Topics in Ship Structures - ocw.snu.ac.kr

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 Elastic-Plastic FractureMechanics


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


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.


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


,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


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


(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


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


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


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


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.


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


Determination of the plastic componentof the crack-mouth-opening 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 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


Schematic comparison of the stress strain behavior of elastic-plastic and nonlinear elastic materials.


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


dJ

dA

U F

*U P U

*

0

P

U dP Nonlinear energy release rate.

Crack length = a+da

Crack length = a



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


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



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

IKJ

E

plastic wakedUJ

dA


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


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


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


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


The J Contour Integral - Proof 15


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.


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


Two arbitrary contours Γ1 and Γ2

around the tip of a crack.

22 4 0i

i

uJ J wdy T ds

x


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


⋯ (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 quasi-static conditions.

18


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


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



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


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


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


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.


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


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.



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




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


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



Therefore

Unit thickness is assumed at the beginning of this derivation The J integral has units of energy/area.

25


0 0

1 12

2 2

PPP

P P

P

dP Pd Pb b



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


0U M d



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 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



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


c

M

0

2 c

cJ Mdb

0

c

c cU Md


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


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


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


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:


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


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



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


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



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



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


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.


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


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.


Computing J for a Growing Crack 37


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.



Ex 3.2) Derive an expression for the applied tearing modulus in the double

cantilever beam (DCB) specimen with a spring in series

38



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


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


Large-scale yielding


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


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