curve (Domingo Sfer et al, [94]).

Although the above discussion deals only with the uniaxial compression

case in pre-peak region (tests in force-control), three deformation stages

can also be qualitatively identified in the same loading cases, the linear

elastic stage, the inelastic stage, and the so-called “localized stage”.

1.2.2 DIFFERENT RESPONSES IN TENSION AND COMPRESSION

Figure 1.3 shows a typical uniaxial tension stress-strain curve. In general

INTRODUCTION

7

the limit of elasticity is observed to be about 60 to 80% of the ultimate

tensile strength. Beyond this level, bond microcracks start growing.

As the uniaxial compressive state tends to arrest the preexisting cracks in

the concrete material, the tension state of stress tends rather to promote the

opening of the same cracks. This is one of the reasons why the behavior of

concrete in tension is quite brittle in nature. In addition, the aggregate-

mortar interface has a significantly lower tensile strength than mortar. This

is the primary reason for the lower tensile strength of concrete materials.

Fig. 1.3 – Uniaxial tensile stress-strain curve (Hurlbut, [55]).

1.2.3 MULTIAXIAL COMPRESSIVE LOADING

A typical stress-strain behavior for concrete under multiaxial loading condi-

tions is shown in Fig. 1.4 (Hurlbut, [55]).

The results are obtained from tests on cylindrical specimens. Concrete

cylinders are submitted to constant lateral pressures, σ2 = σ3. The axial

load, is imposed in terms of strain ε1. Figure 1.4 shows the relationship

between axial stress σ1 and both axial and transverse strains, εz and εr

INTRODUCTION

8

respectively, for various values of confining pressure σ2 = σ3.

The confining pressure significantly affects the deformation behavior of the

specimen. At first, the axial strain at failure (peak value) increases with

confining pressure. However, compared to the uniaxial compression case,

larger strains develop in confined concrete specimens.

Fig. 1.4 – Stress-strain curves under multiaxial compression (Hurlbut, [55]).

Softening behavior can be observed for specimen in unconfined

compression or under low levels of lateral confinement. When lateral

confinement attains a critical value the so-called “softening zone”

disappears and the stress-strain relation is increasing up to the ultimate

strain.

Furthermore, the maximum value of stress increases as confining pressure

is applied. Figure 1.4 shows that the uniaxial strength for the unconfined

specimen is about 19 MPa, but it hugely increases as lateral confinement is

applied. Consequently, concrete and various geotechnical materials are

classified as “pressure-dependent materials”.

INTRODUCTION

9

1.2.4 VOLUME EXPANSION UNDER COMPRESSIVE LOADING

The volumetric strain (namely, the trace of strain tensorzyxv

εεεε ++= )

plotted against the uniaxial compression stress is shown in Fig.1.5

(Domingo Sfer et al., [94]). When concrete specimen is subjected to

increasing uniaxial compression, its-apparent-Poisson’s ratio start to

continuously and significantly increase beyond its well established elastic

value, as plotted in figure 1.7 (Domingo Sfer et al., [94]). On attaining a

certain stress level, called critical stress (0.75 to 0.90 of the ultimate

uniaxial compressive stress), the volume of the concrete starts to increase

rather than continuing to decrease. This inelastic behavior is due to the

composite nature of concrete.

Fig. 1.5 – Volumetric strain vs. stress, under uniaxial compression (Domingo Sfer et al.,

[94]).

Indeed, experimental tests, performed by Shah and Chandra [95], point out

that cement paste itself does not expand under compression loads.

Hardened paste specimens continue to consolidate at an increasing rate

with increased load (figure 1.6).

INTRODUCTION

10

Fig. 1.6 – Mechanical proprieties of Portland cement pastes with water-cement ratios

equal 0.40, 0.47, and o.54 (Shah and Chandra, [95]).

Shah and Chandra [14] observed that increasing the volume fraction of

aggregates significantly reduces the percentage values of that critical stress.

Similarly, increasing the size of aggregate particles or reducing the strength

of bond between aggregate and paste makes concrete more inelastic.

Fig. 1.7 – Poisson’s ratio vs. stress, under uniaxial compression (D. Sfer et al., [94]).

INTRODUCTION

11

Hence, volumetric expansion is observed only when the cement paste is

mixed up with aggregates, consequently the composite nature of concrete

is primarily responsible for the volume dilatation.

1.2.5 STRAIN SOFTENING

Engineering materials like concrete, as well as other natural elements on

concern for engineering purposes like rocks, and soils exhibit a significant

strain-softening behavior beyond the peak stress. Figure 1.8 shows typical

uniaxial compressive stress-strain curves obtained from strain-controlled

tests. Each of these curves has a sharp descending branch beyond the peak

of failure stress.

Fig. 1.8 – Uniaxial compressive stress-strain curve for concrete (Wischers, [107]).

INTRODUCTION

12

It is generally agreed that softening branch of a stress-strain curve does not

reflect a material property, but rather represents the response of the

structure formed by the specimen together with its complete loading

system (van Mier, [102]). This argument is supported by compression tests

of specimens of different heights. The test results in terms of stress and

strain are shown in Fig. 1.9 where the descending branches of the stress-

strain curves are not identical but have slopes decreasing with increasing

specimen heights (van Mier, [102]). On the other hand, however, if the

post-peak displacement rather than strain is plotted against stress, the

stress-displacement curves are almost identical, regardless of the specimen

heights.

Fig. 1.9 - Influence of specimen height on uniaxial stress-strain curve (van Mier, [102]).

This phenomenon can be explained as follows. Since the post-peak strain is

localized in a small region of the specimens (van Mier, [102]). When we

calculate the strains for each specimen, we are using different heights to

divide the same value of displacement (van Mier’s experimental tests

INTRODUCTION

13

[102]). This will result in different strain values. These strain values are not

real in every point of continuous body, but represent some average strains

along the heights of the specimens.

Consequently, as the post-peak deformation is localized, the descending

branch of the stress-strain curve cannot be considered as a material

property.

1.2.6 STIFFNESS DEGRADATION

Figure 1.10 shows a typical uniaxial compressive stress-strain curve of

concrete under cyclic loading. As can be seen, the unloading-reloading

curves are not straight-line segments, but loops of changing size with

decreasing average slopes.

Fig. 1.10 - Cyclic uniaxial compressive stress-strain curve (Sinha et al., [98])

Assuming that average slope is the slope of a straight line connecting the

two turning points of one cycle and that the material behavior upon

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14

unloading and reloading is linearly elastic (outlined line in Fig. 1.10), then

the elastic modulus (or the slope) degrades with increasing straining. This

stiffness degradation behavior is somehow related to damage , which is

significant throughout the post-peak range (Sinha et al., [98]).

1.3 CONSTITUTIVE MODELING OF CONCRETE MATERIALS

The intensive investigations carried out in recent years have led to a better

understanding of the constitutive behavior of concrete under various

loading conditions. Many theories proposed in literature for the prediction

of the concrete behavior such as empirical models, linear elastic, nonlinear

elastic, plasticity based models, models based on endochronic theory of

inelasticity, fracturing models, continuum damage mechanics models,

micromechanics models, etc., are discussed in the following sections.

1.3.1 EMPIRICAL MODELS

Models in which the material constitutive law is, derived through a series

of experimental observations, are called empirical model . The experimental

data is then used to propose functions describing the material behavior by

curve fitting. Many empirical uniaxial and biaxial stress-strain relations are

available in the literature. Stress-strain relations specific for ascending

branch and for different kind of loading are available in the literature:

- compression stress case: Desayi and Krishan [39], Saenz [38], Smith

and Young [99], the European Concrete Committee (CEB) [41],

Attard and Setunge [6], Richard and Abbott [89], Popovics [7], etc.;

- stress-strain relations for reinforced concrete in tension: e.g.,

Carreira and Chu, [23];

- confined concrete: e.g., Mander et al. [78], Attard & Setange [6],

etc.;

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15

- biaxial stress-strain relation: Gerstle [49], Chen [30], etc.;

- triaxial stress conditions: Chen [30], Shina et al. [98], etc. .

1.3.2 LINEAR ELASTIC MODEL

In linear elastic models concrete is treated as linear elastic until it reaches

ultimate strength and subsequently it fails in a brittle way. For concrete in

tension, since the failure strength is small, linear elastic model is quite

accurate and sufficient to predict the behaviour of concrete up to failure

(Babu et al., [7]). Linear elastic stress-strain relation using index notation

can be written as:

klijklijC εσ = (1.4)

where C ijkl represents material stiffness.

Since concrete falls under the category of pressure-sensitive material

whose general response under imposed load is highly nonlinear and

inelastic, this simple linear elastic constitutive law is often inappropriate

when the concrete is subject to external load characterized by elevated

confinements.

1.3.3 NONLINEAR ELASTIC MODEL

Concrete under multiaxial compressive stress states exhibit significant

nonlinearity and linear elastic models fail in these situations. Significant

improvements can be made in this situation using nonlinear constitutive

models. There are two basic approaches followed for nonlinear modelling

namely secant form ulation (total stress-strain) and tangential formulation

(incremental stress-strain), (Babu et al., [7]).

Incremental stress-strain relation using index notation can be written in the

following form (Gerstle, 1981 [49]):

( )klkl

t

ijklijdCd εεσ = (1.5)

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16

where C ijkl

t is the tangent material stiffness.

The secant formulation is a simple extension of linear elastic models

formulated by assuming functional relations as in the following form:

( )klijij

F εσ = (1.6)

The elastic material defined by Eq. (1.6) is termed Cauchy elastic material

(Chen and Han, [31])

Secant formulations are load-path independent. It generally comes

approved in literature what the mechanical behavior of bodies that suffer

irreversible (plastic) strains is function of their load history. Concrete

material falls in this circumstance. For this reason which the secant

formulation is applicable primarily for monotonic or proportional loading

situations.

In the (linear and nonlinear) elasticity based models, a suitable failure

criterion is incorporated for a complete description of the ultimate strength

surface. Failure can be defined as the ultimate load capacity of concrete

and represents the boundary of the work-hardening region. Many failure

criteria are available in the literature for normal, high strength, light weight

and steel fibre concrete (Mohr-Coulomb criterion, [31]; Drucker-Prager,

[31]; Chen and Chen, [27]; Ottosen, [83]; Hsieh-Ting-Chen, [54]; Willam

and Warnke, [106], Menetrey and Willam, [79]; Sankarasubrsmanian and

Rajasekaran, [91], Fan and Wang, [44], etc. The most commonly used

failure criteria are defined in stress space by a number of constants varying

from one to five independent control parameters (Babu et al., [7]).

Accumulate plastic (irreversible) deformations occur in a general concrete

body when certain level of external load-actions are reached. Elastic based

analysis doesn't contemplate, for genesis, the generation of plastic

deformations. If we remove the external load, using these models my body

returns in the original configuration. For this reason that, the elastic models

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17

result to be inadequate to the mathematical modeling of the mechanical

behavior of concrete.

1.3.4 PLASTICITY-BASED MODEL

The classical theory of plasticity was originally developed for metals. The

deformational mechanisms of metals are quite different from those of

concrete, however, from a macroscopic point of view, they have some

similarities, particularly before failure (Chen and Han, [31]). For example,

concrete exhibits a nonlinear stress-strain behavior during loading and has

a significant irreversible strain upon unloading. Especially under

compressive loadings with confining pressure, concrete may show some

ductile behavior. The irreversible deformations of concrete are induced by

microcracking and may be treated by the theory of plasticity (Chen and

Han, [31]).

Any plasticity model must involve three basic assumptions:

(i) an initial yielding surface, within the stress space, defining the

stress level at which plastic deformation begins;

(ii) a hardening rule defining the yielding surface evolution after

beginning of plastic deformations;

(iii) a flow rule, which is related to a plastic potential function, gives an

incremental plastic stress-strain relation.

In plasticity theory the total strain increment tensor is assumed to be the

sum of the elastic and plastic strain increment tensors: p

ij

e

ijijddd εεε += (1.7)

The relationship between incremental stress and incremental strain can be

formulated as in the following form:

kl

ep

ijklij dCd εσ = (2.105)

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18

The coefficient tensor in parentheses represent the elastic-plastic stif fness

tensor in t e rms of tangent moduli . The formulation to research the previous

tensor is treated in detail in various texts regarding the plasticity theory

(e.g., Chan and Han [31], Lubliner [77], Desay and Siriwardane [37]).

There are many researchers who have used plasticity alone to characterize

the concrete behavior (e.g. Chen and Chen [27]; Willam and Warnke [106];

Bazant [12]; Dragon and Mroz [40]; Kotsovos [66]; Ottosen [84]; Hsieh,

Ting, and Chen [54]; Fardis, Alibe, and Tassoulas [45]; Schreyer [93]; Yang

, Dafalias, and Herrmann [109]; Vermeer and de Borst [103]; Chen and

Buyukozturk [28]; Schreyer and Babcock [92]; Han and Chen [52]; Onate

et al. [82]; de Boer and Desenkamp [36]; Lubliner, Oliver et al [76];

Pramono and Willam [87]; Faruque and Chang [46]; Abu-Lebdeh and

Voyiadjis [1]; Karabinis and Kiousis [62]; Este and Willam [42]; Menetrey

and Willam [79]; Feenstra and de Borst [47]; Balan, Filippou, and Popov

[8]; Jiang and Wang [58]; Li and Ansari [73]; Grassl et al. [51]). The main

characteristic of these models is a plasticity yield surface that includes

pressure sensitivity, load-path sensitivity, non-associative flow rule, and

hardening/softening work.

1.3.5 STRAIN SOFTENING AND STRAIN SPACE PLASTICITY.

The stress-strain response after peak (strain softening) depend on many

factors like test equipment, test procedure, sample dimensions and stiffness

of the machine, etc. (Lubliner [77]).

Classical plasticity theories are developed in stress space where stress and

its increments are treated as independent variables. Even though stress

space formulation is commonly accepted in engineering practice this

approach has some inherent disadvantages (Babu et al. [7]):

(i) for strain softening materials, there is no clarity in defining the

INTRODUCTION

19

criteria of loading-unloading.;

(ii) for many structural materials, the slope of the uniaxial stress-strain

curve becomes zero at the ultimate strength point (peak) where

the stress space formulation may not offer reliable results.

These disadvantages of stress space formulation can be eliminated with the

help of strain space formulation. The basic formulation of strain space

plasticity have been discussed in the literature (e.g., Chen and Han [31];

Il’Yushin [56]; Naghdi and Trapp [81]; Casey and Naghdi [24]; Pekau et al.

[86]; Kiousis [63]; Mizono and Hatanaka [80]; Barbagelata [9]; Stevens

[101]; Iwan and Yoder [57]; Dafalias [35]); Runesson et al. [90]; and Lee

[71]; etc.).

1.3.6 FRACTURING AND CONTINUUM DAMAGE MODELS

These models are based on the concept of propagation of microcracks,

which are present in the concrete even before the application of the load.

Damage based models are often used to describe the mechanical behavior

of concrete in tension.

In the earlier class of models, plastic deformation is defined by usual flow

theory of plasticity and the stiffness degradation is modelled by fracturing

theory. A second class of models is based on the use of a set of state

variables quantifying the internal damage resulting from a certain loading

history (Babu et al. [7]). The fundamental assumption in these models is

that the local damage in the material can be represented in the form of

internal damage variables. Then, the tangential stiffness tensor of the

material is directly related to the internal damage.

The models of this category can describe progressive damage of concrete

occurring at the microscopic level, through variables defined at the level of

the macroscopic stress-strain relationship. In 1980s, it was established that

INTRODUCTION

20

damage mechanics could model accurately the strain-softening response of

concrete (Krajcinovic [67] & [68], Lemaitre [69] & [70], Chaboche [25] &

[26]).

Various damage models such as elastic damage, plastic damage are

available in the literature (e.g., Ju [59], Lee and Fenves [72]) or damage

models which use the endochronic theory with continuum damage

mechanics (Voyiadjis [104] & [105]), Wu and Komarakulnanakorn [108]).

1.3.7 MESOMECHANIC ANALYSIS OF CONCRETE BEHAVIOR

Heterogeneous materials like concrete require different levels of

observations to fully understand the mechanism governing their response

behaviors when they are subjected to complex loading cases that activate

non-linear responses. This is particularly when traditional macroscopic

models, based on continuous concept, need observations at meso and,

moreover, micro levels to accurately evaluate and distinguish the rate

sensitivity of the different constituents as well as their influences in the

overall behavior.

Several authors have already recognized the importance of mesostructure

evaluations proposing various mesostructural models for concrete (Granger

et al., [50], Lopez et al., [74], Zhu and Tang, [111], Ciancio et al. [33], Etse

et al. [43], Lorefice et al. [75], Caballero et al. [20], etc.).

Three main features characterize these models:

- it includes a non-regular array of particles representing the largest

aggregates;

- a homogeneous matrix modelling the behavior of mortar plus small

aggregates;

- and the interfaces between the two phases.

INTRODUCTION

21

The mesomechanic level of observation combined with a plasticity theory

allows to numerically evaluate the influence of the composite

mesostructure and a good characterization of mechanical behavior of

concrete. The disadvantage of the mesomechanic model is the complexity

of theory.

1.3.8 MICROPLANE MODELS

Micromechanical models attempt to develop the macroscopic stress-strain

relationship from the mechanics of the microstructure. The microplane

model, first proposed by Budianski (1949), for metals in the name of slip

theory of plasticity and later extended to concrete and other geomaterials

like rocks and soils (Bazant et al. [14], Pande and Sharma [85], Gambarova

and Floris [48], Carol et al. [22], Caner et al. [21], etc.).

Unlike the other constitutive models, which characterize the material

behavior in terms of second order tensors, the microplane model

characterize in terms of stress and strain vectors. The macroscopic strain

and stress tensors are determined as a summation of all these vectors on

planes of various orientations (Microplanes). The main advantage of

microplane models is its conceptual clarity as the model is formulated in

terms of vectors while the disadvantage in the microplane model is the

complexity of theory and the huge computational work.

1.4 MAIN AIMS AND SCOPES OF THE THESIS

In this Thesis, the application of same classical plasticity-based models to

the numerical simulation of the behavior concrete is discussed in some

detail. Emphasis is placed on the underlying concepts of the yield surface,

the hardening rule, and the flow rule which are suitable for modelling the

overall concrete behavior.

INTRODUCTION

22

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BASIC EQUATIONS AND PROCEDURES

30

2. BASIC EQUATIONS

This chapter deals with the formulation of constitutive equations for general

hardening/softening materials, approached through the “incremental theory” or

“flow theory” of plasticity and a typical algorithm for integrating the constitutive

equations is presented in the last part of the chapter.

2.1 STRESS AND STRESS TENSOR

Stress is defined as the intensity of internal forces acting between particles

of a body on ideal internal surfaces. Let us consider a surface area ∆Ω in

the neighbours of a point Po with a unit vector n normal to the area ∆Ω as

shown in Fig. 2.1. Let Fn be the resultant force due to the action across the

area ∆Ω of the material from one side onto the other side of the cut plane n.

Then the stress vector at point Po associated with the cut plane n is defined

by:

∆Ω∆ΩnnnnFFFF

0

lim

→=nt (2.1)

The state of stress at a point defines the stress vector tn as a function of the

normal direction n.

Fig. 2.1 Continuous body.


31

Since we can make an infinite number of cuts through a point, we have an

infinite number of values of tn which, in general, are different from each

other. This infinite number of values of tn characterizes the state of stress at

that point. Fortunately, there is no need to know all the values of the stress

vectors on the infinite number of planes containing the point. If the stress

vectors t1 t2 and t3 on three mutually perpendicular planes are known, the

stress vector on any plane containing this point can be found from

equilibrium conditions at that point.

Fig. 2.2 – Internal forces of continuous body.

Figure 2.3 shows an element OABC with the stress vectors tx ty and tz and tn

acting on its faces OBC, OAC, OAB, and ABC, respectively. Stress vector

tx (ty , tz) represents the stress acting across the cut plane normal to axis x

(y, z) from the negative side onto the positive side.

The unit vector n can be written in the component form:

n = (nx, ny, nz ), (2.2)

and the direction cosines ni are given by:

ni = cos (ei, n). (2.3)


32

Let A be the area of ∆ABC. Then the area of perpendicular to the i-axis,

denoted by Ai, is given by:

Ai = A ni . (2.4)

From equilibrium of the body OABC, we get:

tn A = tx Ax +ty Ay +tz Az (2.5)

and using eq. (2.4), we obtain the well-know Cauchy’s theorem:

tn = tx nx +ty ny +tz nz. (2.6)

Fig. 2.3 – Stress vectors acting on arbitrary plane n and on the coordinate planes.

In general:

σx,τxy,τxz components of tx

τyx , σy,τyz components of ty

τzx,τzy,σz components of tz

and in the compact tensorial form:

tn = σσσσ :::: n (2.7)

where σij denotes the j-th component of the stress vector acting on the i-th

coordinate planes.


33

The nine quantities σij required to define the three stress vector tx ty and tz,

are called the components of the stress tensor, which is given by:

=

zzyzx

yzyyx

xzxyx

ij

σττ

τστ

ττσ

σ (2.8)

It can be shown that the stress tensor ijσ is symmetric (jiij

σσ = ) by means

of considerations of moments equilibrium on a material element.

2.1.1 PRINCIPAL STRESSES AND INVARIANTS OF THE STRESS TENSOR

Suppose that the direction n at a point Po in a body is so oriented that the

shear components of the stress vector tn vanish (Sn =0) and tn = σ n.

The plane n is then called a principal plane at the point, its normal direction

n is called the principal direction, and the scalar normal stress σ is called

the principal stress. At every point in a body, there exist at least three

principal directions. From the definition, we have:

tn = σ n (2.9)

Substituting for tn from Eq. (2.7) leads to:

σσσσ : : : : n = σ n (2.10)

which implies the following three equations:

(σx-σ) nx +τxy ny+τxz nz=0

τxy nx +(σy-σ) ny+τyz nz=0 (2.11)

τxz nx +τyz ny+(σz-σ) nz=0.

These three linear simultaneous equations are homogeneous for nx, ny and

nz. In order to have a non-trivial solution, the determinant of the

coefficients must vanish:

0

σ-σττ

τσ-στ

ττσ-σ

zyzxz

yzyxy

xzxyx

= (2.12)


34

so that this requirement determines the value of σ. There are, in general,

three roots, σ1, σ2 and σ3. Since the basic equation was tn = σ n, these three

possible values of σ are the three possible magnitudes of the normal stress

corresponding to zero shear stress.

Expanding Eq. (2.12) leads to the “characteristics equation”:

032

2

1

3 =−+− III σσσ (2.13)

where

- I1 = sum of diagonal terms of σij;

- zyz

yzy

zxz

xzx

yxy

xyx

2στ

τσ

στ

τσ

στ

τσ++=I (2.14)

- I3 = determinant of σij.

It can be easily shown that:

3231212σσσσσσ ++=I

3213σσσ=I

where σ1, σ2 and σ3 are the roots of Eq. (2.12), namely the principal stress

values.

Quantities I1, I2, I3 are the invariants of the stress tensor, their values are

constant regardless of rotation of the coordinates axis.

2.1.2 STRESS DEVIATION TENSOR AND ITS INVARIANTS

It is convenient in material modeling to decompose the stress tensor into

two parts, one called the spherical or the hydrostatic stress tensor and the

other called the stress deviator tensor. The hydrostatic stress tensor is the

tensor whose elements are pδij where p is the mean stress defined as

follows:

)(3

1

3

1)(

3

1

3

13211

σσσσσσσ ++==++== Ipzyxkk

( 2.15)

The components of the stress deviator tensor sij are defined by subtracting


35

the spherical state of stress from the actual state of stress. We have:

ijij sp += δijσ ( 2.16)

ijij ps δ−= ijσ ( 2.17)

The components of the stress deviator tensor are given by:

+

=

zyzxz

yzyxy

xzxyx

zyzxz

yzyxy

xzxyx

sss

sss

sss

p00

0p0

00p

σττ

τστ

ττσ

( 2.18)

hence:

.zyzxz

yzyxy

xzxyx

zyzxz

yzyxy

xzxyx

sss

sss

sss

p00

0p0

00p

σττ

τστ

ττσ

=

−

( 2.19)

where that δij = 0 and sij = σij for i≠j. It is apparent that by subtracting a

constant value from the normal stresses σx, σy and σz no change in the

principal directions results. In terms of the principal stresses, the stress

deviator tensor ij

s is:

.3

2

1

3

2

1

s00

0s0

00s

p00

0p0

00p

σ00

0σ0

00σ

+

=

( 2.20)

An equation similar to Eq. (2.19) can be considered to obtain the invariants

of the stress deviator tensor sij:

032

2

1

3 =−+− JJsJs σ ( 2.21)

where J1, J2 and J3 are the invariants of the stress deviator tensor. The

invariants J1, J2 and J3 may be expressed in different forms in terms of the

components of Sij or its principal values, s1, s2 and s3, or alternatively, in

terms of the components of the stress tensor σij or its principal values, σ1,

σ2 and σ3 . The following quantities can be defined:

- J1 = is the sum of diagonal terms of sij ( )01

=J ;


36

-

( )

[ ] [ ] [ ]( )232

231

221

222233

222

2112

6

1

2222

1

2

1

σσσσσσ

τττ

−−−

+++++

++=

== yzxzxyijij sssssJ

( 2.22)

- kijkij sssJ2

13 =

It can be shown that the invariants J1, J2 and J3 are related to the invariants

I1, I2 and I3 of the stress tensor σij through the following relations:

)2792(27

1

)3(3

1

0

321

3

13

2

2

12

1

IIIIJ

IIJ

J

+−=

−=

=

( 2.23)

2.1.3 HAIGH-WESTERGAARD STRESS-SPACE

Various geometric representations have been proposed for better pointing

out the stress state described in tensorial terms (see Chen and Han, 1988

[5]).

Among those representations, the Haigh-Westergaard stress-space is very

useful in studying plasticity theory and failure criteria (Lubliner, [13]).

Since the stress tensor σij has six independent components, they can be

considered as positional coordinates in a six-dimensional space. However,

this is too difficult to deal with a six-component space. The simplest

alternative is to take the three principal stresses σ1, σ2 and σ3 as

coordinates, and, represent the stress state at a point in three-dimensional

stress-space. This space is called the Haigh-Westergaard stress space. In

the principal stress space, every point having coordinates σ1, σ2 and

σ3, represents a possible stress state.


37

It is possible that two stress states at a point P differ by the orientation of

their principal axes, but not in the principal stress values and are

consequently represented by the same point in the three-principal stress

space. This implies that this type of stress space representation is focused

primarily on the geometry of stress and not on the orientation of the stress

state with respect to the material body.

Fig. 2.4 – Haigh-Westergaard stress space.

Consider the straight line ON (Fig. 2.4) passing through the origin and

forming the same angle with respect to each of the coordinate axes. Then,

for every point on this line, the state of stress is one for which σ1= σ2 = σ3.

Thus, every point on this line corresponds to a hydrostatic or spherical state

of stress, while the deviatoric stresses are equal to zero. This line is

therefore termed the “hydrostatic axis”. Furthermore, any plane


38

perpendicular to ON is called the “deviatoric plane”. Such plane can be

described by the following equation:

( ) ξσσσ 3321

=++ ( 2.24)

where ξ is the distance from the origin to the plane measured along the

normal ON.

pI

==

++=

3333

1321 σσσξ ( 2.25)

Furthermore the particular deviatoric plane passing through the origin O:

( ) 0321

=++ σσσ ( 2.26)

is called the π-plane.

Fig. 2.5 – State of stress at a point projected on a deviatoric plane.

Let us consider an arbitrary state of stress at a given point with stress

components σ1, σ2 and σ3,this state of stress is represented by the point P =


39

(σ1, σ2, σ3) in the principal stress space in Fig. 2.4. The stress vector OP

can be decomposed into two components, the vector ON in the direction

n=

3

1,

3

1,

3

1 and the vector NP perpendicular to ON . Thus,

ξ=ON ( 2.27)

The components of vector NP are defined as follows:

( ) );(;; 32;1321 ssspppONOPNP =−−−=−= σσσ ( 2.28)

hence, the length ρ of vector NP is given by:

22

32

22

1 2Jsss =++=ρ . (2.29)

The vectors ON and NP represent the hydrostatic components ( ijpδ ) and

the deviatoric stress components ( ijs ), respectively, of the state of stress

( ijσ ) represented by point P in Fig. 2.4.

Figure 2.5, the axes σ1’, σ2’ and σ3’ are the projections of the axes (σ1, σ2

and σ3) on the deviatoric plane, and NP is the projection of vector NP on

the same plane.

Developing some simple geometric considerations, we obtain:

12

3cos s=θρ (2.30)

Substituting for ρ from Eq. (2.29) into Eq. (2.30) results:

2

1

2

3cos

J

s=θ

(2.31)

In a similar manner, the deviatoric stress components s2 and s3,we can also

be obtained in terms of the “lode angle” θ:

2

2

2

3

3

2cos

J

s=

− θπ


40

2

3

2

3

3

2cos

J

s=

+ θπ

(2.32)

The ξ , ρ ,θ coordinates are called Haigh-Westergaard coordinates and

they can be used in alternative to the principal stresses or to the stress

tensor invariants.

In view of Eq. (2.20), (2.31), (2.32), and (2.24), the three principal stresses

of σij are given by:

( )( )

+

−+

=

32cos

32cos

cos

3

22

3

2

1

πθ

πθ

θ

σσσ

J

p

p

p

(2.33)

( )( )

+

−+

=

32cos

32cos

cos

3

2

3

1

3

2

1

πθ

πθ

θ

ρξξξ

σσσ

(2.34)

2.2 YIELD AND FAILURE CRITERIA

Particular surfaces can be described within the stress space and it is

possible alternative representation to describe states of stresses material

resulting in yielding or failure.

2.2.1 YIELD CRITERIA INDEPENDENT OF HYDROSTATIC PRESSURE

The yield criterion defines the elastic limits of a material under combined

states of stress. In general, the elastic limit or yield stress is a function of

the state of stress, ijσ . Hence, the yield condition can generally be

expressed as:

021 =,.....),k,kf(σij (2.35)

where k1, k2… are material constants.

For isotropic materials, the values of the three principal stresses suffice to


41

describe the state of stress uniquely. A yield criterion therefore consists in a

relation of the form:

021321 =,.....),k,k,σ,σf(σ (2.36)

The three principal stresses can be expressed in terms of the combinations

of the three stress invariants ),J,J(I 321 , where I1 is the first invariant of the

stress tensor, J2 and J3 are the second and third invariants of the deviatoric

tensor. Thus, one can replace Eq. (2.36) by:

021321 =,.....),k,k,J,Jf(I (2.37)

Furthermore, these three particular principal invariants are directly related

to Haigh-Westergaard coordinates ),,( θρξ in the stress space:

021 =,.....),k,k,,f( θρξ (2.38)

Yield criteria of materials should be determined experimentally. An

important experimental fact for metals, is that the influence of hydrostatic

pressure on yielding is not appreciable. The absence of a hydrostatic

pressure effect means that the yield function can be reduced to the form:

02132 =,.....),k,k,Jf(J (2.39)

The classical yield criteria used for metal are the Tresca and Von Mises

Criteria [Chen and Han, 1988 [5]).

2.2.1.1 The Tresca Yield Criterion.

The first yield criterion for a combined state of stress for metals was

proposed by Tresca (1864), who suggested that yielding would occur when

the maximum shearing stress at a point reaches a critical value k. In terms

of principal stresses:

k=

−−−

3231212

1;

2

1;

2

1max σσσσσσ (2.40)

where the material constant k may be determined from the simple tension


42

test. Then, 2

0σ=k , in which σ0 is the yield stress in simple tension.

Assuming the ordering of stresses to be σ1 ≥ σ2 ≥ σ3 and using the Eq.

(2.33), we can rewrite:

( )[ ] )600(3

2coscos3

1)(

2

12

21°≤≤=+−=− θπθθσσ kJ (2.41)

obtaining the Tresca criterion in terms of θ,2J coordinates:

( ) 03

sin2)( 02,2 =−+= σπθθ JJf (2.42)

or in terms of the variables ),,( θρξ :

( ) 03

sin2)( 0, =−+= σπθρθρf , (2.43)

Fig. 2.6 – Tresca yield surfaces in principal stress space.

Since the hydrostatic pressure has no effect on the yield surface, Eq. (2.42)

or Eq. (2.43) must be independent by hydrostatic pressure p, the first

invariant I1, or ξ. On the deviatoric plane, Eq. (2.42) or Eq. (2.43) is a

regular hexagon (Fig. 2.8), whose distance from vertices, from Eq. (2.43):

( )3

sin2

0

πθ

σρ

+= (2.44)

while, in a principal stress space, the equations represent the surface in


43

figure 2.6.

Fig. 2.7 – Yield criteria (Tresca and von Mises) in the plane stress state (σ3 = 0).

2.2.1.2 The von Mises Yield Criterion.

The octahedral shear stress is a convenient alternative choice to the

maximum shear stress to formulate a yield criterion for materials which are

pressure independent. The von Mises yield criterion (1913) is based on this

alternative; it states that yielding begins when the octahedral shear stress

reaches a critical value k:

kJoct3

23

22 ==τ (2.45)

which, reduces to the simple form:

0)(2

22 =−= kJJf . (2.46)

Considering Eq. (2.22) and substituting into Eq. (2.46) the following

expression can be derived:

[ ] [ ] [ ] 2232

231

2213,2,1 6)( kf =++= −−− σσσσσσσσσ . (2.47)

In a uniaxial tension test:

- σ1 = σ0 σ2 = 0 σ3 = 0;

- [ ] [ ] [ ] 2232

231

221 6k=++ −−− σσσσσσ (2.48)


44

- [ ] [ ] [ ] 2220

20 60000 k=−+−+− σσ

hence,

3

0σ=k (2.49)

Equation (2.46) represents a circular cylinder whose intersection with the

deviatoric plane is a circle of radius 22J=ρ :

22

222 kJ ==ρ

k2=ρ . (2.50)

Fig. 2.8 –von Mises yield surfaces in principal stress space.

Fig. 2.9 – Yield criteria in a deviatoric plane.


45

If the von Mises and Tresca criteria are made to agree for a simple tension

yield stress, graphically, the von Mises circle circumscribes the Tresca

hexagon as shown in Fig. 2.9. However, if the two criteria are made to

agree for the case of pure shear, the circle will inscribe the hexagon.

2.2.2 FAILURE CRITERION FOR PRESSURE-DEPENDENT MATERIALS

Failure of a material is usually defined in terms of strength limits. As in the

case of the yield criteria, a general form of the failure criteria can be given

by Eq. (2.35) for anisotropic materials and by Eq. (2.36) through (2.39) for

isotropic ones. Yielding of more ductile metals is not affected by

hydrostatic pressure, while failure behavior of many non-metallic

materials, such as soils, rocks, and concrete, is hugely influenced by

hydrostatic pressure.

The general shape of a failure surface 0321 =),J,Jf(I or 0=),,f( θρξ in a

three-dimensional stress space can be described by its cross-section with

the deviatoric planes and its meridians in the meridian planes (Figs. 2.4

and 2.10). The cross sections of the failure surface are the intersection

curves between this surface and a deviatoric plane which is perpendicular

to the hydrostatic axis with ξ = const. The meridians of the failure surface

are the intersection curves between this surface and a plane (the meridian

plane) containing the hydrostatic axis with θ = const (see & 2.1.3).

For an isotropic material the cross-sectional shape (deviatoric planes) of the

failure surface has a threefold symmetry [Chen and Han, 1988 [5]).

Therefore, when performing experiments, it is necessary to explore only

the sector θ = 0° to θ = 60°, the other sector being known by symmetry.

The regular ordering of the principal stresses is 321 σσσ >> . With this

ordering, there are two extreme case:

1) 321 σσσ >= (2.51)


46

Eq. (2.53) represents a stress state corresponding to a hydrostatic

stress state hhh

321 σσσ == , with a further compressive stress

superimposed in one direction. If we substitute Eq. (2.51) into Eq. (

2.31):

2

1

2

3cos

2

1

==J

sϑ

3

πϑ = (2.52)

so the meridian corresponding to θ = 60° is called the compression

meridian.

2) 321 σσσ => (2.53)

Eq. (2.53) represents a state stress corresponding to a hydrostatic

stress state hhh

321 σσσ == , with a tensile stress superimposed in one

direction. In analog manner:

12

3cos

2

1

==J

sϑ

0=ϑ (2.54)

and the meridian corresponding to θ = 0° is called the tensile

meridian.

Furthermore, the meridian determined by θ = 30° is sometimes called the

shear meridian. If we add to a hydrostatic stress state

)(2

1)(

2

1)(

2

1323121

hhhhhh σσσσσσ +++ == , a pure shear state

[ ] [ ]

−−

hhhh1331

2

1;0;

2

1σσσσ , we get:

2

3

2

3cos

2

1

==J

sϑ


47

6

πϑ = (2.55)

Fig. 2.10 – Failure criteria in the meridian planes (a) and in the deviatoric planes (b), for

concretes, (Chen and Han, 1988 [5]).

The failure function for concrete, and other frictional materials, is defined

by experimental data. The available experimental data clearly indicate the

essential features of a failure surface. It is largely accepted that concrete

can be described by a failure surface with curved meridians, indicating that


48

the hydrostatic pressure results in increasing shear capacity of the material

itself (Fig 2.10a). A pure hydrostatic loading cannot cause failure for

ordinary stresses, while an elevated hydrostatic stress state, quite

uncommon in civil engineering, can cause failure of material. The value of

c

t

ρρ

increases with increasing hydrostatic pressure. It is about 0.5 near the

π-plane and reaches a value of about 0.8 for elevated values of hydrostatic

pressure (Fig 2.10 b).

The shape of the failure of concrete, in the deviatoric plane (Fig. 2.10b),

changes from nearly triangular for tensile and small compression stresses to

a bulged shape (near circular) for higher compressive stresses. The

deviatoric sections are convex and θ-dependent (Chen and Han, 1988 [5]).

Based on knowledge concerning the shape of the failure surface of concrete

materials, a variety of failure criteria have been proposed (Mao, 2002 [14]).

In Chapter 5 some of those theories will be discussed. Those criteria are

classified by the number of material constants appearing in the expression

as one-parameter through five-parameter models.

One-parameter models, as the von Mises or Tresca type of failure surface,

is used for pressure independent materials. Because of the limited tensile

capacity of concrete, the von Mises or Tresca surface is an unsuitable

failure model for concrete-like materials.

Among two-parameter models, the Drucker-Prager and Mohr-Coulomb

surfaces are the simplest types of pressure-dependent failure criteria

(Lubliner 1990, [13]). We shall consider in more details in the following

discussion these simple and classical failure criteria. Two-parameters

models with straight lines as the meridians are therefore inadequate for

describing failure surface of concrete in the high-compression range.


49

The refined models, with three, four or five parameters, reproduce all the

important feature of the triaxial failure surface and give a close estimate of

relevant experimental data.

In the present Thesis, some classical models for the elastic-plastic analysis

of concrete, using one, two and three parameters models, will be described

and compared.

2.2.2.1 The Mohr-Coulomb Criterion.

Mohr's criterion (1900), may be considered as a generalized version of the

Tresca criterion. Both criteria are based on the assumption that the

maximum shear stress is the only decisive measure or impending failure.

However, while the Tresca criterion assumes that the critical value of the

shear is constant, Mohr’s failure criterion considers the limiting shears

stress τ in the plane, to be a function of the normal stress σ in the same

plane at the point:

)(στ f= (2.56)

where )(σf is an experimentally determined function.

Fig. 2.11 – Tresca criterion on σ−τ plane.


50

Fig. 2.12 – Mohr’s criterion on σ−τ plane.

In terms of Mohr's graphical representation of the state of stress, Eq (2.56),

means that failure of material will occur if the radius of the large principal

Mohr’s circle is tangent to the envelope curve )(σf , as shown in Fig. 2.12.

In contrast to the Tresca criterion (Fig. 2.11), it is seen that Mohr's criterion

allows for the effect of the mean stress or the hydrostatic stress.

The simplest form of the Mohr envelope )(σf is a straight line, illustrate

in Fig. 2.13. The equation for the straight-line envelope is known

Coulomb's equation (1776 [8]):

)tan(φστ −= c (2.57)

in which c is the cohesion and φ is the angle of internal friction, both are

material constants determined by experiments. Failure criterion associated

with Eq. (2.57), will be referred as the Mohr-Coulomb criterion. In the

special case of frictionless materials, for which φ = 0, this criterion reduces

to the maximum shear stress of Tresca.


51

Fig. 2.13 – Mohr-Coulomb criterion: with straight line as failure envelope.

From Eq. (2.57) and for 321 σσσ ≥≥ , the Mohr-Coulomb criterion can be

written:

φφσσ

σσφσσ tansin2

)()(

2

1cos)(

2

1 31

3131

−++−=− c (2.58)

or rearranging:

1cos2

sin1

cos2

sin131

=−

−+

φφ

σφφ

σcc

(2.59)

if we define:

φφ

sin1

cos2'

−=

cf

c and

φφ

sin1

cos2'

+=

cf

t (2.60)

Eq. (2.59) is further reduced to:

1'

3

'

1

=−ct

ff

σσ (2.61)

where ft’ is the strength in simple tension and fc’ is the strength in simple

compression.


52

Fig. 2.14 – Mohr-Coulomb criterion and Drucker-Prager criterion in principal stress

spase (a), and in a deviatoric plane (b).

It is sometimes convenient to introduce a parameter m, where:

φφ

sin1

sin1'

'

−+

==t

c

f

fm (2.62)

then Eq. (2.61) can be written in the form:

'31c

fm =− σσ for 321 σσσ ≥≥ (2.63)

To demonstrate the shape of the three-dimensional failure surface of the

Mohr-Coulomb criterion, we again use Eq. (2.33) and rewrite Eq. (2.59) in

the following form:

0cossin3

cos3

3sinsin

3

1),,(

2

2121

=−

++

++=

φφπ

θ

πθφθ

cJ

JIJIf

(2.64)

or identically in terms of variables ),,( θρξ :


53

0cos6sin3

cos

3sin3sin2),,(

=−

++

++=

φφπ

θρ

πθρφξθρξ

c

f

(2.65)

with 0° ≤ θ ≤ 60°.

In principal stress space, this gives an irregular hexagonal pyramid (Fig.

2.14a). Its meridians are straight lines, and its cross section in the π-plain is

an irregular hexagon (Fig. 2.14b). Only two characteristic lengths a

required to draw this hexagon, ρc and ρt.

2.2.2.2 The Drucker-Prager Criterion.

As we have seen, the Mohr-Coulomb failure criterion can be considered a

generalized Tresca criterion accounting for the hydrostatic pressure effect.

The Drucker- Prager criterion, formulated in 1952, is a simple modification

of the von Mises criterion, where the influence of a hydrostatic stress

component on failure is introduced by inclusion of an additional term in the

von Mises expression to give:

0),( 2121 =−+= kJIJIf α (2.66)

and using Haigh-Westergaard variables:

026),( =−+= kf ραξρξ (2.67)

where α and k are material constants. Eq. (2.66) reduces to the von Mises

criterion, when α is zero,.

The failure surface of Eq. (2.66) in principal stress space is clearly a right-

circular cone. Its meridian and cross section on the π-plane are shown in

Fig. 2.15.


54

Fig. 2.15 –Drucker-Prager criterion: (a) meridian plane, (b) π plane.

The Mohr-Coulomb hexagonal failure surface is mathematically con-

venient only in problems where it is obvious which one of the six sides is to

be used. If this information is not known in advance, the corners of the

hexagon can cause considerable difficulties resulting in complications to

obtain a numerical solution. The Drucker-Prager criterion, as a smooth

approximation to the Mohr-Coulomb criterion, can be made to match the

latter by adjusting the size of the cone. For example, if the Drucker-Prager

circle is assumed to fit the outer apices of the Mohr-Coulomb hexagon, the

two surfaces coincide along the compression meridian ρc, where 3

πθ = ,

then the constants α and k are related to the constants c and φ :

)sin3(3

cos6,

)sin3(3

sin2

φφ

φφ

α−

=−

=c

k (2.68)

The cone corresponding to the constants in Eq. (2.68) circumscribes the

hexagonal pyramid and represents an outer bound on the Mohr-Coulomb

failure surface (Fig. 2.14). On the other hand, the inner cone passes through


55

the tension meridian ρt, where 0=θ , and will have the constants:

)sin3(3

cos6,

)sin3(3

sin2

φφ

φφ

α+

=+

=c

k (2.69)

Other approximations is possible for 6

πθ = along the shear meridian.

The material constants α and k can be determined from the given tensile

failure stress ft’ and compression failure strength fc’. Substituting stress

states:

- Uniaxial compression:

[ ] [ ] [ ]( )

=++=

−=++=

−===

−−−2'2

322

312

212

'3211

'321

3

1

6

1

00

c

c

c

fJ

fI

f

σσσσσσ

σσσ

σσσ

(2.70)

- Uniaxial tension:

[ ] [ ] [ ]( )

=++=

=++=

===

−−−2'2

322

312

212

'3211

32'

1

3

1

6

1

00

t

t

t

fJ

fI

f

σσσσσσ

σσσ

σσσ

(2.71)

into the failure condition or Eq. (2.66), one gets:

=−+

=−+−

03

03

'

'

'

'

kf

f

kf

f

t

t

c

c

α

α (2.72)

and solving for α and k leads to:

( )

( )

++

−=

+

−=

33

3

'

'

''

''

''

''

t

t

ct

ct

ct

ct

ff

ff

ffk

ff

ffα

(2.73)


56

2.3 LINEAR ELASTIC ISOSTROPIC STRESS-STRAIN RELATION

Elastic materials completely recover their original shape and size after

removing the applied forces. For many materials, the elastic range also

includes a linear relationship between stress and strain. This linear

proportion or the stress-strain relation in general form is given by:

klijklij C εσ = ~ (2.74)

where ijklC is the material elastic constant tensor. It may also be remarked

that Eq. (2.74) is the simplest generalization of the linear dependence of

stress and strain observed in the familiar Hooke's experiment in a simple

tension test, and consequently Eq. (2.74) is often referred to as the

generalized Hooke's law.

We need only two independent elastic constants to describe the behavior

for a homogeneous isotropic linear elastic material. This hypothesis for

concrete can be realistic, at a macroscopic level of investigations. Now Eq.

(2.74) can be written in matrix form as:

[ ] εσ C= (2.75)

where the matrix [ ]C is called the matrix of elastic moduli and in the

hypothesis of homogeneous isotropic linear elastic material, Eq. (2.75) in

explicit form becomes:

−+=

yz

xz

xy

z

y

x

yz

xz

xy

z

y

x

2

2-100000

02

2-10000

002

2-1000

000-1

000-1

000-1

)21)(1(

τ

τ

τ

σ

σ

σ

γγγεεε

ν

ν

νννν

νννννν

ννE

(2.76)

It can be shown that Eq. (2.76), when reduced to the two-dimensional plane


57

stress case (γyz=γxz=0 , τxz=τyz=0 , σz=0), take the following simple form:

−+=

xy

z

y

x

xy

y

x

2

2-1000

0-1

0-1

0-1

)21)(1(

τ

0

σ

σ

γεεε

νννν

νννννν

ννE

(2.77)

Fig. 2.16 – Plane stress case.

It is noted that in the plane stress space, the strain component εz is non-zero

and takes the value:

)(1

)( yxyxz

Eεε

νν

σσν

ε +−

−=+−= (2.78)

The plane stress relations given above are commonly used in many

practical applications. For instance, the analysis of thin, flat plates loaded

in the plane of the plate (x-y plane) are often treated as plane stress

problems (fig. 2.16).

The plane strain conditions (γyz=γxz=0 , τxz=τyz=0 , εz=0) are normally

found in elongated bodies of uniform cross section subjected to uniform

loading along their longitudinal axis (z-axis), such as in the case of tunnels


58

(fig 2.17). Under the conditions of plane strain, Eq. (2.76) can be reduced

to the simple form:

−+=

xy

y

x

xy

z

y

x

0

2

2-1000

0-1

0-1

0-1

)21)(1(

τ

σ

σ

σ

γ

εε

νννν

νννννν

ννE

(2.79)

Fig. 2.17 – Plane strain case.

For this case, the stress component σz has the value:

)( yxz σσνσ += (2.80)

Analysis of bodies of revolution under axisymmetric loading is similar to

that for plane stress and plane strain conditions since this problem is also

two-dimensional. In the usual notation, the nonzero stress components in

the axisymmetric case are σr, σθ, σz, τrθ and the corresponding strains εr,

εθ, εz, γrθ. Equation (2.76) can be reduced:


59

−+=

ϑ

ϑ

ϑ

ϑ

γεεε

νννν

νννννν

ννr

z

r

r

z

r

2

2-1000

0-1

0-1

0-1

)21)(1(

τ

σ

σ

σ

E (2.81)

Fig. 2.18 – Axisymmetric case.

2.4 STRESS-STRAIN RELATION FOR WORK-HARDENING

MATERIALS.

Engineering materials usually exhibit a work-hardening behavior consisting

in an evolution of the initial yielding surface, consequently, stress increases

beyond the initial surface produces both elastic and plastic deformations. At

each stage of plastic deformation, a new yield surface, called the

subsequent loading surface, is established. Only elastic deformations and

no plastic ones develop as the point representing the state of stress tends

moving forward the inner part of such surface.

The classical approach used in plasticity is the incremental theory (or flow

theory) (Chen and Han, 1988 [5]). In this theory the total strain increment

tensor is assumed to be the sum of the elastic and plastic strain increment


60

tensors:

pij

eijij ddd εεε += . (2.82)

According to Hooke’s law, the total stress increment is determined by:

eijijklkl dCd εσ = (2.83)

where ijklC is the material elastic constant tensor.

2.4.1 PLASTIC POTENTIAL AND FLOW RULE

The flow rule is the necessary kinematic assumption postulated for plastic

deformation or plastic flow. It gives the ratio or the relative magnitudes of

the components of the plastic strain increment tensor p

ijdε .

As the elastic strain can be derived directly by differentiating the elastic

potential function Ω, for hyperelastic or Green elastic materials (Lubliner,

1990 [13]):

ij

ij

ij σ

σΩε

∂

∂=

)( (2.84)

von Mises proposed the similar concept of the plastic potential function,

which is a scalar function of the stresses, )( ijg σ (Lubliner 1990, [13]). Then

the plastic flow equations can be written in the form:

ij

pij

gdd

σλε

∂∂

= (2.85)

where λd is a positive scalar factor of proportionality, which is nonzero

only when plastic deformations develop.

)( ijg σ = constant

defines a surface of plastic potential within the nine-dimensional stress

space. The direction cosines of the normal vector to this surface at the point

ijσ on the surface are proportional to the gradient ij

g

σ∂∂

.

In some cases the identity between the yielding function f and the plastic

potential g can be assumed. Hence, the following relationship can be


61

derived:

ij

pij

fdd

σλε

∂∂

= . (2.86)

Equation (2.86) is called the associated flow rule because the flow rule is

connected or associated with the yield criterion, while relation (2.85) is

called a nonassociated flow rule.

The nonlinear volume change during plastic strain is a feature of concrete

materials. Experimental results indicate that under compressive loadings,

inelastic volume contraction occurs at the beginning of yielding and

volume dilatation occurs at about 75 to 90% of the ultimate stress (Shah

and Chandra, 1968 [15]). Inflection points are usually observed (see Fig.

1.5 in Chapter 1). This kind is not compatible with the associated flow rule

(Chen and Han, 1988 [5]) and a plastic potential is needed to define the

flow rule. For the sake of simplicity, a functional form of the Drucker-

Prager type can be assumed:

21JIg += β (2.87)

where β is the dilation angle in the 21

JI − plane. Consequently, the flow

rule takes the following form:

∂

∂

∂∂

+∂

∂

∂∂

=

∂∂

=ijijij

pij

J

J

gI

I

gd

gdd

σσλ

σλε 2

2

1

1

(2.88)

and

+=

∂∂

+∂∂

=∂∂

=221 2 J

sds

J

g

I

gd

gdd

ij

ijijijij

pij βδλδλ

σλε (2.89)

The incremental plastic volume change (inelastic dilatancy), is given by

λβεε ddtrdp

ij

p

v3== (2.90)

To reflect the nonlinear volume chance, a functional form of β may be

defined according to the available experimental data. For the sake of


62

simplicity, a constant value will be assumed for β in the models presented

in the next chapters.

2.4.2 INCREMENTAL STRESS-STRAIN RELATIONSHIP

Loading surface is defined as the subsequent yield surface for an

elastoplastically deformed material; such surface defines the boundary of

the current elastic region. If a stress point lies within this region, no

additional plastic strains develop. On the other hand, if the state of stress is

on the boundary of the elastic region and in the successive time it remains

in the same region, additional plastic deformation will occur, accompanied

by a configuration change of the current loading surface. In other words,

the current loading surface or the subsequent yield surface will change its

current configuration when plastic deformation takes place. Thus, the

loading surface may be generally expressed as a function of the current

state of the stress, and some internal variables of material state such that

0, =k),f(σp

ijij ε (2.91)

where p

ijε is the measure of the plastic strain while ( )p

ijkk ε= is a general

hardening parameter.

For many applications, the quantities p

ijε and k are condensed in only one

parameter, called effective plastic strain p

ε and defined as:

pij

pijp εεε

3

2= ; (2.92)

hence, Eq. (2.91) becomes:

0=),f(σp

ij ε . (2.93)

The total strain increment is assumed as the sum of the elastic and the

plastic one:

pij

eijij ddd εεε +=


63

The elastic strain increment can be obtained inverting the Eq. (2.83):

klijkle

ij dCd σε 1−= (2.94)

and the plastic strains is obtained from the flow rule in Eq. (2.85).

Then the complete strain-stress relations for an elastic-plastic materials are

expressed as:

ij

klijklij

gddCd

σλσε

∂∂

+= −1 (2.95)

where λd is a undetermined factor whose value can be derived to comply

with the following conditions:

λd

<=<=

==>

0000

000

dfand)f(σor)f(σif

dfand)f(σif

ijij

ij

(2.96)

The first expression of Eq. (2.96) is called consistency condition and

rendering explicit the same expression, we obtain:

0),(0),(011 ==⇒= ++

kfandkfdfn

ij

nn

ij

n σσ (2.97)

in which k takes the place of pε of Eq. 2.93 (figure 2.19).

Fig. 2.19 – Graphical representation of Consistency condition.


64

The scalar λd can be determined directly from Eq. (2.96) as described in

the following relationships.

0=∂∂

+∂∂

=p

p

ij

ij

df

dσσ

fdf ε

ε (2.98)

From Eq. (2.82) we can rewrite:

eij

pijij ddd εεε =− (2.99)

and using Eq. (2.83):

( )pklklijklij ddCd εεσ −= (2.100)

Substituting Eqs. (2.100) , (2.92) and (2.85) into the consistency condition

(2.98):

03

2=

∂∂

∂∂

∂∂

+

∂∂

∂∂

= −

ijijpkl

klijkl

ij

ggd

fgddC

σ

fdf

σσλ

εσλε (2.101)

and

03

2=

∂∂

∂∂

∂∂

+

∂∂

∂∂

∂∂

−

ijijpkl

ijkl

ij

klijkl

ij

ggd

fgC

σ

fddC

σ

f

σσλ

εσλε (2.102)

resolving for λd :

ijijpkl

ijkl

ij

klijkl

ij

ggfgC

σ

f

dCσ

f

d

σσεσ

ελ

∂∂

∂∂

∂∂

−

∂∂

∂∂

∂∂

=

3

2 (2.103)

Substituting Eq. (2.103) into Eq. (2.85):

tr

ijijpkl

ijkl

ij

klijkl

ijp

trσ

g

ggfgC

σ

f

dCσ

f

d∂∂

∂∂

∂∂

∂∂

−

∂∂

∂∂

∂∂

=

σσεσ

εε

3

2 (2.104)

then using Eq. (2.100) and rearranging the indices; the following definition

of the stress tensor increment is derived as follows:


65

∂∂

∂∂

∂∂

−

∂∂

∂∂

∂∂

∂∂

−= kl

dedep

tu

rstu

rs

pqkl

pqmn

ijmn

klijklij dggfg

Cσ

f

Cσ

g

σ

fC

dCd ε

σσεσ

εσ

3

2 (2.105)

and in compact form:

[ ] klep

ijklkl

p

ijklijklij dCdCCd εεσ =−= (2.106)

The coefficient tensor in parentheses represents the elastic-plastic stiffness

tensor in terms of tangent moduli.

For a given stress state and loading history, stress increment dσij can be

derived through Eq. (206) as a function of the strain increment dεkl. This

equation is needed in a numerical analysis of plasticity, such as finite

element analysis. If a nonassociated flow rule is used, the elastic-plastic

stiffness tensor is an unsymmetric tensor, while if we adopt an associated

flow rule the some tensor is symmetric as described in the following

elaborations.

The elastic-plastic stiffness tensor is symmetric if the following condition

is satisfied:

ep

klij

ep

ijklCC = (2.107)

using Eq. (2.105), we render explicit the symmetric condition (Eq. 2.107):

dedep

tu

rstu

rs

pqij

pqmn

klmn

klij

dedep

tu

rstu

rs

pqkl

pqmn

ijmn

ijkl

ggfgC

σ

f

Cσ

g

σ

fC

C

ggfgC

σ

f

Cσ

g

σ

fC

C

σσεσ

σσεσ

∂∂

∂∂

∂∂

−

∂∂

∂∂

∂

∂

∂

∂

−

=

∂∂

∂∂

∂∂

−

∂∂

∂∂

∂

∂

∂

∂

−

3

2

3

2

(2.108)


66

eliminating the terms that don't compete the determination of the symmetry,

Eq. (2.108) becomes:

pqij

pqmn

klmnpqkl

pqmn

ijmn Cσ

g

σ

fCC

σ

g

σ

fC

∂∂

∂∂

=∂∂

∂∂

(2.109)

If gf ≠ Eq. (2.109) is an invalid relation and the elastic-plastic stiffness

tensor is an unsymmetric tensor, while if gf = the symmetry is verified:

pqij

pqmn

klmnpqkl

pqmn

ijmn Cσ

f

σ

fCC

σ

f

σ

fC

∂∂

∂∂

=∂∂

∂∂

(2.110)

Opportunely modifying the indices of Eq. (110), we obtain:

mnij

mnpq

klpqpqkl

pqmn

ijmn Cσ

f

σ

fCC

σ

f

σ

fC

∂∂

∂∂

=∂∂

∂∂

(2.111)

and the subsequent condition:

pqkl

pqmn

ijmnpqkl

pqmn

ijmn Cσ

f

σ

fCC

σ

f

σ

fC

∂∂

∂∂

=∂∂

∂∂

(2.112)

we have so shown the symmetry condition of Eq. 107, when an associated

flow rule is adopted.

2.4.3 SOFTENING BEHAVIOR

As discussed in Section 1.1.5, axial compression tests on concrete

specimens usually results in softening behavior. Although, softening

behavior is a consequence of the structural features of the test specimen

rather than an intrinsic mechanical propriety of the material, in the present

Thesis the behavior in the post-failure regime is treated as a particular case

of hardening behavior, where the subsequent yield surface becomes smaller

when the plastic strain take place.

2.5 INTEGRATION SCHEME FOR ELASTO-PLASTIC MODELS.

The incremental constitutive relation for a general elastic-plastic material

has been presented in Section 2.4.2. In particular, Eq. (2.106) relates the


67

stress increment dσij to the total strain increment dεkl. However, in a

numerical analysis, since a finite load increment instead of an infinitesimal

one is applied in each load-step, the relevant increments of stress and strain

have finite sizes. Therefore, the incremental constitutive relation (§ 2.4.2)

has to be integrated numerically. The algorithms to implement this

numerical integration play an important role and, a buy with the algorithms

for solving nonlinear simultaneous equations, constitute the core of an

elastic-plastic numerical analysis. An unsuitable algorithm may lead not

only to an inaccurate stress solution, but may also delay the convergence of

the equilibrium iteration, and even lead to divergence of the iteration. Since

stress computation is usually time-consuming, the global efficiency of an

algorithm is therefore essential.

2.5.1 GENERAL DESCRIPTION OF A GENERAL ELASTOPLASTIC

INTEGRATION

In matrix form, the stress increment ijdσ can be expressed in terms of

elastic strain increment,e

ijdε (as viewed in the section 2.4.2):

( )pklklijkl

eklijklij ddCdCd εεεσ −== (2.113)

or in terms of total strain increment ij

dε , as

klep

ijklij dCd εσ = (2.114)

In a generic load-step ([m+1]th step), we already know, at the end of the

mth load step in which the equilibrium iteration has converged, the

following mechanical quantities:

ij

mσ ,ij

mε , pmε (2.115)

We define a trial stress increment, assuming an elastic response of the

specimen:

klijkl

e

ijC ε∆σ∆ = (2.116)


68

in which the total strain increment can be derived as follows:

kl

m

kl

m

klεεε∆ −= +1

. (2.117)

kl

m ε1+ is the total strain at (m+1)

th step and

kl

mε is the total strain at mth

load-step.

We assume that at the end of (m+1)th load step, the stress point is in elastic

state, satisfying the following condition:

0<+ ),σf(p

me

ijij

m εσ∆ ;

in such case the numerical procedure finishes and we go to the new load-

step ([m+2]th step).

If it occurs that 0>+ ),σf(p

me

ijij

m εσ∆ (inadmissible state) the stress state

enters into an elastic-plastic state in the (m+1)th step.

The most common procedure to integrate elastic-plastic models is

presented in the following integration algorithm:

1) At first step-algorithm, we calculate the following quantities:

eklkl

meij

m σkl

ijkl

σijij

gC

σ

f

σ∆σ∆ σ ++ ∂∂

∂∂

(2.117)

pmp

f

εε∂∂

(2.118)

From Eq. (2.103) we can calculate the first value of λd :

eij

meij

meklkl

meij

m

eij

m

σijijσijijpmpσkl

ijkl

σijij

klijkl

σijij

ggfgC

σ

f

dCσ

f

d

σ∆σ∆σ∆σ∆

σ∆

σσεεσ

ε

λ

++++

+

∂∂

∂∂

∂∂

−

∂∂

∂∂

∂∂

=

3

2

(2.119)

then the new value of the current total stress is:

)(1 λσ∆σ∆σσ d

p

ij

e

ijij

m

ij

m −+=+ (2.120)


69

where )( λσ∆ dp

ij is the plastic corrector (see Figure 2.20) defined in Eq.

(2.100) as:

eij

mσijkl

ijklp

klijklp

ij

gCddCd

σ∆σλελσ∆

+∂∂

==)( (2.121)

Fig. 2.20 – Graphical representation of the elastic predictor and the plastic corrector

(plastic return).

We can calculate the plastic strain increment, using Eq. (2.85) and the new

value of the yielding function can be finally derived:

( ) )ddσf(fp

mp

ij

e

ijij

m λελσ∆σ∆ 1,

+

−+= (2.122)

where eklkl

mσkl

p

m

p

m gd

σ∆σλεε

+

+

∂∂

+=1.

Generally, the yield function in Eq. (2.122), at first (k=1) numerical

iteration assumes a negative value.

We seek that particularly value of λd that brings to zero the yield criterion

(Eq. 2.122). The expression 2.122 describes a function f of an only λd

independent variable.

In numerical analysis, Newton's method (also known as the Newton–

Raphson method) is one of the best known method for finding successively

better approximations to the zeros (or roots) of a real-valued function.


70

Hypothesizing that 1+k

dλ is a root of “f” function ( 0)(1 =+k

df λ ), we can

write (see figure 2.21):

11

10)()()(

)(++

+

−−

=−−

=∂∂

kk

k

kk

kk

k

dd

df

dd

dfdfd

d

f

λλλ

λλλλ

λλ

(2.123)

Fig. 2.21 – An illustration of one iteration of Newton's method.

here, λd

f

∂∂

denotes the derivative of the function f. Then by simple algebra

we can derive:

)(

)(1

k

k

kk

dd

f

dfdd

λλ

λλλ

∂∂

−=+ (2.124)

We will arrest the algorithm process when:

∆λ

λ

λ<

∂∂

)(

)(

k

k

dd

f

df (2.125)

being ∆ a sufficiently small tolerance.


71

Finally, we can know the plastic strain increment ij

pij

gdd

σλε

∂∂

= and the

final plastic corrector )( λσ∆ dp

ij.

Fig. 2.22 – Flow chart of integration scheme for elastoplastic models.


72

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02909, USA.

[3] Bathe, K.-J. (1996), Finite element procedures. Prentice-Hall, Englewood, New

Jersey, USA.

[4] Chen, W. (1982), Plasticity in reinforced concrete. McGraw-Hill, London, England.

[5] W. F. Chen (Author), D. J. Han (Author), Plasticity for Structural Engineers,


[6] Chen, W.F., Constitutive Equations for Engineering Materials, Vol. 1: Elasticity


[7] Cook, R. D. Finite Element Modeling for Stress Analysis J. Wiley & Sons, New

York, 1995.

[8] Coulomb, C. A. (1776). Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture. Mem. Acad. Roy. Div. Sav.,

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[9] Crisfield, M. A. Non-linear Finite Element Analysis of Solids and Structures, Vol. 1-

2.

[10] C. S. Desai, H. J. Siriwardane, Constitutive Law for Engineering Materials, Prentice-Hall (1984).

[11] Drucker, D. (1959). A definition of stable inelastic materials. Journal of Applied Mechanics, 26:101-106.

[12] Etse, G. and Willam, K. (1996). Integration algorithms for concrete plasticity. Engineering Computations, 13(8):38-65.

[13] Jacob Lubliner, Plasticity Theory, Macmillan Publishing, New York (1990).

[14] Mao-hong Yu. Advances in strength theories for materials under complex stress state in the 20th Century. School of Civil Engineering & Mechanics, Xian Jiaotong

University, Xian, 710049, China.

[15] SP Shah, S Chandra . “Critical Stress, Volume Change, and Microcracking of

Concrete”- ACI Journal Proceedings, 1968 - ACI.


73

[16] Shah, S., Swartz, S., and Ouyang, C. (1995). Fracture Mechanics of Concrete.

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[17] Simo, J.C. and Hughes, T.J.R. (1998). Computational inelasticity. Springer, Berlin,

Germany.

[18] Timoshenko, S. and Goodier J.N. Theory of Elasticity. McGrawll-Hill B.C. 1951.

[19] Zienkiewicz, O. and Taylor, R. (1994). The Finite Element Method, volume 1 & 2. McGraw-Hill, London, England, 4th edition.

CO.DRI. INTERACTIVE GRAPHICS

74

3. CO.DRI. INTERACTIVE GRAPHICS

This chapter deals with the mechanical modeling of concrete using the program

CO.DRI. (“Constitutive Drive Interactive Graphics”), originally proposed by

Willam et al.[31]. With a stress/strain history input and material parameters, this

program calculates the resulting stress/strain response using one of the available

material models. The program contains four plasticity material models all described

and calibrated in this chapter.

Numerical simulation in the non-linear range of concrete behavior will be

carried out through the “Constitutive Drive Interactive Graphics” (Willam

and Iordache, [31]) program (also known as Co.Dri.) whose key features

are listed below:

1) interactive simulation of response behavior for plain concrete

under arbitrary input histories in the form of stress, strain

(displacement) and mixed control;

2) interactive sensitivity studies of four classical plasticity models

which have been incorporated into the constitutive driver;

3) interactive comparison with experimental results of 5 load history

tests on concrete which have been included in the data base for

verifications purposes.

3.1 USER OPTIONS

Co.Dri. is a “menu driven” interactive program which prompts the user

with a sequence of questions which are arranged as follows:


75

Window 1: Selection of output type

Fig. 3.1 – Window 1 of Co.Dri.

Window 2: Selection of Load History

Window 2 prompts the user to drive the constitutive model in stress, strain

or mixed control. In other words the user has three options for applying

boundary conditions, either in the form of the history of the three principal

stresses, the three principal strains or a combination of the axial strain with

lateral stress histories (mixed control).


The input histories are either directly defined by the user or by reference to

a particular load history in the database.

Window 3: Selection of Stress State


From window 3 it’s possible to choose among the different options for

stress state as shown in figure 3.3.

Window 4: Selection of Concrete Model


76

The window 4 requests the user to choose one of the four elastoplasticity

model or the Hooke elastic model which are currently incorporated in the

Constitutive Driver.


Window 5: Selection of Experimental Test


Window 6: Selection of Material Parameters



77

The figure 3.6 displays some of the characteristic features of the three

parameters Drucker-Prager (1952) plasticity model (ABAQUS Theory

Manual [1]) (we have for example taken the Drucker-Prager model,

nothing changes for the other models). The values of the material

proprieties are listed in table 3.1.

Table 3.1 – Material proprieties.

mat(1,1) = E Elastic module

mat(2,1) = u Poisson’s ratio

mat(3,1) = yo Hardening/Softening function parameter

mat(4,1) = alpha Yield function parameter

mat(5,1) = beta Plastic potential parameter

mat(6,1) = cp1

Hardening/Softening function parameter mat(7,1) = h

mat(8,1) = yi

mat(9,1) = iend Indicator for the hardening/softening function

mat(10,1) = cp2

Hardening/Softening function parameter

mat(11,1) = yc

mat(12,1) = qc

mat(13,1) = ymax

mat(14,1) = qmax

mat(15,1) = yr

mat(16,1) = qr

mat(17,1) = z

Window 7: Selection Name of Output File

The last information that asks us the interactive program is to declare the

desired name of the output file.



78

3.2 EXPERIMENTAL DATABASE

In the experimental database, 5 tests are loaded. All experiments are stored

in a unified format and labeled to provide easy access using the

Constitutive Driver on one side and characterizing the type of experiment

on the other side. However, the identifier cannot contain all the detailed

characteristics of an experiment. Each experiment is documented by two

plots displaying the stress history and the strain history, the three principal

stresses and strains are recorded vs. time. It is understood, that in quasi-

static experiments 'time' has to be viewed as “pseudo-time” or number of

load-steps.

The database is composed of the load-history tests on concrete with a

nominal strength of f'c = 2.76ksi (19.03N/mm2). The experiments were

performed at the University of Colorado on cylindrical specimen using a

modified Hoek Cell by B.J. Hurlbut (1985, [20]). In those experiments, low

strength concrete was used.

3.2.1 IDENTIFICATION OF EXPERIMENTS

Each experiment is labeled by ten alphanumeric characters. These

descriptors refer to the testing device, the control of the applied load,

characteristics of the load history, the number of the experiment and the

number of repetitions if experiments with identical load histories were

repeated. For example, the label hc03mut.m11 designates an experiment as:

- hc...Hoek Cell.

- 03...Rounded uniaxial compressive strength of the concrete used is 3

ksi.

- m....Mixed boundary conditions: displacement control in the axial

direction and stress control in the radial direction.

- ut...Uniaxial Tension.


79

- m....Monotonic

- 1....experiment 1.

- 1....repetition of experiment with this particular load history, testing

device, concrete etc.

The experimental test data (Hurlbut, 1985 [20]) present in the database are

the following:

- hc03mut.m11: uniaxial direct tension test, NX specimen (no lateral

reading);

- hc03muc.c11: cyclic unconfined compression test, unloading and

reloading cycles in pre and post peak regime (no lateral reading);

- hc03mcc.c11: 100 psi (0.69 MPa) confined compression test,

unloading and reloading cycles in pre and post peak regime;





3.2.2 TEST APPARATUS

The experiments were performed within a servo-controlled MTS

compression - tension test apparatus with 110 kip (where 1kip = 4.448 N,

hence 110 kip = 0.4891 kN) axial load capacity. The axial load can be

applied either in load or in displacement control. The triaxial compression

experiments were performed in a modified Hoek Cell on NX size

specimens (d = 54.74 mm, h = 100 mm) and confining pressure was

applied by manually operated pumps ( Hurlbut, 1985 [20]).


80

Fig. 3.8 – Hoek triaxial cell.

The surfaces were covered with plastic filler material to avoid the intrusion

of the membrane into air voids due to confining pressure is applied during

the experiment.

Due to the application of the axial load with rigid steel end-caps, the

obtained data for the triaxial compression experiments have to be corrected

to eliminate the initial deformations produced by initial confinement. This

was done by assuming linear elastic material behavior during the first load

step when the confining pressure is applied. Hence, zero axial deformation

was assumed during this load step.

The uniaxial compressive strength of specimen was 2.76 ksi after 28 days

of curing time. The experimental data, as stored in the database are

corrected to eliminate experimental errors data in the initial phase of the

experiment. The adopted material properties for the computed

displacements are for Young's modulus 2.800.000 psi (19306 MPa) and 0.2

for Poisson's ratio.


81

3.3 CONSTITUTIVE MODELS

3.3.1 ASSOCIATED VON MISES PLASTICITY MODEL

The von Mises Criterion (1913), also known as octahedral shear stress

theory, is often used to estimate yielding of ductile materials. The features

of the von Mises model, available in Co.Dri. Interactive Graphics, are:

- Yield criterion:

03

)(),( 2

2=−=

qyJkJf n (3.1)

in which 2J is the second invariant of the deviatoric stress tensor and

pij

pijq εε

3

2= is the equivalent plastic strain.

- Isotropic Hardening/Softening law:

)(qyynn

=

3.3.1.1 Quadratic hardening/softening function.

The expression of quadratic hardening/softening law is the following:

qhqcpyqyn

++= 2

02

1)( (3.2)

The parameters y0, cp and h characterize the geometry of curve (fig. 3.9).

Fig. 3.9 – Quadratic Hardening/Softening function (Encinas, 2007 [14]).


82

in which:

- ymax is the maximum dimension of yield surface (failure reached);

- qmax is the equivalent plastic strain as hardening law reaches the

maxim value;

- y0 is the initial dimension of yielding surface.

The parameters of the hardening law are calibrated using the experimental

test data of Hurlbut at el. (1985 [20]), available in the mentioned database.

The boundary conditions assumed in calibration are listed below:

=

=∂

∂

=

maxmax

max

0

)(

0)(

)0(

yqy

qq

y

yy

n

n

n

(3.3)

developing the expressions (3.3), we obtain:

−=

−=

max

max

max0)(2

qcph

q

yycp

(3.4)

3.3.1.2 Simo hardening/softening function.

The exponential expression proposed by Simo et al. (1998, [28]) is the

following:

( )( ) qheyyyqyqcp

in+−−+= − 1

001)( (3.5)

calibrating the function parameters according to the experimental tests

present in the database of Co.Dri. (Hurlbut, 1985).

The boundary conditions used are the followings (figure 3.10):

=

=∂

∂

=

=

rrn

n

n

n

yqy

qq

y

yqy

yy

)(

0)(

)(

)0(

max

maxmax

0

(3.6)


83

Fig. 3.10 – Exponential Simo Hardening/Softening function (Simo at el, 1998, [28]).

in which:

- y0 is the initial dimension of yield criterion;


- qmax is the equivalent plastic strain when the hardening law reaches

the maxim value;

- yr is the dimension of yield surface at the ultimate strength (residual

strength);

- qr is the equivalent plastic strain when hardening law reaches the

residual value;

From the third condition of Equation (3.6):

⇒=∂

∂0)(

maxq

q

yn ( )( ) 0)(

1

max1

0=+−−− −

hceyyp

qcp

i (3.7)

Now, substituting Eq. (3.7) into the second condition of (3.6):

⇒=maxmax

)( yqyn

( )( ) ( )( )max1

max1

0

max1

00max)(1 qceyyeyyyy

p

qcp

i

qcp

i−−+−−+= −−

(3.8)

We define the following quantity:


84

( ) ( )[ ]max1

max1

0max

011

qcp

p

ieqc

yyyyz

−+−

−=−= (3.9)

Finally, we substitute Eq. (3.9) into the last condition of (3.6):

( )[ ]r

qcpqrcp

rqecpezyy

max11

011

−− −−+= (3.10)

The only unknown independent variable in the previous expression is cp1.

Eq. (3.10) is in implicit form, to search the root or solution (cp1) of the

equation a numerical iteration is used.

3.3.1.3 Simo Modified hardening/softening law.

Fig. 3.11 – Modified Simo Hardening/Softening function (Encinas, 2007 [14]).

The Simo Modified expression proposed by Etse (2007, [14]) et al. can be

placed in the following form:

( )( )( )

≤<

≤≤+−−+=

−

−

rc

qcqcp

c

c

qcp

i

n

qqqperey

qqperqheyyyqy

2

1

0001

)( (3.11a & b)

- y0 is the initial dimension of yield criterion;


- qmax is the equivalent plastic strain when the hardening law reaches

the maximum value;


85

- yc is the dimension of yield surface when the expression of

hardening/softening law changes (from 3.11a to 3.11b);

- qc is the equivalent plastic strain when the hardening law reaches the

yc value;

- yr is the dimension of yield surface at the ultimate strength (residual

strength);

- qr is the equivalent plastic strain when the hardening law reaches the

residual value.

We calibrate the function parameters using the Hurlbut experimental test

data (1985). The boundary conditions are the followings:

=

=

=∂

∂

=

=

rrn

ccn

n

n

n

yqy

yqy

qq

y

yqy

yy

)(

)(

0)(

)(

)0(

2

1

max

1

maxmax1

01

(3.12)

Now we can use the expression obtained in the section 3.3.1.2:

( )( )

( )( )[ ]

( )[ ]

−−+=

+−

−=−=

=+−−−

−−

−

−

c

qcpqccp

c

qcpi

qcp

i

qecpezyy

eqcp

yyyyz

hcpeyy

max11

0

max1

max

0max

0

max1

0

11

111

0)1(

(3.13)

and from the last relationship of (3.12), we seek the ultimate constant of the

hardening law:

( )qcqrcp

creyy

−= 2 (3.14)

3.3.1.4 Calibration and validation of the von Mises model.

The one-parameter von Mises criterion is calibrated from a triaxial

compression test with radial confinement σr= - 0,69 MPa (figure 3.12),

such assumption is arbitrary since as a matter of principle, only one


86

parameter defines the von Mises criterion and its value could be different.

Generally for ductile materials, as metals, failure criteria are calibrated with

respect to tension tests.

The Modified Simo hardening/softening law is used and it’s calibrated with

five load paths, one test in direct-tension and four test in compression for

different values of lateral confinement pressure of the fc’ = 19.03 MPa

concrete.

Fig. 3.12 – von Mises failure criterion compared with Hurlbut test data (1985) in the

meridian plane.

Figure 3.12 shows a comparison of the von Mises failure criterion with

Hurlbut test data [20]. Each experimental test is plotted in the I1 - J20.5

plane

through the following transformations:

- direct tension: tfI =

1

3

5.0

2

tf

J =

- direct compression: c

fI −=1

3

5.0

2

cf

J =

- confined compression: lccffI 2

1−−=

3

5.0

2

lccff

J−

=


87

in which ft is the uniaxial tensile strength, fc is the uniaxial

compressive strength, fcc is the confined triaxial compressive

strength and fl is the constant lateral confinement.

The material parameters of the failure surface and the hardening law are

summarized in table 3.2:

Table 3.2 – Material parameters.

hc03mut.m11 hc03muc.c11 hc03mcc.c11 hc03mcc.c21 hc03mcc.c31

Failure

surface

parameter

Yn [MPa] 25.04

Modified Simo

hardening/

softening

law

cp1 210799.64 1252.71 929.54 1557.25 196.22

h [MPa] -154027.24 -4695.74 -937.92 0 -211.26

yi [MPa] 28.22 36.32 29.18 25.04 29.69

cp2 -2589.88 -122.60 -17.08 - -

y0 [MPa] 7.48 8.41 6.72 1.51 0.16

yc [MPa] 8.31 8.05 19.17 - -

qc 0.000129 0.00602 0.01067 - -

Figure (3.13) to (3.17) show the performance of the von Mises model

compared with the results of the direct-tension, direct compression and

triaxial compression tests under various radial confinement pressures σr= -

0.69, -3.45, -13.78 MPa.

Fig. 3.13 – Comparison with Direct-Tension-Test Data (Hurlbut, 1985)


88

For the compression case, the failure criterion overestimates the strength

(figure (3.14)), according to the failure conditions of figure 3.12: the

compression test (hc03muc.c11) is below the compressive meridian of the

failure surface.

Fig. 3.14 – Comparison with Compression Test Data (Hurlbut, 1985)

Fig. 3.15 – Comparison with σr=-0.69 MPa Confined Triaxial Compression Test Data

(Hurlbut, 1985).

One-parameter von Mises criterion doesn't gather the different tensile and

compressive behavior. The model excessively overestimates the uniaxial


89

tensile strength (figure (3.13)). The initial yielding surface is never reached

from the internal stresses such that the behavior of the material is elastic-

linear up to ultimate load-step.


(Hurlbut, 1985).

The model predictions for confined compression test (hc03mcc.c11) with

radial constant confinement σr = -0.69 MPa (fig. 3.15) is excellent (the

model is just calibrated for this test). Figure 3.12 shows as the same

confined compression test (hc03muc.c11) is situated just along the

compressive meridian.

For medium lateral confinement (σr = -3.48 MPa) and high lateral

confinement (σr = -13.78 MPa), the failure criterion underpredicts the

respective strength of the tests (figures (3.16) and (3.17)), according to the

failure conditions of figure 3.12: the confined compression tests with

medium (hc03mcc.c21) and high confinement (hc03mcc.c31), respectively,

are above the compressive meridian of the failure surface.


90


(Hurlbut, 1985).

Further details about the implementation of the von Mises model with

Co.Dri. can be found in the final Appendix.

3.3.2 NON-ASSOCIATED DRUCKER-PRAGER PLASTICITY

MODEL (TWO PARAMETERS).

Drucker Prager model (1952), is typically used for soils, concrete and other

frictional materials. The features of model, available in Co.Dri. Interactive

Graphics, are:

- Yield criterion:

03

)(),,(

12

21=−+=

qyIJkJIf nα (3.15)

1I , is the first invariant of the stress tensor, 2J is the second invariant of the

deviatoric stress tensor and p

ijp

ijq εε3

2= is the equivalent plastic strain.

The parameters α and yf (peak value of )(qyn

function) are the friction and

the cohesion, respectively, for the failure model, also known as material


91

constants. These parameters can be calibrated from experimental data as

described in the previous chapter of this thesis (§ 2.2.2.2).

- Isotropic Hardening/Softening law:

)(qyynn

=

The expressions of isotropic hardening/softening law, available in the

Drucker-Prager elastoplasticity model, are the same of the von Mises

model:

- Quadratic hardening/softening function:

qhqcyqypn

++= 2

02

1)(

- Simo hardening/softening function:

( )( ) qheyyyqyqcp

in+−−+= − 1

001)(

- Modified Simo hardening/softening function:

( )( )( )

≤<

≤≤+−−+=

−

−

rc

qcqcp

c

c

qcp

i

n

qqqperey

qqperqheyyyqy

2

1

0001

)(

The parameters of the hardening law are calibrated using the experimental

test data of Hurlbut at el. (1985), present in the database (§ 3.3.1).

- Plastic potential:

12

21),,( IJkJIg β+= (3.16)

The form of plastic potential is a Drucker-Prager type, in which the

frictional parameter α is opportunely modified to give β. In our models a

constant value of αβ ≠ is assumed. The original plasticity model of

Drucker-Prager (1958) is with α = β in such case the model takes the

diction: “Associated Drucker-Prager plasticity model” not appropriate to

the concrete.

3.3.2.1 Calibration and validation of Drucker-Prager model.


92

The two-parameters Drucker-Prager failure criterion is calibrated using two

experimental test. Usually a direct tension test and a direct compression test

are used for this purpose (figure 3.18).

The Modified Simo hardening/softening law is used and calibrated with

respect to five load paths, one test in direct-tension and four tests in

compression at different levels of confinement of the fc’ = 19.03 MPa

concrete (Hurlbut, 1985).

Fig. 3.18 – Drucker Prager failure criteria compared with Hurlbut test data (1985) in the

meridian plane.

Figure 3.18 shows a comparison of the Drucker-Prager failure criterion

with Hurlbut test data [20]. Each experimental test is plotted in the I1 - J20.5

plane through the following transformations:


1

3

5.0

2

tf

J =


fI −=1

3

5.0

2

cf

J =


93


1−−=

3

5.0

2

lccff

J−

=

The material parameters of the failure surface, flow law and the hardening

law are summarized in table 3.3.



Failure

surface

parameter

α 0.4398

Yf [MPa] 4.53

Modified

Simo

hardening/ softening

law

cp1 258167.76 1252.71 927.84 1557.25 132.68

h [MPa] -35898.1 -893.5 -238.2 0 -102.6

yi [MPa] 5.36 6.91 5.81 4.76 7.27

cp2 -3171.94 -122.60 -29.56 - -

y0 [MPa] 1.42 1.59 0.13 0.18 0

yc [MPa] 1.58 1.53 3.28 - -

qc 0.000105 0.00602 0.01066 - -

Plastic

Potential

parameter

β 0.433 0.1625 0.13 0 -0.16

Figures from (3.19) to (3.23) show the predictions of the Drucker-Prager

plasticity model (two parameters) compared with the results of the direct-

tension experiment and those of triaxial compression tests with radial

confinements σr= -0.69, -3.45, and -13.8 MPa.

The model predictions for direct tension (fig. 3.19) and direct compression

(fig. 3.20) are admirable (the model is just calibrated for these tests). Figure

3.18 shows as the direct tension test (hc03mut.m11) and the direct

compression test (hc03muc.c11) are collocated just along the failure model.


94



For low lateral confinement (σr = -0.69 MPa) the strength prediction of

model is acceptable (figure (3.21)), according to the failure conditions of

figure 3.18.


95


(Hurlbut, 1985).


(Hurlbut, 1985).

The model overestimates the material response as the confinement pressure

increases. For medium and high lateral confinement (σr = -3.48 and -13.79

MPa) the model overestimates the strength (fig. 3.22 and 3.23), according

to the failure conditions of figure 3.18 dealing with the failure envelope of

model with respect to the experimental test (hc03mcc.c21 and

hc03mcc.c31).


96


(Hurlbut, 1985).

Fig. 3.24 – Comparison between associated and non-associated plasticity model for

direct compression case.

It’s interesting to observe resulting by consulting whether associated or

non-associated plasticity model. The concrete behavior is not compatible

with the associated flow rule (Chen and Han, 1988 [6]) and a non-

associated plastic potential is needed to define the flow rule. A non-

associated flow rule is adopted to control the amount of inelastic dilatancy:


97

αβ < (non-associated flow rule)

and thus the lateral deformation behavior in stress-controlled environments

(from fig. 3.24 to fig.3.27).

Fig. 3.25 – Comparison between associated and non-associated plasticity model with

σr=-0.69 MPa Confined Triaxial Compression.




98



Further details about the implementation of the Drucker-Prager model with


3.3.3 NON-ASSOCIATED DRUCKER-PRAGER PLASTICITY

MODEL (THREE PARAMETERS).

The analysis in the previous chapters point out that the behavior of concrete

and granular materials is too complex to be reproduced by plasticity models

initially conceived for ductile materials. However, under the loading

conditions of interest for engineers, rather simple constitutive models lead

to useful design information. These constitutive models are essentially

pressure-dependent plasticity models that have historically been popular in

geotechnics. However, more recently they have also been found to be

useful for modeling some composite materials such as concrete, that

exhibits significantly different yield behavior in tension and compression.

The model described here is an extension of the original Drucker-Prager

model (Drucker and Prager, 1952). The extension of interest includes the


99

use of noncircular yield surfaces in the deviatoric stress plane, and the use

of non-associated flow law as detailing described in this chapter. Such a

failure criterion can be expressed as follows:

0tan =−− dpt β (3.17)

with,

⋅

−−+=3

11

11

2

1

q

r

KKqt (3.18)

Fig. 3.28 – Failure criterion in the meridian plane.

where:

- 1

3

1

3

1Itracep −=−= σσσσ equivalent pressure stress;

- 2

32

3JSSq

ijij== Mises equivalent stress;

- 3

1

3

3

1

2

27

2

9

=

= JSSSrkijkij

third invariant of deviatoric stress;

- ijijij

pS σδ += , is the deviatoric part of the stress tensor σσσσ.

The model provides a noncircular section and associated inelastic flow in

the deviatoric plane, while separate dilation and friction angles in the

meridian plane. Input data parameters (Hurlbut et al., 1985) define the


100

shape of the failure and flow surfaces in the deviatoric plane as well as the

friction and dilation angles for the meridian plane.

Fig. 3.29 – Typical failure surface for the Drucker-Prager model (three parameters), in

the deviatoric plane (ABAQUS Theory Manual, [1]).

In this model we define a deviatoric stress measure t (Eq. (3.18)), in which

K is a material parameter. To ensure convexity of the yield surface,

1778.0 ≤≤ K . This measure of deviatoric stress is used because it allows

the matching of different stress values in tension and compression in the

deviatoric plane, thereby providing flexibility in fitting experimental results

when the material exhibits different failure values in triaxial tension and

compression tests. This function is sketched in Figure 3.29.

Since 1

3

=

q

r in uniaxial tension,

K

qt = in this case; 1

3

−=

q

r in

uniaxial compression and qt = in this other case. When K = 1, the

dependence on the third deviatoric stress invariant is removed; the Mises

circle is recovered in the deviatoric plane: qt = .


101

The parameters β, d and K are the material constants and can be calibrated

from experimental data in the following procedure.

We consider the failure values of three experimental tests in terms of stress

invariants:

333

222

111

,,

,,

,,

rqp

rqp

rqp

(3.19)

Substituting the first test of (3.19) into Eq. (3.17):

dpq

r

KKq =−

−−+ βtan1

11

12

11

3

1

1

1 (3.20)

Now, substituting the second test of (3.19) and Eq. (3.20) into Eq. (3.17),

we obtain:

βtan)(

11

11

2

111

11

2

1

12

3

1

1

1

3

2

2

2

pp

q

r

KKq

q

r

KKq

−

=

−−+−

−−+ (3.21)

Resolving for βtan :

)(

11

11

2

111

11

2

1

tan12

3

1

1

1

3

2

2

2

pp

q

r

KKq

q

r

KKq

−

−−+−

−−+

=β (3.22)

Finally, substituting the third test (3.19) into the failure criterion (3.17) and

using Eq. (3.22), the only material parameter to determine is K:

0tan1

11

12

13

3

3

3

3=−−

−−+ dpq

r

KKq β (3.23)

An isotropic hardening/softening elastoplasticity model is formulated and

the yield surface can be expressed as follows:

( ) 0tan =−− qdpt β (3.24)


102

where )(qd is the Isotropic Hardening/Softening law. As for the previous

models (von Mises and Drucker-Prager) the expressions of isotropic

hardening/softening law are the followings:


qhqcpdqd ++= 2

02

1)(


( )( ) qhedddqdqcp

i+−−+= − 1

001)(


( )( )( )

≤<

≤≤+−−+=

−

−

rc

qcqcp

c

c

qcp

i

qqqpered

qqperqhedddqd

2

1

0001

)(

Equally to the previous models, the parameters of the hardening laws are

calibrated using the experimental test data of Hurlbut at el. (1985), loaded

in the database (§ 3.3.1) of Co.Dri..

Fig. 3.30 – Schematic of hardening for the linear model, in the meridian plane.

The plastic potential, also known as the flow potential, chosen in this

model is:

ψtanptg −= (3.25)


103

where ψ is the dilation angle (see β at Section 2.4.1) in the t–p plane. A

geometrical interpretation of ψ is shown in the t–p diagram of Figure 3.30.

Comparison of Eq. (3.17) and Eq. (3.25) shows that the flow is associated

in the deviatoric plane, because the yield surface and the flow potential

both have the same functional dependence on t.

However, the dilation angle ψ and the material friction angle β may be

different, so the model may not be associated in the t–p plane. For 0=ψ

the material is nondilational and if βψ = , the model is fully associated.

For βψ = and 1=K the original associated Drucker-Prager (1952) model

is recovered.

3.3.3.1 Calibration and validation of Drucker-Prager model.

The three-parameters Drucker-Prager failure criterion is calibrated using

three experimental test. For this reason, a direct tension test a direct

compression test and a confined triaxial compression test (radial stress σr=

-0.69 MPa) are used (figure 3.31).

Fig. 3.31 – Drucker Prager failure criterion (three parameters) compared with Hurlbut

test data (1985), in the meridian plane.


104

Figure 3.31 shows a comparison of the Drucker-Prager failure criterion

(three parameters) with Hurlbut test data [20]. Each experimental test is

plotted in the I1 - J20.5



1

3

5.0

2

tf

J =


fI −=1

3

5.0

2

cf

J =


1−−=

3

5.0

2

lccff

J−

=

The isotropic hardening/softening function is used through the Modified

Simo law, that is calibrated with five load paths, one test in direct-tension

and four test in compression at different levels of confinement of the fc’ =

19.03 MPa concrete (Hurlbut, 1985).

The material parameters defining the failure surface, the nonassociated

flow law and the hardening law are summarized in table 3.4:



Failure

surface

parameter

tgβ 2.1477

df [MPa] 5.4058

K 0.788

Modified Simo

hardening/

softening law

cp1 287156.44 1419.13 2015.57 1769.34 200

h [MPa] -39414 -916.384 -256.79 0 -40.236

di [MPa] 6.05 7.52 5.96 5.36 6.65

cp2 -2996.57 -119.69 -25.83 - -

d0 [MPa] 0.38 1.24 0.77 -34.7 -29.5

dc [MPa] 1.61 1.49 3.95 - -

qc 0.0001126 0.006569 0.007844 - -

Plastic

Potential

parameter

tgψ 2.14 1.15 0.75 0 -0.80


105

Figures from (3.32) to (3.36) show the predictions of the Drucker-Prager

plasticity model (three parameters) compared with the results of the direct-

tension experiments and the triaxial compression tests with radial




For direct tension (fig. 3.32) and direct compression (fig. 3.33), the

Drucker-Prager (three parameters) model predicts the mechanical behavior


106

in an admirable manner. Figure 3.31 shows as the direct tension test

(hc03mut.m11) and the direct compression test (hc03muc.c11) are

collocated just along the failure model.

According to the failure conditions of figure 3.31, for low constant

confinement (σr = -0.69 MPa) the mechanical response of model is

acceptable (figure (3.34)).


(Hurlbut, 1985).

For medium and high lateral pressure (σr = -3.48 and -13.79 MPa) the

model overestimates the confined compression strength (fig. 3.35 and

3.36), according to the failure conditions of figure 3.31 dealing with the

failure envelope of model with respect to the experimental tests

(hc03mcc.c21 and hc03mcc.c31). The model overestimates the material

response as the confinement pressure increases.


107


(Hurlbut, 1985).


(Hurlbut, 1985).

Further details about the implementation of the Drucker-Prager model

(three parameters) with Co.Dri., can be found in the final Appendix.


108

3.3.4 NON-ASSOCIATED BRESLER-PISTER PLASTICITY MODEL

As observed in the previous paragraphs, failure models generally described

by straight line, in the meridian plane, are inadequate for describing the

failure of concrete in the high-confinement range.

The generalized Drucker-Prager surface proposed by Bresler and Pister

(1958 [4]) is a three-parameters model which assumes a parabolic

dependence of 2

J on 1

I , while the deviatoric sections are independent of

θ :

02

2

11=−+− JcIbIa (3.26)

The model provides a circular section in the deviatoric plane and curved

meridian in2

J ,1

I plane.

The parameters a, b and c are material constants and can be defined

according to experimental data.

If we consider the failure values of three experimental tests in terms of

stress invariants:

3

2

3

1

2

2

2

1

1

2

1

1

,

,

,

JI

JI

JI

(3.27)

Substituting the tests of (3.27) into Eq. (3.26), we obtain a

nonhomogeneous system of linear equations:

=−+−

=−+−

=−+−

0

0

0

3

2

32

1

3

1

2

2

22

1

2

1

1

2

12

1

1

1

JIcIba

JIcIba

JIcIba

(3.28)


109

and, in matrix form, one obtains:

=

−

−

−

3

2

2

2

1

2

32

1

3

1

22

1

2

1

12

1

1

1

1

1

1

J

J

J

c

b

a

II

II

II

(3.29)

and, consequently:

bbbbaaaa=MMMM (3.30)

Now, inverting equation (3.30), the vector aaaa of material constants

describing the failure criterion can be determined:

bbbbaaaa -1-1-1-1MMMM= (3.31)

The yield surface, with an isotropic hardening/softening elastoplasticity

model, is the following:

( ) 02

2

11=−+− JcIbIqa (3.32)

in which )(qa is the Isotropic Hardening/Softening law, the expression of

the same law, available in this model, are the same one of the other models

previously viewed:


qhqcpaqa ++= 2

02

1)(


( )( ) qheaaaqaqcp

i+−−+= − 1

001)(


( )( )( )

≤<

≤≤+−−+=

−

−

rc

qcqcp

c

c

qcp

i

qqqperea

qqperqheaaaqa

2

1

0001

)(


110

As in the case of the other introduced models, the parameters of hardening

law are calibrated using the experimental test data of Hurlbut at el. (1985),

present in the database (§ 3.3.1) of the numerical program.

The plastic potential for this model is:

21

'JIbg −= (3.33)

in which '

b is the dilation angle in the 21

JI − plane.

For this model, as for the others, the flow law is associated in the deviatoric

plane, while is nonassociated in the meridian planes. When 0' =b , the

material is nondilational. For bb =' and 0=c the original associated

Drucker-Prager (1952) plasticity model is recovered.

3.3.4.1 Calibration and validation of Bresler-Pister model.

The three-parameters Bresler-Pister failure criterion is calibrated using

three experimental test. For this reason, a direct tension test a direct

compression test and a high confined triaxial compression test (radial stress

σr=-13.79 MPa) are used (figure 3.37).

The Modified Simo law is used as isotropic hardening/softening function,

according the same law with five load paths, one test in direct-tension and

four test in compression at different levels of confinement of the fc’ = 19.03

MPa concrete (Hurlbut 1985).

Figure 3.37 shows a comparison of the Bresler-Pister failure criterion (three

parameters) with Hurlbut test data [20]. Each experimental test is plotted in

the I1 - J20.5



1

3

5.0

2

tf

J =


fI −=1

3

5.0

2

cf

J =


111


1−−=

3

5.0

2

lccff

J−

=

Fig. 3.37 – Bresler and Pister failure criteria (three parameters) compared with Hurlbut

test data (1985), in the meridian plane.



Failure

surface

parameter

af [MPa] 2.81

b 0.4527

c [MPa-1] -0.00122

Modified

Simo hardening/

softening

law

cp1 287156.44 1172.76 1958.52 1769.34 250

h [MPa] -20524.19 -471.12 -160.87 0 0

ai [MPa] 3.15 4.08 3.19 2.81 3.03

cp2 -2998.130138 -107.906 -32.881157 - -

a0 [MPa] 0.19 0.56 0.17 -13.55 -16.54

ac [MPa] 0.83 0.81 1.93 - -

qc 0.0001126 0.00694 0.00784 - -

Plastic

Potential

parameter

b’ 0.35 0.1875 0.15 0 -0.20


112

The material parameters of the failure surface, nonassociated flow and the

hardening law are summarized as in table 3.5.

Figures from (3.38) to (3.42) show the predictions of the Bresler-Pister

plasticity model (three parameters) compared with the results of the direct-

tension experiments and those of triaxial compression tests with radial


Figure 3.37 shows as the tensile test (hc03mut.m11) is located just along

the failure meridian. For this reason, that the model predicts in an excellent

manner the mechanical response of uniaxial tensile case (Fig. 3.38).


The model prediction for direct compression is also excellent (Fig. 3.39).

Figure 3.37 shows that, as for the direct tension case, the direct

compression test (hc03muc.c11) is situated just along the failure model.

Figure 3.37 shows as the confined compression test (hc03mcc.c11) with

radial constant confinement σr = -0.69 MPa is located nearby the


113

compressive meridian. For this reason the model prediction is acceptable

for low confined test (σr = -0.69 MPa, see fig. 3.40).



(Hurlbut, 1985).

For medium lateral confinement (σr = -3.48 MPa) the failure criterion

overestimates the strength (figure (3.41)), according to the failure


114

conditions of figure 3.37: the confined compression test with medium

confinement (hc03mcc.c21) is below the compressive meridian of the

failure surface.


(Hurlbut, 1985).


(Hurlbut, 1985).


115

Figure (3.42) shows the predictions of the Bresler and Pister model

compared with the the triaxial compression test with radial confinement

σr= -13.78 MPa: the model prediction for this load-case is excellent. Figure

3.37 shows as the (σr= -13.78 MPa) confined compression test

(hc03mcc.c31) is situated just along the failure meridian.

Further details about the implementation of the Bresler-Pister model with



116

REFERENCES OF THE THIRD CHAPTER



02909, USA.


Jersey, USA. [4] Bresler B and Pister KS (1958), Strength of concrete under combined stresses, ACI Mater. J., 55, 321–345.






[8] Chinn, J. and Zimmermann, R. (1965). Behavior of plain concrete under various

high triaxial loading conditions. Technical Report, University of Colorado, Boulder, USA.


York, 1995.

[10] Crisfield, M. A. Non-linear Finite Element Analysis of Solids and Structures, Vol.

1-2.

[11] C. S. Desai, H. J. Siriwardane, Constitutive Law for Engineering Materials,

Prentice-Hall (1984). [12] Desayi, P. and Krishnan, S., Equation for the stress-strain curve of concrete, ACI

J., Vol. 61(1964)345-350.

[13] Drucker, D. (1959). A definition of stable inelastic materials. Journal of Applied Mechanics, 26:101-106.

[14] Encinas Galdo J. D. (2007), “Teorias constitutivas elastoplasticas y analisis

computacional del comportamiento mecanico del hormigon”. Proyecto final de

carrera.

[15] Etse, G. (1992). Theoretische und numerische Untersuchung zum di_usen und

lokalisierten Versagen in Beton. PhD thesis, Universitat Karlsruhe, Karlsruhe,

Germany..


117

[16] Este, G., Willam, K.J., A fracture-energy based constitutive formulation for

inelastic behavior of plain concrete, 1994, Journal of Engineering Mechanics, ASCE 120, 1983-2011.

[17] Etse, G. and Willam, K. (1996). Integration algorithms for concrete plasticity.

Engineering Computations, 13(8):38-65.

[18] Etse, G. and Willam, K. (1999). Failure analysis of elastoviscoplastic material models, ASCE-EM, 125(1):60-69.

[19] Gerstle, K.H., Simple formulation of biaxial concrete behavior, ACI Journal, 78(1981)62-68.

[20] Hurlbut, B. J., Experimental and Computational Investigation of Strain-Softening in Concrete, MS thesis, University of Colorado, Boulder,. 1985.

[21] MD Kotsovos, JB Newman, Behavior of Concrete Under Multiaxial Stress by - ACI

Journal Proceedings, 1977 - ACI.

[22] Lee, J., and G. L. Fenves, Plastic-Damage Model for Cyclic Loading of Concrete

Structures, Journal of Engineering Mechanics, vol. 124, no.8, pp. 892–900, 1998.

[23] Lubliner, J., J. Oliver, S. Oller, and E. Oñate, A Plastic-Damage Model for

Concrete, International Journal of Solids and Structures, vol. 25, no.3, pp. 229–326, 1989.


[25] D. Sfer, I. Carol, R. Gettu and G. Etse, "Study of the Behaviour of Concrete Under Triaxial Compression", J. of Engng. Mech., V. 128, No. 2, pp. 156-163 (2002).

[26] SP Shah, S Chandra. Critical Stress, Volume Change, and Microcracking of

Concrete - ACI Journal Proceedings, 1968 - ACI.

[27] Shah, S., Swartz, S., and Ouyang, C. (1995). Fracture Mechanics of Concrete.

John Wiley & Sons, Inc.

[28] Simo, J.C. and Hughes, T.J.R. (1998). Computational inelasticity. Springer, Berlin,

Germany.

[29] Sinha B.P., Gerstle K.H., Stress-Strain Relations for Concrete under Cyclic

Loading, and Tulin L.G., Journal of ACI, Proc., 1964, 61(2), pp. 195-211.

[30] J.G.M. van Mier, Strain-softening of concrete under multiaxial loading conditions,

PhD. thesis, Eindhoven University of Technology, (1984).

[31] Willam, K. and Iordache, M.-M., (1996), ``Constitutive Driver for Cohesive-

Frictional Materials,'' Proc. 4th-ASCE Materials Conference, Washinton D.C., Nov. 10-14, 1996.


118


CONSTITUTIVE MODELS AVAILABLE IN ABAQUS

119

4. CONSTITUTIVE MODELS AVAILABLE IN

ABAQUS

This chapter deals with the mechanical modeling of concrete using the finite

element analysis program ABAQUS (Student Edition). A constitutive model for

concrete, which uses the plasticity theory, is available in ABAQUS. Primarily a

detailed description of the model is treated while the last part of the chapter is

devoted to calibrate and validate a model on experimental tests available in

literature.

A large variety of materials of interest in engineering respond elastically at

least under reasonably limited values of stresses and strains. Elastic

behavior means that the deformation is fully recoverable: the specimen

return to its original shape as the load is removed. If the load exceeds some

limit (the “yield load”), the deformation is no longer fully recoverable.

Plasticity theory models the mechanical response of materials when

nonrecoverable deformations occur. The theory has been developed most

intensively for metals, but it is also applied to soils, concrete, rock, ice, etc.

(ABAQUS Manuals, [1] and [2]). These materials behave in very different

ways. For example, large values of pure hydrostatic pressure cause small

inelastic deformation in metals, but quite small hydrostatic pressure values

may result in significant, nonrecoverable volume change in a concrete

specimen. Nonetheless, the fundamental concepts of plasticity theory are

sufficiently general and models based on these concepts have been

successfully developed for a wide range of materials.

Most of the plasticity models in ABAQUS are based on the “incremental”

theory in which the mechanical strain rate is decomposed into an elastic

part and a plastic (inelastic) part. Incremental plasticity models are usually


120

formulated in terms of (Lubliner, 1990 [20]):

- a yield surface, which generalizes the concept of “yield load” and it

can be used to determine if the material responds purely elastically at

a particular state of stress, temperature, etc;

- a flow rule, which defines the inelastic deformations that occur if the

material point is no longer responding purely elastically;

- evolution laws of the yield surface as the inelastic strains occur

(hardening/softening behavior).

This chapter describes the “Concrete Damaged Plasticity” model provided in

ABAQUS (Student Edition) for the analysis of concrete and other quasi-brittle

materials.

The plasticity model of damage concrete is primarily intended to be used for

analysis of structures under cyclic and/or dynamic loading. The model is also

suitable for the analysis of other quasi-brittle materials, such as rocks, mortars

and ceramics; but it is the behavior of concrete that is used in the remainder of

this section to motivate different aspects of the constitutive theory. Under low

confining pressures, concrete behaves in a brittle manner; the main failure

mechanisms are cracking in tension and crushing in compression (ABAQUS

Manuals [1] and [2]).

Brittle behavior of concrete disappears when confining pressure is sufficiently

large to prevent crack propagation. In these circumstances failure is driven by

the consolidation and collapse of the concrete porous microstructure, leading

to a macroscopic response resembling the aim of a ductile material with

hardening work.

Modelling the behavior of concrete under large hydrostatic pressures is out of

the scope of the plastic-damage model available in ABAQUS (Student

Edition). The constitutive theory in this section aims to capture the effects of

irreversible damage associated with the failure mechanisms that occur in


121

concrete under fairly low confining pressures (less than four or five times the

ultimate compressive stress in uniaxial compression loading, ABAQUS

Manuals [1] and [2]).

The principal features of concrete are summarized in the following

macroscopic properties (ABAQUS Manuals [1] and [2]):

- different yield strengths in tension and compression, with the initial

yield stress in compression being 10 or more higher time than the

initial yield stress in tension;

- softening behavior in tension as opposed to initial hardening

followed by softening in compression;

- different degradation of the elastic stiffness in tension and

compression; and

- stiffness recovery effects during cyclic loading;

The model assumes that the uniaxial tensile and compressive response of

concrete is characterized by damaged plasticity, as shown in Figure 4.1.

Under uniaxial tension the stress-strain response follows a linear elastic

relationship until the value of the failure stress0t

σ is attained. The failure

stress corresponds to the onset of micro-cracking in concrete. Beyond the

failure stress the formation of micro-cracks is represented macroscopically

with a softening stress-strain response, which induces strain localization in

the concrete structure. Under uniaxial compression the response is linear

until the value of initial yield, 0c

σ . In the plastic regime the response is

typically characterized by stress hardening followed by strain softening

beyond the ultimate stress,cu

σ (Figure 4.1.b).

The plastic-damage model in ABAQUS is based on the models proposed

by Lubliner et al. (1989) [20] and by Lee and Fenves (1998) [19].

An additive strain rate decomposition is assumed for the plasticity model:


122

p

ij

e

ijijddd εεε += (4.1)

where ij

dε is the total strain rate, e

ijdε is the elastic part of the strain rate,

and p

ijdε is the plastic part of the strain rate.

Fig. 4.1 – Uniaxial response of concrete in tension (a) and compression (b), ABAQUS

Theory Manual [1].

The stress-strain relations are governed by scalar damaged elasticity:


123

( ) ( ) ( )plelpleld εεεεεεεεDDDDεεεεεεεεDDDDσσσσ −=−−= ::1

0 (4.2)

where, el

0DDDD is the initial (undamaged) elastic stiffness of the material.

( ) eleld

01 DDDDDDDD −= is the degraded elastic stiffness; and d is the scalar stiffness

degradation variable (in figure 4.1, dc is the stiffness degradation variable

for the compression case and dt is the stiffness degradation variable for the

tensile case), which can take values in the range from zero (undamaged

material) to one (fully damaged material).

Within the context of the scalar-damage theory, the stiffness degradation is

isotropic and characterized by a single degradation variable, d. Following

the usual notions of continuum damage mechanics, the effective stress is

defined (Lubliner, 1989 [20]) as:

( )plel εεεεεεεεDDDDσσσσ −= :0

(4.3)

The Cauchy stress is related to the effective stress through the scalar

degradation relation:

( )σσσσσσσσ d−= 1 (4.4)

In the absence of damage, d=0, the effective stress is equivalent to the

Cauchy stress. It is convenient to formulate the plasticity problem in terms

of the effective stress using equation (4.4) to calculate the Cauchy stress.

Our prediction model has been calibrated not considering the part of the

model regarding the damage. Under such hypothesis the effective stress is

equal to the Cauchy stress and the stiffness degradation variable is null.

4.1 YIELD AND FAILURE SURFACE

The yield function, ( )plε~,,,,σσσσFF = , represents a surface in effective stress

space, which determines the states of failure or damage. For the plastic-

damage model ( ) 0~ ≤plε,,,,σσσσF .

The plastic-damage concrete model uses a yield condition based on the


124

yield function proposed by Lubliner et al. (1989) [20] and incorporates the

modifications proposed by Lee and Fenves (1998) [19] to account for

different evolution of strength under tension and compression. In terms of

effective stresses the yield function takes the form:

( ) ( )( ) ( ) 0~ˆˆ~31

1~maxmax

≤−−−+−−

= pl

pl

pl εσσγσεβα

αε

cpqF ,,,,σσσσ (4.5)

where α and γ are material constants;

13

1

3

1Itracep −=−= σσσσ

is the effective hydrostatic stress;

23

2

3JSSq

ijij==

is the Mises equivalent effective stress;

ijijijpS σδ +=

is the deviatoric part of the effective stress tensor σσσσ, and max

σ is the

algebraically maximum eigenvalue of σσσσ. The operator applied to the

quantity max

σ is called “the Macauley bracket” (Lubliner et al., 1989 [20]):

)ˆˆ(2

1ˆmaxmaxmax

σσσ += (4.6)

The function ( )plεβ ~ is given as:

( ) )1()1()~(

)~(~ ααεσεσ

εβ +−−= pl

plp

ltt

cc (4.7)

plc

ε~ and plt

ε~ are the equivalent plastic strains in compression and tension,

respectively:

dtt

cc ∫=0

~~ pl

pl εε ɺ

and

dtt

tt ∫=0

~~ pl

pl εε ɺ

.


125

The effective plastic strain rates are given as:

pl

pl 11

~~ εε ɺɺ −=c

in uniaxial compression;

pl

pl 11

~~ εε ɺɺ =t

in uniaxial tension;

cσ and

tσ are the effective tensile and compressive stresses, respectively.

If we take the tensile and compression stress case:

- compression case:

cIp σ

3

1

3

11

=−=

ccJq σσ === 2

23

133

0ˆmax

=σ : 0)00(

2

1ˆmax

=+=σ

0)00(2

1ˆmax

=−−=− σ

substituting these load conditions into the equation 4.5 we obtain:

( ) 01

1=−−

− cccσσασ

α

then,

ccσσ =

This means that the compressive stress state is automatically

contained into the yield condition (Eq. 4.5).

- tensile case:

tIp σ

3

1

3

11

−=−=

ttJq σσ === 2

23

133

tσσ =

maxˆ

: ttt

σσσσ =+= )(2

1ˆmax

0)(

2

1ˆmax

=−−=−tt

σσσ

if we substitute these load conditions into the equation 4.5 we obtain:

( )( ) 0~

1

1=−++

− ctttσσεβσασ

αpl

resolving for ( )plεβ ~

:

( ) ( ) ( )ασσ

αεβ +−−= 11~

t

cpl


126

proving the equation (4.7).

The coefficient α can be determined from the biaxial and uniaxial

compressive strength, b

σ and c

σ . For the cases of uniaxial and biaxial

compression 0max

=σ and substituting these two load cases into Eq. (4.5),

we obtain:

- biaxial compression case

bIp σ

3

2

3

11

=−= bb

Jq σσ === 2

23

133

0ˆmax

=σ : 0)00(2

1ˆmax

=+=σ 0)00(2

1ˆmax

=−−=− σ

substituting the previous load conditions into the equation 4.5 we

obtain:

( ) 021

1=−−

− cbbσσασ

α

resolving for α , we obtain:

12

1

2 −

−=

−

−=

c

b

c

b

cb

cb

σσ

σσ

σσσσ

α (4.8)

Typical experimental values of the ratio c

b

σσ

for concrete are in the range

from 1.10 to 1.16, yielding values of α between 0.08 and 0.12 (Lubliner et

al., 1989 [20]).

The coefficient γ enters the yield function only for stress states of triaxial

compression, when 0max

<σ . This coefficient can be determined by

comparing the failure conditions along the tensile and compressive

meridians. By definition, the tensile meridian (TM) represents the points in

which stress states satisfying the condition 321max

σσσσ =>= and the

compressive meridian (CM) is the locus of stress states such that


127

321maxσσσσ >== , where

321,, σσσ are the eigenvalues of the effective

stress tensor.

It can be easily shown that,

( ) pqTM

−=3

2max

σ (4.9)

( ) pqCM

−=3

1max

σ (4.10)

along the tensile and compressive meridians, respectively, in the following

manner:

- tensile case, 321max

σσσσ =>= :

3

2

3

1 21

1

σσ +−=−= Ip

2123 σσ −== Jq

( )1

21

213

2

3

2

3

2σ

σσσσ =

++−=− pq

max1σσ = is the algebraically maximum eigenvalue of σσσσ.

- compressive case, 321max

σσσσ >== :

3

2

3

1 31

1

σσ +−=−= Ip

3123 σσ −== Jq

( )1

31

313

2

3

1

3

1σ

σσσσ =

−+−=− pq

max1σσ = is the algebraically maximum eigenvalue of σσσσ.

With 0max

<σ substituting the expressions (4.9) and (4.10) into the failure

criterion (Eq. 4.5), the same criterion becomes:

( ) ( ) ( )TMpqc

σααγγ −=+−

+ 1313

2, (4.11)


128

( ) ( ) ( )CMpqc

σααγγ −=+−

+ 1313

1. (4.12)

Let ( ) ( )CMTMcqqK = for any given value of the hydrostatic pressure p , with

0max

<σ ; then:

32

3

++

=γγ

cK (4.13)

resolving for γ:

( )12

13

−

−=

c

c

K

Kγ (4.14)

A value of 3

2=

cK , which is typical for concrete (Lubliner et al.,1989 [20]),

gives 3=γ .

If 0max

>σ and substituting the equations (4.9) and (4.10) into the failure

surface (Eq. 4.5), the yield conditions along the tensile and compressive

meridians reduce to:

( ) ( ) ( )TMpqc

σααββ −=+−

+ 1313

2, (4.15)

( ) ( ) ( )CMpqc

σααββ −=+−

+ 1313

1. (4.16)

Let ( ) ( )CMTMtqqK = for any given value of the hydrostatic pressure p

with 0max

>σ ; then:

32

3

++

=ββ

tK . (4.17)

The model provides a noncircular section when 1<c

K (figure 4.2). The

tensile meridian and the compression meridian are characterized by a

bilinear form, the change of inclination occurs at the biaxial compression


129

state for the tensile meridian and at the uniaxial compression state for the

compression meridian (fig. 4.7).

Fig. 4.2 – Failure surface in a deviatoric plane, corresponding to different values of Kc,

ABAQUS Theory Manual [1].

Fig. 4.3 – Failure surface in plane stress, ABAQUS Theory Manual [1].


130

Typical failure surfaces are shown in Figure 4.2 in the deviatoric plane and

in Figure 4.3 for plane-stress conditions (σIII = 0).

4.2 HARDENING/SOFTENING LAWS

An isotropically hardening/softening yield surface forms the basis of the

model for the inelastic response. The evolution of the yield (or failure)

surface is controlled by two hardening variables, plc

ε~ and pltε~ , linked to

failure mechanisms under compression and tension loading, respectively.

We refer to plc

ε~ and pltε~ as compressive and tensile equivalent plastic strains

(previously defined), respectively. The hardening/softening behavior is

represented from the laws:

=

=

)~(

)~(

pl

pl

ttt

ccc

εσσ

εσσ (4.18)

were )~(plccc

εσσ = is the isotropic hardening/softening law for the

compression case while )~(plttt

εσσ = is the isotropic softening law for the

tensile case.

4.2.1 COMPRESSIVE BEHAVIOR

We can define the stress-strain behavior of plain concrete in uniaxial

compression outside the elastic range. Compressive stress data are provided

as a tabular function of inelastic strain, inc

ε~ . The compressive inelastic

strain is defined as the total strain minus the elastic strain corresponding to

the undamaged material, el

in ccc

εεε ~~~ −= , where 0

~

E

c

c

σε =e

l , as illustrated in

Figure 4.4. Positive (absolute) values should be given for the compressive

stress and strain.


131

Fig. 4.4 – Definition of the compressive inelastic strain used for the definition of

compression hardening data, ABAQUS Theory Manual [1].

ABAQUS will issue an error message if the calculated plastic strain values

are negative and/or decreasing with increasing inelastic strain, which

typically indicates that the compressive damage curves are incorrect.

Hardening/Softening data are given in terms of an inelastic strain,inc

ε~ ,

instead of plastic strain, plc

ε~ . ABAQUS automatically converts the inelastic

strain values to plastic strain values using the relationship

plc

ε~ =0

)1(

~

Ed

dc

c

c

c

σε

−−i

n . In the absence of compressive damage (dc=0)

inc

ε~ =plc

ε~ as the models calibrate in this Thesis.

4.2.2 TENSILE BEHAVIOR

The post-failure behavior for direct tension is modelled defining the strain-

softening behavior for cracked concrete.

The specification of post-failure behavior generally means giving the post-

failure stress as a function of cracking strain,crt

ε~ . The cracking strain is


132

defined as the total strain minus the elastic strain corresponding to the

undamaged material:

el

cr ttt

εεε ~~~ −= ,

where

0

~

E

t

t

σε =e

l

as illustrated in Figure 4.5.

Fig. 4.5 - Illustration of the definition of the cracking strain used for the definition of

tension softening data, ABAQUS Theory Manual [1].

ABAQUS will issue an error message if the calculated plastic strain values

are negative and/or decreasing with increasing cracking strain, which

typically indicates that the tensile damage curves are incorrect.


133

ABAQUS automatically converts the cracking strain values to plastic strain

values using the relationship plt

ε~ =0

)1(

~

Ed

dt

t

t

c

σε

−−c

r . In the absence of

tensile damage crt

ε~ =plt

ε~ as the work developed in this Thesis.

4.3 NONASSOCIATED FLOW LAW

The plastic-damage model assumes nonassociated potential flow,

ij

ij

Gdd

σλε

∂∂

=pl . (4.19)

In the deviatoric plane the inelastic flow is associated, while separate

dilation and friction angles can be defined in the meridian plane.

The flow potential G chosen for this model is the Drucker-Prager

hyperbolic (Lubliner et al., 1989 [20]) function:

( ) ψψεσ tantan22

0pqG

t−+= (4.20)

where ψ is the dilation angle measured in the p–q plane at high confining

pressure; 0tσ is the uniaxial tensile stress at failure; and ε is a parameter,

referred as the eccentricity, that defines the rate at which the function

approaches the asymptote (figure 4.6). The flow potential tends to a straight

line as the eccentricity tends to zero; the default flow potential eccentricity

is 1.0=ε , which implies that the material has almost the same dilation

angle over a wide range of confining pressure stress values. Increasing the

value of ε provides more curvature to the flow potential, implying that the

dilation angle increases more rapidly as the confining pressure decreases.

This flow potential, which is continuous and smooth, ensures that the flow

direction is defined uniquely. The function asymptotically approaches the

linear Drucker-Prager flow potential at high confining pressure and

intersects the hydrostatic pressure axis at 90° (figure 4.6).


134

Because plastic flow is nonassociated, the use of the plastic-damage

concrete model requires the solution of nonsymmetric equations.

Fig. 4.6 – Schematic Plastic Potential with Drucker-Prager exponential form,

ABAQUS Theory Manual [1].

4.4 CALIBRATION AND VALIDATION OF A DAMAGE

PLASTICITY MODEL.

Failure criterion contained into the damage plasticity model available in

ABAQUS is a four-parameters model, proposed by Lubliner et al, 1989

[20]. With the purpose to calibrate the failure surface, four experimental

tests are necessary:

- direct tension test: 00 ===IIIIItI

f σσσ ;

- direct compression test cIIIIII

f−=== σσσ 00 ;

- biaxial compression test bIIIIII

f−=== σσσ 0 ;

- confined triaxial compression test IIIIII

σσσ >= ;


135

Material parameters of the failure surface and nonassociated flow laws are

calibrated according to the experimental data of Hurlbut et al. (1985) [17]),

as described in the previous sections, and summarized in table 4.1:


hc03mut.m11 hc03muc.c11 hc03mcc.c11 hc03mcc.c21

Failure

surface

parameters

f’c [MPa] 19.03

f’t [MPa] 2.77

α 0.0833

γ 2/3

Plastic

Potential

parameters

Dilation

angle ψ [°] 55 50 40 0

Eccentricity 0 0 0 0

Fig. 4.7 – Lubliner failure criterion (four parameters) compared with Hurlbut test data

(1985) [17], in the meridian plane.

Figure 4.7 shows a comparison of the Lubliner failure curves with Hurlbut

test data [17]. Each experimental test is plotted in the I1 - J20.5

plane through

the following transformations:


136

- direct tension: t

fI =1

3

5.0

2

tf

J =


fI −=1

3

5.0

2

cf

J =

- confined compression: lcc

ffI 21

−−= 3

5.0

2

lccff

J−

=

in which ft is the uniaxial tensile strength, fc is the uniaxial

compressive strength, fcc is the confined triaxial compressive

strength and fl is the constant lateral confinement.

Figure (4.8) to (4.11) show the predictions of the Damage Plasticity model

available in Abaqus compared with the results of the direct-tension test,

direct compression test and the triaxial compression tests with radial

confinements σr= -0.69, -3.45, -13.78 MPa.

Fig. 4.8 – Comparison with Direct-Tension-Test Data (Hurlbut 1985 [17])


137

The model predictions for direct tension (fig. 4.8) and direct compression

(fig. 4.9) are excellent (the model is just calibrated for these tests). Figure

4.7 shows as the direct tension test (hc03mut.m11) and the direct

compression test (hc03muc.c11) are situated just along the tensile meridian

and the compressive meridian, respectively.

Fig. 4.9 – Comparison with Compression Test Data (Hurlbut 1985 [17])

Figure 4.7 shows as the confined compression test (hc03mcc.c11) with

radial constant confinement σr = -0.69 MPa is located above the

compressive meridian. For this reason the model underpredicts the triaxial

compression strength for low confined test (σr = -0.69 MPa, see fig. 4.10).

For medium lateral confinement (σr = -3.48 MPa) the failure criterion

overestimates the strength (figure (4.11)), according to the failure

conditions of figure 4.7: the confined compression test with medium

confinement (hc03mcc.c21) is below the compressive meridian of the

failure surface.


138


(Hurlbut 1985 [17]).


(Hurlbut 1985[17]).

For high confinement pressure the model keeps in error because it accepts

only positive dilation angle while, as pointed out in various models in


139

literature (e.g., Etse - Willam, 1994 [14]) or as seen in the analysis of the

previous chapter, the dilation angle for high confinement is often negative.

The prediction of high-confined compression test (σr = -13.78 MPa) of

Hurlbut is shown in figure 4.12.

For high lateral confinement (σr = -13.78 MPa) the model overestimates

the strength (fig. 4.12), according to the failure conditions of figure 4.7

dealing with the failure envelope of model with respect to the experimental

test (hc03mcc.c31).


(Hurlbut 1985 [17]).


140

REFERENCES OF THE FOURTH CHAPTER



02909, USA.


Jersey, USA.






[7] Chinn, J. and Zimmermann, R. (1965). Behavior of plain concrete under various

high triaxial loading conditions. Technical Report, University of Colorado, Boulder,

USA.


York, 1995.

[9] Crisfield, M. A. Non-linear Finite Element Analysis of Solids and Structures, Vol. 1-

2.

[10] C. S. Desai, H. J. Siriwardane, Constitutive Law for Engineering Materials,

Prentice-Hall (1984).

[11] Drucker, D. (1959). A definition of stable inelastic materials. Journal of Applied

Mechanics, 26:101-106.

[12] Etse, G. (1992). Theoretische und numerische Untersuchung zum di_usen und

lokalisierten Versagen in Beton. PhD thesis, Universitat Karlsruhe, Karlsruhe, Germany..

[13] Este, G., Willam, K.J., A fracture-energy based constitutive formulation for

inelastic behavior of plain concrete, 1994, Journal of Engineering Mechanics, ASCE 120, 1983-2011.



[15] Etse, G. and Willam, K. (1999). Failure analysis of elastoviscoplastic material models, ASCE-EM, 125(1):60-69.


141

[16] Gerstle, K.H., Simple formulation of biaxial concrete behavior, ACI Journal,

78(1981)62-68.

[17] Hurlbut, B. J., Experimental and Computational Investigation of Strain-Softening in Concrete, MS thesis, University of Colorado, Boulder,. 1985.

[18] MD Kotsovos, JB Newman, Behavior of Concrete Under Multiaxial Stress by - ACI Journal Proceedings, 1977 - ACI.

[19] Lee, J., and G. L. Fenves, Plastic-Damage Model for Cyclic Loading of Concrete

Structures, Journal of Engineering Mechanics, vol. 124, no.8, pp. 892–900, 1998.

[20] Lubliner, J., J. Oliver, S. Oller, and E. Oñate, A Plastic-Damage Model for

Concrete, International Journal of Solids and Structures, vol. 25, no.3, pp. 229–326, 1989.


[22] Shah, S., Swartz, S., and Ouyang, C. (1995). Fracture Mechanics of Concrete. John Wiley & Sons, Inc.

[23] Sinha B.P., Gerstle K.H., Stress-Strain Relations for Concrete under Cyclic

Loading, and Tulin L.G., Journal of ACI, Proc., 1964, 61(2), pp. 195-211.

[24] Simo, J.C. and Hughes, T.J.R. (1998). Computational inelasticity. Springer, Berlin, Germany.

[25] J.G.M. van Mier, Strain-softening of concrete under multiaxial loading conditions,

PhD. thesis, Eindhoven University of Technology, (1984).


FAILURE CRITERIA FOR CONCRETE UNDER TRIAXIAL STRESSES

142

5. FAILURE CRITERIA FOR CONCRETE

UNDER TRIAXIAL STRESSES

This chapter presents a survey of the strength theories of concrete, under complex

stress states, pointing out the relationships among them. In the second part, the

chapter introduces the features of passive confinement with steel and FRP materials

within the framework of the theory of plasticity.

About one-hundred years ago (in 1900), the well-known Mohr-Coulomb

strength theory (see Chen and Han, 1988 [7]) was established for cohesive-

frictional materials. A considerable amount of theoretical and experimental

research on strength theory of materials under complex stress states has

been carried out in the 20th Century.

For many concrete structures, such as dams or nuclear power plants,

concrete is subjected to multiaxial stresses. Even though most of the

concrete design codes employ uniaxial strength for ultimate design, there is

a need for studying concrete strength under multiaxial stresses. This is

especially true nowadays when engineers are striving to design structures

more economically, as the uniaxial strength has not utilized the full

potential of concrete.

Concrete is a highly nonlinear material, so it is quite difficult to build a

mathematical strength model. Much of the research on concrete strength

under multiaxial stresses has been mainly numerically and experimentally

based, and various models based on experimental data have been proposed

in the past years.

Considerable efforts have been devoted to the formulation of strength

theories and to their correlation with test data, but no single model or


143

criterion has emerged which is fully adequate. Hundreds of models or

criteria have been proposed. The most common failure criteria are defined

by a number of constants varying from one to five independent parameters,

e.g.:

- two parameters: Leon [24], Hoek and Brown [18], Etse [12], etc.;

- three parameters: Bresler and Pister [4], Willam-Warnke [37],

Menetrey-Willam [29], etc.;

- four parameters: Ottosen [32], Hsieh, Ting, and WF Chen [19], de

Boer et al. [9], etc.;

- five parameters: Willam-Warnke [37], Song-Zhao-Peng [36], Jiang

[21], etc..

5.1 SOME CLASSICAL FAILURE CRITERIA

Some of the above mentioned models will be discussed in the following

sections. Calibration of those models with respect to a set of experimental

result on concrete specimens under various level of confinement will be

also proposed.

5.1.1 LEON FAILURE CRITERION

The isotropic Leon (1935 [24]) failure criterion (two parameters) for

frictional materials like concrete and rocks was originally formulated as a

parabolic expression of Mohr's failure envelope, which may be cast in

terms of the major (σI) and minor (σIII) principal stresses as:

0,(('

2

'=−

++

−==

L

c

IIII

L

c

IIII

IIIIc

fm

fFF

σσσσσσ )) ))σ

)

σ

)

σ

)

σ

)

(5.1)

with tension being positive, L

m and L

c represent the frictional parameter

and the cohesion of the Leon model.

For plasticity studies, yield and failure criteria are usually expressed in


144

terms of stress invariants and using the Haigh-Westergaard coordinates (§

2.2). Using the following transformation relationships presented in section

2.1.3:

( )3

2cos3

2

3

1

cos3

2

3

1

πθρξσ

θρξσ

++=

+=

III

I

(5.2)

equation (5.1) can be transformed as follows:

03

2

'1sin

2

3cos2/1

3

2

'

sin2/1cos2

3

'

22

=+−

−

+

+

ξθθρ

θθρ

cc

c

f

m

f

m

f(5.3)

where θρξ ,, are the Haigh – Westergaard coordinates (§2.1.3), which are

directly related to the stress tensor invariants:

2/3

23

2

1

2/273cos

2

3

JJ

J

I

=

=

=

θ

ρ

ξ

(5.4)

where I1 is the first invariant of the stress tensor, J2 and J3 are the second

and third invariants of the deviatoric tensor (§ 2.1).

Equation (5.3) defines a failure criterion with curved meridians (figure 5.1)

and noncircular cross sections on the deviatoric planes.

Leon criterion is a two – parameters model, which implies that two load

condition cases are strictly required for its directed calibration:

- direct compression stress case:

'0

cIIIIIIf−=== σσσ (5.5)

Substituting this load condition into the failure criterion as

expressed in Eq. (5.1), we have:


145

0'

'2

'

'

=−

−+

L

c

c

L

c

c cf

fm

f

f (5.6)

resolving for the cohesion parameter cL:

LLmc −=1 (5.7)

- direct tension stress case:

0' ===

IIIIItIf σσσ (5.8)

Substituting into the failure surface as formulated in Eq. (5.1):

0'

'2

'

'

=−

+

L

c

t

L

c

t cf

fm

f

f (5.9)

Fig. 5.1 – Leon failure criterion [24] (two parameters) compared with Hurlbut test data

(1985) [20], in the I1-J20.5

plane.

Now, using the relation (5.7) into Eq. (5.9) and resolving for the frictional

coefficient mL, we obtain:


146

01'

'2

'

'

=−+

+

L

c

t

L

c

t mf

fm

f

f (5.10)

then,

+

−

=

'

'

2

'

'

1

1

c

t

c

t

L

f

f

f

f

m (5.11)

Calibrating the Leon failure model with the experimental test data of

Hurlbut (1985 [20]), the values of the constant materials calculated with the

relations (5.11) and (5.7) are the followings:

143.0

857.0

=

=

L

L

c

m (5.12)

Figure 5.1, based on the comparison with respect to Hurlbut’s cylindrical

tests, shows that Leon criterion underpredicts the confined triaxial

compression strength, as the lateral confinement increases.

5.1.2 HOEK AND BROWN FAILURE CRITERION

A modification of Leon criterion was proposed by Hoek and Brown (1980

[18]):

0,(('

2

'=−

+

−==

HB

c

I

HB

c

IIII

IIIIc

fm

fFF

σσσσσ )) ))σ

)

σ

)

σ

)

σ

)

(5.13)

Both parabolic failure criteria are described by two strength parameters:

- the uniaxial compressive strength '

cf ;

- and the friction parameter HB

m ;

whereby, the cohesion parameter is not an independent variable at ultimate

strength.

The model can be calibrated in same way of the previous Leon model,


147

considering only the two followings elementary stress states:

- direct compression stress case:

'0

cIIIIIIf−=== σσσ (5.14)

substituting Eq. (5.14) into the failure criterion Eq. (5.13), we

have:

0

2

'

'

=−

HB

c

c cf

f (5.15)

which implies that the cohesion parameter cHB:

1=HB

c (5.16)

- direct tension stress case:

0' ===

IIIIItIf σσσ (5.17)

substituting into the failure surface of Eq. (5.13):

01'

'2

'

'

=−

+

c

t

HB

c

t

f

fm

f

f (5.18)

and resolving for the frictional coefficient mHB, we obtain:

''

2'2'

tc

tc

HBff

ffm

−= (5.19)

The failure criterion may be expressed in terms of the mean normal stress

31I=σ , the deviatoric stress

22J=ρ , and the polar angle θ as:

01cos2

3

'sin2/1cos

2

3

'

22

=−

++

+

θρσθθ

ρ

ccf

m

f (5.20)

Similarly to Leon, Hoek and Brown (5.20) define a failure criterion with

curved meridians (figure 5.2) and noncircular cross sections on the

deviatoric planes.


148

The constant materials calibrated in accord with Hurlbut [20] experimental

tests are the followings:

1

844.6

=

=

L

L

c

m (5.21)

Figure 5.2 shows that the Leon modified model performed by Hoek and

Brown produces some good results. The strength measures of envelope

model, in comparison with experimental tests, are acceptable for low and

middle levels of confinement. Only when high ranges of hydrostatic

pressure is applied to the specimens, that the model underpredicts the

triaxial strength (figure 5.2).

Fig. 5.2 – Hoek and Brown [18] failure criterion (two parameters) compared with

Hurlbut test data (1985) [20], in the I1-J20.5

plane.


149

5.1.3 WILLAM AND WARNKE (THREE PARAMETERS) FAILURE

CRITERION

The Willam-Warnke [37] yield criterion is a function that is used to predict

when failure occur in concrete and other cohesive-frictional materials such

as rock, soil, and ceramics. The early version of the three – parameter

surface developed by Willam and Warnke retains the linear p–q relation,

but deviatoric sections exhibit lode angle θ – dependence:

meridiantensileqbpb

meridianecompressivqapa

0

0

10

10

=++

=++ (5.22)

where, a0, a1, b0 and b1 are material constants;

13

1

3

1Itracep −=−= σσσσ

is the spherical stress;

23

2

3JSSq

ijij==

is the Mises equivalent stress;

ijijijpS σδ +=

is the deviatoric part of the stress tensor σσσσ.

Since the two meridians must intersect the hydrostatic axis at the same

point, it fallows that:

Aba ==00

. (5.23)

The three parameters can be determined by three typical tests:

- Uniaxial tension: '

3

1ttfp −=

'

ttfq = °= 0θ

- Uniaxial compression: '

3

1ccfp =

'

ccfq = °= 60θ

- Confined compression: 3

2'

lcc

cc

ffp

+=

lccccffq −= '

°= 60θ .


150

Substituting these load conditions into the failure criterion (Eq. 5.22), we

have:

meridianecompressivp

p

a

a

q

q

cc

c

cc

c

−

−=

1

0

1

1 (5.24)

then,

meridianecompressivp

p

q

q

a

a

cc

c

cc

c

−

−

=

−1

1

0

1

1 (5.25)

with Aba ==00

:

meridiantensileq

pbb

t

t−−

= 0

1 (5.26)

Using the well-known Hurlbut [20] experimental tests, the constant

materials assume the following values:

MPaA 813.3= 534.01

−=a 067.11

−=b (5.27)

Fig. 5.3 – Willam-Warnke [37] failure criterion (three parameters) compared with

Hurlbut test data (1985) [20], in the I1-J20.5

plane.


151

Once the two meridians have been determined from a set of experimental

data, the cross section can be constructed by connecting the meridians and

using appropriate curves.

If we use an appropriate function in a generic deviatoric section, Willam

and Warnke failure criterion, is convex and smooth everywhere. This is

achieved by using a portion of an elliptic curve proposed by Willam and

Warnke [37] (fig. 5.4). Due to the threefold symmetry, it is only necessary

to consider the part °≤≤° 600 θ .

Fig. 5.4 – Elliptic approximation of Willam-Warnke failure criterion [34] (three

parameters), in the deviatoric sections.

The equation of the ellipse, in terms of polar coordinates (q, θ), can be

expressed in terms of the parameters qt and qc:

( )222

2222

)12(cos)1(4

45cos)1(4)12(cos)1(2

−+−

−+−−+−=

ee

eeeeeqq

c θθθ

θ (5.28)


152

where

ctqqe /=

In which qt represents the semiminor axis and qc the semimajor axis of the

ellipse.

Two limiting cases of Eq. (5.28) can be observed. First, for 1/ ==ct

qqe ,

the ellipse degenerates into a circle (similar to the deviatoric trace of the

von Mises (1913) or Drucker-Prager (1952) models as in figure 5.5).

Second, when the ratio 5.0/ ==ct

qqe , the deviatoric trace becomes nearly

triangular (similar to the deviatoric section of the Rankine criterion (1876),

see figure 5.5).

Fig. 5.5 – Schematic representations of failure criteria in the deviatoric planes, Chen

and Han (1988 [7]).

The triaxial failure criterion, approximated by the elliptic description in the

deviatoric region (Willam and Warnke 1975 [37]), generate a Cl-continuous


153

surface with the complete elimination of corners in the deviatoric trace.

With the advantage that numerical applications whit plasticity based

models don't suffer convergence problems when the yield surface is smooth

and convex (Etse and Willam, 1994 [13]).

The triangular deviatoric curve has corners at the compressive meridians

(3

πθ = in fig. 5.5), in which the first derivate of function is discontinue.

Therefore, both convexity and smoothness of the failure curve (Eq. 5.28)

can be assured for 1/5.0 ≤=<ct

qqe .

5.1.4 WILLAM AND WARNKE (FIVE PARAMETERS) FAILURE

CRITERION

The Willam-Warnke five-parameter [37] model has curved tensile and

compressive meridians expressed by quadratic parabolas of the form:

meridiantensileqbqbpb

meridianecompressivqaqapa

0

0

2

210

2

210

=+++

=+++ (5.29)

where p is the spherical stress and q is the Mises equivalent stress defined

in the previous section; a0, a1, a2, b0, b1, b2 are material constants.

The two meridians must intersect the hydrostatic axis at the same point

again and it follows that:

Aba ==00

. (5. 30)

The five parameters can be determined by five typical tests:

- Uniaxial tension 00' ==>=

IIIIItIf σσσ :

'

3

1ttfp −=

'

ttfq = °= 0θ

- Uniaxial compression '

0cIIIIIIf−=== σσσ :

'

3

1ccfp =

'

ccfq = °= 60θ


154

- Biaxial compression '

0bIIIIIIf−=== σσσ :

'

3

2bbfp =

'

bbfq = °= 0θ

- Confined uniaxial compression '

ccIIIlIIIff −=−== σσσ :

3

2'

lcc

cc

ffp

+=

lccccffq −= '

°= 60θ

- Confined biaxial compression ,

bcIIIIIaIff ==>= σσσ :

3

2 ''

bca

bc

ffp

+=

'

bcacffq −= °= 0θ

Using these load conditions into the failure criterion (Eqs. 5.22), we have:

meridiantensile

p

p

p

b

b

b

qq

qq

qq

bc

b

t

bcbc

bb

tt

−

−

−

=

2

1

0

2

2

2

1

1

1

(5.31)

inverting:

meridiantensile

p

p

p

qq

qq

qq

b

b

b

bc

b

t

bcbc

bb

tt

−

−

−

=

−1

2

2

2

2

1

0

1

1

1

(5.32)

While along the compressive meridian:

meridianecompressivAp

Ap

a

a

qq

qq

cc

c

cccc

cc

−−

−−=

2

1

2

2

(5.33)

then,

meridianecompressivAp

Ap

qq

qq

a

a

cc

c

cccc

cc

−−

−−

=

−1

2

2

2

1 (5.34)

In literature typical failure data are suggested, e.g., Chen and Han (1988

[7]):

1. Uniaxial compressive strength: '

cf

2. Uniaxial tensile strength: ''

1.0ctff =


155

3. Biaxial compressive strength: ''

15.1cbff =

4. Confined biaxial compressive strength:

'95.1

cbcfp −=

'393.3

cbcfq =

5. Confined uniaxial compressive strength:

'9.3

cccfp −=

'239.4

cccfq =

Based on Mills et al. tests (1970 [30]), with MPafc

3.22' = concrete, the

five parameters of the failure function are now determined and the values

of the constant materials are the followings:

MPaA 107.4= 702.01

−=b 1

200137.0

−−= MPab

467.01

−=a 1

2000693.0

−−= MPaa

Fig. 5.6 – Willam-Warnke failure criterion [37] (five parameters) compared with Mills

et al. test data (1970) [30], in the I1-J20.5

plane.

The failure criterion is approximated by the elliptic function in the

deviatoric plane (Willam and Warnke 1975 [37]), generating a Cl-


156

continuous surface with the complete elimination of corners in the

deviatoric trace (figure 5.7).

Fig. 5.7 – Willam Warnke failure criterion in comparison with some experimental data.

5.1.5 OTTOSEN FOUR-PARAMETER MODEL

Ottosen (l977 [32]) suggested the following criterion for concrete material

involving all three stress invariants θ,3,3

2

1 JqI

p =−= :

012 =−−+ pbqaq λ (5.35a)

where λ is a function of θ3cos :

( ) ( )

( ) ( )

≤=

−−

≥=

=−

−

03cos,3coscos3

1

3cos

03cos,3coscos3

1cos

212

1

1

212

1

1

θθθπ

θθθλ

forkRkkk

forkQkkk

(5.35b)

In Eq. (5.35a,b) a, b, k1 and k2 are constant materials. For convenience in

the calibration of the model, we introduce the functions ( ) ( )θθ ,,,22

kRkQ .

Equations (5.35a, b) define a failure surface with curved meridians and

noncircular cross sections on the deviatoric planes (figure 5.8). The

meridians described by Eq. 5.35a are quadratic parabolas which are convex


157

if 0>a and 0>b . The cross sections have the geometric properties of

symmetry and convexity, and have changing shapes from nearly triangular

to nearly circular with increasing hydrostatic pressure. The model

encompasses several earlier models as special cases, e.g., the von Mises

(1913) model for 0== ba and .const=λ and the Drucker Prager (1952)

model if 0=a , 0≠b and .const=λ

The four parameters in the failure criterion may be determined on the basis

of two typical uniaxial concrete tests (compression strength,'

cf , and tension

strength '

tf ) and two typical biaxial an triaxial data:

1. Uniaxial compressive strength ( 13cos3

−=⇒= θπ

θ ): '

cf

2. Uniaxial tensile strength ( 13cos0 =⇒= θθ ): ''

1.0ctff =

3. Biaxial compressive strength ( 13cos0 =⇒= θθ ): ''

15.1cbff =

4. Confined uniaxial compressive strength ( 13cos3

−=⇒= θπ

θ ):

'9.3

cccfp −=

'239.4

cccfq =

values suggested by Chen and Han, 1988 [7].

The steps of the calibration are the followings:

- substituting the uniaxial compression test into the failure surface

(Eq. 5.35a, b):

resolving for b:

c

ccc

p

qRkaqb

11

2 −+= (5.36)

- substituting the confined compression test into Eq. 5.35a, b and

using Eq. 5.36:

011

2 =−−+cccc

pbqRkaq 011

2 =−−+cccc

pbqRkaq


158

011

1

2

1

2 =−−+

−+cc

c

ccc

ccccccp

p

qRkaqqRkaq

inverting for a:

cccccc

cccccccccccc

pqpq

pqRkpqRkppa

22

11

−

−+−= (5.37)

- substituting the uniaxial compression test into Eq. 5.35a, b:

011

2 =−−+tttt

pbqQkaq

using eqs. (5.36) and (5.37) and inverting for k1, we obtain:

)

(

))((

2222

2222

22

2222

1

ccctccccctcctcccccctccccc

cttccccttcccctcccccctcccc

ccccccct

tccctccctccctc

pqpqRpqpqRpqpqRpqpqR

pqQpqpqQpqpqpqRpqpqR

pqpqpp

pqppqppqppqp

k

+−+−

+−+−

−−−

−++−

=

(5.38)

- finally, substituting the biaxial compression test into Eq. 5.35a, b:

011

2 =−−+bbbb

pbqQkaq (5.39)

using eqs. (5.36) , (5.37) and (5.38) into Eq. (5.39) which reduces

to only k2 –dependence. Hence we have one implicit equation and

one incognita and the equation system becomes easily resolvable.

The values obtained for the parameters from Mills et al. test (1970 [30])

data are the following:

884.02

=k 00122.01

=k

10000010075.0

−= MPaa 00163.0=b

Figure 5.8 shows the comparison of the failure criterion with Mills et al.

triaxial data in meridian planes.

In general, the four-parameter failure criterion is valid for a wide range of

stress combinations. However, the expression for the λ-function is quite

involved. Hsieh et al. (1982) proposed a simpler form which can also fit the


159

experimental data very well and presented in the next section.

Fig. 5.8 – Ottosen failure criterion (four parameters) compared with Mills et al. test data

(1970) [30], in the I1-J20.5

plane.

5.1.6 HSIEH – TING – CHEN FOUR-PARAMETER MODEL

Hsieh et al. (1982 [19]) proposed a λ-function with the simple form

)cos( cb += θλ for °≤ 60θ , where b are c are constants. Replacing λ in

Eq. (5.35a) of Ottosen model by this expression the failure function is in

the following form:

01)cos(2 =−−++ dpqcbaq θ (5.40)

a, b, c, d are material constants.

To determine the four material parameters, a, b, c and d, we use some of the

triaxial tests of Mills and Zimmerman (1970 [30]). As for the Ottosen

criterion, the parameters are determined from the following four failure


160

states and in absence of experimental data some classical failure values are

proposed by Chen and Han (1988 [7]):

1. Uniaxial compressive strength ( 13cos3

−=⇒= θπ

θ ): '

cf

2. Uniaxial tensile strength ( 13cos0 =⇒= θθ ): ''

1.0ctff =

3. Biaxial compressive strength ( 13cos0 =⇒= θθ ): ''

15.1cbff =

4. Confined uniaxial compressive strength ( 13cos3

−=⇒= θπ

θ ):

'9.3

cccfp −=

'239.4

cccfq =

Fig. 5.9 – Hsieh – Ting – Chen failure criterion (four parameters) compared with Mills

et al. test data (1985) [30], in the I1-J20.5

plane.

To calibrate the failure criterion we use the following procedure:


161

=

−°

−°

−°

−°

1

1

1

1

60cos

60cos

0cos

0cos

2

2

2

2

d

c

b

a

pqqq

pqqq

pqqq

pqqq

cccccccc

cccc

bbbb

tttt

(5.41)

inverting the nonhomogeneous system of linear equation (5.41), we obtain

the material constants:

=

−°

−°

−°

−°

=

−

0.00162789

0.00034078

0.00085945

MPa0.00000101

1

1

1

1

60cos

60cos

0cos

0cos 1-

1

2

2

2

2

cccccccc

cccc

bbbb

tttt

pqqq

pqqq

pqqq

pqqq

d

c

b

a

(5.42)

In Fig. 5.9 the compressive

=3

πθ and the tensile ( )0=θ meridians are

shown. The failure criterion and the experiments of Launay and Gachon

(1970 [23]) are compared in the deviatoric plane in Fig. 5.10.

Fig. 5.10 – Comparison of Hsieh – Ting – Chen failure criterion with Launay and

Gachon triaxial data (1970) [23], in the deviatoric plane.


162

The four-parameter criterion satisfies the convexity requirement for alls

stress conditions, it still has edges along compressive meridians (Fig. 5.10)

where continuous derivatives along the edges do not exist. Continuous

derivatives used in a general constitutive relation with a general load case

would result in a better convergence during iteration in a numerical

analysis. While for classical load cases (e.g., uniaxial compression case or

confined compressive case) such problems don't occur. Thus, smoothness

everywhere of the yield surface is a desirable property.

5.1.7 EXTENDED LEON MODEL (ELM) PROPOSED BY ETSE

To describe the triaxial concrete strength, the failure criterion by Leon

(1935 [24]) and its extension by Hoek and Brown (1980 [18]) is adopted by

Etse (1992 [12]).

The principal advantages of this criterion are the followings (Etse and

Willlam, 1994 [13]):

- it retains simplicity while not sacrificing accuracy;

- it provides continuous transition between failure in direct tension and

triaxial compression;

- it reduces calibration of the failure criterion to two strength

parameters that are readily available from uniaxial-tension and

uniaxial-compression data.

The isotropic Leon (1935) failure criterion for concrete and the subsequent

modification by Hoek and Brown (1980) presented in the sections 5.1.1

and 5.1.2 respectively have the followings expressions in terms of Haigh –

Westergaard coordinates:


163

- Leon model:

03

2

'1sin

2

3cos2/1

3

2

'

sin2/1cos2

3

'

22

=+−

−

+

+

ξθθρ

θθρ

cc

c

f

m

f

m

f

(5.43)

- Hoek and Brown model:

01cos2

3

'sin2/1cos

2

3

'

22

=−

++

+

θρσθθ

ρ

ccf

m

f (5.44)

The main disadvantage of both the Leon and the Hoek-Brown criteria are

the corners in the deviatoric trace complicating the numerical

implementation to the simulation of mechanical behavior of concrete (Fig.

5.12).

Considering the tension and compression meridians of Hoek and Brown

model (Eq. 5.44), respectively, when ( )0=θ and

=3

πθ

:

013

2

''2

32

=−

++

t

cc

t

f

m

fρσ

ρ (5.45)

016''2

32

=−

++

c

cc

c

f

m

f

ρσ

ρ (5.46)

Etse approximates the triaxial failure criterion of Hoek and Brown with the

elliptic description of the Willam and Warnke model (1975 [37]) in the

deviatoric region in order to generate a Cl-continuous surface. A smooth

surface is obtained by replacing the deviatoric radius vector with the

following function proposed by Willam and Warnke:


164

( )222

2222

)12(cos)1(4

45cos)1(4)12(cos)1(2

−+−

−+−−+−=

ee

eeeeec θ

θθρθρ (5.47)

and where the eccentricity is defined by the ratio ct

e ρρ /= . The polar

coordinate ( )θρ defines the elliptic variation of the model (to see section

5.1.3).

Fig. 5.11 – Deviatoric View of ELM (Etse 1992).

Fig. 5.12 – Deviatoric Sections of Hoek and Brown model (1980).


165

Hence, the discontinuous failure surface (Eq. 5.44) may be approximated

by the following smooth approximation as a function of the three scalar

invariants σ , ρ , and θ :

( ) ( )01

6''2

3),,(

2

=−

++

=

θρσ

θρθρσ

ccf

m

fF (5.48)

It is worth noticing that for e = 1, the influence of the polar angle θ

disappears, and the deviatoric shape of the failure surface becomes circular

along the line of the generalized Drucker – Prager criterion (1952). The

eccentricity must satisfy the condition 15.0 ≤< e .

Fig. 5.11 illustrates the deviatoric sections of the smooth failure criterion at

various levels of hydrostatic pressure. The difference between the smooth

failure criterion as compared to the original Leon-Hoek and Brown criteria

is best appreciated in the deviatoric planes. (difference between Fig. 5.11

and Fig. 5.12).

Fig. 5.13 – ELM failure criterion (Etse 1992 [12]) compared with Hurlbut test data

(1985) [20], in the I1-J20.5

plane.


166

The smooth failure criterion of Etse describes a C1-continuous curvilinear

trace as opposed to the highly polygonal shape of the Hoek and Brown

criterion in the deviatoric region.

5.2 APPLICATION OF PLASTICITY BASED MODELS TO

PASSIVE CONFINEMENT

In the present section, the influence of passive confinement is studied in the

field of the elastoplasticity models. The models presented in the previous

chapters (see §3 & §4) have shown that the same models are suitable to

describe the mechanical behavior in case of constant – active confinement.

In the present section, it is aimed to show if the elastoplasticity models are

also capable to describe the response of specimens in passive confinement.

Concrete cylinders enclosed by steel and by fibre reinforced polymers

(FRP), are representative examples of passively confined structures.

Fig. 5.14 Stresses acting on (a) the confining material (steel or CFRP) and (b) the

enclosed concrete, Grassl (2002) [17].

Triaxial compression stress states are usually activated by lateral expansion

when the specimens are passively confined. The amount of lateral

expansion determines the confinement stress, which is called passive

confinement. The confining material, in the circumferential direction,


167

activates radial stresses, as shown in Fig. 5.14. It is assumed that only the

concrete core is loaded in the axial direction. Additionally, the confining

stress is assumed to be constant in the circumferential direction, provided

that both the concrete and the confining material have a homogenous

surface.

By means of these simplifications, the relation of the t

σ stresses, in the

enclosing material, to the radial stresses acting on the concrete specimen,

rσ , may be derived by stress equilibrium in the lateral direction as:

tdrtr

σϕϕσπ

2sin0

=∫ (5.49)

so that the lateral stress, r

σ , results as:

trD

tσσ

2= (5.50)

in the case of passive confinement with transversal and discontinuous

armature, the Eq. (5.50) becomes:

t

s

rsD

Aσσ

2= (5.51)

in which s

A is the transversal area and s is the step.

Furthermore, the radial strainr

ε :

r

rr

∆ε = (5.52)

is equal to the axial strain of the confining material (steel or FRP):

rtr

r

U

Uε

π∆π∆

ε ===2

2 (5.53)

The main difference between steel and FRP as confining materials (see Fig.

5.15) is:

- the ratio of the maximum stress;

- the difference of the elasticity modulus and


168

- the post-peak behavior.

The behavior of the confining steel was idealized by an elastic–plastic

stress–strain relation, where the Young’s modulus is assumed to be E = 200

GPa and the yield stressy

σ , which correspond to a stress–stiffness ratio of

Eyy/σε = .

Fig. 5.15 Difference of mechanical behavior between steel and FRP, Grassl (2002) [17].

The response of the FRP is idealized by means of an elastic-brittle stress–

strain relation with the Young’s modulus that is assumed to be E (varying

according to the type of FRP material: Carbon FRP (160 – 300 GPa),

Aramid FRP (50 – 90 GPa), Glass FRP (25 – 80 GPa) and the ultimate

stress is u

σ , which results E

u

u

σε = .

The plasticity models are founded upon three fundamental assumptions

(Chen and Han, 1988 [7]):

(i) an initial yielding surface, within the stress space, defining the

stress level at which plastic deformation begins;

(ii) an isotropic hardening/softening rule defining the yielding surface


169

evolution after beginning of plastic deformations;

(iii) a flow rule, which is related to a plastic potential function, gives an

incremental plastic stress-strain relation.

Prediction models introduced in the previous chapters (§3 and §4) are

appropriate only when constant confinement is applied to the cylindrical

specimens. The reasons for these features can be listed in the following

manner:

1) isotropic hardening/softening law, characterizing the same models, is

strongly dependent on the level of lateral confinement, for this

reason that in the chapters regarding the plasticity based models, the

hardening/softening law is calibrated in accord with various confined

test. Then, for levels of confinement which the experimental tests are

not available, a linear interpolation of the isotropic law, by means of

the value of confinement, is used.

2) As for the hardening/softening law the dilation angle of plastic flow

(see chapter 3 & 4) depends on the level of lateral confinement.

Assuming decreasing values as confinement increases thin to assume

negative values for high hydrostatic pressure.

5.2.1 STEEL CONFINEMENT

Figure 5.15 shows the mechanical behavior in uniaxial tension of steel

material. It is worth noticing that if Eyy/σε = quantity is sufficiently

small, we can hypothesize that concrete cylinders enclosed by steel can be

studied as constant – confined compression tests with the following lateral

constant stress:

yrD

tσσ

2= (5.54)

in which y

σ is the yield and/or failure uniaxial stress of steel.


170

Typical value of yield stress is MPay

350=σ which correspond to a yield

strain of 00175.0/ == Eyy

σε .

Fig. 5.16 – Typical uniaxial compressive stress-strain curve (Domingo Sfer et al, 2002

[35]).

Comparing the yield strain value of the considered steel with the radial

strain of a typical uniaxial compressive test we observe that these values

are comparable.

This implicates that when we confine a cylindrical specimen with steel, all

the capacity strength of confining material is mobilized. For this reason that

we don't commit reasonable errors if we consider a constant confinement to

predicts the failure condition when confining steel material is used.

Various authors have been proposed, according to experimental test-data,

empirical or semi-empirical approaches to describe the strength of a

confined concrete, e.g.:

- Richart et al., 1928 [33] (see Fig. 5.17):

lcccfff 1.4

0+= (5.55)


171

- Newman and Newman, 1972 [31] (see Fig. 5.18):

+=

86.0

0

07.31

c

l

cccf

fff (5.56)

- Mander et al., 1988 [28] (see Fig. 5.19):

−++−=

00

0294.71254.2254.1

c

l

c

l

cccf

f

f

fff (5.57)

where ccf and

0cf are the confined and unconfined concrete strength,

respectively and lf is the lateral confinement, based on equilibrium (Eq.

5.54):

ylD

tf σ

2= (5.58)

Fig. 5.17 – Comparison between Richart formula (1928) with experimental steel

confinement data available in literature.


172

Fig. 5.18 – Comparison between Newman and Newman formula (1972) with

experimental steel confinement data available in literature.

Fig. 5.19 – Comparison between Mander et al. formula (1988) with experimental steel



173

Figures from 5.17 to 5.19 show the comparison of these empirical formule

with some steel-confined concrete (cylindrical specimens).

The comparison with some available tests has shown that the relationships

proposed by Richart et al. [33], Newman and Newman [31] and Mander

[28] to describe the mechanical properties of concrete subjected to triaxial

compression capture in a excellent mode the steel-confined compressive

strength (see Figs. 5.17, 5.18 and 5.19).

Figures from 5.20 to 5.26 show the comparison of classical failure criteria

for concrete, with some steel confined (cylindrical tests). Note that all

failure criteria have been normalized by '

cf (uniaxial compressive

strength).

Fig. 5.20 – Comparison between Drucker – Prager criterion (1952) with experimental

steel confinement data available in literature.

The Drucker-Prager model overestimates the material response as the

confinement pressure increases (Fig. 5.20) according to the results obtained


174

in the previous Chapter in which we had found that failure models

generally described by straight line, in the meridian plane, are inadequate

for describing the failure of concrete in high-confinement range (see § 3.3.2

and 3.3.3).

Fig. 5.21 – Comparison between Bresler – Pister (1958, [4]) criterion with experimental


The comparison between Bresler and Pister model with some available

tests has shown that the same model is appropriate to describe the

mechanical properties of concrete subjected to steel-confined triaxial

compression (Fig. 5.21).

Figure 5.22, based on the comparison with respect to concrete cylinders

enclosed by steel, shows that Leon criterion underpredicts the confined

triaxial compression with steel-passive confinement, as lateral confining

steel-material increases.


175

Fig. 5.22 – Comparison between Leon criterion (1935 [24]) with experimental steel


Fig. 5.23 – Comparison between Hoek and Brown (1980, [18]) criterion with



176

Figure 5.23 shows that the Leon modified model (performed by Hoek and

Brown, 1980) produces some good results. The strength measures of

envelope model, in comparison with experimental tests, are acceptable for

low and middle levels of steel confinement. Only when high ranges of

hydrostatic pressure produced by steel confinement is applied to the

specimens, that the model underpredicts the triaxial strength (figure 5.23).

Fig. 5.24 – Comparison between Willam Warnke (1975, [37]) criterion with


Willam-Warnke (five parameters), Ottosen (four parameters) and Hsieh et

al. (four parameters) are models characterized by the same features to

modeling the confined triaxial compressive strength (from figure 5.24 to

5.26). The strength values of the failure criteria, in comparison with

experimental tests, are acceptable for low and middle levels of steel

confinement. The models overestimate the triaxial strength as high ranges

of confining steel is applied to the specimens (figures 5.24, 5.25 and 5.26).


177

Fig. 5.25 – Comparison between Ottosen (1978, [32]) criterion with experimental steel


Fig. 5.26 – Comparison between Hshie et al. (1982, [19]) criterion with experimental



178

5.2.2 FRP CONFINEMENT

The response of FRP is represented by elastic-brittle stress–strain relation

(figure 5.15). For this reason passive confinement, realized through FRP,

produces a radial stress gradually increasing as the radial strain of specimen

increases.

Plasticity based models with isotropic hardening/softening rule and

constant dilation angle of plastic potential as introduced in the previous

chapters result to be inadequate to predict the mechanical response of

specimens in FRP passive confinement.

Nevertheless the incremental elastic – plasticity algorithm introduced in the

Chapter 2 (Basic Equations and Procedure), can be improved adding a new

return – cycle in the calculation scheme (figure 5.27).

Fig. 5.27 – Flow chart of elastoplastic procedure modified to simulate the FRP passive

confinement.


179

The new algorithm inserted to valley of the incremental elastoplastic

relationships (for more details see chapter 2) is the following:

- we calculate the radial strain at the first step of incremental

elastoplastic calculation I

rε∆ , with the radial confinement equal

to the calculated value of the previous elastoplastic cycle 1−i

rσ ;

- now, we may be derived by stress equilibrium in the lateral

direction a new radial action in the following manner:

I

rFRP

I

rE

D

tε∆σ∆

2= (5.59)

then, the new radial confinement is:

I

r

i

rσ∆σ +−1

(5.60)

- with the previous level of confinement (Eq. 5.60), we again do the

incremental elastoplastic iteration and we obtain the subsequent

lateral strain II

rε∆ . It is opportune to observe that:

I

r

II

r

i

r

I

r

i

rε∆ε∆σσ∆σ <⇒>+ −− 11

(5.61)

- we may be derived by stress equilibrium in the lateral direction a

new radial confinement in the following manner:

II

rFRP

II

rE

D

tε∆σ∆

2= (5.62)

then, the subsequent radial confinement is:

II

r

i

rσ∆σ +−1

(5.63)

with:

II

r

i

r

I

r

i

rσ∆σσ∆σ +>+ −− 11

(5.64)

In this way to proceed, we can observe that:

III

r

II

rσ∆σ∆ <

IV

r

III

rσ∆σ∆ >

V

r

IV

rσ∆σ∆ <


180

etc.

- the algorithm will finish when:

( )14

1

1

∆σ∆

σ∆σ∆≤

−−

−

IN

r

IN

r

IN

r (5.65)

being ∆ a sufficiently small tolerance.

And we will go to the next incremental elastoplastic cycle with

IN

r

i

r

i

rσ∆σσ += −1

.

In this way we can use the plasticity based models (originally used for

predicting the behavior of concrete under constant confined compression)

to the numerical simulation of non-constant confined compression using

FRP passive confinement.


181

REFERENCES OF THE FIRST CHAPTER

[1] Abu-Lebdeh TM and Voyiadjis GZ (1993), Plasticity-damage model for concrete under cyclic multiaxial loading, J. Eng. Mech., 119-7, 1465–1484.

[2] Ahmad, S.H., Shah, S.P. Complete triaxial stress–strain curves for concrete, J. Struct. Div. Proc. ASCE 108 (1982) 728–742.


Jersey, USA.

[4] Bresler B and Pister KS (1958), Strength of concrete under combined stresses, ACI

Mater. J., 55, 321–345.

[5] Buyukozturk O, Nilson AH, and Slate FO (1971), Stress-strain response and

fracture of a concrete model in biaxial loading, ACI Mater. J., 68-8, 590–599.

[6] Cedolin L, Crutzen YRJ, and Dei Poli S (1977), Triaxial stress-strain relationship for concrete, J. Eng. Mech. Div., 103-3, 423–439.

[7] Chen, W.-F., and Han, D. J. (1988). Plasticity for structural engineers. Springer-Verlag, New York, N.Y.

[8] Dafalias YF and Popov EP (1975), A model of nonlinearly hardening materials for

complex loading, Acta Mech., 21, 173–192.

[9] de Boer R and Desenkamp HT (1989), Constitutive equations for concrete in failure

state, J. Eng. Mech. Div., 115-8, 1591–1608.

[10] Drucker, D. (1959). A definition of stable inelastic materials. Journal of Applied

Mechanics, 26:101-106.

[11] Eid, R., “Behavior of Circular Reinforced Concrete Columns Confined with Transverse Steel Reinforcement and FRP Sheets,” ISIS Canada 11th Annual

Conference, May 2006, Calgary, Canada.

[12] Etse, G. (1992). "Theoretische und numerische Untersuchung zum diffusen und

lokalisierten Versagen in Beton," PhD thesis, University of Karlsruhe, Karlsruhe, Germany.

[13] Etse, G. and Willam, K. (1994). Fracture energy formulation for inelastic behavior of plain concrete. ASCE-EM, 120:1983-2011.



[15] Faruque MO and Chang CJ (1990), A constitutive model for pressure sensitive

materials with particular reference to plain concrete, Int. J. Plast., 6-1, 29–43.


182

[16] Geniev GA et al (1978), Strength of Lightweight Concrete and Porous Concrete

under Complex Stress State in Russian, Moscow Building Press.

[17] Grassl Peter (2002) Modeling of dilation of concrete and its effect intriaxial compression. Chalmers University of Technology, Department of Structural

Engineering, Concrete Structures, SE-412 96 Gloteborg, Sweden

[18] Hoek, E., and Brown, E. T. (1980). "Empirical strength criterion for rock masses."

J. Geotech. Engrg., ASCE, 106(9), 1013-1035.

[19] Hsieh SS, Ting EC, and Chen WF (1982), A Plasticity-fracture model for concrete,

Int. J. Solids Struct., 18-3, 181–197.

[20] Hurlbut, B. (1985). "Experimental and computational investigation of strain-softening in concrete," MSc thesis, University of Colorado, Boulder, Colo.

[21] Jiang JJ and Wang HL (1998), Five-parameter failure criterion of concrete and its application, Strength Theory, Science Press, Beijing, New York, 403–408.

[22] Labbane M, Saha NK, and Ting EC (1993), Yield criterion and loading function

for concrete plasticity, Int. J. Solids Struct., 30-9, 1269–1288.

[23] Launay, P., and Gachon H. (1972). "Strain and ultimate strength of concrete under

triaxial stress." Am. Concrete Inst. Spec. Publ. 34, paper 13, American Concrete Institute (ACI), D..etroit, Mich.

[24] Leon, A. (1935). "Uber die Scherfestigkeit des Betons." Beton und Eisen, Berlin, Germany, 34(8) (in German).

[25] Li JK (1997), Experimental research on behavior of high strength concrete under

combined compressive and shearing loading in Chinese, China Civil Engrg. J., 30-3,

74–80.

[26] Li, Y.-F., and Fang, T.-S. 2004. A constitutive model for concrete confined by steel reinforcement and carbon fiber reinforced plastic sheet. Structural Engineering and

Mechanics, 18: 21–40.

[27] Lubliner Jacob, Plasticity Theory, Macmillan Publishing, New York (1990).

[28] Mander JB, Priestley MJN, Park R. Theoretical stress-strain model for confined

concrete. J Str Eng ASCE 1988;114(8):18041826.

[29] Menetrey P and Willam KJ (1995), Triaxial failure criterion for concrete and its

generalization, Colloid J. USSR, 92-3, 311–318.

[30] Mills LL and Zimmerman RM (1970), Compressive Strength of plain concrete

under multiaxial loading conditions, ACI Mater. J., 67-10, 802–807.


183

[31] Newman, K., & Newman J.B, 1972, Failure theories and design criteria for plain

concrete, Proceedings, International Engineering Materials Conference

[32] Ottosen NS (1977), A failure citerion for concrete, J. Eng. Mech., 103-4, 527–535.

[33] Richart, F. E., Brandtzaeg, A., and Brown, R. L., A Study of the Failure of

Concrete under Combined Compressive Stresses, Bulletin No. 185, Engineering Experimental Station, University of Illinois, Urbana, 1928, 104 pp.

[34] Schreyer HL (1989), Smooth limit surfaces for metals: Concrete and geotechnical

materials, J. Eng. Mech. Div., 115-9, 1960–1975.

[35] Sfer, D., Carol, I., Gettu R. and Etse G., "Study of the Behaviour of Concrete

Under Triaxial Compression", J. of Engng. Mech., V. 128, No. 2, pp. 156-163 (2002).

[36] Song YP and Zhao GF (1996), A general failure criterion for concretes under

multi-axial stress, China Civil Eng. Journal, 29-2, 25–32.

[37] Willam KJ and Warnke EP (1975), Constitutive model for the triaxial behavior of concrete, Int. Assoc. Bridge. Struct. Eng. Proc., 191–31.

[38] Wu HC (1974), Dual failure criterion for plain concrete, J. Eng. Mech. Div., 100-6, 1167–1181.

[39] Yang BL, Dafalias YF, and Herrmann LR (1983), A bounding surface plasticity

model for concrete, J. Eng. Mech. Div., 111-3, 359–380.

[40] Zienkiewicz, O. and Taylor, R. (1994). The Finite Element Method, volume 1 & 2.

McGraw-Hill, London, England, 4th edition.

SUMMARY AND CONCLUSION

184

6. SUMMARY AND CONCLUSIONS

Numerical methods along with sophisticated material models provide a

powerful tool for the analyses of concrete structures. Among the large

number of constitutive models for concrete, plasticity-based models are

commonly used for a wide range of applications.

In the first part of this work, five constitutive models for plain concrete,

formulated in the framework of plasticity theory, are discussed:

- von Mises (one parameter) plasticity model (1913);

- Drucker-Prager (two parameters) plasticity model (1952);

- Drucker-Prager (three parameters) plasticity model (1952);

- Bresler-Pister (three parameters) plasticity model (1958);

- Lubliner et al. (four parameters) plasticity model (1989) available in

ABAQUS (student edition).

The main steps of the development and validation of constitutive models

for concrete are:

- the development of a robust and efficient algorithmic formulation of

the material models; an elastoplasticity integration scheme

(incremental stress-strain relationship) is used for the numerical

integration of the evolution equations. The non-linear system of

equations, are solved by means of Newton-Raphson algorithm;

- the model performances are investigated with respect to experimental

SUMMARY AND CONCLUSION

185

data available within literature. The influence of material and model

parameters on the numerical results is investigated through these

experiments. It is found that some of calibrated models are capable

to describe concrete behavior for a broad range of loading conditions

while other models (e.g. von Mises model) result to be less suitable

for these purposes.

In the second part, plasticity models are used to study the behavior of

concrete in compression on passively confined structures. For passively

confined structures, the behavior differs significantly for steel and FRP-

confined cylinders.

When we confine a cylindrical specimen with steel, all the capacity

strength of confining material is mobilized, in failure regime. For this

reason that we can consider a constant-confinement case to predict the

failure condition of concrete specimen when confining steel material is

used. Consequently, the plasticity models introduced, in the previous

chapters, result to be adequate.

Instead, plasticity-based models with isotropic hardening/softening rule and

constant dilation angle of plastic potential as introduced in the previous

chapters result to be inadequate to predict the mechanical response of

specimens in FRP passive confinement.

APPENDIX:

“CODRI.F”

CODRI.F

187

c----------------------------------------------------------------------c c ELASTIC-PLASTIC CONSTITUTIVE MODEL c c FOR FRICTIONAL-COHESIVE MATERIALES c c----------------------------------------------------------------------c c CONTROL TYPE: MIXED - STRAIN CONTROL - STRESS CONTROL c c STATES: PLANE STRAIN - PLANE STRESS - AXISYMMETRIC STATE c c----------------------------------------------------------------------c c Variables c c----------------------------------------------------------------------c c c c sal - output type c c = 1 sigma vs. epsilon c c = 2 square root(J2) vs. I1 c c con - control type c c = 1 strain c c = 2 mixed c c = 3 stress c c est - indicative of state c c = 1 plane strains c c = 2 plane stresses c c = 3 axisymmetric c c mod - indicative of model c c inc - number of increment c c ninc - number of total increments c c mat - material proprieties c c e - recorded strains c c s - recorded stresses c c de - recorded strain increment c c ds - recorded stress increment c c dem - calculated strain increment c c dsm - calculated stress increment c c em - calculated strains c c sm - calculated stresses c c x - recorded volumetric stress c c y - recorded deviatoric stress c c xm - calculated volumetric stress c c ym - calculated deviatoric stress c c c c----------------------------------------------------------------------c c Version: 08.03.08 c c Modified by: A.Caggiano c c----------------------------------------------------------------------c program codri c----------------------------------------------------------------------c c Principal Program c----------------------------------------------------------------------c implicit none character arcsal*32 integer*4 sal,con,est,mod,inc,ninc,csal,esal,i real*8 s,e,sm,em,ds,de,dsm,dem,sigini,mat,x,y,xm,ym,q dimension s(3,1500),e(3,1500),sm(3,1500),em(3,1500), . ds(3),de(3),dsm(3),dem(3),sigini(4),mat(17) c-----Initial zero resetting of calulated stresses and strains sm(1,1)=0.d0 sm(2,1)=0.d0 sm(3,1)=0.d0 em(1,1)=0.d0 em(2,1)=0.d0 em(3,1)=0.d0 c-----Menu call menu(sal,con,est,mod) c-----Imputh dates call entrada(s,e,mat,ninc) csal=7 esal=1 do while(esal.ne.0) c-----entry of the name of output file write(*,'(" Output File: ",\)') read (*,'(a32)')arcsal open(unit=csal,file=arcsal,status='NEW',iostat=esal) end do rewind(csal) if (sal.eq.1) then write(csal,'(12a13)')'DEF 1','TEN 1', . 'DEF 2','TEN 2', . 'DEF 3','TEN 3', . 'DEFM 1','TENM 1', . 'DEFM 2','TENM 2', . 'DEFM 3','TENM 3' elseif (sal.eq.2) then write(csal,'(4a13)')'X','Y','Xm','Ym' endif if (sal.eq.1) then write(csal,'(12(E13.4))')e(1,1), s(1,1),

CODRI.F

188

. e(2,1), s(2,1),

. e(3,1), s(3,1),

. em(1,1), sm(1,1),

. em(2,1), sm(2,1),

. em(3,1), sm(3,1) elseif (sal.eq.2) then x=(s(1,1)+s(2,1)+s(3,1))/3.d0 y=((s(1,1)-s(2,1))**2+(s(2,1)-s(3,1))**2+(s(3,1) . -s(1,1))**2)/6. y=sqrt(y) xm=(sm(1,1)+sm(2,1)+sm(3,1))/3.d0 ym=((sm(1,1)-sm(2,1))**2+(sm(2,1)-sm(3,1))**2+ . (sm(3,1)-sm(1,1))**2) ym=ym/6. ym=sqrt(ym) write(csal,'(4(E13.4))')x,y,xm,ym endif do inc=2,ninc sigini(1) = sm(1,inc-1) sigini(2) = sm(2,inc-1) sigini(3) = 0.d0 sigini(4) = sm(3,inc-1) c-----Calculation of the imputh increases of stresses and strains do i=1,3 ds(i)=s(i,inc)-s(i,inc-1) de(i)=e(i,inc)-e(i,inc-1) end do call modelos(dsm,dem,ds,de,sigini,mat,inc,con,est,mod,q) c-----Calculation of the output increases of stresses and strains using the mo dels do i=1,3 sm(i,inc)=sm(i,inc-1)+dsm(i) em(i,inc)=em(i,inc-1)+dem(i) end do c-----Output Stress-strain if(sal.eq.1)then write(csal,'(12(E13.4))')e(1,inc), s(1,inc), . e(2,inc), s(2,inc), . e(3,inc), s(3,inc), . em(1,inc), sm(1,inc), . em(2,inc), sm(2,inc), . em(3,inc), sm(3,inc) c-----Output square root(J2) vs. I1 elseif(sal.eq.2)then x=(s(1,inc)+s(2,inc)+s(3,inc))/3.d0 y=((s(1,inc)-s(2,inc))**2+(s(2,inc)-s(3,inc))**2+(s(3,inc) . -s(1,inc))**2)/6. y=sqrt(y) xm=(sm(1,inc)+sm(2,inc)+sm(3,inc))/3.d0 ym=((sm(1,inc)-sm(2,inc))**2+(sm(2,inc)-sm(3,inc))**2+ . (sm(3,inc)-sm(1,inc))**2) ym=ym/6. ym=sqrt(ym) write(csal,'(4(E13.4))')x,y,xm,ym endif end do c-----Closing of output file close(unit=csal) write(*,*) c-----End of the program write(*,*)'PROGRAMA FINALIZADO' write(*,*) end program c----------------------------------------------------------------------c c SUBROUTINES c----------------------------------------------------------------------c subroutine modelos(dsm,dem,ds,de,sigini,mat,inc,con,est,mod,q) c----------------------------------------------------------------------c c Calculation of the output increases of stresses and strains using the models c----------------------------------------------------------------------c implicit none real*8 dsm,dem,ds,de,sigini,mat,sal,q integer*4 inc,con,est,mod,i dimension dsm(3),dem(3),ds(3),de(3),sigini(4),mat(17) c-----Linear Elastic Model (imod=1) if (mod.eq.1)then call elalin(dsm,dem,ds,de,sigini,mat,inc,con,est) c-----von Mises Model (imod=2) elseif(mod.eq.2)then call vonMises(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c-----No-associated Linear Drucker-Prager Model [two parameters model](imod=3) elseif(mod.eq.3)then call linealdp (dsm,dem,ds,de,sigini,mat,inc,con,est,q) c-----No-associated Linear Drucker-Prager Model [three parameters model](imod=

CODRI.F

189

4) elseif (mod.eq.4)then call dp3(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c-----No-associated Bresler and Pister [three parameters model](imod=5) elseif (mod.eq.5)then call bp(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c-----Error message else write(*,*) stop ' NONEXITENT MODEL' endif end subroutine c----------------------------------------------------------------------c subroutine menu(sal,con,est,mod) c----------------------------------------------------------------------c c Options menu c----------------------------------------------------------------------c implicit none integer*4 sal,con,est,mod write(*,'(7(a65,/))') +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ', +'ÛÛ ÛÛ', +'ÛÛ ÛÛ', +'ÛÛ OPTIONS FOR THE CONTROL ÛÛ', +'ÛÛ ÛÛ', +'ÛÛ ÛÛ', +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ' write(*,'(7(a65,/))') +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ', +'ÛÛ ÛÛ', +'ÛÛ OUTPUT ÛÛ', +'ÛÛ (1) Sigma vs. epsilon ÛÛ', +'ÛÛ (2) square root(J2) vs. I1 ÛÛ', +'ÛÛ ÛÛ', +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ' write(*,'(16x," ",\)') read (*,*) sal write(*,*) write(*,'(8(a65,/))') +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ', +'ÛÛ ÛÛ', +'ÛÛ TYPE OF CONTROL ÛÛ', +'ÛÛ (1) Strains ÛÛ', +'ÛÛ (2) Mixed ÛÛ', +'ÛÛ (3) Stresses ÛÛ', +'ÛÛ ÛÛ', +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ' write(*,'(16x," ",\)') read (*,*) con write(*,*) write(*,'(8(a65,/))') +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ', +'ÛÛ ÛÛ', +'ÛÛ STATES ÛÛ', +'ÛÛ (1) Plane strain ÛÛ', +'ÛÛ (2) Plane stress ÛÛ', +'ÛÛ (3) Axisymmetric ÛÛ', +'ÛÛ ÛÛ', +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ' write(*,'(16x," ",\)') read (*,*) est write(*,*) write(*,'(14(a65,/))') +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ', +'ÛÛ ÛÛ', +'ÛÛ MODELS ÛÛ', +'ÛÛ (1) Linear Elastic ÛÛ', +'ÛÛ (2) von Mises (one parameter) ÛÛ', +'ÛÛ (3) Linear D.-P. (two parameters) ÛÛ', +'ÛÛ (4) Linear D.-P. (three parameters) ÛÛ', +'ÛÛ (5) Bresler and Pister (three parameters) ÛÛ', +'ÛÛ ÛÛ', +'ÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛÛ' write(*,'(16x," ",\)') read (*,*) mod write(*,*) end subroutine c----------------------------------------------------------------------c subroutine entrada(s,e,dmt,ninc) c----------------------------------------------------------------------c c Reading inputh dates c----------------------------------------------------------------------c implicit none real*8 s, e, dmt

CODRI.F

190

integer*4 ninc, cdat, edat, cmat, emat, ii, i character arcdat*32,lineas*80 dimension s(3,1500),e(3,1500),dmt(17),lineas(9) cdat=5 edat=1 do while(edat.ne.0) write(*,'(" Database Experiment: ",\)') read (*,'(a32)')arcdat open (unit=cdat,file=arcdat,status='OLD',iostat=edat) end do rewind(cdat) do i=1,9 read (cdat,'(a80)',iostat=edat)lineas(i) if (edat.ne.0) stop 'Error en encabezado de archivo de datos' end do write(*,'(9(a80))')lineas c-----Reading of stresses, strains and number of increases ii=1 do read(cdat,'(6e13.0)',iostat=edat) + s(1,ii),e(1,ii),s(2,ii),e(2,ii),s(3,ii),e(3,ii) if (edat.gt.0) stop 'Error en archivo de datos de ensayos' if (edat.lt.0) exit ii=ii+1 end do ninc=ii-1 write(*,'(" Number of recorded increments: ",\)') write(*,*)ninc close(unit=cdat) write(*,*) c-----Reading of material proprieties cmat=10 emat=1 do while(emat.ne.0) write(*,'(" File of Material Parameters: ",\)') read (*,'(a32)')arcdat open (unit=cmat,file=arcdat,status='OLD',iostat=emat) end do rewind(cmat) do i=1,17 read (cmat,'(e13.0)',iostat=emat)dmt(i) if (emat.gt.0) stop 'Error en archivo datos de materiales' if (emat.lt.0) exit write(*,'(1x,e13.5)')dmt(i) end do close(unit=cmat) write(*,*) end subroutine c----------------------------------------------------------------------c c----------------------------------------------------------------------c c COMMON SUBRUTINES IN THE CONSTITUTIVE MODELS c c----------------------------------------------------------------------c c----------------------------------------------------------------------c c----------------------------------------------------------------------c subroutine elast(mat,ee,dd) c----------------------------------------------------------------------c c Elastic matrix c c----------------------------------------------------------------------c c mat - material proprieties c c ee - stiffness matrix c c dd - inverse stiffness matrix c c e - Young's module c c poi - Poisson's ratio c c----------------------------------------------------------------------c implicit none real*8 mat, ee, dd, e, poi integer*4 i, j dimension mat(17),ee(4,4),dd(4,4) c-----Material Parameters e=mat(1) poi=mat(2) c-----Definition elements of the stiffness matrix do j=1,4 do i=1,4 ee(i,j) = 0.d0 dd(i,j) = 0.d0 end do end do ee(1,1)=e*(1.-poi)/((1.+poi)*(1.-2*poi)) ee(1,2)=e*poi/((1.+poi)*(1.-2*poi)) ee(2,1)=ee(1,2) ee(2,2)=ee(1,1) ee(3,3)=e/(2.*(1.+poi)) ee(1,4)=ee(1,2)

CODRI.F

191

ee(2,4)=ee(1,2) ee(4,1)=ee(1,2) ee(4,2)=ee(1,2) ee(4,4)=ee(1,1) c-----Definition elements of the inverse stiffness matrix dd(1,1)=1/e dd(1,2)=-poi/e dd(2,1)=dd(1,2) dd(2,2)=dd(1,1) dd(3,3)=2.*(1.+poi)/e dd(1,4)=dd(1,2) dd(2,4)=dd(1,2) dd(4,1)=dd(1,2) dd(4,2)=dd(1,2) dd(4,4)=dd(1,1) end subroutine c----------------------------------------------------------------------c c----------------------------------------------------------------------c subroutine predel(sigi,epsi,sigfin,sigini,sigp,ds,de, . ee,dd,poi,con,est) c----------------------------------------------------------------------c c Elastic predictor c----------------------------------------------------------------------c c ds - Imposed stress increment c de - Imposed strain increment c sigi - Elastic stress increment (elastic predictor) c epsi - Strain increment c sigp - Plastic stress increment (plastic corrector) c sigfin - Final stress c sigini - Initial stress c ee - Elastic matrix c dd - Inverse elastic matrix c poi - Poisson's ratio c con - type of control c est - tensional state c----------------------------------------------------------------------c implicit none real*8 ds,de,epsi,sigi,sigp,sigfin,sigini,ee,dd,poi integer i,j,con,est dimension de(3),ds(3),epsi(4),sigi(4), * sigp(4),sigfin(4),sigini(4), * ee(4,4),dd(4,4) c STRAIN CONTROL if(con.eq.1)then c imposed strains epsi(1)=de(1) epsi(2)=de(2) epsi(3)=0.d0 if (est.eq.2)then epsi(4)=-poi/(1-poi)*(de(1)+de(2)) !plane stress elseif(est.eq.1)then epsi(4)=0.d0 !plane strain elseif(est.eq.3)then epsi(4)=de(3) !axisymmetry else stop ' ERROR al seleccionar ESTADO TENSIONAL' endif c elastic predictor do i=1,4 sigi(i)=0.d0 do j=1,4 sigi(i)=sigi(i)+ee(i,j)*epsi(j) end do end do c final stress do i=1,4 sigfin(i)=sigini(i)+sigi(i) end do c MIXED CONTROL elseif(con.eq.2)then !PLANE STRAIN if(est.eq.1)then c elastic predictor sigi(1)=ds(1)+sigp(1) c imposed stain epsi(2)=0.d0 epsi(3)=0.d0 epsi(4)=de(3) epsi(1)=sigi(1)/ee(1,1) do j=2,4 epsi(1)=epsi(1)-ee(1,j)*epsi(j)/ee(1,1) end do c elastic predictor do i=2,4

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sigi(i)=0.d0 do j=1,4 sigi(i)=sigi(i)+ee(i,j)*epsi(j) end do end do !PLANE STRESS else if(est.eq.2)then c elastic predictor sigi(2)=0.d0 + sigp(2) c imposed strain epsi(1)=de(1) epsi(3)=0.d0 epsi(4)=de(3) epsi(2)=sigi(2)/ee(2,2)-ee(2,1)*epsi(1)/ee(2,2) do j=3,4 epsi(2)=epsi(2)-ee(2,j)*epsi(j)/ee(2,2) end do c elastic predictor do i=3,4 sigi(i)=0.d0 do j=1,4 sigi(i)=sigi(i)+ee(i,j)*epsi(j) end do end do sigi(1)=0.d0 do j=1,4 sigi(1)=sigi(1)+ee(1,j)*epsi(j) end do !AXISYMMETRIC else if(est.eq.3)then c elastic predictor sigi(1)=ds(1)+sigp(1) sigi(2)=ds(2)+sigp(2) sigi(3)=sigp(3) c imposed strain epsi(4)=de(3) do i=1,3 epsi(i)=0.d0 do j=1,3 epsi(i)=epsi(i)+dd(i,j)*(sigi(j)-ee(j,4)*epsi(4)) end do end do c elastic predictor sigi(4)=0.d0 do i=1,4 sigi(4)=sigi(4)+ee(4,i)*epsi(i) end do else stop ' ERROR to select the TENSIONAL STATE' endif c Final stress do i=1,4 sigfin(i)=sigini(i)+sigi(i) end do C STRESS CONTROL elseif(con.eq.3)then c elasti predictor sigi(1)=ds(1)+sigp(1) sigi(2)=ds(2)+sigp(2) sigi(3)=0.d0 +sigp(3) if (est.eq.2)then sigi(4)=0.d0+sigp(4) !plane stress elseif(est.eq.1)then sigi(4)=poi*(ds(1)+ds(2))+sigp(4) !plane strain elseif(est.eq.3)then sigi(4)=ds(3)+sigp(4) !axisymmetric else stop ' ERROR to select TENSIONAL STATE' endif do i=1,4 epsi(i)=0.d0 do j=1,4 epsi(i)=epsi(i)+dd(i,j)*sigi(j) end do end do do i=1,4 sigfin(i)=sigini(i)+sigi(i) end do else stop ' ERROR to select TYPE OF CONTROL' endif end subroutine c----------------------------------------------------------------------c subroutine incten(dsm,ds,sigfin,sigini,poi,con,est)

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c----------------------------------------------------------------------c c Calculated increments of stress c----------------------------------------------------------------------c c dsm - calculated increment of stress c ds - imposed increment of stress c sigfin - final stress c sigini - initial stress c poi - Poisson's ratio c con - type of control c est - tensional state c----------------------------------------------------------------------c implicit none real*8 dsm,ds,sigfin,sigini,poi integer con,est,i dimension dsm(3),ds(3),sigfin(4),sigini(4) C---- STRAIN CONTROL if(con.eq.1)then dsm(1)=sigfin(1)-sigini(1) dsm(2)=sigfin(2)-sigini(2) dsm(3)=sigfin(4)-sigini(4) C---- MIXED CONTROL elseif(con.eq.2)then !PLANE STRAIN if(est.eq.1)then dsm(1)=ds(1) dsm(2)=sigfin(2)-sigini(2) dsm(3)=sigfin(4)-sigini(4) !PLANE STRESS else if(est.eq.2)then dsm(1)=sigfin(1)-sigini(1) dsm(2)=0.d0 dsm(3)=sigfin(4)-sigini(4) !AXISYMMETRY else if(est.eq.3)then dsm(1)=ds(1) dsm(2)=ds(2) dsm(3)=sigfin(4)-sigini(4) else stop ' ERROR to select the TENSIONAL STATE' endif c STRESS CONTROL elseif(con.eq.3)then dsm(1)=ds(1) dsm(2)=ds(2) if (est.eq.2)then dsm(3)=0.d0 !plane stress elseif(est.eq.1)then dsm(3)=poi*(ds(1)+ds(2)) !plane strain elseif(est.eq.3)then dsm(3)=ds(3) !axisymmetry else stop ' ERROR to select the TENSIONAL STATE' endif else stop ' ERROR to select the TYPE OF CONTROL' endif end subroutine c----------------------------------------------------------------------c c----------------------------------------------------------------------- c FUNCTIONS c----------------------------------------------------------------------- c======================================================================= real*8 function norma(vector) c----------------------------------------------------------------------- c Vector norm c----------------------------------------------------------------------- implicit none real*8 vector dimension vector(4) norma= dsqrt(vector(1)*vector(1)+ + vector(2)*vector(2)+ + vector(3)*vector(3)+ + vector(4)*vector(4) ) return end c----------------------------------------------------------------------- c----------------------------------------------------------------------c c LINEAR ELASTIC MODEL c c----------------------------------------------------------------------c c Variables c c----------------------------------------------------------------------c c c c inc - number of increase c c est - indicative of state c

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c = 1 plane strain c c = 2 plane stress c c = 3 axisymmetric c c con - control type c c = 1 strain c c = 2 mixed c c = 3 stress c c mat - material proprieties c c poi - Poisson's ratio c c ee - stiffness matrix c c dd - inverse stiffness matrix c c ds - Imposed stress increment c c de - Imposed strain increment c c dsm - Calculated stress increment (output) c c dem - Calculated strain increment (output) c c epsi - Strain increment c c sigi - Elastic stress increment c c sig - Final stress state c c sigini - Initial stress state c c sigp - plastic corrector (null) c c c c----------------------------------------------------------------------c c----------------------------------------------------------------------c subroutine elalin(dsm,dem,ds,de,sigini,mat,inc,con,est) c----------------------------------------------------------------------c c Calculation of the stress and strain increments c c----------------------------------------------------------------------c implicit none real*8 de,ds,dem,dsm,mat,ee,dd,e, * sigp,epsi,sigi,sig,poi,sigini integer*4 inc, con, est, i, j dimension de(3), ds(3), dem(3), dsm(3), mat(16), sigini(4), * sig(4), sigp(4), sigi(4), epsi(4), * ee(4,4), dd(4,4) c-----Material proprieties e=mat(1) !stiffness module poi=mat(2) !Poisson's ratio c-----Elastic matrix call elast(mat,ee,dd) c-----Plastic corrector (null) sigp(1) = 0.d0 sigp(2) = 0.d0 sigp(3) = 0.d0 sigp(4) = 0.d0 c-----Elastic stress and strain increments call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est) c-----Calculated stress and strain increments call incten(dsm,ds,sig,sigini,poi,con,est) dem(1)=epsi(1) dem(2)=epsi(2) dem(3)=epsi(4) end subroutine c----------------------------------------------------------------------c c VON MISES PLASTICITY MODEL c c----------------------------------------------------------------------c c Variables c c----------------------------------------------------------------------c c c c inc - number of increment c c est - indicative of state c c = 1 plane strain c c = 2 plane stress c c = 3 axisymmetric c c con - control type c c = 1 strain c c = 2 mixed c c = 3 stress c c mat - material proprieties c c e - Elastic module c c poi - Poisson's ratio c c cp - Hardening/Softening parameter of quadratic function c c cp1 - Hardening/Softening parameter of Simo's function c c cp2 - Hard./Soft.parameter of Modified Simo's function c c h - Hard./Soft.parameter of Simo and Mod Simo's function c c yi - Hard./Soft.parameter of Simo and Mod Simo's function c c iend - indicator for the hardening/softening function c c = 1 Hard./Soft. Quadratic c c = 2 Hard./Soft. exponential SIMO c c = 3 Hard./Soft. modified exponential SIMO c c y0 - Hard./Soft.parameter of Q-S-MS function c c ymax - Hard./Soft.parameter of Q-S-MS function c c qmax - Hard./Soft.parameter of Q-S-MS function c c yc - Hard./Soft.parameter of Q-S-MS function c c qc - Hard./Soft.parameter of Q-S-MS function c

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c yr - Hard./Soft.parameter of Q-S-MS function c c qr - Hard./Soft.parameter of Q-S-MS function c c ee - Elastic matrix c c dd - Inverse elastic matrix c c func - yield function c c ds - Imposed stress increment (input) c c de - Imposed strain increment (input) c c dsm - Calculated stress increment (output) c c dem - Calculated strain increment (output) c c epsi - strain increments c c sigi - Elastic stress increment (elastic predictor) c c sigini - Auxiliary stress state c c sig - Calculated stress c c rm - Gradient function of plastic potential c c rn - Gradient function of yield function c c en - plastic corrector direction c c dsigp - variation of plastic corrector c c sigp - plastic corrector at anterior iteration c c sigpnew - plastioc corrector: actual iteration c c norma1 - norm of dsigp c c norma2 - norm of sigpnew c c itr - number of iteration (stress control) c c maxitr - maxim number of iterations (stress control) c c maxitc - maxim number of iterations (palstic corrector) c c errrel - maxim relative error between two iterations c c minmin - tolerance to the zero c c c c----------------------------------------------------------------------c c----------------------------------------------------------------------c subroutine vonMises(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c----------------------------------------------------------------------c c Calculation of stress and strain increments c c for Associated Plastic Potential c c----------------------------------------------------------------------c implicit none real*8 de, ds, dem, dsm, mat, ee, dd, dq,cp, * sigp,sigpnew,dsigp,epsi,sigi,sig, e,sigini,ymax,qmax, * poi, alpha, beta, func, yi, q, dlam, cp1, yc, qc,yr, * norma, norma1, norma2, minmin, errrel, cp2, h, y0,qr integer*4 inc,con,est,itr,maxitr,maxitc,i,j,iend dimension de(3),ds(3),dem(3),dsm(3),mat(16), * ee(4,4),dd(4,4),epsi(4),sigi(4),sig(4), * sigp(4),sigpnew(4),dsigp(4),sigini(4) c-----Tolerances minmin=1./10.**14 errrel=0.0001 maxitr=100 maxitc=200 c-----Material and model data e = mat(1) poi = mat(2) y0 = mat(3) alpha= mat(4) beta = mat(5) cp1 = mat(6) h = mat(7) yi = mat(8) iend = mat(9) cp2 = mat(10) yc = mat(11) qc = mat(12) ymax = mat(13) qmax = mat(14) yr = mat(15) qr = mat(16) c-----Elastic Matrix call elast(mat,ee,dd) c-----Plastic corrector reset sigp(1) = 0.d0 sigp(2) = 0.d0 sigp(3) = 0.d0 sigp(4) = 0.d0 if(inc.eq.1)q=0 c-----START TO THE ITERATION (for mixed and stress control) do itr=1,maxitr call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est) c-----Flow Condition call vmfunc(sig,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp) c-----Calculated stress in elastic regime if(func.le.minmin)exit c-----Plastic Corrector call vonres(epsi,sig,ee,q,dlam,y0,yi,h,cp1,iend, + sigpnew,dq,cp2,yc,qc,cp) c---- Final stress update

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sig(1)=sig(1)-sigpnew(1) sig(2)=sig(2)-sigpnew(2) sig(3)=sig(3)-sigpnew(3) sig(4)=sig(4)-sigpnew(4) c Exit condition for strain control if(con.eq.1) exit c Exit condition for stress and mixed control do i=1,4 dsigp(i)=sigpnew(i)-sigp(i) end do norma1=norma(dsigp) norma2=norma(sigpnew) do i=1,4 sigp(i)=sigpnew(i) end do if(dabs(norma1/norma2).lt.errrel) exit c-----END OF ITERACION (for stress and mixed control) end do if(itr.ge.maxitr) write(*,*)' the model does not converge', . ' using the stress control' c-----State variable update q=q+dq c-----Real stress increments call incten(dsm,ds,sig,sigini,poi,con,est) c-----Real strain increments dem(1)=epsi(1) dem(2)=epsi(2) dem(3)=epsi(4) end subroutine c----------------------------------------------------------------------c subroutine vonres(epsi,st,ee,q,dlam,y0,yi,h,cp1,iend, + sigp,dq,cp2,yc,qc,cp) c----------------------------------------------------------------------c c Function :von Mises c Plastic corrector for associated plastic potential c Non-linear Hardening/Softening c ---------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension epsi(4),ee(4,4),sigp(4) dimension sn(4),st(4),en(4),rn(4) c-----Plastic potential gradient call vmgrad (st,rn) c-----Plastic potential direction call vmdir (rn,en,ee) c-----Initial value for the plastic multiplier call vmdlmt (st,rn,en,q,h,y0,yi,cp1,func,dlma,iend,cp2,yc,qc,cp) c-----Development of non-linear consistence condition iter=1 dlma=0.d0 10 call vmdlam(st,rn,sn,en,dq,q,h,y0,yi,cp1,dlma,dlam,iend, + cp2,yc,qc,cp) if(dabs((dlam-dlma)/dlam).gt.1.d-9.and.iter.le.50)then dlma=dlam call vmgrad(sn,rn) call vmdir(rn,en,ee) iter=iter+1 goto 10 endif sigp(1)=dlam*en(1) sigp(2)=dlam*en(2) sigp(4)=dlam*en(4) sigp(3)=dlam*en(3) return end c----------------------------------------------------------------------c c-----------------------------------------------------------------------c subroutine vmfunc(s,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp) c-----------------------------------------------------------------------c c Function : von Mises Yield Function c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension s(4) trs1=s(1)+s(2)+s(4) trs2=s(1)*s(1) +s(2)*s(2) +s(4)*s(4) +2.d0*s(3)*s(3) c-----Quadratic Hardening/Softening if(iend.eq.1)then yn=0.5d0*cp*q*q+h*q+y0 c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q c-----Modified Exponential SIMO Hardening/Softening

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elseif(iend.eq.3.and.q.lt.qc)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q elseif(iend.eq.3.and.q.ge.qc) then yn=yc*dexp(cp2*(q-qc)) else write(*,*)'Hardening/Softening not available',iend stop endif func=dsqrt(.5d0*trs2-trs1*trs1/6.d0)-yn/dsqrt(3.d0) return end c-----------------------------------------------------------------------c subroutine vmgrad (s,rn) c-----------------------------------------------------------------------c c Function:Gradient of associated von Mises plastic potential c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) dimension s(4), rn(4) trs1=s(1)+s(2)+s(4) trs2=s(1)**2+s(2)**2+s(4)**2+2.d0*s(3)**2 trd2=trs2-trs1**2/3.d0 den=dsqrt(2.d0/3.d0*trd2) rn(1)=(-trs1/3.d0+s(1))/den rn(2)=(-trs1/3.d0+s(2))/den rn(4)=(-trs1/3.d0+s(4))/den rn(3)= s(3)/den c-----Engineering strain rn(3)=2.d0*rn(3) return end c-----------------------------------------------------------------------c subroutine vmdir(rn,en,ee) c-----------------------------------------------------------------------c c Function:Direction of the von Mises plastic stress c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) dimension rn(4), en(4), ee(4,4) c en(1)=ee(1,1)*rn(1)+ee(1,2)*rn(2)+ee(1,4)*rn(4) en(2)=ee(2,1)*rn(1)+ee(2,2)*rn(2)+ee(2,4)*rn(4) en(4)=ee(4,1)*rn(1)+ee(4,2)*rn(2)+ee(4,4)*rn(4) c-----Engineering strain en(3)=ee(3,3)*rn(3) return end c-----------------------------------------------------------------------c subroutine vmdlmt(st,rn,en,q,h,y0,yi,cp1,func,dlmt,iend,cp2,yc, + qc,cp) c-----------------------------------------------------------------------c c Function :Consistenze condition c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 irhd dimension st(4), rn(4), en(4) trne=rn(1)*en(1)+rn(2)*en(2)+rn(4)*en(4)+ + rn(3)*en(3) trm2=rn(1)*rn(1)+rn(2)*rn(2)+rn(4)*rn(4)+ + rn(3)*rn(3)/2.d0 c-----Quadratic Hardening/Softening if(iend.eq.1)then yn=0.5d0*cp*q*q+h*q+y0 dfdq=(-cp*q-h)/dsqrt(3.d0) c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h )/dsqrt(3.d0) c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.q.lt.qc) then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h )/dsqrt(3.d0) elseif(iend.eq.3.and.q.ge.qc) then yn=yc*dexp(cp2*(q-qc)) dfdq=-(yc*(dexp(cp2*(q-qc)))*cp2)/dsqrt(3.d0) else write(*,*)'Hardening/Softening not available',iend stop endif dq=dsqrt(2.d0/3.d0*trm2) dlmt=func/(trne-dfdq*dq) return end c-----------------------------------------------------------------------c subroutine vmdlam (st,rn,sn,en,dq,q,h,y0,yi,cp1,dlmt,dlam,

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+ iend,cp2,yc,qc,cp) c-----------------------------------------------------------------------c c Function:Consistence condition c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension st(4),rn(4),sn(4),en(4), sigp(4) tre1=en(1) +en(2) +en(4) tre2=en(1)*en(1)+en(2)*en(2)+en(4)*en(4)+ + en(3)*en(3)+en(3)*en(3) trs1=st(1) +st(2) +st(4) trs2=st(1)*st(1)+st(2)*st(2)+st(4)*st(4)+ + st(3)*st(3)+st(3)*st(3) trse=st(1)*en(1)+st(2)*en(2)+st(4)*en(4)+ + st(3)*en(3)+st(3)*en(3) trm2=rn(1)*rn(1)+rn(2)*rn(2)+rn(4)*rn(4)+ + rn(3)*rn(3)/2.d0 dq=dsqrt(2.d0/3.d0*trm2) w0=(0.5d0*trs2-trs1*trs1/6.d0) w1=(-trse+trs1*tre1/3.d0) w2=(0.5d0*tre2-tre1*tre1/6.d0) c-----Newton-Raphson iteration dlam=dlmt iter=0 10 iter=iter+1 qn1=q+dlam*dq c-----Quadratic Hardening/Softening if(iend.eq.1)then yn1=(0.5d0*cp*qn1*qn1+h*qn1+y0)/dsqrt(3.d0) dyn=(cp*qn1*dq+h*dq)/dsqrt(3.d0) c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1)/dsqrt(3.d0) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq )/dsqrt(3.d0) c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.qn1.lt.qc)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1)/dsqrt(3.d0) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq )/dsqrt(3.d0) elseif(iend.eq.3.and.qn1.ge.qc) then yn1=yc*dexp(cp2*(qn1-qc))/dsqrt(3.d0) dyn=yc*dexp(cp2*(qn1-qc))*cp2*dq/dsqrt(3.d0) else write(*,*)'Hardening/Softening not available',iend stop endif fn1=dsqrt(dlam*dlam*w2+dlam*w1+w0)-yn1 dfn= (dlam*2.d0*w2+w1)/2.d0/dsqrt(dlam*dlam*w2+dlam*w1+w0)-dyn dlam=dlam-fn1/dfn if(dabs(fn1/dfn).gt.1.d-10.and.iter.le.50) goto 10 if(dlam.lt.0.d0) write(*,*)' dlam < null !! ',dlam dq=dq*dlam sn(1)=st(1)-dlam*en(1) sn(2)=st(2)-dlam*en(2) sn(4)=st(4)-dlam*en(4) sn(3)=st(3)-dlam*en(3) return end c----------------------------------------------------------------------c c DRUCKER-PRAGER PLASTICITY MODEL c c (two parameters) c c----------------------------------------------------------------------c c Variables c c----------------------------------------------------------------------c c c c inc - number of increment c c est - indicative of state c c = 1 plane strains c c = 2 plane stresses c c = 3 axisymmetric c c con - control type c c = 1 strains c c = 2 mixed c c = 3 stresses c c mat - material proprieties c c e - Elastic module c c poi - Poisson's ratio c c alfa - yield function parameter c c beta - plastic potential parameter c c cp - Hardening/Softening parameter of quadratic function c c cp1 - Hardening/Softening parameter of Simo's function c c cp2 - Hard./Soft.parameter of Modified Simo's function c c h - Hard./Soft.parameter of Simo and Mod Simo's function c c yi - Hard./Soft.parameter of Simo and Mod Simo's function c

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c iend - indicator for the hardening/softening function c c = 1 Hard./Soft. Quadratic c c = 2 Hard./Soft. exponential SIMO c c = 3 Hard./Soft. modified exponential SIMO c c y0 - Hard./Soft.parameter of Q-S-MS function c c ymax - Hard./Soft.parameter of Q-S-MS function c c qmax - Hard./Soft.parameter of Q-S-MS function c c yc - Hard./Soft.parameter of Q-S-MS function c c qc - Hard./Soft.parameter of Q-S-MS function c c yr - Hard./Soft.parameter of Q-S-MS function c c qr - Hard./Soft.parameter of Q-S-MS function c c ee - Elastic matrix c c dd - Inverse elastic matrix c c func - yield function c c ds - Imposed stress increment (input) c c de - Imposed strain increment (input) c c dsm - Calculated stress increment (output) c c dem - Calculated strain increment (output) c c epsi - strain increments c c sigi - Elastic stress increment (elastic predictor) c c sigini - Auxiliary stress state c c sig - Calculated stress c c rm - Gradient function of plastic potential c c rn - Gradient function of yield function c c en - plastic corrector direction c c dsigp - variation of plastic corrector c c sigp - plastic corrector at anterior iteration c c sigpnew - plastioc corrector: actual iteration c c norma1 - norm of dsigp c c norma2 - norm of sigpnew c c itr - number of iteration (stress control) c c maxitr - maxim number of iterations (stress control) c c maxitc - maxim number of iterations (palstic corrector) c c errrel - maxim relative error between two iterations c c minmin - tolerance to the zero c c c c----------------------------------------------------------------------c c----------------------------------------------------------------------c subroutine linealdp(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c----------------------------------------------------------------------c c Calculation of stress and strain increments c c for No-Associated Plastic Potential c c----------------------------------------------------------------------c implicit none real*8 de,ds,dem,dsm, mat,ee,dd,dq,norma,norma1,norma2, * sigp,sigpnew,dsigp,epsi,sigi,sig,e,sigini,ymax,qmax, * poi,alpha,beta,func,yi,q,dlam,cp1,yc,qc,yr,cp, * minmin,errrel,cp2,h,y0,qr integer*4 inc,con,est,itr,maxitr,maxitc,i,j,iend dimension de(3),ds(3),dem(3),dsm(3),mat(16), * ee(4,4),dd(4,4),epsi(4),sigi(4),sig(4), * sigp(4),sigpnew(4),dsigp(4),sigini(4) c-----Tolerances minmin=1./10.**14 errrel=0.0001 maxitr=100 maxitc=200 c-----Material and model data e = mat(1) poi = mat(2) y0 = mat(3) alpha= mat(4) beta = mat(5) cp1 = mat(6) h = mat(7) yi = mat(8) iend = mat(9) cp2 = mat(10) yc = mat(11) qc = mat(12) ymax = mat(13) qmax = mat(14) yr = mat(15) qr = mat(16) c-----Elastic Matrix call elast(mat,ee,dd) c-----Plastic corrector reset sigp(1) = 0.d0 sigp(2) = 0.d0 sigp(3) = 0.d0 sigp(4) = 0.d0 if(inc.eq.2)q=0 c-----START TO THE ITERATION (for mixed and stress control) do itr=1,maxitr

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call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est) c-----Flow Condition call vcfunc(sig,alpha,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp) c-----Calculated stress in elastic regime if(func.le.minmin)exit c-----Plastic Corrector call vcresp(epsi,sig,ee,q,dlam,y0,yi,alpha,beta,h,cp1,iend, + sigpnew,dq,cp2,yc,qc,cp) c---- Final stress update sig(1)=sig(1)-sigpnew(1) sig(2)=sig(2)-sigpnew(2) sig(3)=sig(3)-sigpnew(3) sig(4)=sig(4)-sigpnew(4) c Exit condition for strain control if(con.eq.1) exit c Exit condition for stress and mixed control do i=1,4 dsigp(i)=sigpnew(i)-sigp(i) end do norma1=norma(dsigp) norma2=norma(sigpnew) do i=1,4 sigp(i)=sigpnew(i) end do if(dabs(norma1/norma2).lt.errrel) exit c-----END OF ITERACION (for stress and mixed control) end do if(itr.ge.maxitr) write(*,*)' The model does not converge', . ' using the stress control' c-----State variable update q=q+dq c-----Real stress increments call incten(dsm,ds,sig,sigini,poi,con,est) c-----Real strain increments dem(1)=epsi(1) dem(2)=epsi(2) dem(3)=epsi(4) end subroutine c----------------------------------------------------------------------c subroutine vcresp(epsi,st,ee,q,dlam,y0,yi,alpha,beta,h,cp1,iend, + sigp,dq,cp2,yc,qc,cp) c----------------------------------------------------------------------c c Function :No-associated Drucker Prager. c Plastic corrector for plastic potential function c Non-linear Hardening/Softening c ---------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension epsi(4),ee(4,4),sigp(4) dimension sn(4),st(4),em(4),rm(4) c-----Plastic potential gradient call vcgrad(st,rm,beta) c-----Plastic potential direction call vcsdir(rm,em,ee) c-----Initial value for the plastic multiplier call vcdlmt(st,rm,em,q,h,y0,yi,alpha,cp1,func,dlma,iend, + cp2,yc,qc,cp) c-----Development of non-linear consistence condition iter=1 dlma=0.d0 10 call vcdlam(st,rm,sn,em,dq,q,h,y0,yi,alpha,cp1,dlma,dlam,iend, + cp2,yc,qc,cp) if(dabs((dlam-dlma)/dlam).gt.1.d-9.and.iter.le.50)then dlma=dlam call vcgrad(sn,rm,beta) call vcsdir(rm,em,ee) iter=iter+1 goto 10 endif sigp(1)=dlam*em(1) sigp(2)=dlam*em(2) sigp(4)=dlam*em(4) sigp(3)=dlam*em(3) return end c-----------------------------------------------------------------------c subroutine vcfunc(s,alpha,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp) c-----------------------------------------------------------------------c c Function : Drucker-Prager Yield Function (two parameters) c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension s(4)

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trs1=s(1) +s(2) +s(4) trs2=s(1)**2+s(2)**2+s(4)**2+s(3)**2+s(3)**2 trd2=trs2-trs1*trs1/3.d0 c-----Quadratic Hardening/Softening if(iend.eq.1)then yn=0.5d0*cp*q*q+h*q+y0 c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.q.lt.qc)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q elseif(iend.eq.3.and.q.ge.qc) then yn=yc*dexp(cp2*(q-qc)) else write(*,*)'Hardening/Softening not available',iend stop endif func=alpha*trs1+dsqrt(.5d0*trd2)-yn/dsqrt(3.d0) return end c-----------------------------------------------------------------------c subroutine vcgrad(s,rm,beta) c-----------------------------------------------------------------------c c Function:Gradient of Drucker-Prager plastic potential c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) dimension s(4),rm(4) trs1=s(1) +s(2) +s(4) trs2=s(1)**2+s(2)**2+s(4)**2+s(3)**2+s(3)**2 trd2=trs2-trs1*trs1/3.d0 den=2.d0*dsqrt(.5d0*trd2) rm(1)=beta+(s(1)-trs1/3.d0)/den rm(2)=beta+(s(2)-trs1/3.d0)/den rm(4)=beta+(s(4)-trs1/3.d0)/den rm(3)=1.d0*(s(3) )/den c-----Engineering strain rm(3)=2.d0*rm(3) return end c-----------------------------------------------------------------------c subroutine vcsdir(rm,em,ee) c-----------------------------------------------------------------------c c Function:Direction of the Drucker-Prager plastic stress c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) dimension rm(4),em(4),ee(4,4) em(1)=ee(1,1)*rm(1)+ee(1,2)*rm(2)+ee(1,4)*rm(4) em(2)=ee(2,1)*rm(1)+ee(2,2)*rm(2)+ee(2,4)*rm(4) em(4)=ee(4,1)*rm(1)+ee(4,2)*rm(2)+ee(4,4)*rm(4) c-----Engineering strain em(3)=ee(3,3)*rm(3) return end c-----------------------------------------------------------------------c subroutine vcdlmt(st,rm,em,q,h,y0,yi,alpha,cp1,func,dlmt,iend, + cp2,yc,qc,cp) c-----------------------------------------------------------------------c c Function :Consistenze condition c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 irhd dimension st(4),rm(4),rn(4),em(4) call vcgrad(st,rn,alpha) trne=rn(1)*em(1)+rn(2)*em(2)+rn(4)*em(4)+ + rn(3)*em(3) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+ + rm(3)*rm(3)/2.d0 c-----Quadratic Hardening/Softening if(iend.eq.1)then yn=0.5d0*cp*q*q+h*q+y0 dfdq=(-cp*q-h)/dsqrt(3.d0) c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h )/dsqrt(3.d0) c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.q.lt.qc)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h )/dsqrt(3.d0) elseif(iend.eq.3.and.q.ge.qc) then yn=yc*dexp(cp2*(q-qc)) dfdq=-(yc*(dexp(cp2*(q-qc)))*cp2)/dsqrt(3.d0)

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else write(*,*)'Hardening/Softening not available',iend stop endif dq=dsqrt(2.d0/3.d0*trm2) dlmt=func/(trne-dfdq*dq) return end c-----------------------------------------------------------------------c subroutine vcdlam(st,rm,sn,em,dq,q,h,y0,yi,alpha,cp1,dlmt,dlam, + iend,cp2,yc,qc,cp) c-----------------------------------------------------------------------c c Function:Consistence condition c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension st(4),rm(4),sn(4),em(4), sigp(4) tre1=em(1) +em(2) +em(4) tre2=em(1)*em(1)+em(2)*em(2)+em(4)*em(4)+ + em(3)*em(3)+em(3)*em(3) trs1=st(1) +st(2) +st(4) trs2=st(1)*st(1)+st(2)*st(2)+st(4)*st(4)+ + st(3)*st(3)+st(3)*st(3) trse=st(1)*em(1)+st(2)*em(2)+st(4)*em(4)+ + st(3)*em(3)+st(3)*em(3) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+ + rm(3)*rm(3)/2.d0 dq=dsqrt(2.d0/3.d0*trm2) w0=(0.5d0*trs2-trs1*trs1/6.d0) w1=(-trse+trs1*tre1/3.d0) w2=(0.5d0*tre2-tre1*tre1/6.d0) c-----Newton-Raphson iteration dlam=dlmt iter=0 11 iter=iter+1 qn1=q+dlam*dq c-----Quadratic Hardening/Softening if(iend.eq.1)then yn1=(0.5d0*cp*qn1*qn1+h*qn1+y0)/dsqrt(3.d0) dyn=(cp*qn1*dq+h*dq)/dsqrt(3.d0) c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1)/dsqrt(3.d0) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq )/dsqrt(3.d0) c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.qn1.lt.qc)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1)/dsqrt(3.d0) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq )/dsqrt(3.d0) elseif(iend.eq.3.and.qn1.ge.qc) then yn1=yc*dexp(cp2*(qn1-qc))/dsqrt(3.d0) dyn=yc*dexp(cp2*(qn1-qc))*cp2*dq/dsqrt(3.d0) else write(*,*)'Hardening/Softening not available',iend stop endif fn1=alpha*(trs1-dlam*tre1)+dsqrt(dlam*dlam*w2+dlam*w1+w0) + -yn1 dfn=-alpha*tre1 + +(dlam*2.d0*w2+w1)/2.d0/dsqrt(dlam*dlam*w2+dlam*w1+w0) + -dyn dlam=dlam-fn1/dfn if(dabs(fn1/dfn).gt.1.d-10.and.iter.le.200) goto 11 if(dlam.lt.0.d0) write(*,*)' dlam < null !! ',dlam dq=dq*dlam sn(1)=st(1)-dlam*em(1) sn(2)=st(2)-dlam*em(2) sn(4)=st(4)-dlam*em(4) sn(3)=st(3)-dlam*em(3) call vcfunc(sn,alpha,y0,yi,h,(q+dq),cp1,func,iend,cp2,yc,qc,cp) if(dabs(func).gt.1.d-4)then write(*,*)' false dlam, func= ',func endif return end c----------------------------------------------------------------------c c DRUCKER-PRAGER PLASTICITY MODEL c c (three parameters) c c----------------------------------------------------------------------c c Variables c c----------------------------------------------------------------------c c c c inc - number of increment c c est - indicative of state c

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c = 1 plane strains c c = 2 plane stresses c c = 3 axisymmetric c c con - control type c c = 1 strains c c = 2 mixed c c = 3 stresses c c mat - material proprieties c c e - Elastic module c c poi - Poisson's ratio c c alfa - yield function parameter c c z - yield function parameter c beta - plastic potential parameter c c cp - Hardening/Softening parameter of quadratic function c c cp1 - Hardening/Softening parameter of Simo's function c c cp2 - Hard./Soft.parameter of Modified Simo's function c c h - Hard./Soft.parameter of Simo and Mod Simo's function c c yi - Hard./Soft.parameter of Simo and Mod Simo's function c c iend - indicator for the hardening/softening function c c = 1 Hard./Soft. Quadratic c c = 2 Hard./Soft. exponential SIMO c c = 3 Hard./Soft. modified exponential SIMO c c y0 - Hard./Soft.parameter of Q-S-MS function c c ymax - Hard./Soft.parameter of Q-S-MS function c c qmax - Hard./Soft.parameter of Q-S-MS function c c yc - Hard./Soft.parameter of Q-S-MS function c c qc - Hard./Soft.parameter of Q-S-MS function c c yr - Hard./Soft.parameter of Q-S-MS function c c qr - Hard./Soft.parameter of Q-S-MS function c c ee - Elastic matrix c c dd - Inverse elastic matrix c c func - yield function c c ds - Imposed stress increment (input) c c de - Imposed strain increment (input) c c dsm - Calculated stress increment (output) c c dem - Calculated strain increment (output) c c epsi - strain increments c c sigi - Elastic stress increment (elastic predictor) c c sigini - Auxiliary stress state c c sig - Calculated stress c c rm - Gradient function of plastic potential c c rn - Gradient function of yield function c c en - plastic corrector direction c c dsigp - variation of plastic corrector c c sigp - plastic corrector at anterior iteration c c sigpnew - plastioc corrector: actual iteration c c norma1 - norm of dsigp c c norma2 - norm of sigpnew c c itr - number of iteration (stress control) c c maxitr - maxim number of iterations (stress control) c c maxitc - maxim number of iterations (palstic corrector) c c errrel - maxim relative error between two iterations c c minmin - tolerance to the zero c c c c----------------------------------------------------------------------c c----------------------------------------------------------------------c subroutine dp3(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c----------------------------------------------------------------------c c Calculation of stress and strain increments c c for No-Associated Plastic Potential c c----------------------------------------------------------------------c implicit none real*8 de,ds,dem,dsm, mat,ee,dd,dq,norma,norma1,norma2, * sigp,sigpnew,dsigp,epsi,sigi,sig,e,sigini,ymax,qmax, * poi,alpha,beta,func,yi,q,dlam,cp1,yc,qc,yr,cp,z, * minmin,errrel,cp2,h,y0,qr,rm,t integer*4 inc,con,est,itr,maxitr,maxitc,i,j,iend dimension de(3),ds(3),dem(3),dsm(3),mat(17),rm(4), * ee(4,4),dd(4,4),epsi(4),sigi(4),sig(4), * sigp(4),sigpnew(4),dsigp(4),sigini(4) c-----Tolerances minmin=1./10.d0**14 errrel=0.0001 maxitr=100 maxitc=200 c-----Material and model data e = mat(1) poi = mat(2) y0 = mat(3) alpha= mat(4) beta = mat(5) cp1 = mat(6) h = mat(7) yi = mat(8)

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iend = mat(9) cp2 = mat(10) yc = mat(11) qc = mat(12) ymax = mat(13) qmax = mat(14) yr = mat(15) qr = mat(16) z = mat(17) c-----Elastic Matrix call elast(mat,ee,dd) c-----Plastic corrector reset sigp(1) = 0.d0 sigp(2) = 0.d0 sigp(3) = 0.d0 sigp(4) = 0.d0 if(inc.eq.2)q=0 c-----START TO THE ITERATION (for mixed and stress control) do itr=1,maxitr call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est) c-----Flow Condition call vcfunc3(sig,alpha,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp,z + ,t) c-----Calculated stress in elastic regime if(func.le.minmin)exit c-----Plastic Corrector call vcresp3(func,epsi,sig,ee,q,dlam,y0,yi,alpha,beta,h,cp1,iend, + sigpnew,dq,cp2,yc,qc,cp,z,rm) c---- Final stress update sig(1)=sig(1)-sigpnew(1) sig(2)=sig(2)-sigpnew(2) sig(3)=sig(3)-sigpnew(3) sig(4)=sig(4)-sigpnew(4) c Exit condition for strain control if(con.eq.1) exit c Exit condition for stress and mixed control do i=1,4 dsigp(i)=sigpnew(i)-sigp(i) end do norma1=norma(dsigp) norma2=norma(sigpnew) do i=1,4 sigp(i)=sigpnew(i) end do if(dabs(norma1/norma2).lt.errrel) exit c-----END OF ITERACION (for stress and mixed control) end do if(itr.ge.maxitr) write(*,*)' The model does not converge', . ' using the stress control' c-----State variable update q=q+dq c-----Real stress increments call incten(dsm,ds,sig,sigini,poi,con,est) c-----Real strain increments dem(1)=epsi(1) dem(2)=epsi(2) dem(3)=epsi(4) end subroutine c----------------------------------------------------------------------c subroutine vcresp3(func,epsi,st,ee,q,dlam,y0,yi,alpha,beta,h,cp1, + iend,sigp,dq,cp2,yc,qc,cp,z,rm) c----------------------------------------------------------------------c c Function :No-associated Drucker Prager. c Plastic corrector for plastic potential function c Non-linear Hardening/Softening c ---------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension epsi(4),ee(4,4),sigp(4) dimension sn(4),st(4),em(4),rm(4) c-----Plastic potential gradient call vcgrad3(st,rm,beta,z,dtrd2s1,dtrd3s1,dqjs1,dts1,dpds1, + dfds1) c-----Plastic potential direction call vcsdir3(rm,em,ee) c-----Initial value for the plastic multiplier call vcdlmt3(st,rm,em,q,h,y0,yi,alpha,cp1,func,dlma,iend, + cp2,yc,qc,cp,z,yn,dfdq,dq) c-----Development of non-linear consistence condition 10 call vcdlam3(st,rm,sn,em,dq,q,h,y0,yi,alpha,cp1,dlma,dlam,iend, + cp2,yc,qc,cp,z, + fn1,tl,pl,yn1,dfn) if(dabs((dlam-dlma)/dlam).gt.1.d-10.and.iter.le.50)then dlma=dlam

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call vcgrad3(sn,rm,beta,z,dtrd2s1,dtrd3s1,dqjs1,dts1,dpds1, + dfds1) iter=iter+1 goto 10 endif sigp(1)=dlam*em(1) sigp(2)=dlam*em(2) sigp(4)=dlam*em(4) sigp(3)=dlam*em(3) return end c-----------------------------------------------------------------------c subroutine vcfunc3(s,alpha,y0,yi,h,q,cp1,func,iend,cp2,yc,qc,cp,z + ,t) c-----------------------------------------------------------------------c c Function : Drucker-Prager Yield Function (three parameters) c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension s(4) trs1=s(1) +s(2) +s(4) trs2=s(1)*s(2)+s(2)*s(4)+s(4)*s(1)-s(3)*s(3) trs3=s(1)*s(2)*s(4)-s(3)*s(3)*s(4) trd2=1/3.d0*(trs1*trs1-3*trs2) trd3=1/27.d0*(2*trs1*trs1*trs1-9*trs1*trs2+27*trs3) p=-trs1/3.d0 qj=dsqrt(trd2*3.d0) t=1/2.d0*qj*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3)) c-----Quadratic Hardening/Softening if(iend.eq.1)then yn=0.5d0*cp*q*q+h*q+y0 c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.q.lt.qc)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q elseif(iend.eq.3.and.q.ge.qc) then yn=yc*dexp(cp2*(q-qc)) else write(*,*)'Hardening/Softening not available',iend stop endif func=t-alpha*p-yn return end c-----------------------------------------------------------------------c subroutine vcgrad3(s,rm,beta,z,dtrd2s1,dtrd3s1,dqjs1,dts1,dpds1, + dfds1) c-----------------------------------------------------------------------c c Function:Gradient of Drucker-Prager plastic potential c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) dimension s(4),rm(4) trs1=s(1) +s(2) +s(4) trs2=s(1)*s(2)+s(2)*s(4)+s(4)*s(1)-s(3)*s(3) trs3=s(1)*s(2)*s(4)-s(3)*s(3)*s(4) trd2=1/3.d0*(trs1*trs1-3*trs2) trd3=1/27.d0*(2*trs1*trs1*trs1-9*trs1*trs2+27*trs3) p=-trs1/3.d0 qj=dsqrt(trd2*3.d0) t=1/2.d0*qj*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3)) dtrd2s1=1/3.d0*(2*trs1-3*(s(2)+s(4))) dtrd3s1=1/27.d0*(6*trs1*trs1-9*trs2-9*trs1*(s(2)+s(4))+27* + (s(2)*s(4))) dqjs1=1/2.d0/(dsqrt(trd2*3.d0))*3.d0*dtrd2s1 dts1=1/2.d0*(dqjs1*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3))+qj* + (-(1-1/z)*(-3*(1/qj)**4*dqjs1*(27/2.d0*trd3)+ + (1/qj)**3*27/2.d0*dtrd3s1))) dpds1=-1/3.d0 dfds1=dts1-beta*dpds1 rm(1)=dfds1 dtrd2s2=1/3.d0*(2*trs1-3*(s(1)+s(4))) dtrd3s2=1/27.d0*(6*trs1*trs1-9*trs2-9*trs1*(s(1)+s(4))+27* + (s(1)*s(4))) dqjs2=1/2.d0/(dsqrt(trd2*3.d0))*3.d0*dtrd2s2 dts2=1/2.d0*(dqjs2*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3))+qj* + (-(1-1/z)*(-3*(1/qj)**4*dqjs2*(27/2.d0*trd3)+ + (1/qj)**3*27/2.d0*dtrd3s2))) dpds2=-1/3.d0 dfds2=dts2-beta*dpds2 rm(2)=dfds2 dtrd2s4=1/3.d0*(2*trs1-3*(s(2)+s(1)))

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dtrd3s4=1/27.d0*(6*trs1*trs1-9*trs2-9*trs1*(s(2)+s(1))+27* + (s(2)*s(1))) dqjs4=1/2.d0/(dsqrt(trd2*3.d0))*3.d0*dtrd2s4 dts4=1/2.d0*(dqjs4*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3))+qj* + (-(1-1/z)*(-3*(1/qj)**4*dqjs4*(27/2.d0*trd3)+ + (1/qj)**3*27/2.d0*dtrd3s4))) dpds4=-1/3.d0 dfds4=dts4-beta*dpds4 rm(4)=dfds4 dtrd2s3=1/3.d0*(-3*(-2.d0*s(3))) dtrd3s3=1/27.d0*(-9*trs1*(-2.d0*s(3))+27*(-s(4)*2.d0*s(3))) dqjs3=1/2.d0/(dsqrt(trd2*3.d0))*3.d0*dtrd2s3 dts3=1/2.d0*(dqjs3*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3))+qj* + (-(1-1/z)*(-3*(1/qj)**4*dqjs3*(27/2.d0*trd3)+ + (1/qj)**3*27/2.d0*dtrd3s3))) dpds3=0.d0 dfds3=dts3-beta*dpds3 rm(3)=dfds3 c-----Engineering strain rm(3)=2.d0*rm(3) return end c-----------------------------------------------------------------------c subroutine vcsdir3(rm,em,ee) c-----------------------------------------------------------------------c c Function:Direction of the Drucker-Prager plastic stress c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) dimension rm(4),em(4),ee(4,4) em(1)=ee(1,1)*rm(1)+ee(1,2)*rm(2)+ee(1,4)*rm(4) em(2)=ee(2,1)*rm(1)+ee(2,2)*rm(2)+ee(2,4)*rm(4) em(4)=ee(4,1)*rm(1)+ee(4,2)*rm(2)+ee(4,4)*rm(4) c-----Engineering strain em(3)=ee(3,3)*rm(3) return end c-----------------------------------------------------------------------c subroutine vcdlmt3(st,rm,em,q,h,y0,yi,alpha,cp1,func,dlmt,iend, + cp2,yc,qc,cp,z,yn,dfdq,dq) c-----------------------------------------------------------------------c c Function :Consistenze condition c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 irhd dimension st(4),rm(4),rn(4),em(4) call vcgrad3(st,rn,alpha,z,dtrd2s1,dtrd3s1,dqjs1,dts1,dpds1, + dfds1) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+rn(3)*em(3) trne=rn(1)*em(1)+rn(2)*em(2)+rn(4)*em(4)+rm(3)*rm(3)/2.d0 c-----Quadratic Hardening/Softening if(iend.eq.1)then yn=0.5d0*cp*q*q+h*q+y0 dfdq=(-cp*q-h) c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h ) c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.q.lt.qc)then yn=y0+(yi-y0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-y0)*cp1*dexp(-cp1*q)+h ) elseif(iend.eq.3.and.q.ge.qc) then yn=yc*dexp(cp2*(q-qc)) dfdq=-(yc*(dexp(cp2*(q-qc)))*cp2) else write(*,*)'Hardening/Softening not available',iend stop endif dq=dsqrt(2.d0/3.d0*trm2) dlmt=func/(trne-dfdq*dq) return end c-----------------------------------------------------------------------c subroutine vcdlam3(st,rm,sn,em,dq,q,h,y0,yi,alpha,cp1,dlmt,dlam, + iend,cp2,yc,qc,cp,z, + fn1,tl,pl,yn1,dfn) c-----------------------------------------------------------------------c c Function:Consistence condition c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension st(4),rm(4),sn(4),em(4), sigp(4)

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trs1=st(1) +st(2) +st(4) trs2=st(1)*st(2)+st(2)*st(4)+st(4)*st(1)-st(3)*st(3) trs3=st(1)*st(2)*st(4)-st(3)*st(3)*st(4) trd2=1/3.d0*(trs1*trs1-3*trs2) trd3=1/27.d0*(2*trs1*trs1*trs1-9*trs1*trs2+27*trs3) p=-trs1/3.d0 qj=dsqrt(trd2*3.d0) t=1/2.d0*qj*(1+1/z-(1-1/z)*(1/qj)**3*(27/2.d0*trd3)) tre1=em(1) +em(2) +em(4) tre2=em(1)*em(2)+em(2)*em(4)+em(4)*em(1)-em(3)*em(3) tre3=em(1)*em(2)*em(4)-em(3)*em(3)*em(4) trde2=1/3.d0*(tre1*tre1-3*tre2) trde3=1/27.d0*(2*tre1*tre1*tre1-9*tre1*tre2+27*tre3) pe=-tre1/3.d0 qje=dsqrt(trde2*3.d0) te=1/2.d0*qje*(1+1/z-(1-1/z)*(1/qje)**3*(27/2.d0*trde3)) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+rm(3)*rm(3)/2.d0 dq=dsqrt(2.d0/3.d0*trm2) c-----Newton-Raphson iteration dlam=dlmt iter=0 11 iter=iter+1 qn1=q+dlam*dq c-----Quadratic Hardening/Softening if(iend.eq.1)then yn1=(0.5d0*cp*qn1*qn1+h*qn1+y0) dyn=(cp*qn1*dq+h*dq) c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq ) c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.qn1.lt.qc)then yn1=(y0+(yi-y0)*(1.d0-dexp(-cp1*qn1))+h*qn1) dyn=( (yi-y0)*cp1*dq*dexp(-cp1*qn1)+h*dq ) elseif(iend.eq.3.and.qn1.ge.qc) then yn1=yc*dexp(cp2*(qn1-qc)) dyn=yc*dexp(cp2*(qn1-qc))*cp2*dq else write(*,*)'Hardening/Softening not available',iend stop endif trs1l=st(1)+st(2)+st(4)-dlam*(em(1)+em(2)+em(4)) trs2l=trs2+dlam**2*tre2-dlam*(st(1)*em(2)+em(1)*st(2)+ + st(2)*em(4)+em(2)*st(4)+st(4)*em(1)+em(4)*st(1)) + +2.d0*dlam*st(3)*em(3) trs3l=trs3-dlam*(st(1)*st(2)*em(4)+st(1)*em(2)*st(4)+ + st(2)*em(1)*st(4))+dlam**2*(st(1)*em(2)*em(4)+ + st(2)*em(4)*em(1)+st(4)*em(1)*em(2))-dlam**3*tre3 + +st(3)*st(3)*dlam*em(4)-dlam**2.d0*em(3)*em(3)*st(4) + +2.d0*st(3)*dlam*em(3)*st(4)-2.d0*st(3)*dlam*em(3)*dlam*em(4) trd2l=1/3.d0*(trs1l*trs1l-3*trs2l) trd3l=1/27.d0*(2*trs1l*trs1l*trs1l-9*trs1l*trs2l+27*trs3l) pl=-trs1l/3.d0 qjl=dsqrt(trd2l*3.d0) tl=1/2.d0*qjl*(1+1/z-(1-1/z)*(1/qjl)**3*(27/2.d0*trd3l)) dtrs1l=-(em(1)+em(2)+em(4)) dtrs2l=2*dlam*tre2-(st(1)*em(2)+em(1)*st(2)+st(2)*em(4)+em(2)* + st(4)+st(4)*em(1)+em(4)*st(1))+2.d0*st(3)*em(3) dtrs3l=-(st(1)*st(2)*em(4)+st(1)*em(2)*st(4)+st(2)*em(1)*st(4))+ + 2*dlam*(st(1)*em(2)*em(4)+st(2)*em(4)*em(1)+st(4)*em(1)* + em(2))-3*dlam**2*tre3 + +st(3)*st(3)*em(4)-2*dlam*em(3)*em(3)*st(4) + +2.d0*st(3)*em(3)*st(4)-4.d0*st(3)*em(3)*dlam*em(4) dtrd2l=1/3.d0*(2*trs1l*dtrs1l-3*dtrs2l) dtrd3l=1/27.d0*(6*trs1l*trs1l*dtrs1l-9*dtrs1l*trs2l-9*trs1l*dtrs2l + +27*dtrs3l) dqjl=1/2.d0/qjl*3.d0*dtrd2l dtl=1/2.d0*((dqjl*(1+1/z-(1-1/z)*(1/qjl)**3*(27/2.d0*trd3l))+qjl* + (-(1-1/z))*((-3)*1/qjl**4*dqjl*(27/2.d0*trd3l)+ + (1/qjl)**3*27/2.d0*dtrd3l))) dpl=-1/3.d0*dtrs1l fn1=tl-alpha*pl-yn1 dfn=dtl-alpha*dpl-dyn dlam=dlam-fn1/dfn if(dabs(fn1/dfn).gt.1.d-10.and.iter.le.200) goto 11 if(dlam.lt.0.d0) write(*,*)' dlam < null !! ',dlam dq=dq*dlam sn(1)=st(1)-dlam*em(1) sn(2)=st(2)-dlam*em(2) sn(4)=st(4)-dlam*em(4) sn(3)=st(3)-dlam*em(3) return end

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c----------------------------------------------------------------------c c BRESLER-PISTER PLASTICITY MODEL c c (three parameters) c c----------------------------------------------------------------------c c Variables c c----------------------------------------------------------------------c c c c inc - number of increment c c est - indicative of state c c = 1 plane strains c c = 2 plane stresses c c = 3 axisymmetric c c con - control type c c = 1 strains c c = 2 mixed c c = 3 stresses c c mat - material proprieties c c e - Elastic module c c poi - Poisson's ratio c c b - yield function parameter c c c - yield function parameter c c bb - plastic potential parameter c c cc - plastic potential parameter c c cp - Hardening/Softening parameter of quadratic function c c cp1 - Hardening/Softening parameter of Simo's function c c cp2 - Hard./Soft.parameter of Modified Simo's function c c h - Hard./Soft.parameter of Simo and Mod Simo's function c c yi - Hard./Soft.parameter of Simo and Mod Simo's function c c iend - indicator for the hardening/softening function c c = 1 Hard./Soft. Quadratic c c = 2 Hard./Soft. exponential SIMO c c = 3 Hard./Soft. modified exponential SIMO c c a0 - Hard./Soft.parameter of Q-S-MS function c c amax - Hard./Soft.parameter of Q-S-MS function c c qmax - Hard./Soft.parameter of Q-S-MS function c c ac - Hard./Soft.parameter of Q-S-MS function c c qc - Hard./Soft.parameter of Q-S-MS function c c ar - Hard./Soft.parameter of Q-S-MS function c c qr - Hard./Soft.parameter of Q-S-MS function c c ee - Elastic matrix c c dd - Inverse elastic matrix c c func - yield function c c ds - Imposed stress increment (input) c c de - Imposed strain increment (input) c c dsm - Calculated stress increment (output) c c dem - Calculated strain increment (output) c c epsi - strain increments c c sigi - Elastic stress increment (elastic predictor) c c sigini - Auxiliary stress state c c sig - Calculated stress c c rm - Gradient function of plastic potential c c rn - Gradient function of yield function c c en - plastic corrector direction c c dsigp - variation of plastic corrector c c sigp - plastic corrector at anterior iteration c c sigpnew - plastioc corrector: actual iteration c c norma1 - norm of dsigp c c norma2 - norm of sigpnew c c itr - number of iteration (stress control) c c maxitr - maxim number of iterations (stress control) c c maxitc - maxim number of iterations (palstic corrector) c c errrel - maxim relative error between two iterations c c minmin - tolerance to the zero c c c c----------------------------------------------------------------------c c----------------------------------------------------------------------c subroutine bp(dsm,dem,ds,de,sigini,mat,inc,con,est,q) c----------------------------------------------------------------------c c Calculation of stress and strain increments c c for No-Associated Plastic Potential c c----------------------------------------------------------------------c implicit none real*8 de,ds,dem,dsm, mat,ee,dd,dq,norma,norma1,norma2, * sigp,sigpnew,dsigp,epsi,sigi,sig,e,sigini,amax,qmax, * poi,b,c,bb,cc,func,yi,q,dlam,cp1,ac,qc,ar,cp, * minmin,errrel,cp2,h,a0,qr,rm integer*4 inc,con,est,itr,maxitr,maxitc,i,j,iend dimension de(3),ds(3),dem(3),dsm(3),mat(17),rm(4), * ee(4,4),dd(4,4),epsi(4),sigi(4),sig(4), * sigp(4),sigpnew(4),dsigp(4),sigini(4) c-----Tolerances minmin=1./10.d0**14 errrel=0.0001 maxitr=100

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maxitc=200 c-----Material and model data e = mat(1) poi = mat(2) a0 = mat(3) b = mat(4) !failure function c = mat(5) !failure function bb = mat(6) !plastic potential cc = mat(7) !plastic potential cp1 = mat(8) h = mat(9) yi = mat(10) iend = mat(11) cp2 = mat(12) ac = mat(13) qc = mat(14) amax = mat(15) qmax = mat(16) qr = mat(17) c-----Elastic Matrix call elast(mat,ee,dd) c-----Plastic corrector reset sigp(1) = 0.d0 sigp(2) = 0.d0 sigp(3) = 0.d0 sigp(4) = 0.d0 if(inc.eq.2)q=0 c-----START TO THE ITERATION (for mixed and stress control) do itr=1,maxitr call predel(sigi,epsi,sig,sigini,sigp,ds,de,ee,dd,poi,con,est) c-----Flow Condition call bpfunc(sig,b,c,a0,yi,h,q,cp1,func,iend,cp2,ac,qc,cp) c-----Calculated stress in elastic regime if(func.le.minmin)exit c-----Plastic Corrector call bpresp(func,epsi,sig,ee,q,dlam,a0,yi,b,c,bb,cc,h,cp1,iend, + sigpnew,dq,cp2,ac,qc,cp,rm) c---- Final stress update sig(1)=sig(1)-sigpnew(1) sig(2)=sig(2)-sigpnew(2) sig(3)=sig(3)-sigpnew(3) sig(4)=sig(4)-sigpnew(4) c Exit condition for strain control if(con.eq.1) exit c Exit condition for stress and mixed control do i=1,4 dsigp(i)=sigpnew(i)-sigp(i) end do norma1=norma(dsigp) norma2=norma(sigpnew) do i=1,4 sigp(i)=sigpnew(i) end do if(dabs(norma1/norma2).lt.errrel) exit c-----END OF ITERACION (for stress and mixed control) end do if(itr.ge.maxitr) write(*,*)' The model does not converge', . ' using the stress control' c-----State variable update q=q+dq c-----Real stress increments call incten(dsm,ds,sig,sigini,poi,con,est) c-----Real strain increments dem(1)=epsi(1) dem(2)=epsi(2) dem(3)=epsi(4) end subroutine c----------------------------------------------------------------------c subroutine bpresp(func,epsi,st,ee,q,dlam,a0,yi,b,c,bb,cc,h,cp1, + iend,sigp,dq,cp2,ac,qc,cp,rm) c----------------------------------------------------------------------c c Function :No-associated Bresler Pister. c Plastic corrector for plastic potential function c Non-linear Hardening/Softening c ---------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension epsi(4),ee(4,4),sigp(4) dimension sn(4),st(4),em(4),rm(4) c-----Plastic potential gradient call bpgrad(st,rm,bb,cc,dtrd2s1,dtrj2s1,dfds1 ) c-----Plastic potential direction call bpsdir(rm,em,ee) c-----Initial value for the plastic multiplier

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call bpdlmt(st,rm,em,q,h,a0,yi,b,c,cp1,func,dlma,iend, + cp2,ac,qc,cp,an,dfdq,dq) c-----Development of non-linear consistence condition 10 call bpdlam(st,rm,sn,em,dq,q,h,a0,yi,b,c,cp1,dlma,dlam,iend, + cp2,ac,qc,cp,fn1,an1,dfn) if(dabs((dlam-dlma)/dlam).gt.1.d-10.and.iter.le.50)then dlma=dlam call bpgrad(sn,rm,bb,cc,dtrd2s1,dtrj2s1,dfds1) iter=iter+1 goto 10 endif sigp(1)=dlam*em(1) sigp(2)=dlam*em(2) sigp(4)=dlam*em(4) sigp(3)=dlam*em(3) return end c-----------------------------------------------------------------------c subroutine bpfunc(s,b,c,a0,yi,h,q,cp1,func,iend,cp2,ac,qc,cp) c-----------------------------------------------------------------------c c Function : Bresler Pister Yield Function (three parameters) c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension s(4) trs1=s(1) +s(2) +s(4) trs2=s(1)*s(2)+s(2)*s(4)+s(4)*s(1)-s(3)*s(3) trd2=1/3.d0*(trs1*trs1-3*trs2) trj2=dsqrt(trd2) c-----Quadratic Hardening/Softening if(iend.eq.1)then an=0.5d0*cp*q*q+h*q+a0 c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then an=a0+(yi-a0)*(1.d0-dexp(-cp1*q))+h*q c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.q.lt.qc)then an=a0+(yi-a0)*(1.d0-dexp(-cp1*q))+h*q elseif(iend.eq.3.and.q.ge.qc) then an=ac*dexp(cp2*(q-qc)) else write(*,*)'Hardening/Softening not available',iend stop endif func=-an+b*trs1-c*trs1**2.d0+trj2 return end c-----------------------------------------------------------------------c subroutine bpgrad(s,rm,bb,cc,dtrd2s1,dtrj2s1,dfds1) c-----------------------------------------------------------------------c c Function:Gradient of Bresler-Pister plastic potential c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) dimension s(4),rm(4) trs1=s(1) +s(2) +s(4) trs2=s(1)*s(2)+s(2)*s(4)+s(4)*s(1)-s(3)*s(3) trd2=1/3.d0*(trs1*trs1-3*trs2) trj2=dsqrt(trd2) dtrd2s1=1/3.d0*(2*trs1-3*(s(2)+s(4))) dtrj2s1=1/2.d0/dsqrt(trd2)*dtrd2s1 dfds1=bb-cc*2.d0*trs1+dtrj2s1 rm(1)=dfds1 dtrd2s2=1/3.d0*(2*trs1-3*(s(1)+s(4))) dtrj2s2=1/2.d0/dsqrt(trd2)*dtrd2s2 dfds2=+bb-cc*2.d0*trs1+dtrj2s2 rm(2)=dfds2 dtrd2s4=1/3.d0*(2*trs1-3*(s(1)+s(2))) dtrj2s4=1/2.d0/dsqrt(trd2)*dtrd2s4 dfds4=+bb-cc*2.d0*trs1+dtrj2s4 rm(4)=dfds4 dtrd2s3=1/3.d0*(-3*(-2.d0*s(3))) dtrj2s3=1/2.d0/dsqrt(trd2)*dtrd2s3 dfds3=+dtrj2s3 rm(3)=dfds3 c-----Engineering strain rm(3)=2.d0*rm(3) return end c-----------------------------------------------------------------------c subroutine bpsdir(rm,em,ee) c-----------------------------------------------------------------------c c Function:Direction of the Bresler-Pister plastic stress c-----------------------------------------------------------------------c

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implicit real*8 (a-h,o-z) dimension rm(4),em(4),ee(4,4) em(1)=ee(1,1)*rm(1)+ee(1,2)*rm(2)+ee(1,4)*rm(4) em(2)=ee(2,1)*rm(1)+ee(2,2)*rm(2)+ee(2,4)*rm(4) em(4)=ee(4,1)*rm(1)+ee(4,2)*rm(2)+ee(4,4)*rm(4) c-----Engineering strain em(3)=ee(3,3)*rm(3) return end c-----------------------------------------------------------------------c subroutine bpdlmt(st,rm,em,q,h,a0,yi,b,c,cp1,func,dlmt,iend, + cp2,ac,qc,cp,an,dfdq,dq) c-----------------------------------------------------------------------c c Function :Consistenze condition c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 irhd dimension st(4),rm(4),rn(4),em(4) call bpgrad(st,rn,b,c,dtrd2s1,dtrj2s1,dfds1) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+rn(3)*em(3) trne=rn(1)*em(1)+rn(2)*em(2)+rn(4)*em(4)+rm(3)*rm(3)/2.d0 c-----Quadratic Hardening/Softening if(iend.eq.1)then an=0.5d0*cp*q*q+h*q+a0 dfdq=(-cp*q-h) c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then an=a0+(yi-a0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-a0)*cp1*dexp(-cp1*q)+h ) c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.q.lt.qc)then an=a0+(yi-a0)*(1.d0-dexp(-cp1*q))+h*q dfdq=-( (yi-a0)*cp1*dexp(-cp1*q)+h ) elseif(iend.eq.3.and.q.ge.qc) then an=ac*dexp(cp2*(q-qc)) dfdq=-(ac*(dexp(cp2*(q-qc)))*cp2) else write(*,*)'Hardening/Softening not available',iend stop endif dq=dsqrt(2.d0/3.d0*trm2) dlmt=func/(trne-dfdq*dq) return end c-----------------------------------------------------------------------c subroutine bpdlam(st,rm,sn,em,dq,q,h,a0,yi,b,c,cp1,dlmt,dlam, + iend,cp2,ac,qc,cp,fn1,an1,dfn) c-----------------------------------------------------------------------c c Function:Consistence condition c Non-linear Hardening/Softening c-----------------------------------------------------------------------c implicit real*8 (a-h,o-z) integer*4 iend dimension st(4),rm(4),sn(4),em(4), sigp(4) trs1=st(1) +st(2) +st(4) trs2=st(1)*st(2)+st(2)*st(4)+st(4)*st(1)-st(3)*st(3) trd2=1/3.d0*(trs1*trs1-3*trs2) trj2=dsqrt(trd2) tre1=em(1) +em(2) +em(4) tre2=em(1)*em(2)+em(2)*em(4)+em(4)*em(1)-em(3)*em(3) trde2=1/3.d0*(tre1*tre1-3*tre2) trje2=dsqrt(trde2) trm2=rm(1)*rm(1)+rm(2)*rm(2)+rm(4)*rm(4)+rm(3)*rm(3)/2.d0 dq=dsqrt(2.d0/3.d0*trm2) c-----Newton-Raphson iteration dlam=dlmt iter=0 11 iter=iter+1 qn1=q+dlam*dq c-----Quadratic Hardening/Softening if(iend.eq.1)then an1=(0.5d0*cp*qn1*qn1+h*qn1+a0) dan=(cp*qn1*dq+h*dq) c-----Exponential SIMO Hardening/Softening elseif(iend.eq.2)then an1=(a0+(yi-a0)*(1.d0-dexp(-cp1*qn1))+h*qn1) dan=( (yi-a0)*cp1*dq*dexp(-cp1*qn1)+h*dq ) c-----Modified Exponential SIMO Hardening/Softening elseif(iend.eq.3.and.qn1.lt.qc)then an1=(a0+(yi-a0)*(1.d0-dexp(-cp1*qn1))+h*qn1) dan=( (yi-a0)*cp1*dq*dexp(-cp1*qn1)+h*dq ) elseif(iend.eq.3.and.qn1.ge.qc) then an1=ac*dexp(cp2*(qn1-qc))

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dan=ac*dexp(cp2*(qn1-qc))*cp2*dq else write(*,*)'Hardening/Softening not available',iend stop endif trs1l=st(1)+st(2)+st(4)-dlam*(em(1)+em(2)+em(4)) trs2l=trs2+dlam**2*tre2-dlam*(st(1)*em(2)+em(1)*st(2)+ + st(2)*em(4)+em(2)*st(4)+st(4)*em(1)+em(4)*st(1)) + +2.d0*dlam*st(3)*em(3) trd2l=1/3.d0*(trs1l*trs1l-3*trs2l) trj2l=dsqrt(trd2l) dtrs1l=-(em(1)+em(2)+em(4)) dtrs2l=+2.d0*dlam*tre2-(st(1)*em(2)+em(1)*st(2)+ + st(2)*em(4)+em(2)*st(4)+st(4)*em(1)+em(4)*st(1)) + +2.d0*st(3)*em(3) dtrd2l=1/3.d0*(2.d0*trs1l*dtrs1l-3.d0*dtrs2l) dtrj2l=1/2.d0/dsqrt(trd2l)*dtrd2l fn1=-an1+b*trs1l-c*trs1l**2.d0+trj2l dfn=-dan+b*dtrs1l-c*2.d0*trs1l*dtrs1l+dtrj2l dlam=dlam-fn1/dfn if(dabs(fn1/dfn).gt.1.d-10.and.iter.le.200) goto 11 if(dlam.lt.0.d0) write(*,*)' dlam < null !! ',dlam dq=dq*dlam sn(1)=st(1)-dlam*em(1) sn(2)=st(2)-dlam*em(2) sn(4)=st(4)-dlam*em(4) sn(3)=st(3)-dlam*em(3) return end

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