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International Journal of Fracture 104: 131143, 2000.

2000 Kluwer Academic Publishers. Printed in the Netherlands.

Numerical Calculation of Stress Intensity Factors in Functionally

Graded Materials

G. ANLAS, M. H. SANTARE and J. LAMBROSDepartment of Mechanical Engineering, University of Delaware, Newark, DE 19716, U.S.A.; Permanentaddress: Bogazici University, Bebek, Istanbul, Turkey.

Received 3 August 1999; accepted 7 January 2000

Abstract. The finite element method is studied for its use in cracked and uncracked plates made of functionally

graded materials. The material property variation is discretized by assigning different homogeneous elastic prop-

erties to each element. Finite Element results are compared to existing analytical results and the effect of mesh

size is discussed. Stress intensity factors are calculated for an edge-cracked plate using both the strain energy

release rate and the J-contour integral. The contour dependence of J in an inhomogeneous material is discussed.

An alternative, contour independent integral J is calculated and it is shown numerically that J, the strain energyrelease rate G, and the limit of J as approaches the crack tip (where is the contour of integration) are

all approximately equal. A simple method, using a relatively coarse mesh, is introduced to calculate the stress

intensity factors directly from classical J-integrals by obtaining lim0 J.

Key words: J-integrals, stress intensity factors, uncracked FGM plate.

1. Introduction

There is a class of materials which have continuously varying mechanical properties. Although

this type of material exists commonly in nature, there has been recent interest in manufacturing

them for specific engineering applications. Such materials are called functionally graded ma-

terials because their properties have a spatial gradient. Functionally graded materials (FGMs)

are very attractive for demanding applications such as thermal and wear protective coatings,

yet their behavior needs to be better understood to fully exploit their characteristics.

Analytical work on functionally graded materials goes back as early as the late 1960s when

soil was modeled as a nonhomogeneous material by Gibson (1967). More recently, Delale and

Erdogan (1983) analytically studied the crack problem in an infinite plane where the elastic

properties varied exponentially in the direction of the crack. They showed that the asymptotic

cracktip stress field possesses the same square root singularity seen in homogeneous materials.

In 1987, Eischen studied the crack-tip-singular behavior of the stress field in a nonhomo-

geneous infinite plane by using an eigenfunction expansion technique. He verified that the

leading term of the asymptotic expansion for stresses was square-root singular. This result was

further confirmed by Jin and Noda (1994) for materials with piecewise differentiable propertyvariations. In 1994, Konda and Erdogan studied the behavior of an infinite cracked plane with

exponential properties gradients in both the in-plane directions. Using a similar technique,

Erdogan and Wu (1997) studied various far-field loadings of an infinite FGM strip. They used

an exponentially varying Youngs Modulus, E, but kept the Poissons ratio, , constant. This

approximation was based on the earlier work of Delale and Erdogan (1983) who showed that

the effect of a variation of is negligible. The work by Erdogan and Wu (1997) is one of the

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132 G. Anlas, N. H. Santare and J. Lambros

few fracture solutions available for a finite width FGM. For this reason, their results are used

as the basis of comparison in the current study. For a comprehensive review of fracture of

functionally graded materials, see Erdogan (1985).

In 1996, Jin and Batra studied the crack tip stress field, strain energy release rate and stress

intensity factors in a ceramic-metal FGM. In two separate studies, Gu and Asaro (1997a, b)

analyzed fracture and crack deflection in functionally gradient materials.

In contrast to the above-described growing body of analytical studies, there are relatively

few experimental and numerical investigations of fracture of functionally graded materials. In

a recent publication, Gu et al. (1999) discussed a methodology for numerical determination

of stress intensity factors in FGMs. They studied the effect of material nonhomogeneity in

numerical computations of the J-integral. They concluded that the traditional version of the

J-integral can provide accurate results provided that it is evaluated very close to the crack

tip, using very small elements ( 105b, where b is the crack length). Li et al. (2000),show an experimental-numerical hybrid method for evaluating the strain energy release rate in

laboratory-scale FGMs. This method doesnt require a highly refined mesh but does require

experimental data. Experimental studies have in general been hampered by difficulties asso-

ciated with material fabrication even though some new experimental techniques have been

developed recently (see Lambros et al., 1999; Butcher et al., 1999; Parameswaran and Shukla,1998).

The focus of this paper is on the calculation and comparison of the stress intensity fac-

tors obtained for a cracked FGM plate by using the several different numerical techniques.

However, in the next section we first, briefly discuss finite element solutions for uncracked

FGM plates to validate the approximations and assumptions used. Subsequently, a technique

similar to that of Gu et al. (1999) is used to evaluate the J-integral numerically. The strain

energy release rate G and the J-integral are calculated for an edge-cracked FGM subject to

far field loading. A modified path independent integral, J similar to that proposed by Honein

and Herrmann (1997) is computed for the nonhomogeneous case. The results are compared to

the analytical solutions of Erdogan and Wu (1997). The relationship between the accuracy of

the finite element method (FEM) and mesh refinement is also investigated. To our knowledge,

this is the first numerical implementation and assessment of the path independent J-integral

for FGMs.

2. Study of uncracked FGM plate

The finite element method has been used extensively in solving problems involving homoge-

neous materials. The error introduced by geometric discretization of the domain, necessary in

FEM, has also received close scrutiny. However, when modeling an inhomogeneous material

(in this context, defined as one which possesses a continuous spatial variation of E and/or ) a

material property discretization is introduced in addition to the geometric one. In this section,

we investigate the accuracy of the FEM, when a material property discretization is introduced,

by comparing FEM results to simple analytical solutions. (Note that it is possible to avoidmaterial discretization by allowing a specific material property variation when formulating

the stiffness matrix. This case is being investigated in a separate work.)

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Numerical Calculation of Stress Intensity Factors 133

Figure 1. Edge-cracked plate under uniform traction or displacement loading.

2.0.1. Finite element solution

The finite element solutions of a rectangular plate made of a functionally graded material

under uniform traction or constant displacement loadings are obtained using ABAQUS (1997).

Assume that the material gradient is in the x-direction, and is exponential according to

(x) = 1ex (1)where is the shear modulus of the material, 1 the shear modulus at x = 0, and is amaterial constant which represents the length scale over which the properties change (the units

for are 1/length). The Poissons ratio, , is taken to be constant. The normal stress values are

calculated on the line of symmetry. The results evaluated on the line of symmetry correspond

to the results of an infinite strip, so they can be compared to those obtained analytically in the

paper by Erdogan and Wu (1997).

2.1. UNIFORM TRACTION LOADING

The plate considered has height twice its width, and is symmetric with respect to its midline,

the y = 0 axis. The geometry of the problem is given in Figure 1 with b = 0. Using symmetry,only the upper half is considered in the finite element model. The upper edge is loaded by a

uniform traction, yy = 0. The lower edge has a zero displacement boundary condition

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Table 1. Percent error in FEM results for uniform traction loading, 40

elements used in the x-direction.

x/ h E2/E1 = 2 E2/E1 = 5 E2/E1 = 10 E2/E1 = 20

0.0 0.37 1.31 2.38 3.86

0.1 0.04 0.14 0.52 1.24

0.2 0.05 0.08 0.35 0.87

0.3 0.05 0.15 0.15 0.06

0.4 0.09 0.36 0.61 0.84

0.5 0.08 0.47 0.94 1.55

0.6 0.05 0.43 0.96 1.76

0.7 0.02 0.18 0.59 1.35

0.8 0.08 0.28 0.27 0.04

0.9 0.18 0.8 1.81 3.65

1.0 0.16 3.18 21.28 50.07

in the y-direction to account for symmetry. The plate is divided into a uniform mesh with

40 elements along the x-direction and 10 elements along the y-direction, resulting in 400

elements (900 deg of freedom (dof)).The material is functionally graded, and the modulus of elasticity changes exponentially

along the x-axis as, E(x) = E1ex , where E1 is the elastic modulus at x = 0. From equation(1), it can be shown that E1 = 21(1 + ). For the finite element solution, this change ismodeled discretely by assigning each of the 40 regions along the x-axis the value of E at

the centroid of the region, calculated according to the exponential relation given above. The

Poissons ratio is taken as = 0.3. Note that the use of a uniform rectangular mesh makesthe assignment of material properties quite straightforward. For a radially focused mesh, as

is commonly used in fracture problems, the material property discretization may be more

involved.

A two dimensional continuum element with four nodes and four integration points is used

for the plane strain problem. Stresses are calculated at the nodes of the 40 elements on y = 0(the axis of symmetry). Since each row of elements has a discrete modulus, there will be

significantly different nodal stresses for each node shared by adjacent elements. The reported

results for yy /0 are the numerical averages in these cases. The results for E2/E1 = 2, 5,10 and 20 are compared to the analytical results from Erdogan and Wu (1997) in Table 1

where E2 is the modulus of elasticity at x = h. For an assumed exponential gradient as inEquation (1), there are two independent parameters needed to establish the material property;

either the modulus at two locations, as used here, or the modulus at one point and the length

scale .

The table shows that in this case, the error involved is less than 1% for most of the nodes. Ingeneral, the average error increases with increasing E2/E1. The errors at the end points x = 0and x = h are larger due to the discretization of the material property in the element. Thereis no adjacent element to mitigate this discrepancy as there is at the other points (recall that

the material property assignment is made at the centroid of the elements, not at the nodes).

Table 2 shows the same comparison for a 100 by 25 element mesh. It is seen that these results

are even closer to the analytical results.

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Table 2. Percent error in FEM results for uniform

traction loading, 100 elements used in the x-direction.

x/ h E2/E1 = 2 E2/E1 = 5 E2/E1 = 10

0.0 0.10 0.70 1.35

0.1 0.04 0.31 0.76

0.2 0.03 0.22 0.55

0.3 0.05 0.05 0.05

0.4 0.04 0.26 0.48

0.5 0.08 0.43 0.82

0.6 0.09 0.43 0.93

0.7 0.01 0.23 0.63

0.8 0.04 0.19 0.13

0.9 0.09 0.54 1.39

1.0 0.04 1.43 10.40

2.2. UNIFORM DISPLACEMENT LOADING

For the case of uniform displacement loading, the boundary condition at the top and the bottom

of the plate is an applied constant displacement, V0, in the y-direction. The numerical results

for normalized stress, yy h(1 2)/E1V0 for uniform displacement loading are compared toanalytical results from Erdogan and Wu (1997), and the percent error is tabulated in Table 3.

The comparison of the FEM results to analytical results shows that the error is less than

0.1% in most of the cases, when 400 elements are used to discretize the domain. With the

same discretization, the results of constant displacement loading are more accurate than the

corresponding ones for uniform traction loading (compare with Table 1). One reason is that the

analytical solution for the uniform traction problem used in the comparison, is exact only on

the line of symmetry, y = 0. The displacement solution on the other hand, is exact everywherein the domain. The comparison of analytical and FE results in Table 4 shows that with 100

elements used in the direction of property change, instead of the 40, FE results are almost

identical to the analytical solution.

3. Study of Edge-cracked FGM Plate

In this section, the problem of a nonhomogeneous finite plate with an edge crack is stud-

ied. The problem geometry is shown in Figure 1 for either a uniform traction or a uniform

displacement applied in the y-direction.

The domain is discretized into uniform meshes of 20 by 20 elements (about 880 dof), 100

by 25 elements (about 5250 dof) and 200 by 50 elements (about 20800 dof) respectively. Afour-node, two dimensional plane strain element is used. Stress intensity factors are calculated

using the strain energy release rate, G, the J-contour integral as r 0 (we will denote thislimiting value as J (1)), and the contour integral J which will be defined subsequently. The

numerically evaluated stress intensity factors are then compared to analytical results given by

Erdogan and Wu (1997). The effect of mesh size, crack length, , and the relation among G,

J (1), and J are discussed.

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Table 3. Percent error in FEM results for constant displacement

loading, 40 elements used in the x-direction.

x/ h E2/E1 = 2 E2/E1 = 5 E2/E1 = 10 E2/E1 = 20

0.0 0.9 2 2.9 3.8

0.1 0 0 0.08 0.07

0.2 0 0 0.06 0.05

0.3 0 0 0.1 0.04

0.4 0 0 0.04 0.06

0.5 0 0 0.06 0.09

0.6 0 0 0.05 0.07

0.7 0 0.03 0.04 0.07

0.8 0 0.03 0.05 0.04

0.9 0 0.02 0.05 0.08

1.0 0.85 2 2.84 3.7

Table 4. Percent error in FEM results for con-stant displacement loading, 100 elements used in the

x-direction.

x/ h E2/E1 = 2 E2/E1 = 5 E2/E1 = 10

0.0 0.30 0.80 1.20

0.1 0.00 0.00 0.08

0.2 0.00 0.00 0.00

0.3 0.00 0.00 0.00

0.4 0.00 0.00 0.00

0.5 0.00 0.00 0.03

0.6 0.00 0.00 0.01

0.7 0.00 0.00 0.02

0.8 0.00 0.00 0.00

0.9 0.00 0.00 0.01

1.0 0.35 0.80 1.14

In the general case, when a crack advances at a fixed displacement, the strain energy release

rate is defined as follows:

G = dUdA

(2)

where U is the strain energy, and A is the crack surface area. By using a node release technique

in the finite element program, dU/dA can be approximated by U/A. U is the change

in strain energy as a result of an increment in crack growth, holding the external boundary

conditions constant. The smaller the increment A (A = tb, where t is the thickness ofthe plate), the more accurate this result will be. The stress intensity factor can be calculated

using the following relation for plane strain (Jin and Batra, 1996),

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Figure 2. J-integral vs. contour, b/ h = 0.2, uniform traction loading.

K

2

l =GEtip

1 2 , (3)where Etip is the Youngs Modulus at the location of the crack tip.

For homogeneous materials, an alternative measure of the strain energy release rate is the

J-integral, which is defined by Rice (1968),

J =

W n1 ijni

uj

x1

ds, (4)

where in a two-dimensional cracked body, is any path beginning at the bottom crack face

and ending on the top crack face. Here W is the strain energy density, uj are the displacement

components and ni are the components of the outward unit normal to . In the absence of

body forces, thermal strains and crack surface tractions, the J-integral is path independent for

homogeneous materials. It is well known that in the case of homogeneous materials, G = J.For a nonhomogeneous material, in general, the J-integral is not path independent. In this

study, the contour integral, J, is calculated using ABAQUS. The change of J with contour

number is shown in Figure 2 for E2/E1 = 2, 5, and 10. In this and subsequent figures, thecontour numbers represent incrementally larger contours around the crack tip where the size

and the increment is governed by the mesh refinement. Each contour is placed symmetrically

around the crack tip and includes an integer number of elements. For a nonhomogeneous

material, G = J because J is path dependent. However, it can be shown that G = J as 0 (Gu et al., 1999). Tohgo et al. (1996) also show that in FGMs the J-integral is pathdependent and its value for a path adjacent to the crack tip is identical to that for homogeneous

materials.

4. J-integral

In the case of nonhomogeneous materials, the strain energy density W is not only a function

of (x), but it also depends on x explicitly, i.e., W = W((x),x), due to material propertygradients. This results in an extra term in the classical J-integral which needs to be subtracted

from Equation (4) to obtain the contour independent integral J as

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J =

W n1 ijni

uj

x1

ds

A

W,1q dA, (5)

where is any path beginning at the bottom crack face and ending on the top crack face and A

is the area enclosed by that contour. In Equation (5), W,1 denotes partial differentiation of W

with respect to the explicit x dependence. The

J integral will yield a zero value on any closed

contour in a homogeneous as well as an inhomogeneous material, and therefore when used

for fracture problems will always be path independent. This result was confirmed analytically

in the work of Honein and Herrmann (1997). In the following, we will numerically implement

the path-independent J and compare the calculated results with those derived through other

methods.

The finite element formulation of the contour integral J is constructed as suggested by

Li et al. (1985). In the absence of thermal stresses, and crack-face tractions, the following

discretized form is obtained from Equation (5):

J =

A

Np

=1

(ijui,1 W 1,i )q1,i W,1q1

det

xk

l

p

p, (6)

where p are the weight functions of the corresponding Gauss integration points. The quan-

tities in { } are all evaluated at the integration points for each element within the contourchosen. N is the number of integration points per element and q1 is a device used to facilitate

the evaluation of a contour integral in finite elements. In general, a nodal value of 0 or 1 is

assigned to Q1 and q1 can be defined within an element as follows:

q1 =4

i=1Ni Q1i , (7)

where Ni are the interpolation functions for the elements. In this study, for the calculation

of q1, a plateau function representation of q1 is used (see Li et al., 1985). Recall that W,1 in

equation (12) is not the total derivative but rather the partial with respect to x1 which willonly exist if the material property E is an explicit function of x1, (e.g., E = E(x1)). If thematerial property variation is given explicitly, it is simple to analytically derive an expression

for W/x1. In the case of equation (1), W/x1 = W(,x1)

5. Calculation of the Stress Intensity Factors

We analyze edge-cracked strips with b/ h = 0.4, and E2/E1 = 2.0 and E2/E1 = 0.5 (referto Fig. 1). The stress intensity factors, KI are calculated in three different manners; using the

strain energy release rate G, the contour integral J, and the path independent contour integral

J as follows:

K2I(1 2)Etip

= G = J = J (1) lim0

J. (8)

The graphs ofJ and J are plotted in Figs. 3 and 4 to show their variation with contour number.

Clearly, J is contour dependent as was seen previously in Fig. 2. In contrast, J is contour

independent after the first two contours thus numerically verifying the path independence of

Equation (5).

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Figure 3. J and J vs. contour, b/ h = 0.4, E2/E1 = 2.0, 20 by 20 mesh.

The values for J and J from the first contour are disregarded since they are generallynot considered accurate for most finite element meshes. Using the results of the J-integrals

calculated from contour 2 to contour 20, and fitting a fourth order polynomial to these data

points, a value for J (1) can be obtained. Extending the curve to larger number of contours

produces very little change. J (1) can be approximated numerically as the intercept of the

polynomial curve fit at n = 1, i.e. the crack tip. From Fig. 3 we can see that the limitingvalue of J, called J (1), is approximately equal to the path independent integral J for a 200

by 50 mesh. In addition, the stress intensity factors can be calculated from energy quantities

using Equation (14). All results are normalized using the procedure outlined by Erdogan and

Wu (1997) and compared to the exact results obtained by them. For uniform traction the

normalized stress intensity factor is KI = KI/0

b where 0 is the applied traction and

b the crack length. For the case of applied displacement, 0,

KI

=KI/0

b where 0

=E1V0/ h(1 2).For b/ h = 0.4, using the 20 by 20 mesh and J, we have computed the normalized stress

intensity factor as 1.9244. The exact result calculated by Erdogan and Wu (1997) is 1.9573.

Using the same mesh, G computed from U/A is not reliable because the mesh is very

coarse. In addition, J (1) is not close to J (see Fig. 3 for a comparison of J and J (1)). When

the 100 by 25 mesh is used however, J and J (1) are almost equal, and G is 4% lower. In this

case, the normalized stress intensity factors are 1.9313, 1.9324 and 1.898 respectively. When

the even finer, 200 by 50 mesh is used, the normalized stress intensity factors are 1.9458,

1.9461 and 1.931, respectively.

A similar computation is carried out for E2/E1 = 0.5. The plots for, J and J (1) are givenin Fig. 5 for a 200 by 50 mesh. Note that in this case J decreases with increasing contour

number since now E2/E1 < 1. The normalized stress intensity factor is 2.231 when calculatedusing J, and 2.229 when calculated using J (1). The exact result is 2.2598 (Erdogan and Wu,

1997). The results show that, computationally J (1) J and the use ofJ (1) gives satisfactoryresults in stress intensity factor calculations with relatively coarse meshes.

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6. Comparison of numerically calculated stress intensity factors

We have shown numerically that J is path independent. However, this quantity can be cum-

bersome to compute. We have also shown that, J (1) is a good approximation to J, which is

in agreement with Gu et al. (1999) and Tohgo et al. (1996). In the previous section, we have

also shown that the stress intensity factors can be approximately calculated using J (1) from a

coarse, uniform mesh, when the element size is in the order of

102b, where b is the crack

length. In this section, we study the accuracy of J (1). To generate the numerical results shownin this section, we use a uniform mesh of 100 by 25, with 8-noded elements (15 500 dof).Note that although this mesh is more refined than those used in previous sections, it is still

relatively coarse and does not focus on the crack tip. Figs. 6 and 7 show the normalized stress

intensity factors for different crack lengths and E2/E1 = 2.0 and E2/E1 = 10 respectively.Numerical results are generated using G and J (1), where G is calculated using the node

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Figure 6. Comparison of the normalized stress intensity factors for uniform traction loading, E2/E1 = 2.0.

Figure 7. Comparison of the normalized stress intensity factors for uniform traction loading, E2/E1 = 10.

release technique described in section3. The exact results are again taken from Erdogan and

Wu (1997).

Even with this relatively coarse mesh, we see that J (1) gives very good results. On average,

the error increases with increasing . However, the interdependence of parameters on , b/ h,

and mesh refinement is rather complex. Ideally, we would like to conduct a full field compar-

ison between the numerical and analytical results. However, the analytical results published

to date do not provide this detail. One local field parameter we do have access to is the crack

opening displacements from Erdogan and Wu (1997). The crack surface displacements foruniform traction loading are calculated for an edge crack with b/ h = 0.2 and plotted in Fig. 8for E2/E1 = ratios of 2.0, 5.0, and 10.0. These results are in very good agreement with theexact results presented by Erdogan and Wu (1997).

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Figure 8. Crack surface displacement for b/ h = 0.2. Edge crack under uniform traction loading,V(s) = E1(x, 0)/[2h0(1 )].

7. Concluding remarks

In this study, stresses in the uncracked plate are calculated under both uniform traction and

uniform displacement loadings. The results are compared to exact solutions in Tables 14. Tra-

ditional finite elements give fairly accurate results for the uncracked case with the assignment

of properties at the centroid of each element.

In addition, we have used the J contour integral results of ABAQUS in the calculation of

stress intensity factors for an edge cracked plate made of a functionally graded material. We

have numerically demonstrated the path independence of the contour integral J, a modified

J-integral in Equation (11), and shown that J = J (1) where J (1) = lim0 J. We comparedour results for normalized stress intensity factors to the analytical results presented by Erdogan

and Wu (1997). This comparison showed that even using a relatively coarse, uniform mesh,the results obtained from J and J (1) are very close to the analytical ones. However, for

the same mesh, the normalized stress intensity factor results obtained using the node release

technique for G are far less accurate. Clearly, the accuracy of the node release technique can

be improved with mesh refinement. Nevertheless, significant mesh refinement is not needed

to obtain accuracy using J (1), a quantity which can be calculated using many existing finite

element codes. The method presented here is quite simple and easy to implement compared

with the analytical and other computational methods.

Acknowledgement

The authors would like to thank NSF for support of this research through grant CMS-9712831.

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