ch2-LEFM- Fracture Mechanics
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Transcript of ch2-LEFM- Fracture Mechanics
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Fracture Mechanics
Brittle fracture
Fracture mechanics is used to formulate quantitatively
• The degree of Safety of a structure against brittle fracture
• The conditions necessary for crack initiation, propagation
and arrest
• The residual life in a component subjected to
dynamicfatigue loading
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Fracture mechanics identifies three primary factors that control the susceptibility
of a structure to brittle failure!
1. Material Fracture Toughness. Material fracture toughness may be defined
as the ability to carry loads or deform plastically in the presence of a notch!
"t may be described in terms of the critical stress intensity factor, #"c, under
a variety of conditions! $These terms and conditions are fully discussed in
the follo%ing chapters!&
'! Crack Size. Fractures initiate from discontinuities that can vary from
e(tremely small cracks to much larger %eld or fatigue cracks! Furthermore,
although good fabrication practice and inspection can minimi)e the si)e and number of cracks, most comple( mechanical components cannot be
fabricated %ithout discontinuities of one type or another!
*! Stress Level. For the most part, tensile stresses are necessary for brittle
fracture to occur! These stresses are determined by a stress analysis of the
particular component!
+ther factors such as temperature, loading rate, stress concentrations,
residual stresses, etc!, influence these three primary factors!
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Fracture at the tomic level
T%o atoms or a set of atoms are bonded
together by cohesive energy or bond energy! T%o atoms $or sets of atoms& are said to be
fractured if the bonds bet%een the t%o atoms
$or sets of atoms& are broken by e(ternally
applied tensile load
Theoretical -ohesive Stress
"f a tensile force .T/ is applied to separate the
t%o atoms, then bond or cohesive energy is
$'!0&
1here is the equilibrium spacing bet%een t%o atoms!
"deali)ing force2displacement relation as one
half of sine %ave
$'!'&
o(
Td(∞
φ = ∫
( o
(
-T sin$ &πλφ =
+ +
xo
BondEnergy
CohesiveForce
λ
EquilibriumDistance xo
Distance
Repulsion
Attraction
Tension
Compression k
BondEnergy
Distance
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Theoretical -ohesive Stress $-ontd!&
ssuming that the origin is defined at and for small
displacement relationship is assumed to be linear such
that 3ence force2displacement
relationship is given by
$'!'&
Bond stiffness .k/ is given by
$'!*&
"f there are n bonds acing per unit area and assuming
as gage length and multiplying eq! '!* by n then .k/
becomes young/s modulus and beecomes cohesive
stress such that
$'!4&
+r $'!5&
"f is assumed to be appro(imately equal to the atomic
spacing
+ +
xo
BondEnergy
Cohesive
Force
λ
EquilibriumDistance xo
Distance
Repulsion
Attraction
Tension
Compression k
BondEnergy
Distance
o(
((
sin$ &πλπ
≈
λ
-
(T T
π≈
λ
-T
k
π
= λo(
o(
-T
-σ
c
o
6(λσ = π
c
6σ =
π
λ
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Theoretical -ohesive Stress $-ontd!&
+ +
xo
BondEnergy
CohesiveForce
λ
EquilibriumDistance xo
Distance
Repulsion
Attraction
Tension
Compression k
BondEnergy
Distance
The surface energy can be estimated as
$'!7&
The surface energy per unit area is
equal to one half the fracture energy because t%o surfaces are created %hen a
material fractures! 8sing eq! '!4 in to
eq!'!7
$'!9&
( )(0's - -
:
sin d(λ
πλ∫ λ
γ ≈ σ = σπ
s
-
o
6(γ σ =
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Fracture stress for realistic material"nglis $0;0*& analy)ed for the flat plate %ith an
elliptical hole %ith major a(is 'a and minor a(is 'b,
subjected to far end stress The stress at the tip of
the major a(is $point & is given by
$'!
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Fracture stress for realistic material $contd!&
1hen a >> b eq! '!0: becomes
$'!00&
For a sharp crack, a >>> b, and stress at the crack tip tends to
ssuming that for a metal, plastic deformation is )ero and the sharpest
crack may have root radius as atomic spacing then the stress is
given by
$'!0'&
1hen far end stress reaches fracture stress , crack propagates and
the stress at reaches cohesive stress then using eq! '!9
$'!0*&
This %ould
a'
σ = σ ÷ρ
:ρ = ∞
o(ρ =
o
a'( σ = σ ÷
-σ = σf σ = σ
0 '
sf
6
4a
γ σ = ÷
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?riffith/s 6nergy balance approach
•First documented paper on fracture
$0;':&
•-onsidered as father of FractureMechanics
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?riffith laid the foundations of modern fracture mechanics by
designing a criterion for fast fracture! 3e assumed that pre2
e(isting fla%s propagate under the influence of an applied stressonly if the total energy of the system is thereby reduced! Thus,
?riffith@s theory is not concerned %ith crack tip processes or the
micromechanisms by %hich a crack advances!
?riffith/s 6nergy balance approach $-ontd!&
a
!
"
B
σ
σ
?riffith proposed that .There is a simpleenergy balance consisting of the decrease
in potential energy %ith in the stressed
body due to crack e(tension and this
decrease is balanced by increase in surfaceenergy due to increased crack surface/
?riffith theory establishes theoretical strength of
brittle material and relationship bet%een fracture
strength and fla% si)e .a/ f σ
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a
!
"
B
σ
σ
?riffith/s 6nergy balance approach $-ontd!&
The initial strain energy for the uncracked plate
per thickness is
$'!04&
+n creating a crack of si)e 'a, the tensile force
on an element ds on elliptic hole is rela(ed
from to )ero! The elastic strain energyreleased per unit %idth due to introduction of a
crack of length 'a is given by
$'!05&
'
i
8 d'6
∫ σ=
a0
a ':
8 4 d( v∫ = − σ× ×
d(σ×
%here displacement
v a sin6
σ= × θusin g ( a cos= × θ
' '
a
a8
6
πσ=
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?riffith/s 6nergy balance approach $-ontd!&
a
!
"
B
σ
σ
6(ternal %ork = $'!07&
The potential or internal energy of the body is
Aue to creation of ne% surface increase in
surface energy is
$'!09&
The total elastic energy of the cracked plate is
%8 Fdy,δ∫ =
%here F= resultant force = area
=total relative displacement
σ ×
δ
p i a %8 =8 8 28
s8 = 4aγ γ
' ' '
t s
a8 d Fdy 4a
'6 6 δ∫ ∫
σ πσ= + − + γ
#$
#
Displacement% v
Crack beginsto gro& 'rom
length (a)
Crack islonger by anincrement (da)
v
? iffi h/ 6 b l h $- d &
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?riffith/s 6nergy balance approach $-ontd!&
Cracklength% a
Elastic *trainenergy released
Total energy
#otential energyrelease rate + ,
*yr'ace energy-unitextension ,
Cracklength% a
ac
.nstable*table
(a)
(b)
(a) /ariation o' Energy &ith Crack length
(b) /ariation o' energy rates &ith crack length
The variation of %ith crack
e(tension should be minimum
Aenoting as during fracture
$'!0;& for plane stress
$'!':&
for plane strain
t8
'
t
s
d8 ' a
: 4 :da 6
πσ= ⇒ − + γ =
f σσ0 '
sf
'6
a
γ σ = ÷
π
0 '
sf '
'6
a$0 &
γ σ = ÷π − ν
The ?riffith theory is obeyed by
materials %hich fail in a completely
brittle elastic manner, e.g. glass,
mica, diamond and refractory
metals!
? iffi h/ 6 b l h $- d &
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?riffith/s 6nergy balance approach $-ontd!&
?riffith e(trapolated surface tension values of soda lime glass
from high temperature to obtain the value at room temperature as
8sing value of 6 = 7'?Ca,The value of as :!05 From the e(perimental study on spherical vessels he
calculated as :!'5 D :!'<
3o%ever, it is important to note that according to the ?riffiththeory, it is impossible to initiate brittle fracture unless pre2
e(isting defects are present, so that fracture is al%ays considered
to be propagation2 $rather than nucleation2& controlledE this is a
serious short2coming of the theory!
'
s :!54 m !γ =0 '
s'6γ ÷π MCa m!
0 '
sc
'6a
γ σ = ÷π MCa m!
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Modification for Ductile Materials
For more ductile materials $e.g. metals and plastics& it is found that
the functional form of the ?riffith relationship is still obeyed, i.e.
. 3o%ever, the proportionality constant can be used to
evaluate γ s $provided 6 is kno%n& and if this is done, one finds thevalue is many orders of magnitude higher than %hat is kno%n to be
the true value of the surface energy $%hich can be determined by
other means&! For these materials plastic deformation accompanies
crack propagation even though fracture is macroscopically brittleE
The released strain energy is then largely dissipated by producing
locali)ed plastic flo% at the crack tip! "r%in and +ro%an modified
the ?riffith theory and came out %ith an e(pression
1here γ prepresents energy e(pended in plastic %ork! Typically for
cleavage in metallic materials γ p=0:4 m' and γ s=0 m'! Since γ p>>
γ s %e have
0 '
s pf
'6$ &a
γ + γ σ = ÷π
0 '
pf '6
aγ σ = ÷π
0 '
f aσ µ
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Strain Energy Release RateThe strain energy release rate usually referred to
Gote that the strain energy release rate is respect to crack length and
most definitely not time! Fracture occurs %hen reaches a critical
value %hich is denoted !
t fracture %e have so that
+ne disadvantage of using is that in order to determine it is
necessary to kno% 6 as %ell as ! This can be a problem %ith somematerials, eg polymers and composites, %here varies %ith
composition and processing! "n practice, it is usually more
convenient to combine 6 and in a single fracture toughness
parameter %here ! Then can be simply determined
e(perimentally using procedures %hich are %ell established!
d8?
da=
c?
c? ?=0 '
cf
0 6?
H a
σ = ÷π c? f σ
c?
c? c# '
c c# 6?=c#
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I"G6J 6IST"- FJ-T8J6 M6-3G"-S $I6FM&
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I"G6J 6IST"- FJ-T8J6 M6-3G"-S $I6FM&For I6FM the structure obeys 3ooke/s la% and global behavior is linear
and if any local small scale crack tip plasticity is ignored
The fundamental principle of fracture mechanics is that the stress field around a
crack tip being characteri)ed by stress intensity factor # %hich is related to both
the stress and the si)e of the fla%! The analytic development of the stress intensity
factor is described for a number of common specimen and crack geometries belo%!
The three modes of fracture
Mode ! Opening mod e" %here the crack surfaces separate symmetrically %ith
respect to the plane occupied by the crack prior to the deformation $results from
normal stresses perpendicular to the crack plane&E
Mode ! Sliding mod e" %here the crack surfaces glide over one another in
opposite directions but in the same plane $results from in2plane shear&E and
Mode ! Tearing mod e" %here the crack surfaces are displaced in the crack plane and parallel to the crack front $results from out2of2plane shear&!
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"n the 0;5:s "r%in K9L and co%orkers introduced the concept of stress intensity
factor, %hich defines the stress field around the crack tip, taking into account
crack length, applied stress σ and shape factor H$ %hich accounts for finite si)e
of the component and local geometric features&! The #iry stress function."n stress analysis each point, (,y,), of a stressed solid undergoes the stressesE
σ( σy, σ), τ(y, τ(),τy)! 1ith reference to figure '!*, %hen a body is loaded andthese loads are %ithin the same plane, say the (2y plane, t%o different loading
conditions are possible
I"G6J 6IST"- FJ-T8J6 M6-3G"-S $-ontd!&
Crack#lane
ThicknessB
ThicknessB
σ
σ
σ
σ
σ0 σ0
σ0 σ0a
#lane *tress #lane *train
y
!
σ
σ
σ
σyy
0! plane stress (PSS), %hen the
thickness of the body is
comparable to the si)e of the
plastic )one and a free
contraction of lateral surfacesoccurs, and,
'! plane strain (PSN), %hen the
specimen is thick enough to
avoid contraction in the
thickness )2direction!
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"n the former case, the overall stress state is reduced to the three
componentsE σ(, σy, τ(y, sinceE σ), τ(), τy)= :, %hile, in the latter
case, a normal stress, σ), is induced %hich prevents the )displacement, ε) = % = :! 3ence, from 3ooke@s la%
σ) = ν $σ(σy&%here ν is Coisson@s ratio!For plane problems, the equilibrium conditions are
"f φ is the iry/s stress function satisfying the biharmoniccompatibility -onditions
∂∂
+ ∂
∂ =
∂
∂ +
∂
∂ =
σ τ σ τ x xy y xy
x y y x: : E
∇ = 4
: φ
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Then
For problems %ith crack tip 1estergaard introduced iry/s stress
function as
1here Z is an analytic comple( function
' ' '
( y (y' ', ,
y ( (y
∂ φ ∂ φ ∂ φσ = σ = τ = −
∂ ∂ ∂
JeK L y "mKNLN−=
φ = +
Z z z y z z x iy= +JeK L "mK L E =
nd are 'nd and 0st integrals of Z(z)
Then the stresses are given by
N,N= −
'@
( '
'@
y '
'@
(y
@
JeKNL y "mKN Ly
JeKNL y "mKN L(
y "mKN L(y
%here N =dN d)
∂ φσ = = −
∂
∂ φσ = = +∂∂ φ
τ = = −∂
+ i d l i M d "
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+pening mode analysis or Mode "
-onsider an infinite plate a crack of length 2a subjected to a bia(ial
State of stress! Aefining
Boundary -onditions • t infinity
• +n crack faces
( ) ( y (yO ) O , := ∞ σ = σ = σ τ =
( ) ( (ya ( aEy : :− < < = σ = τ =
( )' ' )N
) aσ=
−
By replacing ) by z+a , origin shifted to crack tip!
Z z a
z z a=
−
+
σ
b '
nd %hen Oz|: at the vicinity of the crack tip
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nd %hen O z|: at the vicinity of the crack tip
# " must be real and a constant at the crack tip! This is due to a
Singularity given by
The parameter # " is called thestress intensity factor for opening
mode "!
Z a
az
K
z
K a
I
I
= =
=
σ
π
σ π
' '
0
z
Since origin is shifted to crack
tip, it is easier to use polar
-oordinates, 8sing
Further Simplification gives
z ei= θ
# *θ θ θ
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"(
"y
"(y
# *cos 0 sin sin
' ' '' r
# *cos 0 sin sin
' ' '' r
# *sin cos cos' ' '' r
θ θ θ σ = − ÷ ÷ ÷ π θ θ θ σ = + ÷ ÷ ÷ π
θ θ θ τ = ÷ ÷ ÷ π
( )"ij ij "#
"n general f and # H a' r
%here H = configuration factor
σ = θ = σ ππ
From 3ooke/s la%, displacement field can be obtained as
'
"
'"
'$0 & r 0u # cos sin
6 ' ' ' '
'$0 & r 0v # sin cos6 ' ' ' '
− ν θ κ − θ = + ÷ ÷ π
− ν θ κ − θ = + ÷ ÷ π
%here u, v = displacements in (, y directions
$* 4 & for plane stress problems
* for plane strain problems0
κ = − ν
− ν κ = ÷+ ν
The vertical displacements at any position along ( a(is
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The vertical displacements at any position along (2a(is
$θ = 0) is given by
The strain energy required for creation of crack is given by the
%ork done by force acting on the crack face %hile rela(ing the
stress σ to )ero
( )
( )
' '
'' '
v a ( for plane stress6
$0 &v a ( for plane strain
6
σ= −
σ − ν= − x
v
x
y
( ) ( )
a
'a a' ' ' '
a a: :
' '
0 8 Fv
'For plane stress For plane strain
$0 &8 4 a ( d( 8 4 a ( d(
6 6
a
6
∫ ∫
=
σ σ − ν= σ× − = σ× −
πσ ' ' '
a
' ' '
" "
'
""
a $0 &
6
The strain energy release rate is given by ? d8 da
a $0 &a? = ? =
6 6
# ? =6
πσ − ν
=
πσ πσ − ν
' '
"" # $0 & ? =
6− ν
Slidi d l i M d '
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Sliding mode analysis or Mode '
For problems %ith crack tip under shear loading, iry/s stress
function is taken as
8sing ir/s definition for stresses
"" yJeKNL
−
φ = −
'@
( '
'@
y '
'@
(y
' "mKNL y JeKN Ly
y JeKN L(
JeKNL y "mKN L(y
∂ φσ = = +
∂
∂ φσ = = −
∂
∂ φτ = − = −
∂
8sing a 1estergaard stress function of the form
( ):
' '
)N
) a
τ=
−
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Boundary -onditions • t infinity• +n crack faces
( ) ( y (y :O ) O :,= ∞ σ = σ = τ = τ
( ) ( (ya ( aEy : :
− < < = σ = τ =1ith usual simplification %ould give the stresses as
""(
""y
""(y
# *cos cos ' cos cos
' ' ' '' r
# *cos sin cos' ' '' r
# *cos 0 sin sin
' ' '' r
θ θ θ θ σ = + ÷ ÷ ÷ ÷ ÷ π
θ θ θ σ = ÷ ÷ ÷ π θ θ θ τ = − ÷ ÷ ÷ π
Aisplacement components are given by
( )[ ]
( )[ ]
""
""
# r u $0 &sin ' cos
6 ' '
# r v $0 &cos ' cos
6 ' '
θ = + ν κ + + θ ÷π
θ = + ν κ − + θ ÷
π
# a= τ π
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"" o
'
""
' '
"
"
# a
# ? = for plane stress
6
# $0 & ? = for plane strain
6
= τ π
− ν
Tearing mode analysis or Mode *
"n this case the crack is displaced along )2a(is! 3ere
the displacements u and v are set to )ero and hence
( y (y y(
(y y( y) )y
y)()
' ''
' '
:
% % and
( y
the equilibrium equation is %ritten as
:( y
Strain displacement relationship is given by
% %
% :( y
ε = ε = γ = γ =
∂ ∂γ = γ = γ = γ =
∂ ∂
∂τ∂τ+ =
∂ ∂
∂ ∂
+ = ∇ =∂ ∂
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(y y)
N
if % is taken as
0% "mK L
?
Then
"mKN LE JeKN L
−=
′ ′τ = τ =
8sing 1estergaard stress functionas
( )
( )
:
' '
:
) y) (y
y) :
)N) a
%here is the applied boundary shear stress
%ith the boundary conditions
on the crack face a ( aEy : :
on the boundary ( y ,
τ=−
τ
− < < = σ = τ = τ =
= = ∞ τ = τ
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"""()
"""y)
( y (y
"""
""" o
The stresses are given by
# sin
'' r #
cos'' r
:
and displacements are given by# 'r
% sin? '
u v :
# a
θ τ = ÷π
θ τ = ÷π
σ = σ = τ =
θ = ÷π = =
= τ π