Milano Lectures

7/27/2019 Milano Lectures

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Introduction to Fractureand Damage Mechanics

Wolfgang Brocks

Five Lectures

at

Politecnico di Milano

Milano, March 2012


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Contents

I. Linear elastic fracture mechanics (LEFM)

Stress field at a crack tip Stress intensity approach (IRWIN) Energy approach (GRIFFITH) J-integral Fracture criteria fracture toughness Terminology

II. Plasticity

Fundamentals of incremental plasticity Finite plasticity (deformation theory) Plasticity, damage, fracture

Porous metal plasticity (GTN Model)

III. Small Scale Yielding

Plastic zone at the crack tip Effective crack length (Irwin) Effective SIF Dugdale model Crack tip opening displacement (CTOD) Standards: ASTM E 399, ASTM E 561

IV. Elastic-plastic fracture mechanics (EPFM)

Deformation theory of plasticity J as energy release rate HRR field, CTOD Deformation vs incremental theory of plasticity R curves Energy dissipation rate

V. Damage Mechanics

Deformation, damage and fracture Crack tip and process zone Continuum damage mechanics (CDM) Micromechanisms of ductile fracture Micromechanical models Porous metal plasticity (GTN Model)


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Seite 1

Concepts of Fracture Mechanics

Part I: L inear Elastic Fracture Mechanics

W. Brocks

Christian Albrecht University

Material Mechanics

Milano_2012 2

Overview

I. Linear elastic fracture mechanics

(LEFM)

II. Small Scale Yielding

III. Elastic-plastic fracture mechanics (EPFM)


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Seite 2

Milano_2012 3

LEFM

Stress Field at a crack tip

Stress intensity approach (Irwin)

Energy approach (Griffith)

J-integral

Fracture criteria fracture toughness

Terminology

Milano_2012 4

Stress Field at a Crack Tip

boundary conditions

( 0, 0) ( , ) 0

( 0, 0) ( , ) 0

yy

xy r

x y r

x y r

= = = =

= = = =

Hookes law of linear elasticity

21 2

ij ij kk ijG

= +

Inglis [1913], Westergaard[1939], Sneddon [1946], ...

Williams series [1957]

0 =Airys stress function

( )

( )

2 cos cos 2

sin sin 2

r A B

C D

+ = + + +

+ + +

21 15 3 5 3 3104 2 4 2 4 2 4 2

9 3 5 9 15 51 14 2 4 2 4 2 4 2

cos cos sin cos 4 cos

cos cos sin cos ( )

rr

A CA

r r

A r C r r

= + + + +

+ + + + + O


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Seite 3

Milano_2012 5

Fracture Modes

Milano_2012 6

LEFM: Stress Intensity Approach

Irwin [1957]: mode I

1 1

II( , ) ( )2

j j ji ii

Kr f

rT

= +

T = non-singular T-stress

Rice [1974]: effect on plastic zone

General asymptotic solution

I II III

I II III

1( , ) ( ) ( ) ( )

2ij ij ij ij

r K f K f K f r

= + + stresses

displacements I II IIII II III

1( , ) ( ) ( ) ( )

2 2i i i i

ru r K g K g K g

G

= + +

I (geometry)K a Y =

KI = stress intensity factor


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Milano_2012 7

Angular Funct ions in LEFM

Milano_2012 8

A.A. Gri ff ith [1920]

LEFM: Energy Approach

Elastic strain energy of a panel of thickness

B under biaxial tension in a circular

domain of radius r

( )( ) ( )2 2

2 2e

0 1 1 2 116

BrU

G

= + +

elliptical hole, axes a,b

( ) ( ) ( )

( )( ) ( ) ( )

22 2e

rel

22 2 2 2 2

1 132

2 1 1

BU a b

G

a b a b

= + + +

+ + + +

( )2 2

e

rel 18

a BU

G

= +crack

insert hole: fixed grips

> energy releasee e e

0 relU U U=


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Milano_2012 9

Fracture of Brittle Materials

( )( )

e

rel sep0

2U U

B aCrack extends if

= =

G

L

eee rel

(2 ) (2 )v

UU

B a B aEnergy release rate

sep

c2

(2 )

U

B a

= =

Separation energy (SE)

(energy per area)

= =Ge c( ) 2afracture criterion

fracture stress

=

c

cE

a

Irwin [1957]:

2e IK

E=

G

Milano_2012 10

Path-Independent Integrals

is some (scalar, vector, tensor) field quantity being

steadily differentiable in domain Band divergence free

, : 0 inii

x

= =

B

( )i

x

, 0i idv n da

= = B B

Gau theorem

singularity Sin B: 0 = SB B B

0

(.) (.) (.) (.) (.) 0 +

= + + + = v v? >SB B BB B

(.) (.) and (.) (.)+

= = v v? >B B B B

i in da n da

= > >SB B

path independence


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Energy Momentum Tensor

Eshelby [1965]: energy momentum tensor

, ,

,

with 0ij ij k i ij j

k j

wP w u P

u

= =

( )i j

w

u x

energy density

displacement field

Material forces acting on singularities (defects) in the continuum,

e.g. dislocations, inclusions, ...

i ij jF P n da

= >B

and

the J-vector ,i i jk k j iJ wn n u ds

= >0

t

i j ijw d

=

=

Milano_2012 12

J-Integral

The J-integral ofCherepanov [1967] and Rice [1986]

is the 1st component of the J-vector

2 ,1ij j iJ w dx n u ds

= >

Conditions :

9 time independent processes

9 no volume forces

9 homogeneous material

9 plane stress and strain fields, no dependence on x3

9 straight and stress free crack faces parallel to x1

Equilibrium

Small (linear) strains

Hyperelastic material

, 0ij i =

( )1 , ,2ij i j j iu u = +

ij

ij

w

=


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J as Energy Release Rate

L

ee

crack v

UJ

A

= =

G

crack

C(T), SE(B)

M(T), DE(T)2

B a

A

B a

=

2 2 2e e e e I II III

I II III2

K K KJ

E E G= = + + = + +

G G G G

mixed mode

mode I

Milano_2012 14

Fracture Criteria

Brittle fracture:

predominantly elastic catastrophic failure

Mode I

stress Intensity factor I IcK K=

energy release ratee

I Ic=G G2

e IcIc c

K

E

= =

G

J-integral IcJ J=2

IcIc Ic

KJ

E= =

G

2

plane stress

plane strain1

E

E E

=


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Milano_2012 15

ASTM E 1823

Crack extension, a an increase in crack size.

Crack-extension force, G the elastic energy per unit of new

separation area that is made available at the front of an ideal

crack in an elastic solid during a virtual increment of forward

crack extension.

Crack-tip plane strain a stress-strain field (near the crack tip)

that approaches plane strain to a degree required by an

empirical criterion.

Crack-tip plane stress a stress-strain field (near the crack tip)

that is not in plane strain.

Fracture toughness a generic term for measures of resistance to

extension of a crack.

Standard Terminology Relating to Fatigue and Fracture Testing

Milano_2012 16

ASTM E 1823 ctd.

Plane-strain fracture toughness, KIc the crack-extension

resistance under conditions of crack-tip plane strain in Mode I for

slow rates of loading under predominantly linear-elastic

conditions and negligible plastic-zone adjustment. The stress

intensity factor, KIc, is measured using the operational procedure

(and satisfying all of the validity requirements) specified in Test

Method E 399, that provides for the measurement of crack-

extension resistance at the onset (2% or less) of crack extensionand provides operational definitions of crack-tip sharpness, onset

of crack-extension, and crack-tip plane strain.

Plane-strain fracture toughness, JIc the crack-extension

resistance under conditions of crack-tip plane strain in Mode I

with slow rates of loading and substantial plastic deformation.

The J-integral, JIc, is measured using the operational procedure

(and satisfying all of the validity requirements) specified in Test

Method E 1820, that provides for the measurement of crack-

extension resistance near the onset of stable crack extension.


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ASTM E 399

Characterizes the resistance of a material to fracture in a neutral environment inthe presence of a sharp crack under essentially linear-elastic stress and severe

tensile constraint, such that (1) the state of stress near the crack front

approaches tritensile plane strain, and (2) the crack-tip plastic zone is smallcompared to the crack size, specimen thickness, and ligament ahead of the

crack;

Is believed to represent a lower limiting value of fracture toughness;

May be used to estimate the relation between failure stress and crack size for a

material in service wherein the conditions of high constraint described abovewould be expected;

Only if the dimensions of the product are sufficient to provide specimens of thesize required for valid KIc determination.

Standard Test Method for Linear-Elastic Plane-Strain FractureToughness KIc of Metallic Materials

2

Ic

YS

2.5K

W a

Specimen size YS 0.2 % offset yield strength

Milano_2012 18

ASTM E 399 ctd.

Specimen conf igurations

SE(B): Single-edge-notched and fatigue precracked beam loaded in

three-point bending; support span S = 4 W, thickness B = W/2,

W = width;

C(T): Compact specimen, single-edge-notched and fatigue precracked

plate loaded in tension; thickness B = W/2;

DC(T): Disk-shaped compact specimen, single-edge-notched and

fatigue precracked disc segment loaded in tension; thickness

B = W/2;

A(T): Arc-shaped tension specimen, single-edge-notched and fatigue

precracked ring segment loaded in tension; radius ratio unspecified;

A(B): Arc-shaped bend specimen, single-edge-notched and fatigue

precracked ring segment loaded in bending; radius ratio forS/W = 4 and for S/W = 3;


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Specimen Configurations

Bend Type

C(T) SE(B)

W widthB thickness

a crack length

b = W-a ligament width


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Inelastic Deformation and Damage

Part I: Plasticity

W. Brocks


Material Mechanics

Milano_2012 2

Outline

Fundamentals of incremental plasticity

Finite plasticity (deformation theory)

Plasticity, damage, fracture Porous Metal Plasticity (GTN Model)


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Plasticity

Inelastic deformation of metals at low temperatures and slow(= quasistatic) loading, i. e. time and rate independent material

behaviour

Microscopic mechanisms: motion of dislocations, twinning.

Phenomenological theory on macro-scale in the framework of

continuum mechanics.

Incremental Theory of Plasticity

,ij ij ij ijt t

Material behaviour is non-linear and plastic (permanent)deformations depend on loading history.

Constitutive equations are established incrementally forsmall changes of loading and deformation

t> 0 is no physical time but a scalar loading parameter, and hence

ij and ij are no velocities but rates

Milano_2012 4

Uniaxial Tensile Test

linear elasticity: Hooke0

:R E true stress-strain curve

0:R plasticity: nonlinear--curve

permanent strain

e p p

E

yield condition F p F 0( ) , (0)R R R

R0 yield strength (0, Y)

RF(p) uniaxial yield curve

loading / unloading p

p

0 , 0 loading

0 , 0 unloading


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Milano_2012 5

3D Generalisation

Additive decomposition of strain ratese p

ij ij ij

Total plastic strainsp p

0

t

ij ijd

Plastic incompressibility:

plastic deformations are isochoric

plastic yielding is not affected by hydrostatic stress

h

1h 3

ij ij ij

kk

deviatoric stress

hydrostatic stress

p 0kk

Yield condition pp 2( , ) ( ) 0iij ij ij ijj ij

Hardening:

kinematic: tensorial variable back stresses

isotropic: scalar variable accumulated plastic strain

ij

p

Milano_2012 6

Yield Surface

0 elastic

0 elast

0 inadm

ic

issible


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Associated flow rule normality rule

Elasto-Plasticity

Loading condition

p

p

0 , 0

0, 0

ij ij

ij

ij ij

ij

loading

unloading

p

ij

ij

Consistency conditionp

p0ij ij

ij ij

e 12 1

ij ij kk ijG

Hookes law of elasticity

Stability

Drucker [1964]

p p 0ij ij Equivalence of dissipation rates

Milano_2012 8

von Mises Yield Condition

Yield condition - isotropic hardening

p 2 2

F p( , ) ( ) 0ij ij R

32 2

3 ( )ij ij ij

J von Mises [1913, 1928]

equivalent stress

Loading condition0

0

ij ij

ij ij

loading

unloading

2 2 2 2 2 21 11 22 22 33 33 11 12 23 132 3

J2-theory

p

ij ij Flow rule plastic multiplier

from uniaxial test


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Milano_2012 9

Prandtl-Reuss

e p e e p1h3

p F

1 1 3

2 3 2ij ij ij ij kk ij ij ij ij ij

G K T R

Total strain rates (Prandtl [1924], Reu [1930])

Equivalence of dissipation rates

p p p2p3

0

t

ij ij d

equivalent plastic strain

Hookes law of elasticity + associated flow rule e pij ij ij

p p

ij ij ij ij

pp

F p

3 3

2 2 ( )R

plastic multiplier

Milano_2012 10

Finite Plasticity

additive decomposition of total strainse p

ij ij ij Hencky [1924]

p

ij ij plastic strains

h

p

1 3 1

2 2 3ij ij ij

G S K

total strains

power law ofRamberg & Osgood [1945]

0 0 0

n

uniaxial

1p

0 0 0

3

2

n

ij ij

3D


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Milano_2012 11

Deformation Theory

For radial (proportional) loading,the Hencky equations can be derived by integration of the

Prandtl-Reu equations.

This has do hold for every point of the continuum and excludes

stress redistribution,

unloading.

0

( ) ( )ij ijt t

Finite plasticity actually describes a hyperelastic material having

a strain energy density

ij

ij

w

0

t

ij ijw d

so that

In elastic-plastic fracture mechanics (EPFM), finite plasticity +

Ramberg-Osgood Power law are adressed as

Deformation Theory of plasticity.

Milano_2012 12

Plasticity and Fracture

uniaxial tensile test


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Fracture of a Tensile Bar

Milano_2012 14

Fracture surface

round tensile specimen of Al 2024 T 351


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Milano_2012 15

Deformation, Damage, Fracture

Deformation

Cohesion of matter is conserved

Elastic: atomic scale

Reversible change of atomic distances

Plastic: crystalline scale

Irreversible shift of atoms, dislocation movement

Damage

Laminar or volumetric discontinuities on the micro scale (micro-cracks, microvoids, micro-cavities)

Damage evolution is an irreversible process, whosemicromechanical causes are very similar to deformationprocesses but whose macroscopic implications are muchdifferent

Fracture

Laminardiscontinuities on the macro scale leading to globalfailure (cleavage fracture, ductile rupture)

Milano_2012 16

Damage Models

Damage models describe evolution of degradation phenomena

on the microscale from initial (undamaged or predamaged) state

up to creation of a crack on the mesoscale (material element)

Damage is described by means ofinternal variables in the

framework ofcontinuum mechanics.

Phenomenological models

Change of macroscopically observable properties are

interpreted by means of the internal variable(s);

Concept of effective stress: Kachanov [1958, 1986], Lemaitre& Chaboche [1992], Lemaitre [1992].

Micromechanical models

The mechanical behaviour of a representative volume element

(RVE) with defect(s) is studied;

Constitutive equations are formulated on a mesoscale by

homogenisation of local stresses and strains in the RVE.


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Ductile Damage

Nucleation, growth and coalescence of microvoids at inclusions or

second-phase particles

Void growth is strain controlled, and depends on hydrostatic stress

Milano_2012 18

SI

SIII

SII

Porous Metal Plasticity

Plastic Potential ofGurson, Tvergaard & Needleman (GTN model)

including scalardamage variable f*(f), (f= void volume fraction)

22h

1 2

* * *

p

pF p

32

F

3( , , ) 2 cosh 1 0

( ) 2 ( )ij f f

Rq

Rfq q

Pores are assumed

to be present from the beginning, f0,

or nucleate as a function of plastic

equivalent strain, fn, n, sn

Evolution equation of damage

Volume dilatation caused by void growth

p1 kkf f

p

kk


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Seite 1


Part II: Small Scale Yielding

W. Brocks

Christian Albrecht UniversityMaterial Mechanics

Milano_2012 2

Overview


II. Small Scale Yielding (SSY)

III. Elastic-plastic fracture mechanics (EPFM)


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Seite 2

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SSY

Plastic zone at the crack tip

Effective crack length (Irwin)

Effective SIF

Dugdale model

Crack tip opening displacement (CTOD)

Standards: ASTM E 399, ASTM E 561

Milano_2012 4

Yielding in the Ligament (I)

Irwin [1964]: extension of LEFM to small plastic zones

Small Scale Yielding, SSY

K dominated stress field mode I

(a) plane stress 3

I1 2

0

2

zz

xx yy

K

r

= =

= = = =

Yield condition (both: von Mises and Tresca) 0 p, 0yy R r r =

2

Ip

0

1

2

Kr

R

=

0 =Yielding in the ligament

perfectly plastic material F p 0( )R R =

rp =radiusof plastic zone


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Yielding in the Ligament (II)

(b) plane strain

( )

3

I3 1 2

0

22

2

zz

zz yy

K

r

= =

= = + = =

Yield condition (both: von Mises and Tresca)

( ) 0 p1 2 , 0yy R r r =

( )22

Ip

0

1 2

2

Kr

R

=

smaller plastic zone dueto triaxiality of stress state

Cut-off of largest principle stress at R0 (plane stress)p p

Iyy p 0 p

0 0

2( ) 2

2 r

r r

Kr dr dr r R r

= = =

equilibrium ?

Milano_2012 6

Effective Crack Length

effIeff I eff eff ( )

aK K a a Y

W

= =

effective SIF

pr aeff pa a r= +

total diameterof plastic zone

( )

2

I2p p

0

1 plane stress2 ,

2 1 2 plane strain

Kd r

R

= = =

2Ieff

ssy ssy

KJ

E= =

Geffective J

no singularityat the crack tip


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Example SSY (I)

I

aK a Y

W

=

F

BW =

ASTM E 399

C(T)

aY

W

Milano_2012 8

Example SSY (II)

2

p

02

d a aY

W W R W

=

plane stress

plane strain

effIeff eff

aK a Y

W

=

plane stress

plane strain


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CTOD (Irwin)

I

2

1 plane stress( , ) 4

1 plane strain2y

K ru r

E

=

t p2 ( , )yu r =Wells [1961]

( )( )21 1 2 0.36

plane strain plane stress

t t0,36

( ) ( )

2Irwin It 2

0

1 plane stress4

1 1 2 plane strain

K

ER

=

Criterion for crack initiation: t c =

Milano_2012 10

Shape of Plastic Zone

Yield condition (von Mises)

( ) ( ) ( ) ( )2 222 2 2 21 11 22 22 33 33 11 12 23 132 3 = + + + + +

p0r

R =

( )33 11 22

0 plane stress

plane strain

=

+

( ) ( )

2 232I

p 2230 2

1 sin cos plane stress1( )

2 sin 1 2 1 cos plane strain

Kd

R

+ + =

+ +

II( , ) ( )2

ij ij

Kr f

r

=LEFM:


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Plastic Zone

plane stress

plane strain

3D: dog bone model

Hahn & Rosenfield [1965]

plane stressplane strain

Milano_2012 12

Dugdale Model

0 p( ,0) , 0yy r R r d =

p

0 0

1 cos sec 12 2

d c aR R

= =

p2 2 2c a d= +

(1) (2)

I I 0

2, arccos

aK c K R c

c

= = Superposition

0

cos2

a

c R

=

no singularity (1) (2)I I 0K K+ =!

no restriction

with respect to

plastic zone size!

2 22

Irwin

p p

0 0

1.23 1.238

d c a d R R

=

0 1:R plane stress!

strip

yield

model


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CTOD (Dugdale)

Dugdale 0t

0

8lnsec

2

Ra

E R

=

0

0

( , 0) 4 ln sec2

y

Ru x a y a

E R

= = =

Crack opening profile

t 2 ( , 0)yu x a y = = =Definition of CTOD

2Irwin

t

0

4a

ER

=

for Griffith crack, plane stress

no dependenceon geometry!

Milano_2012 14

Barenblatt Model

Idea:

Singularity at the crack tip is unphysical

Griffith [1920]: Energy approach

Irwin [1964]: Effective crack length

Dugdale [1960]: Strip yield model

Barenblatt [1959]: Cohesive zone

Stress distribution (x) is unknown and cannot be measured

Cohesive model: traction-separation law ()


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Energy Release Rate

= =

+= = = +

G

G G G

L

rel

2 2p e pIeff I

ssy

v

UU

B a B a

a rK K

E a E

= > =sep e

c c 2U

B a

Separation energyCohesive model:

c

c

0

( ) d

=

c=GFracture criterion:

local criterion!

Milano_2012 16

ASTM E 399: Size Condition

Characterizes the resistance of a material to fracture in a neutralenvironment in the presence of a sharp crack under essentially linear-elastic stress and severe tensile constraint, such that

(1) the state of stress near the crack front approaches tritensile planestrain, and

(2) the crack-tip plastic zone is small compared to the crack size,

specimen thickness, and ligament ahead of the crack;

Standard Test Method for Linear-Elastic Plane-Strain Fracture

Toughness KIc of Metallic Materials

2

Ic

YS

2.5K

W a

Specimen size YS 0.2 % offset yield strength

( )22

Ip

YS

1 2 Kd

=

( )

( )

21 2

0.022.5

pd B

W a W a


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ASTM E 561

Determination of the resistance to fracture under Mode I loadingusing M(T), C(T), or crack-line wedge-loaded C(W) specimen;continuous record of toughness development in terms ofKR plottedagainst crack extension.

Materials are not limited by strength, thickness or toughness, so long asspecimens are of sufficient size to remain predominantly elastic.

Plot ofcrack extension resistance KR as a function of effectivecrack extension ae.

Measurement of physical crack size by direct observation and thencalculating the effective crack size ae by adding the plastic zone

size; Measurement of physical crack size by unloading compliance and then

calculating the effective crack size ae by adding the plastic zone size;

Measurement of the effective crack size ae directly by loadingcompliance.

Standard Test Method forK

-R Curve Determination

Milano_2012 18

FM Test Specimens (Tension Type)

M(T)

DE(T)


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Seite 1


Part III: Elastic-Plastic Fracture

W. Brocks


Material Mechanics

Milano_2012 2

Overview


II. Small Scale Yielding

III. Elastic-plastic fracture mechanics

(EPFM)


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Seite 2

Milano_2012 3

EPFM

Deformation theory of plasticity

J as energy release rate

HRR field, CTOD

Deformation vs incremental theoryof plasticity

R curves

Energy dissipation rate

Milano_2012 4

EPFM

Analytical solutions and analyses in

Elastic-Plastic Fracture Mechanics,

i.e. fracture under large scale yielding (LSY)conditions

are based on Deformation Theory of Plasticity

which actually describes hyperelastic materials

ij

ij

w

=

p

0 0

t t

ij ij ij ijw d d

= =

=

e stands for linear elastic

p stands for nonlinear

e p e p

L LU U U F dv F dv= + = + L

2 pe p I

crack v

K UJ J J

E A

= + =

in the following, the superscripts


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J as Stress Intensity Factor

Power law ofRamberg & Osgood [1945]

e p

0 0 0 0 0

n

= + = +

uniaxial

1p

0 0 0

3

2

n

ij ij

=

3D

Hutchinson [1868],

Rice & Rosengren [1968]

singular stress and strain fields at the crack tip (HRR field) mode I1

1

p 10

0

( )

( )

nij ij

nn

nij ij ij

K r

Kr

+

+

=

=

1

1

0

0 0

n

n

JK

I

+ =

p 1( )

ij i jr =O

Milano_2012 6

HRR Angular Functions

xx

yy


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Seite 4

Milano_2012 7

CTOD

HRR displacement field

111

0

0 0

( )

n

nn

i i

n

Ju r u

I

++ =

Crack Tip Opening Displacement, t , Shih [1981]

t t t t2 ( , ) , ( , ) ( , )y x yu r r u r u r = =

t

0

n

Jd

=

( )

1

0 nn nd D=

Milano_2012 8

Path Dependence ofJ

FE simulation

C(T) specimen

plane strain

stationary crack

incremental theoryof plasticity

ASTM E 1820: reference value

far field value

el plJ J J= +

e pJ J J= +

2e IKJ

E=

( )IK a Y a W =

( )

pp U

JB W a

=


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Stresses at Crack Tip

incremental theory of plasticity

large strain analysis

1 1

1 1

0 0 t

n nij r r

J

+ +

HRR

Milano_2012 10

R-Curves in FM

Different from quasi-brittle fracture, ductile crack extension is

deformation controlled: R-curves J(a), (a)R curve a plot of crack-extension resistance as a function of

stable crack extension (ASTM E 1820)

p2( 1) ( ) ( )e p p

( ) ( ) ( 1) ( 1)

( 1) ( 1)

( ) 1i i iI

i i i i

i i

U aKJ a J J J

E b B b

= + = + +

JR-curve: J(a)

recursion formula

measure F, VL, a


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ASTM E 1823

Crack extension, a an increase in crack size.

Crack-extension resistance, KR, GR ofJR a measure of the resistance

of a material to crack extension expressed in terms of the stress-intensity factor, K; crack-extension force, G; or values ofJ derived

using the J-integral concept.

Crack-tip opening displacement (CTOD), the crack displacementresulting from the total deformation (elastic plus plastic) at variously

defined locations near the original (prior to force application) crack tip.

J-R curve a plot of resistance to stable crack extension, ap.

R curve a plot of crack-extension resistance as a function of stable

crack extension, ap orae.Stable crack extension a displacement-controlled crack extension

beyond the stretch-zone width. The extension stops when the applied

displacement is held constant.

Standard Terminology Relating to Fatigue and Fracture Testing

Milano_2012 12

ASTM E 1820

Determination of fracture toughness of metallic materials using the parameters K,

J, and CTOD ().

Assuming the existence of a preexisting, sharp, fatigue crack, the material fracture

toughness values identified by this test method characterize its resistance to

(1) Fracture of a stationary crack

(2) Fracture after some stable tearing

(3) Stable tearing onset

(4) Sustained stable tearing

This test method is particularly useful when the material response cannot be

anticipated before the test.

Serve as a basis for material comparison, selection and quality assurance;

rank materials within a similar yield strength range;

Serve as a basis for structural flaw tolerance assessment; awareness of

differences that may exist between laboratory test and field conditions is

required.

Standard Test Method for Measurement of Fracture Toughness


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ASTM E 1820 (ctd.)

Fracture after some stable tearing is sensitive to material

inhomogeneity and to constraint variations that may be induced

to planar geometry, thickness differences, mode of loading, and

structural details;

J-R curve from bend-type specimens, SE(B), C(T), DC(T), has

been observed to be conservative with respect to results from

tensile loading configurations;

The values ofc, u, Jc, and Ju, may be affected by specimen

dimensions

Cautionary statements

Milano_2012 14

CTOD R-Curve

Schwalbe [1995]: 5(a)

ASTM E 2472

particularly for thin panels


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ASTM E 2472

Determination of the resistance against stable crack extension

of metallic materials under Mode I loading in terms of critical

crack-tip opening angle (CTOA) and/orcrack opening

displacement (COD) as 5 resistance curve.

Materials are not limited by strength, thickness or toughness, as

long as and , ensuring low constraint conditions in M(T) and

C(T) specimens.

Standard Test Method for Determination of Resistance toStable Crack Extension under Low-Constraint Conditions

Milano_2012 16

Limitations

For extending crack

J becomes significantly path dependent

J looses its property of being an energy release rate

J is a cumulated quantity of global dissipation

JR curves are geometry dependent

bending

tension

BAM Berlin


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( ) = + +

e pexsep

WU U U

B a B aEnergy balance

=

pp U

RB a

global plastic dissipation rate

=sep

c

U

B alocal separation rate

Dissipation Rate

= = + = +

psep pdiss

c

UU UR R

B a B a B a

Dissipation rate

Turner (1990)

p cRcommonly: geometry dependence ofJR curves

( )

( )

p

pp

M(T), DE(T)

( )

C(T), SE(B)

dJW a

daR a

W a dJJ

da

= +

Milano_2012 18

R(a)

stationaryvalue

accumulatedplastic work


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W. Brocks

Christian Albrecht UniversityMaterial Mechanics

Inelastic Deformation and Damage

Part II: Damage Mechanics

Milano_2012 2

Outline

Deformation, Damage and Fracture

Crack Tip and Process Zone

Damage Models

Micromechanisms of Ductile Fracture Micromechanical Models

Porous Metal Plasticity (GTN Model)


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l'endommagement, comme le diable, invisible mais redoutable

Surface or volume-like discontinuities on the materials micro-level (microcracks, microvoids)

Damage evolution is irreversible (dissipation!)

Damage causes degradation (reduction of performance)

Examples for processes involving damage phenomena:ductile damage in metals, creep damage, fiber cracking orfiber-matrix delamination in reinforced composites, corrosion,fatigue

Damage - Definition

Milano_2012 4

Observable Effects

Physical Appearance of Damage

volume defects (microvoids, microcavities)

surface defects (microcracks)

Mascroscopic Effects of Damage

decreases elasticity modulus decreases yield stress

decreases hardness

increases creep strain rate

decreases sound-propagation velocity

decreases density

increases electrical resistance


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Damage Models

Damage models describe evolution of degradation phenomenaon the microscale from initial (undamaged or predamaged) stateup to creation of a crack on the mesoscale (material element)

Damage is described by means ofinternal variables in theframework ofcontinuum mechanics.

Phenomenological models

Change of macroscopically observable properties areinterpreted by means of the internal variable(s);

Concept of effective stress: Kachanov [1958, 1986], Lemaitre& Chaboche [1992], Lemaitre [1992].

Micromechanical models

The mechanical behaviour of a representative volume element(RVE) with defect(s) is studied;

Constitutive equations are formulated on a mesoscale byhomogenisation of local stresses and strains in the RVE.

Milano_2012 6

Continuum Damage Mechanics (CDM)

AA

J . Lemaitre, R. DesmoratEngineering Damage Mechanics

Springer, 2005

D. KrajcinovicDamage MechanicsElsevier, 1996

Kachanov [1958]

Hult [1972]

Lemaitre [1971]

Lemaitre & Chaboche [1976]


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Effective Area

voidsV V

RVE

VD f

V

= =

D(n)

=A

cracks

ARVE

Volume density of microvoids

Surface density of microcracks or intersections

of microvoids with plane of normal n

DA A A= "Effective" area

D =A

D

AIsotropic Damage Scalar Damage Variable

ifD(n) does not depend on n

( )1A D A =

Milano_2012 8

Anisotropic Damage

Tensorial Damage Variables

rank 2 tensorD

rank 4 tensor D

with symmetries , most general case

metric tensor (mn) defines the reference configuration

( )A A= n 1 n D

( ) ( ) ( )A A= mn mn I D

D

ijkl = Dijlk = Djikl = Dklij

AA

nmn

munchanged


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Effective Stress (I)

( )A A A = = S mn S mn S mn I D

( ) ( )11

orij ik jl ijkl kl

D

= = S S I D

Anisotropic Damage

1 D=

or1 1

ij

ijD D

= =

SS

Isotropic Damage

1D

3D

rank 4 tensor D

A A = m S n m S n projection of stress vector on m

Milano_2012 10

Effective Stress (II)

Rank 2 tensorDrequires additional conditions:

Symmetry of the effective stress:

is not symmetric!

Compatibility with the thermodynamics framework: existence ofstrain potentials and principle of strain equivalence,

Symmetrisation (not derived from a potential)

Different effect of the damage on the hydrostatic and deviatoricstress

( )1

= S S 1 D

( ) ( )1 1

12

= +

S S 1 1 S D D

( ) ( )1 2h

h

with 11 D

= + =

S S 1

H H H D


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11

Concept of effective stress (III)

1 1 1

11 2 h

1 11

1 2

4 2 1

9 1 9 1 3

2 1

31 31

D D D

D D

= + +

= +

1

1

2

2

2

2

10 0

10 0

10 0 ; 0 0

10 0

10 0

1

i j i j

DD

DD

D

D

= =

e e e eD H

Example: Rank 2 Tensor

Transversal isot ropic damage (2=3):Uniaxial loading 1,

effective von Mises stress

Milano_2012 12

Thermodynamics of Damage

1. Definition ofstate variables, the actual value of each definingthe present state of the corresponding mechanism involved

2. Definition of a state potential from which derive the state lawssuch as thermo-elasticity and the definition of the variablesassociated with the internal state variables

3. Definition of a dissipation potential from which derive the lawsof evolution of the state variables associated with the dissipativemechanism

Check 2nd Principle of Thermodynamics !


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Variables

-YDDamage anisotropic

-YDDamage isotropicXAKinematic hardening

RpIsotropic hardening

-SEpPlasticity

sTemperature/Entropy

SEThermoelasticity

internalobservable

conjugate

variable

State variableMechanism

p,p R

Milano_2012 14

State Potential

( )e e p, or , , ,D p = + +E A D

* = supE

1

SE

= supE

e

1

SEe

e

+ 1

S E

p

p

Gibbs specific free enthalpy taken as state potential

**p e pe

*

s

= = + = +

=

E E E ES S

State laws of thermoelasticitycan be deducted

Helmholtz specific free energy


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R = *

p

X = *

A

Y= *

Dor Y =

*

D

Dissipation Potential

Definition ofconjugate variables

( )pgrad

0R p

+ + q

S E X A Y

D

2nd Principle of Thermodynamics (Clausius-Duhem inequality)

( ), , , or ,R YS X Y

Evolution equations for internal variables (kinetic laws) are derivedfrom a dissipation potential , which is a convex function of theconjugate variables

Milano_2012 16

( )

( ) ( )

p

or

pR

DY Y

= =

=

=

= = = =

ES S

AX

Y Y

D

Normality Rule

Normality rule of generalised standard materials

Nice and consistent theoretical framework but

wherefrom to get the dissipation potential ?

flow rule


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Principle of Strain Equivalence

Strain constitutive equations of a damage material are derivedfrom the same formalism as for a non-damaged material except

that the stress is replaced by the effective stress

( )

( ) ( )

2*

e

1

2 1 2 1

ij ij kk

E D E D

+=

Example: State potential for linear isotropic elasticity

*

e 1eij ij kk ij

ij E E

+= =

Elastic strain

( ) ( )2*

e h2 1 3 1 22 3

YD E

= = + +

Energy density release rate Y1

h 3

23

kk

ij ij

=

=

Milano_2012 18

Local and Micromechanical Approaches

Cleavage (brittlefracture)

Microcrack formation and coalescence

Stress controlled

Ritchie, Knott & Rice [1973]: RKR model, Beremin [1983]

Ductile tearing

Nucleation, growth and coalescence of microvoids at inclusions orsecond-phase particles

Strain controlled, void growth dependent on hydrostatic stress

Rice & Tracey [1973], Gurson [1977], Beremin [1983],Tvergaard&Needleman [1982, 1984, ...],Thomason [1985, 1990], Rousselier[1987]

Creep damage

Nucleation, growth and coalescence of micropores at grain boundaries

Stress or strain controlled

Hutchinson [1983], Rodin & Parks [1988], Sester& Riedel [1995]


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Cleavage

Mechanisms of microcrack initiation

Broberg [1999]

Coalescence of microcracks

Milano_2012 20

Failure mechanisms: Nucleation, growth and coalescence of voids

Voids nucleate at secondary phase particles due to particle/matrixdebonding and/or particle fracture

Localisation of plastic deformation is prior to failure

fracture surface of Al 2024

Ductile Fracture


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Void Nucleation

Void nucleation at coarse particles in Al 2024 T 351

Milano_2012 22

Ductile Crack Extension (I)

Ductile crack extensionin an Al alloy

Schematic view of processzone with unit cells

Broberg [1999]


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Ductile Crack Extension (I)

Milano_2012 24

Models of Void Growth (I)

McClintock [1968]

coalescence 2x xr = A

power law n =

void growth

( ) ( )( ) ( ) ( )

00

3 11 3 3sinh

2 1 2 4ln

xx yy xx yyzx

x x

nd

d nr

+ = +

A

( ) ( )

( )( ) ( )( )

0 0

f

1 ln

sinh 1 2 3

x x

xx yy

n r

n

=

+

A

fracture strain

( )

( )0 0[ln ]

1ln

x x

zx zx

x x

d rd

r = =

A

A

damage


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Models of Void Growth (II)

Rice & Tracey [1969]

h20.283exp3

rD

r

= =

h T

= triaxiality

void-volume fraction for tensile test

Milano_2012 26

Particle cracking

Particle-matrix debonding

in Al-TiAl MMC

Void Nucleation


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Representative Volume Element (Unit Cell)

In-situ observation and FEsimulation of voidnucleation by particledecohesion and fracture

Milano_2012 28

Unit Cell Simulations

Debonding of matrix at a particle

Cracking of particle


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Evolution of void volume fraction can be computed from

simple geometrical RVEs Critical volume fractions can be obtained from plastic collapseof the cell (function of triaxiality!)

Procedure can be applied independently of the aggregate(void, particle, evolving object)

0.0 0.1 0.2 0.30.00

0.02

0.06

0.10

0.00

0.05

0.10

0.15

0.20

0.30

necking

Volume fraction

f -2 E1

fc

Ev

T=2

Representative Volume Element (Unit Cell)

Milano_2012 30

Mesoscopic Response

FE simulation of void growth:

Mesoscopic stress-strain curves


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31

Uncoupled models: Example

Rice & Tracey [1969]

p 30.283 exp

2

drd T

r

=

round tensile bar:

Coupled / Uncoupled Models

0,0 0,2 0,4 0,6 0,8 1,0

0

5

10

15

20

25

30

35

(2)

F

[kN]

u2/2 [mm]

von Misescoupled model

(1)

GTN

notched bar

Milano_2012 32

32

Porous Metal Plasticity

Additional scalar internal variable in the yield potential, which is afunction ofporosity f

Porosity equals the void volume fraction in an RVE:

voids

RVE

Vf

V

=

Yield potential formulation is obtained from homogenisation

( ) ( )

( )

RVE RVERVE RVE

, ,

RVE RVE

1 1

1 1

2

jij ij ij

V V

ij ij i j j i

V V

dV n dS V V

dV u u dV V V

= =

= = +

mesoscopicstresses andstrains

( )p, 0ij = ( )p

, , 0ij

f =

Evolution equation of void growth is derived from plasticincompressibility of matrix

( ) p1 kkf f = p

0kk

volume dilatationdue to void growth


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33

Gurson and Rousselier Model

Gurson [1977], Tvergaard & Needleman [1984]

2* *2h

1 2 3

p p( ) (

32 cosh 1 0

2 )q f

Rfq q

R

= + =

Rousselier [1987]

( ) ( )1

p p

h

1

exp( 1)

1 01 ( )

fDR Rf f

= + =

damage variable f*(f)

p p p2p 3 ij ij

E = =

Milano_2012 34

34

0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

Gurson

Rousselier

0R

h 0R

Comparison


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35

Extensions

Tvergaard & Needleman

damage function

( )c*

c c c

for

for

f ff

f

f

ff f f

=

+

p

growth nucl n p(1 ) kkf f f f A = + = +

nn

n

2

p

n

n

1exp22

fs

As

=

Chu & Needleman [1980]

void nucleation

Milano_2012 36

36

0

c

n

n

n

1

2

2

3 1

0

0.12

0.05

0.05

0.15

1.51.0

2.25

f

f

f

s

qq

q q

=

=

=

=

=

==

= =

Effect o f Triaxiality


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Milano_2012 37

Tensile Test: GTN model

Simulation of deformation and damage in a round tensile bar

Milano_2012 38

SE(B): GTN model


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39

Simulation with the GTN model

0 50 100 1500

200

400

600

u3 [mm]

F [kN] Experiment

Simulation

1

2

3

axial forc

punch force

2

1

x-axis 3

Punch Test

Milano_2012 40

Punch Test


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Milano_2012 41

Summary (I)

Ductile crack extension and fracture can be modelled on various

length scales:

(1) Micromechanics: void nucleation, growth and coalescence

(2) Continuum mechanics: constitutive equations with damage

(3) Cohesive surfaces: traction-separation law

(4) Elastic-plastic FM: R-curves for Jor CTOD

The models require determination of respective parameters:

(1) Microstructural characteristics: volume fraction, shape, distance of

particles, ...

(2) Initiation: f0, fn, n, sn, coalescence: fc, final fracture: ff, ....

(3) Shape of TSL, cohesive strength c , separation energy c

(4) J(a) or (a)

Summary (II)

The models have specific favourable and preferential applications:

(1) Effects of nucleation mechanism, stress triaxiality, void /

particle shape, void / particle spacing, ...

(2) Constraint effects, inhomogeneous materials, damage

evolution, ...

(3) Large crack growth, residual strength of structures

(4) Standard FM assessment of engineering structures

Acknowledgement:

FE simulations by Dr. Dirk Steglich,

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