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Reference : Fracture Mechanics by T.L. Anderson
Lecture Note of Eindhoven University of Technology
2017. 10
by Jang, Beom Seon
Topics in Ship Structures
08 Elastic-Plastic FractureMechanics
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Contents
1. Crack-Tip –Opening displacement
2. The J Contour Integral
3. Relationships Between J and CTOD
4. Crack-Growth Resistance Curves
5. J -Controlled Fracture
6. Crack-Tip Constraint Under Large-Scale Yielding
7. Scaling Model for Cleavage Fracture
8. Limitations of Two-Parameter Fracture Mechanics
2
0. INTRODUCTION
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Definition of CTOD
Structural steels has higher toughness than KIc values characterized by
LEFM.
The crack faces moves apart prior to fracture; plastic deformation blunts an
initially sharp crack.
Crack-Tip-opening Displacement (CTOD) : a measure of fracture toughness.
3
1. Crack-Tip-opening displacement
Crack-tip-opening displacement (CTOD). Estimation of CTOD from the displacementof the effective crack in the Irwin plastic zonecorrection.
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Relationship between CTOD and KI and G : Irwin plastic zone
Effective crack length = a+ry
The displacement ry behind the effective crack
tip for plane stress.
The Irwin plastic zone correction for plane
stress.
CTOD to be related with KI or G .
4
1. Crack-Tip-opening displacement
,yr r
eff ya a r
(3 ) / (1 ), (plane stress)2(1 )
Ev v G
v
3 1
( 1) 41
2 2 2 22
2(1 )
y y y
y I I I
v vr r rvu K K K
E E
v
242 I
y
YS
Ku
E
2
1
2
Iy
YS
Kr
4
YS
G
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Relationship between CTOD and KI and G : Irwin plastic zone
For plane strain
The Irwin plastic zone correction for plane strain
CTOD can be related with The Irwin plastic
5
1. Crack-Tip-opening displacement
(3 4 ), (plane strain)2(1 )
Ev G
v
2
( 1) 4(1 ) 4 4
2 2 2 2 22
2(1 ) (1 )
y y y y
y I I I I
r r r rvu K K K K
E E E
v v
242
3
Iy
YS
Ku
E
2
1
6
Iy
YS
Kr
4
3 YS
G
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Relationship between CTOD and KI and G : Strip-yield model
CTOD in a through crack in an infinite plate subject to a remote tensile
stress for plate stress.
Series expansion of the “ln sec” term gives
6
1. Crack-Tip-opening displacement
Estimation of CTOD from the
strip-yield model.
24 42
2 4
y I Iy I I
YS YS
r K Ku K K
E E E
8ln sec
2
YS
YS
a
E
/ 0YS
2
I
YS YS
K
E
G2
I
YS YS
K
m E m
G 1 for plane stress
2 for plane strain
m
m
82
2
y
y I
ru K
E
22 2 2
2
8,
8
,
YS I
YS YS YS
I
a Ka
E E E
here K a
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Two most common definitions of CTOD
Two definitions, (a) and (b) are equivalent if the crack blunts in a semicircle.
CTOD can be estimated from a similar triangles construction:
Where r is the rotational factor, a dimensionless constant between 0 and 1.
The hinge model is inaccurate when displacements are primarily elastic.
7
1. Crack-Tip-opening displacement
Alternative definitions of CTOD: (a) displacement at the
original crack tip and (b) displacement at the intersection of a
90° vertex with the crack flanks.
The hinge model for estimating CTOD
from three-point bend specimens.
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Two most common definitions of CTOD
A typical load (P) vs. displacement (V) curve from a CTOD
The shape of the load-displacement curve is similar to a stress-strain
curve.
At a given point on the curve, the displacement is separated into
elastic and plastic components by constructing a line parallel to the
elastic loading line.
The dashed line represents the path of unloading for this specimen,
assuming the crack does not grow during the test.
The CTOD in this specimen is estimated by
The plastic rotational factor rp is approximately
0.44 for typical materials and test specimens.
8
1. Crack-Tip-opening displacement
Determination of the plastic componentof the crack-mouth-opening displacement
2
I
YS
K
m E
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Introduction
The J contour integral has enjoyed great success as a
fracture characterizing parameter for nonlinear
materials.
An elastic material : a unique relationship between
stress and strain.
An elastic-plastic material : more than one stress
value for a given strain if the material is unloaded or
cyclically loaded.
Nonlinear elastic behavior may be valid for an elastic-
plastic material, provided no unloading occurs.
The deformation theory of plasticity , which relates
total strains to stresses in a material, is equivalent to
nonlinear elasticity.
The nonlinear energy release rate J could be written
as a path independent line integral an energy
parameter.
J uniquely characterizes crack-tip stresses and strains
in nonlinear materials a stress intensity parameter.
9
2. The J Contour Integral
Schematic comparison of the stress strain behavior of elastic-plastic and nonlinear elastic materials.
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Nonlinear Energy Release Rate
Rice [4] presented J is a path-independent contour integral for the analysis
of cracks.
J is equal to the energy release rate in a nonlinear elastic body that
contains a crack.
The energy release rate for nonlinear elastic materials.
Π : the potential energy, A : the crack area, U : strain energy stored in the body,
F : the work done by external forces.
For load control,
U* the complimentary strain energy
10
2. The J Contour Integral
dJ
dA
U F
*U P U
*
0
P
U dP Nonlinear energy release rate.
Crack length = a+da
Crack length = a
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Nonlinear Energy Release Rate
If the plate is in load control, J is given by
If the crack advances at a fixed displacement, F = 0, and J is given by
J for load control is equal to J for displacement control.
J in terms of load
J in terms of displacement
11
2. The J Contour Integral
Nonlinear energy release rate.
*
P
d dUJ
dA dA
d dUJ
dA dA
0 0
P P
P P
J dP dPa a
0 0
PJ Pd d
a a
U F U
* 1/ 2 , * 0, 0dU dU dPd dU dU *
0
P
U dP
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Nonlinear Energy Release Rate
More general version of the energy release rate. For the special case of a
linear elastic material, J= G . For linear elastic Mode I,
Caution when applying J to elastic-plastic material.
The energy release rate : the potential energy that is released from a
structure when the crack grows in an elastic material.
However, much of the strain energy absorbed by an elastic-plastic material
is not recovered. A growing crack in an elastic-plastic material leaves a
plastic wake.
Thus, the energy release rate concept has a somewhat different
interpretation for elastic-plastic materials.
J indicates the difference in energy absorbed by specimens with
neighboring crack sizes.
12
2. The J Contour Integral
2
IKJ
E
plastic wakedUJ
dA
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J as a Path-Independent Line Integral
Consider an arbitrary counterclockwise path (Γ) around the tip of a
crack
The strain energy density is defined as
13
2. The J Contour Integral
Arbitrary contour around
the tip of a crack.
The traction is a stress vector at a given point on the contour.
where nj are the components of the unit vector normal to Γ.
J : a path-independent integral
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The J Contour Integral - Proof Evaluate 𝐽 along Γ∗
Using divergence theorem, the line integral can be converted into an areal integral.
Using the chain rule and the definition of strain energy density, the first term in
square bracket in Eq. (a). Here, w = strain energy density.
Applying the strain-displacement relationship and ij= ji
Invoking the equilibrium condition
Thus, J=0 for any closed contour
14
2. The J Contour Integral
Γ∗: closed contour𝐴∗: Area enclosed by Γ∗
⋯ (a) “To be proved in the next slide.”
0
* ** 0i
ijA A
j
uw w wJ dxdy dxdy
x x x x x
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The J Contour Integral - Proof 15
2. The J Contour Integral
1 2
cos( 90) sin( 90)s s s
n s n s y x
n i j
i j i j
A
C
B
θ
cos sins s s
x y
r i j
i j
x i
s
y j
1 1 2 2* * *
1 2 1 2* *
1
1
* i i i ii ij j i i
i i i ii i i i
A
i
u u u uJ wdy T ds wdy n ds wdy n ds n ds
x x x x
u u u uwwdy dy dx dxdy dxdy dxdy
x x x x x y x
w
x x
2* *
2
i i ii ij
A Aj
u u uwdxdy dxdy
x x x x x x
2 11 2
R C
F Fdxdy F dx F dy
x y
1 2,x x x y
Using divergence theorem, the line integral can be converted into an areal
integral.
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The J Contour Integral - Proof Consider now two arbitrary contours.
If Γ1 and Γ2 are connected by segments along the crack face (Γ3 and Γ4), a
closed contour is formed.
The total J along the closed contour is equal to the sum of contributions
from each segment:
On the crack face, Ti =dy = 0.
Thus, J1 = −J3.
“Therefore, any arbitrary (counterclockwise) path around a crack will yield the same
value of J; J is path-independent.”
16
2. The J Contour Integral
Two arbitrary contours Γ1 and Γ2
around the tip of a crack.
22 4 0i
i
uJ J wdy T ds
x
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J as a Nonlinear Elastic Energy Release Rate Consider a two-dimensional cracked body bounded by the curve Γ.
Under quasi-static conditions and in the absence of body forces, the potential
energy is
Consider the change in potential energy resulting from a virtual extension of the
crack:
When the crack grows, the coordinate axis moves. Thus a derivative with respect to
crack length can be written as
(b)를 (a)에 적용
By applying the same assumptions
17
2. The J Contour Integral
⋯ (b)
⋯ (a)
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J as a Nonlinear Elastic Energy Release Rate Invoking the principle of virtual work gives
Thus,
Applying the divergence theorem and multiplying both sides by –1 leads to
Therefore, the J contour integral is equal to the energy release rate for a linear or
nonlinear elastic material under quasi-static conditions.
18
2. The J Contour Integral
2 11 2
R C
F Fdxdy F dx F dy
x y
2 1, 0F w F
xA
wdxdy wdy wn ds
x
s y x n i j
x y
y xn n
s s
n i j i j
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J as a Stress Intensity Parameter
• Ramberg-Osgood eq. : Inelastic stress-strain relationship for uniaxial deformation
• In order to remain path independent, stress–strain must vary as 1/r near the crack
tip.
• At distances very close to the crack tip, well within the plastic zone, elastic strains
are small in comparison to the total strain, and the stress strain behavior reduces to
a simple power law.
• k1 and k2 are proportionality constants, which are defined more precisely below. For
a linear elastic material, n = 1.
19
2. The J Contour Integral
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J as a Stress Intensity Parameter
The actual stress and strain distributions are obtained by applying the appropriate
boundary conditions HRR singularity named after Hutchinson, Rice, and Rosengren.
The J integral defines the amplitude of the HRR singularity.
In : an integration constant that depends on n.
𝜎𝑖𝑗 and 𝜀𝑖𝑗 : dimensionless functions of n and θ.
20
2. The J Contour Integral
Effect of the strain-hardening exponent on the
HRR integration constant.Angular variation of dimensionless
stress for n = 3 and n = 13
plane stress
plane strain
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The Large Strain Zone
The HRR singularity predict infinite stresses as r → 0, The large strains at
the crack tip cause the crack to blunt, which reduces the stress triaxiality
locally. The blunted crack tip is a free surface; thus xx must vanish at r = 0.
HRR singularity does not consider the effect of the blunted crack tip on the
stress fields, nor the large strains that are present near the crack tip.
21
2. The J Contour Integral
Large-strain crack-tip finite element results.
yy/0 reaches a peak when x0/J is
unity twice the CTOD.
The HRR singularity is invalid within
this region, where the stresses are
influenced by large strains and crack
blunting.
However, as long as there is a region
surrounding the crack tip, the J integral uniquely characterizes crack-
tip conditions, and a critical value of
J is a size independent measure of
fracture toughness.
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Laboratory Measurement of J
Linear Elastic : J= G, G is uniquely related to the stress intensity factor.
Nonlinear : The principle of superposition no longer applies, J is not
proportional to the applied load.
One option for determining J is to apply the line integral J integral in test
panels by attaching an array of strain gages in a contour around the crack
tip.
This method can be applied to finite element analysis.
22
2. The J Contour Integral
Schematic of early experimental measurements
of J, performed by Landes and Begley.
A series of test specimens of the
same size, geometry, and material
and introduced cracks of various
lengths.
Multiple specimens must be tested
and analyzed to determine J.
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Laboratory Measurement of J
J directly from the load displacement curve of a single specimen.
Double-edge-notched tension panel of unit thickness.
dA = 2da = −2db
Assuming an isotropic material that obeys a
Ramberg- Osgood stress-strain law.
From the dimensional analysis,
Φ is a dimensionless function. For fixed material properties,
23
2. The J Contour Integral
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Laboratory Measurement of J
If plastic deformation is confined to the ligament between the crack tips, we can
assume that b is the only length dimension that influences Δp.
If plastic deformation is confined to the ligament between the crack tips b is the
only length dimension that influences Δp.
Taking a partial derivative with respect to the ligament length
Integrating by part
24
2. The J Contour Integral
0 0 0 0
0 0 0
1 1 1
2 2 2
1 1 12 2 2
2 2 2
P P
P P P PP P
P P P P
P b
P
P P P P P P P
dP P dP P dP dPb b P b
P dP P P Pd Pd Pb b b
fg dx fg f gdx
0 0
PP
P P PdP P Pd
P
0
P
PPd
0
P
PdP
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Laboratory Measurement of J
Therefore
Unit thickness is assumed at the beginning of this derivation The J integral has units of energy/area.
25
2. The J Contour Integral
0 0
1 12
2 2
PPP
P P
P
dP Pd Pb b
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Laboratory Measurement of J
An edge-cracked plate in bending Example
Ω = Ωnc (angular displacement under no crack)+ Ωc (angular displacement when cracked)
If the crack is deep, Ωc >> Ωnc.
The energy absorbed by the plate
J for the cracked plate in bending can be written as
26
2. The J Contour Integral
0U M d
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Laboratory Measurement of J
If the material properties are fixed, dimensional analysis leads to
Integration by parts
If the crack is relatively deep Ωnc should be entirely elastic,
while Ωc may contain both elastic and plastic contributions.
27
2. The J Contour Integral
2 3 2 20 0 0
2 0 0 0
2 2 1
2 2 2
2
c
M M Mc
M
M M
c c c
M M MJ dM F dM F MdM
b b b b b b
M MF M F dM M dM Md
b b b b b
fg dx fg f gdx
2 32
c
M
M MF
b b b
c
M
0 0
cM
c cM dM MdM
0
2 c
cJ Mdb
( )
( )0 0
2 2c el p
c el pJ Md Mdb b
2
0
2 pIp
KJ Md
E b
or
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Laboratory Measurement of J
General Expression
In general, the J integral for a variety of configurations can be written in
the following form
η : dimensionless constant. Note the above Eq. contains the actual
thickness, while the above derivations assumed a unit thickness for
convenience.
For a deeply cracked plate in pure bending, η = 2, it can be separated into
elastic and plastic components.
28
2. The J Contour Integral
c
M
0
2 c
cJ Mdb
0
c
c cU Md
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General
For linear elastic conditions, the relationship between CTOD and G is given
by
Sinc J= G for linear elastic material behavior, in the limit of small-scale
yielding,
where m is a dimensionless constant that depends on the stress state and
material properties. It can be shown that it applies well beyond the validity
limits of LEFM.
Consider, for example, a strip-yield zone ahead of a crack tip,
29
3. Relationships between J and CTOD
Contour along the boundary of the
strip-yield zone ahead of a crack tip
If the damage zone is long and slender
(𝜌 >> δ), the first term in the J contour
integral vanishes because dy = 0.
-
m=1 for plane stress
m=2 for plane strain
① ②
③④
0
0
𝜌
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General Since the only surface tractions within 𝜌 are in the y direction
Define a new coordinate system with the origin at the tip of the strip-yield
zone (𝑋 = 𝜌 − 𝑥)
Since the strip-yield model assumes yy= YS.
Thus the strip-yield model, which assumes plane stress conditions and a
non-hardening material, predicts that m = 1 for both linear elastic and
elastic plastic conditions
30
3. Relationships between J and CTOD
1 11 1 12 2 13 3
2 21 1 22 2 23 3 22
3 31 1 32 2 33 3
0
0
yy
T n n n
T n n n
T n n n
𝜌
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General
Many materials with high toughness do not fail catastrophically at a particular value
of J or CTOD. Rather, these materials display a rising R curve, where J and CTOD
increase with crack growth.
In the initial stages, there is a small amount of apparent crack growth due to
blunting. As J increases, the material at the crack tip fails locally and the crack
advances further.
Because the R curve is rising, the initial crack growth is usually stable, but an
instability can be encountered later.
One measure of fracture toughness JIc is defined near the initiation of stable crack
growth.
31
4. Crack-Growth Resistance Curves
Schematic J resistance curve for a ductile material
The definition of JIc is somewhat
arbitrary.
The slope of the R curve is indicative of
the relative stability of the crack growth;
a material with a steep R curve is less
likely to experience unstable crack
propagation.
For J resistance curves, the slope is
usually quantified by a dimensionless
tearing modulus:
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Stable and Unstable Crack Growth
Instability occurs when the driving force curve is tangent to the R curve.
Load control is usually less stable than displacement control.
In most structures, between the extremes of load control and
displacement control. The intermediate case can be represented by a
spring in series with the structure, where remote displacement is fixed
Driving force can be expressed in terms of an applied tearing modulus:
ΔT : the total remote displacement, Cm (the system compliance), Δ(line
displacement)
32
4. Crack-Growth Resistance Curves
The slope of the driving force curve
for a fixed ΔT is
Stable crack growth
Unstable crack growth
Schematic J driving force/R curve diagram which
compares load control and displacement control.
&R app RJ J T T
&R app RJ J T T
mC
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Stable and Unstable Crack Growth
For most structure,
If the structure is held at a fixed remote displacement T
assuming & J depends only on load and crack length.
For load control, Cm= and for displacement control, Cm= 0 , T= .
33
4. Crack-Growth Resistance Curves
0T m
P a
d da dP C dPa P
P a
J JdJ da dP
a P
T TP a
J J J P
a a P a
1
T
m
P a a
J J JC
a a P a P
mC
1
P
m
P a
PC dP
a P a
T P
J J
a a
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Stable and Unstable Crack Growth
The point of instability in a material with a rising R curve depends
on the size and geometry of the cracked structure; a critical value of
J at instability is not a material property if J increases with crack
growth.
However, It is usually assumed that the R curve, including the JIC
value, is a material property, independent of the configuration. This
is a reasonable assumption, within certain limitations.
34
4. Crack-Growth Resistance Curves
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Computing J for a Growing Crack
The cross-hatched area represents the energy
that would be released if the material were
elastic.
In an elastic-plastic material, only the elastic
portion of this energy is released; the
remainder is dissipated in a plastic wake.
The energy absorbed during crack growth in
an nonlinear elastic-plastic material exhibits a
history dependence.
35
4. Crack-Growth Resistance Curves
The geometry dependence of a J resistance curve is influenced by the way in which J is calculated.
The equations derived in Section 3.2.5 are based on the pseudo energy release rate
definition of J and are valid only for a stationary crack.
There are various ways to compute J for a growing crack.
The deformation J : used to obtain experimental J resistance curves.
.
Schematic load-displacement curve for a specimen with a crack that grows to a1 from an initial length ao.
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Computing J for a Growing Crack
The cross-hatched area represents the energy
that would be released if the material were
elastic.
In an elastic-plastic material, only the elastic
portion of this energy is released; the
remainder is dissipated in a plastic wake.
The energy absorbed during crack growth in
an nonlinear elastic-plastic material exhibits a
history dependence.
36
4. Crack-Growth Resistance Curves
The geometry dependence of a J resistance curve is influenced by the way in which J is calculated.
The equations derived in Section 3.2.5 are based on the pseudo energy release rate
definition of J and are valid only for a stationary crack.
There are various ways to compute J for a growing crack; the deformation J and the
far-field J,
The deformation J
used to obtain experimental J resistance curves.
Schematic load-displacement curve for a specimen with a crack that grows to a1
from an initial length ao.
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Computing J for a Growing Crack 37
4. Crack-Growth Resistance Curves
The deformation J - continued
the J integral for a nonlinear elastic body with a growing crack is given by
where b is the current ligament length.
The calculation of UD(p) is usually performed incrementally, since the
deformation theory load displacement curve depends on the crack size.
A far-field J
For a deeply cracked bend specimen, J contour integral in a rigid, perfectly
plastic material
By the deformation theory
The J integral obtained from a contour integration is path-dependent when a
crack is growing in an elastic-plastic material, however, and tends to zero as the
contour shrinks to the crack tip.
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Stable and Unstable Crack Growth
Ex 3.2) Derive an expression for the applied tearing modulus in the double
cantilever beam (DCB) specimen with a spring in series
38
4. Crack-Growth Resistance Curves
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Stationary Cracks
Small-scale yielding,
K uniquely characterizes crack-tip
conditions, despite the fact that the
1/ 𝑟 singularity does not exist all the
way to the crack tip.
Similarly, J uniquely characterizes
crack-tip conditions even though the
deformation plasticity and small
strain assumptions are invalid within
the finite strain region.
Elastic-plastic conditions
J is still approximately valid, but there
is no longer a K field.
As the plastic zone increases in size
(relative to L), the K-dominated zone
disappears, but the J-dominated zone
persists in some geometries.
The J integral is an appropriate
fracture criterion
39
5. J-Controlled Fracture
22
small-scale yielding
Elastic-plastic conditions
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Stationary Cracks
Large-scale yielding
the size of the finite strain zone
becomes significant relative to L,
and there is no longer a region
uniquely characterized by J.
Single-parameter fracture
mechanics is invalid in large-scale
yielding, and critical J values exhibit
a size and geometry dependence.
For a given material,5 dimensional
analysis leads to the following
functional relationship for the
stress distribution within this region:
40
5. J-Controlled Fracture
Large-scale yielding
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J-Controlled Crack Growth
J-controlled conditions exist at the tip of a stationary crack (loaded
monotonically and quasistatically), provided the large strain region
is small compared to the in-plane dimensions of the cracked body
41
5. J-Controlled Fracture