OPen INteractive Structural Lab
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 ElasticPlastic Fracture
Mechanics
Contents
1. CrackTip –Opening displacement 2. The J Contour Integral
3. Relationships Between J and CTOD 4. CrackGrowth Resistance Curves 5. J Controlled Fracture
6. CrackTip Constraint Under LargeScale Yielding 7. Scaling Model for Cleavage Fracture
8. Limitations of TwoParameter Fracture Mechanics
2
0. INTRODUCTION
Definition of CTOD
Structural steels has higher toughness than K
_{Ic}values characterized by LEFM.
The crack faces moves apart prior to fracture; plastic deformation blunts an initially sharp crack.
CrackTipopening Displacement (CTOD) : a measure of fracture toughness.
3
1. CrackTipopening displacement
Cracktipopening displacement (CTOD). Estimation of CTOD from the displacement of the effective crack in the Irwin plastic zone correction.
Relationship between CTOD and K
_{I}and G
:Irwin plastic zone
Effective crack length = a+r
_{y} The displacement r
_{y}behind the effective crack tip for plane stress.
The Irwin plastic zone correction for plane stress.
CTOD to be related with K
_{I}or G .
4
1. CrackTipopening displacement
y
,
r r
eff y
a a r
(3 ) / (1 ), (plane stress)
2(1 )
v v G E
v
3 1
( 1) 1 4
2 2 2 2 2
2(1 )
y y y
y I I I
v v
r v r r
u K K K
E E
v
4
22
_{y} ^{I}YS
u K
E
1 2
2
I y
YS
r K
4
YS
G
Relationship between CTOD and K
_{I}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. CrackTipopening displacement
(3 4 ), (plane strain)
2(1 ) v G E
v
2
( 1) 4(1 ) 4 4
2 2 2 2 2
22(1 ) (1 )
y y y y
y I I I I
r v r r r
u K K K K
E E E
v v
4
22 3
I y
YS
u K
E
1 26
I y
YS
r K
4
3
_{YS}
G
Relationship between CTOD and K
_{I}and G : Stripyield 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. CrackTipopening displacement
Estimation of CTOD from the stripyield model.
4 4 2
2 2 4
y I I
y I I
YS YS
r K K
u K K
E E E
^{} ^{} ^{} ^{}
8 ln sec 2
YS
YS
a E
/
_{YS}0
2 I
YS YS
K
E
G
^{I}^{2}YS YS
K
m E m
G 1 for plane stress 2 for plane strain m
m
2 8
2
y
y I
u K r
E
2
2 2 2
2
8 ,
8 ,
YS I
YS YS YS
I
a a K
E E E
here K a
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. CrackTipopening 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 threepoint bend specimens.
Two most common definitions of CTOD
A typical load ( P ) vs. displacement ( V ) curve from a CTOD
The shape of the loaddisplacement curve is similar to a stressstrain 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 r
_{p}is approximately 0.44 for typical materials and test specimens.
8
1. CrackTipopening displacement
Determination of the plastic component of the crackmouthopening displacement 2
I YS
K m E
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 elasticplastic 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 cracktip 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.
Nonlinear Energy Release Rate
Rice [4] presented J is a pathindependent 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
J d
dA
U F
U P U
*
*
0
U P dP Nonlinear energy release rate.
Crack length = a+da Crack length = a
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 dU
J dA dA
d dU
J dA dA
_{}
0 0
P P
P P
J dP dP
a a
0 0
J Pd P d
a a
U F U
* 1 / 2 , * 0, 0
dU dU dPd dU dU
^{*}0
U
P dP
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 elasticplastic 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 elasticplastic material is not recovered. A growing crack in an elasticplastic material leaves a plastic wake.
Thus, the energy release rate concept has a somewhat different interpretation for elasticplastic materials.
J indicates the difference in energy absorbed by specimens with neighboring crack sizes.
12
2. The J Contour Integral
2
K
IJ E
plastic wake
J dU
dA
_{}
J as a PathIndependent 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 n
_{j}are the components of the unit vector normal to Γ .
J : a pathindependent integral
The J Contour Integral  Proof
Evaluate𝐽along 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 straindisplacement relationship and _{ij}= _{ji}
Invoking the equilibrium condition
Thus, J=0 for any closed contour
14
2. The J Contour Integral
Γ^{∗}: closed contour
𝐴^{∗}: Area enclosed by Γ^{∗}
⋯ (a) “To be proved in the next slide.”
0
* *
*
_{ij} ^{i}0
A A
j
u
w w w
J dxdy dxdy
x x x x x
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 sin
s s s
x y
r i j
i j
xi
s
yj
1 1 2 2
* * *
1 2 1 2
* *
1 1
*
_{i} ^{i} _{ij} _{j} ^{i} _{i} ^{i} _{i} ^{i}i i i i
i i i i
A
i
u u u u
J wdy T ds wdy n ds wdy n ds n ds
x x x x
u u w u u
wdy dy dx dxdy dxdy dxdy
x x x x x y x
w
x x
* 2 *
2
i i i
i ij
A A
j
u u w u
dxdy dxdy
x x x x x x
2 1
1 2
R C
F F
dxdy F dx F dy
x y
1
,
2x x x y
Using divergence theorem, the line integral can be converted into an areal
integral.
The J Contour Integral  Proof
Consider now two arbitrary contours.
If Γ
_{1 }and Γ
_{2}are connected by segments along the crack face (Γ
_{3}and Γ
_{4}), 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, T
_{i}=dy = 0.
Thus, J
_{1}= −J
_{3}.
“Therefore, any arbitrary (counterclockwise) path around a crack will yield the same value of J; J is pathindependent.”
16
2. The J Contour Integral
Two arbitrary contours Γ^{1 }and Γ^{2 } around the tip of a crack.
2
2 4 _{i}
u
^{i}0
J J wdy T ds
x
J as a Nonlinear Elastic Energy Release Rate
Consider a twodimensional cracked body bounded by the curve Γ.
Under quasistatic 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)
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 quasistatic conditions.
18
2. The J Contour Integral
2 1
1 2
R C
F F
dxdy F dx F dy
x y
^{F}^{2} ^{} ^{w F}^{,} ^{1} ^{}^{0}A x
wdxdy wdy wn ds
x
^{n}^{ }^{s} ^{y}^{i} ^{x}^{j}x y
y x
n n
s s
n i j i j
J as a Stress Intensity Parameter
•
RambergOsgood eq. : Inelastic stressstrain 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.
• k_{1} and k_{2} are proportionality constants, which are defined more precisely below. For a linear elastic material, n = 1.
19
2. The J Contour Integral
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.
I_{n} : an integration constant that depends on n.
𝜎_{𝑖𝑗} and _{𝑖𝑗} 𝜀 : dimensionless functions of n and θ.
20
2. The J Contour Integral
Effect of the strainhardening exponent on the
HRR integration constant. Angular variation of dimensionless stress for n = 3 and n = 13
plane stress
plane strain
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
Largestrain cracktip finite element results.
_{yy/}
_{0}reaches a peak when x
_{0}/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.
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.
Laboratory Measurement of J
J directly from the load displacement curve of a single specimen.
Doubleedgenotched tension panel of unit thickness.
dA = 2da = −2 db
Assuming an isotropic material that obeys a
Ramberg Osgood stressstrain law.
From the dimensional analysis,
Φ is a dimensionless function. For fixed material properties,
23
2. The J Contour Integral
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 Δ_{p}.
If plastic deformation is confined to the ligament between the crack tips b is the only length dimension that influences Δ_{p}.
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 1
2 2 2
2 2 2
P P
P P P P
P P
P P P P
P b
P
P P P P P P P
dP P dP P dP dP
b b P b
P dP P P Pd Pd P
b b b
fg dx fg f gdx
0 0
P P
PdP P P ^{} Pd P
P 0
P
Pd P
0 P
PdP
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 1
2 2 2
P P
P
P P
P
dP Pd P
b b
Laboratory Measurement of J
An edgecracked plate in bending Example
Ω = Ω_{nc} (angular displacement under no crack)+ Ω_{c} (angular displacement when cracked)
If the crack is deep, Ω_{c} >> Ω_{nc}.
The energy absorbed by the plate
J for the cracked plate in bending can be written as
26
2. The J Contour Integral
U
0^{}M d
Laboratory Measurement of J
If the material properties are fixed, dimensional analysis leads to Integration by parts
If the crack is relatively deep Ω_{nc} should be entirely elastic, while Ω_{c} may contain both elastic and plastic contributions.
27
2. The J Contour Integral
2 3 2 2
0 0 0
2 0 0 0
2 2 1
2 2 2
2
c
M M M
c M
M M
c c c
M M M
J dM F dM F MdM
b b b b b b
M M
F M F dM M dM Md
b b b b b
fg dx fg f gdx
2 3
2
c M
M M
b F b b
c
M
0 0
M c
cM cdM ^{} MdM
0
2
^{c}J Md
cb
( )
( )
0 0
2
^{c el}2
^{p}c el p
J Md Md
b b
or J K E
^{I}^{2} b 2
_{0}^{}^{p}Md
^{p}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}J Md
cb
^{U}^{c} ^{}
^{0}^{}^{c}^{Md}^{}^{c}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 smallscale 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 stripyield zone ahead of a crack tip,
29
3. Relationships between J and CTOD
Contour along the boundary of the stripyield 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
④ ③
0 0
𝜌
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 stripyield zone (𝑋 = 𝜌 − 𝑥)
Since the stripyield model assumes
_{yy}=
_{YS.} Thus the stripyield model, which assumes plane stress conditions and a nonhardening 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
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 J_{Ic}is defined near the initiation of stable crack growth.
31
4. CrackGrowth Resistance Curves
Schematic J resistance curve for a ductile material
The definition of J_{Ic} 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:
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:
Δ_{T} : the total remote displacement, C_{m} (the system compliance), Δ(line displacement)
32
4. CrackGrowth 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 R
J J T T
&
R app R
J J T T
Cm
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, C_{m}= and for displacement control, C_{m}= 0 , _{T=} .
33
4. CrackGrowth Resistance Curves
T m 0
P a
d da dP C dP
a P
P a
J J
dJ da dP
a P
T P a T
J J J P
a _{} a P a _{}
1
T
m
P a a
J J J
a a P a C P
Cm
1
P
m
P a
C dP P
a P a
T P
J J
a _{} a
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 J
_{IC}value, is a material property, independent of the configuration. This is a reasonable assumption, within certain limitations.
34
4. CrackGrowth Resistance Curves
Computing J for a Growing Crack
The crosshatched area represents the energy that would be released if the material were elastic.
In an elasticplastic 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 elasticplastic material exhibits a history dependence.
35
4. CrackGrowth 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 loaddisplacement curve for a specimen with a crack that grows to a_{1} from an initial length a_{o}.
Computing J for a Growing Crack
The crosshatched area represents the energy that would be released if the material were elastic.
In an elasticplastic 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 elasticplastic material exhibits a history dependence.
36
4. CrackGrowth 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 farfield J,
The deformation J
used to obtain experimental J resistance curves.
Schematic loaddisplacement curve for a specimen with a crack that grows to a_{1} from an initial length a_{o}.
Computing J for a Growing Crack
^{37}4. CrackGrowth 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 U_{D(p)} is usually performed incrementally, since the deformation theory load displacement curve depends on the crack size.
A farfield 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 pathdependent when a crack is growing in an elasticplastic material, however, and tends to zero as the contour shrinks to the crack tip.
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. CrackGrowth Resistance Curves
Stationary Cracks
Smallscale yielding,
K uniquely characterizes cracktip conditions, despite the fact that the 1/ 𝑟 singularity does not exist all the way to the crack tip.
Similarly, J uniquely characterizes cracktip conditions even though the deformation plasticity and small
strain assumptions are invalid within the finite strain region.
Elasticplastic 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 Kdominated zone disappears, but the Jdominated zone persists in some geometries.
The J integral is an appropriate fracture criterion
39
5. JControlled Fracture
2 ^{2}
smallscale yielding
Elasticplastic conditions
Stationary Cracks
Largescale yielding
the size of the finite strain zone becomes significant relative to L , and there is no longer a region uniquely characterized by J .
Singleparameter fracture
mechanics is invalid in largescale 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. JControlled Fracture
Largescale yielding
JControlled 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 inplane dimensions of the cracked body
41