Introduction to Fracture and Damage Mechanics

Wolfgang Brocks

Five Lectures at

Politecnico di Milano

Milano, March 2012

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)

Seite 1

Concepts of Fracture Mechanics

Part I: Linear Elastic Fracture Mechanics

W. Brocks

Christian Albrecht University Material Mechanics

Milano_2012 2

Overview

I. Linear elastic fracture mechanics (LEFM)

II. Small Scale YieldingIII. Elastic-plastic fracture mechanics (EPFM)

Seite 2

Milano_2012 3

LEFM

Stress Field at a crack tipStress intensity approach (Irwin)Energy approach (Griffith)J-integralFracture criteria – fracture toughnessTerminology

Milano_2012 4

Stress Field at a Crack Tip

boundary conditions( 0, 0) ( , ) 0

( 0, 0) ( , ) 0yy

xy r

x y r

x y rθθ

θ

σ σ θ π

σ σ θ π

≤ = = = =

≤ = = = =

Hooke’s law of linear elasticity

21 2ij ij kk ijG νσ ε ε δ

ν⎛ ⎞= +⎜ ⎟−⎝ ⎠ Inglis [1913], Westergaard

[1939], Sneddon [1946], ...Williams’ series [1957]

0ΔΔΦ =Airy’s stress function

( )( )

2 cos cos 2

sin sin 2

r A B

C D

λΦ λθ λ θ

λθ λ θ

+ ⎡= + + +⎣⎤+ + + ⎦

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 ( )

rrA C A

r rA r C r r

θ θ θ θ

θ θ θ θ

σ θ− −= − + − + + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

+ + + + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ O

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Fracture Modes

Milano_2012 6

LEFM: Stress Intensity Approach

Irwin [1957]: mode I

1 1II( , ) ( )

2j j ji iiKr f

rTσ θ θ δ

πδ= +

T = non-singular T-stressRice [1974]: effect on plastic zone

General asymptotic solution

I II IIII II III

1( , ) ( ) ( ) ( )2ij ij ij ijr K f K f K f

rσ θ θ θ θ

π⎡ ⎤= + +⎣ ⎦stresses

displacements I II IIII II III

1( , ) ( ) ( ) ( )2 2i i i i

ru r K g K g K gG

θ θ θ θπ⎡ ⎤= + +⎣ ⎦

I (geometry)K a Yσ π∞=

KI = stress intensity factor

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Angular Functions in LEFM

Milano_2012 8

A.A. Griffith [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 2e0 1 1 2 1

16BrU

Gπ σ κ λ λ∞ ⎡ ⎤= − + + −⎣ ⎦

elliptical hole, axes a,b

( ) ( ) ( )

( )( ) ( ) ( )

22 2e

rel

22 2 2 2 2

1 132

2 1 1

BU a bG

a b a b

π σ κ λ

λ λ

∞ ⎡= + − + +⎣

⎤+ − − + + + ⎦

( )2 2

erel 1

8a BU

Gπ σ κ∞= +crack

insert hole: fixed grips> energy release

e e e0 relU U U= −

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Fracture of “Brittle” Materials

( ) ( )∂− ≥

∂erel sep 0

2U U

B aCrack extends if

⎛ ⎞ ∂∂= − =⎜ ⎟∂ ∂⎝ ⎠

GL

eee rel

(2 ) (2 )v

UUB a B a

Energy release rate

sepc2

(2 )U

B aγ Γ

∂= =

∂Separation energy (SE)(energy per area)

Γ γ= =G ec( ) 2afracture criterion

fracture stressΓ

σπ′

=ccE

a

Irwin [1957]:2

e IKE

=′

G

Milano_2012 10

Path-Independent Integrals

is some (scalar, vector, tensor) field quantity being steadily differentiable in domain B and divergence free

, : 0 iniixϕϕ ∂

= =∂

B

( )ixϕ

, 0i idv n daϕ ϕ∂

= =∫ ∫B B

Gauß’ theorem

singularity S in B : 0 = − SB B B

0

(.) (.) (.) (.) (.) 0− +∂ ∂ ∂∂ ∂

= + + + =∫ ∫ ∫ ∫ ∫SB B BB B

(.) (.) and (.) (.)+ −∂ ∂ ∂ ∂

= = −∫ ∫ ∫ ∫B B B B

i in da n daϕ ϕ∂ ∂

=∫ ∫SB B

path independence

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Energy Momentum Tensor

Eshelby [1965]: energy momentum tensor

, ,,

with 0ij ij k i ij jk j

wP w u Pu

δ ∂= − =

∂

( )i j

wu x

energy densitydisplacement field

Material forces acting on singularities (defects) in the continuum, e.g. dislocations, inclusions, ...

i ij jF P n da∂

= ∫B

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

Γ

σ⎡ ⎤= −⎣ ⎦∫0

t

ij ijw dτ

σ ε τ=

= ∫

Milano_2012 12

J-Integral

The J-integral of Cherepanov [1967] and Rice [1986]is the 1st component of the J-vector

2 ,1ij j iJ wdx n u dsΓ

σ⎡ ⎤= −⎣ ⎦∫

Conditions:

time independent processesno volume forceshomogeneous materialplane stress and strain fields, no dependence on x3

straight and stress free crack faces parallel to x1

• Equilibrium

• Small (linear) strains

• Hyperelastic material

, 0ij iσ =

( )1, ,2ij i j j iu uε = +

ijij

wσε∂

=∂

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J as Energy Release Rate

L

ee

crack v

UJA

⎛ ⎞∂= = −⎜ ⎟∂⎝ ⎠

G

crack

C(T), SE(B)

M(T), DE(T)2

B aA

B a

∂⎧⎪∂ ⎪ ∂= ⎨ ∂∂ ⎪⎪ ∂⎩

2 2 2e e e e I II III

I II III 2K K KJE 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 rate eI Ic=G G

2e Ic

Ic cKE

Γ= =′

G

J-integral IcJ J=2Ic

Ic IcKJE

= =′

G

2

plane stress

plane strain1

EE E

ν

⎧⎪′ = ⎨⎪ −⎩

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ASTM E 1823

Crack extension, Δa – 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 extension and 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 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 plane strain, and (2) the crack-tip plastic zone is small compared 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 above would be expected;

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

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

2

Ic

YS

2.5 KW aσ⎛ ⎞

− ≥ ⎜ ⎟⎝ ⎠

Specimen size σYS 0.2 % offset yield strength

Milano_2012 18

ASTM E 399 ctd.

Specimen configurationsSE(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 precrackedplate 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 for S/W = 4 and for S/W = 3;

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Specimen Configurations

Bend Type

C(T) SE(B)

W widthB thicknessa crack lengthb = W-a ligament width

Seite 1

Inelastic Deformation and Damage

Part I: Plasticity

W. Brocks

Christian Albrecht University

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” for small 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 σ-ε-curve

permanent strain

e p pE

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

R0 yield strength (σ0, σY)

RF(εp) uniaxial yield curve

loading / unloading p

p

0 , 0 loading

0 , 0 unloading

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3D Generalisation

Additive decomposition of strain ratese p

ij ij ij

Total plastic strainsp p

0

t

ij ij d

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 1

2 1ij ij kk ij

G

Hooke’s 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 ijJ 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 1323

“J2-theory”

p

ij ij Flow rule plastic multiplier

from uniaxial test

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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], Reuß [1930])

Equivalence of dissipation rates

p p p2p3

0

t

ij ij d

equivalent plastic strain

Hooke’s law of elasticity + associated flow rulee p

ij 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 of Ramberg & 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-Reuß 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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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, whose micromechanical causes are very similar to deformation processes but whose macroscopic implications are much different

Fracture

Laminar discontinuities on the macro scale leading to global failure (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 of internal variables in the

framework of continuum 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 of Gurson, Tvergaard & Needleman (GTN model)

including scalar damage 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, εn, sn

Evolution equation of damage

Volume dilatation caused by void growth

p1 kkf f

p

kk

Seite 1

Concepts of Fracture Mechanics

Part II: Small Scale Yielding

W. Brocks

Christian Albrecht University Material Mechanics

Milano_2012 2

Overview

I. Linear elastic fracture mechanics (LEFM)II. Small Scale Yielding (SSY)III. Elastic-plastic fracture mechanics (EPFM)

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SSY

Plastic zone at the crack tipEffective crack length (Irwin)Effective SIFDugdale modelCrack 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 yyK

r

σ σ

σ σ σ σπ

= =

= = = =

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

2

Ip

0

12

KrRπ

⎛ ⎞⇒ = ⎜ ⎟

⎝ ⎠

0θ =Yielding in the ligament

perfectly plastic material F p 0( )R Rε =

rp = “radius” of plastic zone

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Yielding in the Ligament (II)

(b) plane strain

( )3

I3 1 2

022

2

zz

zz yyK

r

ε ενσ σ ν σ σ νσπ

= =

= = + = =

Yield condition (both: von Mises and Tresca)

( ) 0 p1 2 , 0yy R r rν σ− = ≤ ≤

( ) 22

Ip

0

1 22

KrR

νπ

− ⎛ ⎞⇒ = ⎜ ⎟

⎝ ⎠smaller plastic zone due to triaxiality of stress state

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

Iyy p 0 p

0 0

2( ) 22 r

r rKr dr dr r R rσ

ππ= = =∫ ∫

equilibrium ?

Milano_2012 6

Effective Crack Length

effIeff I eff eff( ) aK K a a Y

Wσ π∞

⎛ ⎞= = ⎜ ⎟⎝ ⎠

effective SIF

pr aeff pa a r= +

total “diameter” of plastic zone

( )

2

I2p p

0

1 plane stress2 ,

2 1 2 plane strainKd rR

β βπ ν

⎧⎛ ⎞ ⎪= = = ⎨⎜ ⎟ −⎝ ⎠ ⎪⎩

2Ieff

ssy ssyKJE

= =′

Geffective J

no singularity at the crack tip

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Example SSY (I)

IaK a YW

σ π∞⎛ ⎞= ⎜ ⎟⎝ ⎠

FBW

σ∞ =

ASTM E 399

C(T)

aYW⎛ ⎞⎜ ⎟⎝ ⎠

Milano_2012 8

Example SSY (II)

2p

02d a aYW W R W

σβ ∞⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

plane stress

plane strain

effIeff eff

aK a YW

σ π∞⎛ ⎞= ⎜ ⎟⎝ ⎠

plane stress

plane strain

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CTOD (Irwin)

I2

1 plane stress( , ) 4

1 plane strain2yK ru rE

πνπ

⎧= ⎨ −⎩

t p2 ( , )yu rδ π=Wells [1961]

( ) ( )21 1 2 0.36ν ν− − ≈plane strain plane stresst t0,36δ δ≈

( )( )2

Irwin It 2

0

1 plane stress41 1 2 plane strain

KER

δν νπ

⎧⎪= ⎨ − −⎪⎩

Criterion for crack initiation: t cδ δ=

Milano_2012 10

Shape of Plastic Zone

Yield condition (von Mises)

( ) ( ) ( ) ( )2 222 2 2 2111 22 22 33 33 11 12 23 132 3σ σ σ σ σ σ σ σ σ σ⎡ ⎤= − + − + − + + +⎣ ⎦

p0r

Rσ =

( )3311 22

0 plane stressplane strain

σν σ σ⎧

= ⎨ +⎩

( ) ( )

2 232I

p 2230 2

1 sin cos plane stress1( )2 sin 1 2 1 cos plane strain

KdR

θ θθ

π θ ν θ

⎧ + +⎛ ⎞ ⎪= ⎨⎜ ⎟+ − +⎝ ⎠ ⎪⎩

II( , ) ( )2ij ijKr f

rσ θ θ

π=LEFM:

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Plastic Zone

plane stressplane strain

3D: dog bone model

Hahn & Rosenfield [1965]

plane stressplane strain

Milano_2012 12

Dugdale Model

0 p( ,0) , 0yy r R r dσ = ≤ ≤

p0 0

1 cos sec 12 2

d c aR R

π σ π σ∞ ∞⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞

= − = −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

p2 2 2c a d= +

(1) (2)I I 0

2, arccos aK c K R cc

σ π ππ∞= = −Superposition

0

cos2

ac R

π σ∞⎛ ⎞⇒ = ⎜ ⎟

⎝ ⎠no singularity (1) (2)

I I 0K K+ =!

no restrictionwith respect toplastic zone size!

2 22Irwin

p p0 0

1.23 1.238

d c a dR R

π σ σ∞ ∞⎛ ⎞ ⎛ ⎞≈ ≈ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠0 1:Rσ∞ plane stress!

stripyield

model

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CTOD (Dugdale)

Dugdale 0t

0

8 lnsec2

R aE R

π σδπ

∞⎛ ⎞= ⎜ ⎟

⎝ ⎠

0

0

( , 0) 4 lnsec2y

Ru x a y aE R

π σπ

∞⎛ ⎞= = = ⎜ ⎟

⎝ ⎠Crack opening profile

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

2Irwint

0

4aERσδ ∞=

for Griffith crack, plane stress

no dependence on 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 σ(x) is unknown and cannot be measured

Cohesive model: traction-separation law σ(δ)

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Energy Release Rate

∂∂⎛ ⎞= − =⎜ ⎟∂ ∂⎝ ⎠

+= = ≈ = +

′ ′

G

G G G

L

rel

2 2p e pIeff I

ssy

v

UUB a B a

a rK KE a E

Γ Γ γ∂

= > =∂sep e

c c 2UB a

Separation energyCohesive model:c

c0

( )dδ

Γ σ δ δ= ∫

cΓ=GFracture criterion:

local criterion!

Milano_2012 16

ASTM E 399: Size Condition

Characterizes the resistance of a material to fracture in a neutral environment 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 plane strain, 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

2.5 KW aσ⎛ ⎞

− ≥ ⎜ ⎟⎝ ⎠

Specimen size σYS 0.2 % offset yield strength

( ) 22I

pYS

1 2 Kdν

π σ− ⎛ ⎞

= ⎜ ⎟⎝ ⎠

( )( )

21 20.02

2.5pd B

W a W aνπ

−⇒ ≤ ≈

− −

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ASTM E 561

• Determination of the resistance to fracture under Mode I loading …using M(T), C(T), or crack-line wedge-loaded C(W) specimen; continuous record of toughness development in terms of KR plotted against crack extension.

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

• Plot of crack extension resistance KR as a function of effective crack extension Δae.

• Measurement of physical crack size by direct observation and then calculating 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 loading compliance.

Standard Test Method for K-R Curve Determination

Milano_2012 18

FM Test Specimens (Tension Type)

M(T)

DE(T)

Seite 1

Concepts of Fracture Mechanics

Part III: Elastic-Plastic Fracture

W. Brocks

Christian Albrecht University Material Mechanics

Milano_2012 2

Overview

I. Linear elastic fracture mechanics (LEFM)II. Small Scale YieldingIII. Elastic-plastic fracture mechanics

(EPFM)

Seite 2

Milano_2012 3

EPFM

Deformation theory of plasticityJ as energy release rateHRR field, CTODDeformation vs incremental theory of plasticityR curvesEnergy 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

ijij

wσε∂

=∂

p

0 0

t t

ij ij ij ijw d dτ τ

σ ε τ σ ε τ= =

= ≈∫ ∫

“e” stands for “linear elastic”“p” stands for “nonlinear”

e p e pL LU U U F dv F dv= + = +∫ ∫

L

2 pe p I

crack v

K UJ J JE A

⎛ ⎞∂= + = − ⎜ ⎟′ ∂⎝ ⎠

in the following, the superscripts

Seite 3

Milano_2012 5

J as Stress Intensity Factor

Power law of Ramberg & Osgood [1945]

e p

0 0 0 0 0

nε ε ε σ σαε ε ε σ σ

⎛ ⎞= + = + ⎜ ⎟

⎝ ⎠uniaxial

1p

0 0 0

32

nij ijε σσαε σ σ

− ′⎛ ⎞= ⎜ ⎟

⎝ ⎠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

n nn

ij ij ij

K r

K r

σ

σ

σ σ θ

ε ε αε ε θσ

−+

−+

=

⎛ ⎞≈ = ⎜ ⎟

⎝ ⎠

11

00 0

n

n

JKIσ σ

ασ ε

+⎛ ⎞= ⎜ ⎟

⎝ ⎠ p 1( )ij ij rσ ε −=O

Milano_2012 6

HRR Angular Functions

xxσ

yyσ

Seite 4

Milano_2012 7

CTOD

HRR displacement field11

10

0 0

( )

nn

ni i

n

Ju r uI

αε θασ ε

++

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Crack Tip Opening Displacement, δt , Shih [1981]

t t t t2 ( , ) , ( , ) ( , )y x yu r r u r u rδ π π π= − =

t0

nJdδσ

=

( )1

0n

n nd Dαε=

Milano_2012 8

Path Dependence of J

FE simulationC(T) specimenplane strainstationary crackincremental theoryof plasticity

ASTM E 1820: reference value“far field” value

el plJ J J= +

e pJ J J= +2

e IKJE

=′

( )IK a Y a Wσ π∞=

( )p

p UJB W aη

=−

Seite 5

Milano_2012 9

Stresses at Crack Tip

• incremental theory of plasticity• large strain analysis

1 11 1

0 0 t

n nij r r

Jσσ σ δ

− −+ +⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

HRR

Milano_2012 10

R-Curves in FM

Different from quasi-brittle fracture, ductile crack extension is deformation controlled: R-curves J(Δa), δ(Δa)

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 iIi i i i

i i

U aKJ a J J JE b B b

η Δ Δγ−

− −− −

⎛ ⎞⎛ ⎞= + = + + −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′ ⎝ ⎠⎝ ⎠

JR-curve: J(Δa)

recursion formula

measure F, VL, Δa

Seite 6

Milano_2012 11

ASTM E 1823

Crack extension, Δa – an increase in crack size.Crack-extension resistance, KR, GR of JR – 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 of J derived using the J-integral concept.

Crack-tip opening displacement (CTOD), δ – the crack displacement resulting 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, Δap. R curve – a plot of crack-extension resistance as a function of stable

crack extension, Δap or Δae.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 (δ).

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 tearingThis 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

Seite 7

Milano_2012 13

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 of δc, δu, Jc, and Ju, may be affected by specimen dimensions

Cautionary statements

Milano_2012 14

CTOD R-Curve

Schwalbe [1995]: δ5(Δa)ASTM E 2472

particularly for thin panels

Seite 8

Milano_2012 15

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/or crack opening displacement (COD) as δ5 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 to Stable 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

Seite 9

Milano_2012 17

( )∂ ∂= + +

∂ ∂e pex

sepW U U UB a B a

Energy balance

∂=

∂

pp UR

B aglobal plastic dissipation rate

Γ∂

=∂sep

c

UB a

local separation rate

Dissipation Rate

Γ∂∂ ∂

= = + = +∂ ∂ ∂

psep pdiss

c

UU UR RB a B a B a

Dissipation rateTurner (1990)

Γ ⇒pcRcommonly: geometry dependence of JR curves

( )

( )

p

pp

M(T), DE(T)( )

C(T), SE(B)

dJW adaR a

W a dJ Jda

Δγ

η η

⎧−⎪⎪= ⎨ −⎪ +

⎪⎩

Milano_2012 18

R(Δa)

stationary value

accumulated plastic work

Seite 1

W. Brocks

Christian Albrecht University Material Mechanics

Inelastic Deformation and Damage

Part II: Damage Mechanics

Milano_2012 2

Outline

Deformation, Damage and FractureCrack Tip and Process ZoneDamage ModelsMicromechanisms of Ductile FractureMicromechanical ModelsPorous Metal Plasticity (GTN Model)

Seite 2

Milano_2012 3

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 or fiber-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

Seite 3

Milano_2012 5

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 of internal variables in the framework of continuum mechanics.

Phenomenological modelsChange 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 modelsThe 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.

Milano_2012 6

Continuum Damage Mechanics (CDM)

AΔAΔ

J. Lemaitre, R. DesmoratEngineering Damage MechanicsSpringer, 2005

D. KrajcinovicDamage MechanicsElsevier, 1996

Kachanov [1958]Hult [1972]Lemaitre [1971]Lemaitre & Chaboche [1976]

Seite 4

Milano_2012 7

Effective Area

voidsV V

RVE

VD fVΔΔ

= =

D(n) =ΔAcracks

ΔARVE

Volume density of microvoids

Surface density of microcracks or intersections of microvoids with plane of normal n

DA A A= −Δ Δ Δ"Effective" area

D =ΔAD

ΔAIsotropic Damage ⇒ Scalar Damage Variable

if D(n) does not depend on n

( )1A D AΔ Δ= −

Milano_2012 8

Anisotropic Damage

Tensorial Damage Variables

• rank 2 tensor D

• rank 4 tensor D

with symmetries , most general case

metric tensor (mn) defines the reference configuration

( )A A= − ⋅n 1 nΔ ΔD

( ) ( ) ( )A A= − ⋅⋅mn mnΔ ΔI D

Dijkl = Dijlk = Djikl = Dklij

AΔAΔ

nmn

munchanged

Seite 5

Milano_2012 9

Effective Stress (I)

( )A A AΔ Δ Δ⋅⋅ = ⋅⋅ = ⋅⋅ − ⋅⋅S mn S mn S mnI D

( ) ( ) 11 or ij ik jl ijkl klDσ δ δ σ−−= ⋅⋅ − = −S S I D

Anisotropic Damage

1 D=

−σσ

or1 1

ijijD D

σσ= =

− −SS

Isotropic Damage

1D

3D

rank 4 tensor D

A A⋅ ⋅ = ⋅ ⋅m S n m S nΔ Δ projection of stress vector on m

Milano_2012 10

Effective Stress (II)

Rank 2 tensor D requires additional conditions:

• Symmetry of the effective stress: is not symmetric!

• Compatibility with the thermodynamics framework: existence of strain 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

( ) ( )1 112

− −⎡ ⎤= ⋅ − + − ⋅⎣ ⎦S S 1 1 SD D

( ) ( ) 1 2h

h

with 11 D

−′′= ⋅ ⋅ + = −−

S S 1ση

H H H D

Seite 6

Milano_2012 1111

Concept of effective stress (III)

1 1 11

1 2 h

1 11

1 2

4 2 19 1 9 1 32 13 1 31

D D D

D D

σ σ σση

σ σσ σ

= + +− −

= + ≥− −

11

22

2

2

1 0 01

0 010 0 ; 0 0

10 0

10 01

i j i j

DD

DD

D

D

⎛ ⎞⎜ ⎟−⎜ ⎟

⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟= =⎜ ⎟ −⎜ ⎟⎜ ⎟⎝ ⎠ ⎜ ⎟

⎜ ⎟⎜ ⎟−⎝ ⎠

e e e eD H

Example: Rank 2 Tensor

Transversal isotropic damage (2=3):

Uniaxial loading σ1,

effective von Mises stress

Milano_2012 12

Thermodynamics of Damage

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

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

3. Definition of a dissipation potential from which derive the laws of evolution of the state variables associated with the dissipative mechanism

Check 2nd Principle of Thermodynamics !

Seite 7

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Variables

-YDDamage anisotropic-YDDamage isotropicXAKinematic hardeningRpIsotropic hardening-SEpPlasticitysθTemperature/EntropySEThermoelasticity

internalobservable

conjugate variable

State variableMechanism

p ,p Rε σ≡ ≡

Milano_2012 14

State Potential

( )e e p, or , , ,D p = + +E A θψ θ ψ ψ ψD

ψ * = supE

1ρ

S ⋅⋅E−ψ⎡

⎣⎢

⎤

⎦⎥

= supEe

1ρ

S ⋅⋅Ee −ψ e⎡

⎣⎢

⎤

⎦⎥ +

1ρ

S ⋅⋅Ep −ψ p −ψθ

Gibbs specific free enthalpy taken as state potential

**p e pe

*

s

ψψρ ρ

ψθ

∂∂= = + = +

∂ ∂

∂=∂

E E E ES S

State laws of thermoelasticity can be deducted

Helmholtz specific free energy

Seite 8

Milano_2012 15

R = −ρ∂ψ *

∂p

X = −ρ∂ψ *

∂A

−Y = −ρ∂ψ *

∂Dor −Y = −ρ

∂ψ *

∂D

Dissipation Potential

Definition of conjugate variables

( )p grad 0R p ⋅⋅⋅ − + ⋅⋅ + ⋅⋅ − ≥

qS E X A Y θθ

D

2nd Principle of Thermodynamics (Clausius-Duhem inequality)

( ), , , or ,R YS X YΦ θ

Evolution equations for internal variables (kinetic laws) are derived from a dissipation potential Φ, which is a convex function of the conjugate variables

Milano_2012 16

( )

( ) ( )

p

or

pR

DY Y

∂ ∂= − =

∂ − ∂

∂= −

∂∂

= −∂∂ ∂ ∂ ∂

= − = = − =∂ − ∂ ∂ − ∂

ES S

AX

Y Y

Φ Φλ λ

Φλ

Φλ

Φ Φ Φ Φλ λ λ λD

Normality Rule

Normality rule of “generalised standard materials”

Nice and consistent theoretical framework – but wherefrom to get the dissipation potential Φ ?

flow rule

Seite 9

Milano_2012 17

Principle of Strain Equivalence

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

that the stress is replaced by the effective stress

( )( ) ( )

2*e

12 1 2 1

ij ij kk

E D E Dσ σν σνρψ

+= −

− −

Example: State potential for linear isotropic elasticity

*e 1e

ij ij kk ijij E E

ψ ν νε ρ σ σ δσ∂ +

= = −∂

Elastic strain

( ) ( )2*

e h2 1 3 1 22 3

YD Eψ σ σρ ν ν

σ⎡ ⎤∂ ⎛ ⎞= = + + −⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎢ ⎥⎣ ⎦

Energy density release rate Y1

h 3

23

kk

ij ij

σ σ

σ σ σ

=

′ ′=

Milano_2012 18

Local and Micromechanical Approaches

Cleavage (“brittle” fracture)Microcrack formation and coalescenceStress controlledRitchie, Knott & Rice [1973]: RKR model, Beremin [1983]

Ductile tearingNucleation, growth and coalescence of microvoids at inclusions or second-phase particlesStrain controlled, void growth dependent on hydrostatic stressRice & Tracey [1973], Gurson [1977], Beremin [1983], Tvergaard & Needleman [1982, 1984, ...], Thomason [1985, 1990], Rousselier[1987]

Creep damageNucleation, growth and coalescence of micropores at grain boundariesStress or strain controlledHutchinson [1983], Rodin & Parks [1988], Sester & Riedel [1995]

Seite 10

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Cleavage

Mechanisms of microcrack initiationBroberg [1999]

Coalescence of microcracks

Milano_2012 20

• Failure mechanisms: Nucleation, growth and coalescence of voids• Voids nucleate at secondary phase particles due to particle/matrix

debonding and/or particle fracture• Localisation of plastic deformation is prior to failure

fracture surface of Al 2024

Ductile Fracture

Seite 11

Milano_2012 21

Void Nucleation

Void nucleation at coarse particles in Al 2024 T 351

Milano_2012 22

Ductile Crack Extension (I)

Ductile crack extension in an Al alloy

Schematic view of process zone with “unit cells”Broberg [1999]

Seite 12

Milano_2012 23

Ductile Crack Extension (I)

Milano_2012 24

Models of Void Growth (I)

McClintock [1968]

coalescence 2 x xr =

power law nσ ε=

void growth

( ) ( )( ) ( ) ( )

00

3 11 3 3sinh2 1 2 4ln

xx yy xx yyzx

x x

ndd nr

σ σ σ σηε σ σ

∞ ∞ ∞ ∞

∞

⎡ ⎤⎛ ⎞+ −−⎢ ⎥⎜ ⎟= +

⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦

( ) ( )( )( ) ( )( )

0 0

f

1 ln

sinh 1 2 3

x x

xx yy

n r

nε

σ σ σ∞ ∞

−=

⎡ ⎤− +⎣ ⎦fracture strain

( )( )0 0

[ln ]1

lnx x

zx zxx x

d rd

rη η= = ≤∫ ∫

damage

Seite 13

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Models of Void Growth (II)

Rice & Tracey [1969]

h20.283exp3

rDr

σε σ

⎛ ⎞= = ⎜ ⎟⎝ ⎠

h Tσσ

= triaxiality void-volume fraction for tensile test

Milano_2012 26

Particle crackingParticle-matrix debonding

in Al-TiAl MMC

Void Nucleation

Seite 14

Milano_2012 27

Representative Volume Element (Unit Cell)

In-situ observation and FE simulation of void nucleation by particle decohesion and fracture

Milano_2012 28

Unit Cell Simulations

Debonding of matrix at a particle

Cracking of particle

Seite 15

Milano_2012 29

• Evolution of void volume fraction can be computed from simple geometrical RVEs

• Critical volume fractions can be obtained from plastic collapse of 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

neckingVolume 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

Seite 16

Milano_2012 3131

Uncoupled models: Example

Rice & Tracey [1969]

p 30.283 exp2

dr d Tr

ε ⎛ ⎞= ⎜ ⎟⎝ ⎠

round tensile bar:

Coupled / Uncoupled Models

0,0 0,2 0,4 0,6 0,8 1,00

5

10

15

20

25

30

35

(2)

F [k

N]

u2/2 [mm]

von Mises coupled model

(1)GTN

notched bar

Milano_2012 3232

Porous Metal Plasticity

Additional scalar internal variable in the yield potential, which is a function of porosity f

Porosity equals the void volume fraction in an RVE:

voids

RVE

VfV

ΔΔ

=

Yield potential formulation is obtained from homogenisation

( ) ( )

( )RVE RVERVE RVE

, ,RVE RVE

1 1

1 12

jij ij ijV V

ij ij i j j iV V

dV n dSV V

dV u u dVV V

Δ Δ

Δ Δ

Σ σ σΔ Δ

Ε εΔ Δ

∂

= =∂

= = +

∫∫∫ ∫∫

∫∫∫ ∫∫∫

“mesoscopic”stresses andstrains

( )p, 0ijΦ σ ε = ⇒ ( )p, , 0ij fΦ Σ Ε =

Evolution equation of void growth is derived from plastic incompressibility of “matrix”

( ) p1 kkf f Ε= − p 0kkΕ ≠volume dilatation due to void growth

Seite 17

Milano_2012 3333

Gurson and Rousselier Model

Gurson [1977], Tvergaard & Needleman [1984]2

* *2h1 2 3

p p( ) (32 cosh 1 02 )

q fR

fq qRΣ ΣΦε ε

⎛ ⎞= + − − =⎜ ⎟⎜ ⎟

⎝ ⎠

Rousselier [1987]

( ) ( )1

p p

h

1

exp( 1)

1 01 ( )

fDR Rf f

σε ε σ

Σ ΣΦ⎛ ⎞

= + − =⎜ ⎟− −⎝ ⎠

damage variable f*(f)p p p2

p 3 ij ijEε Ε Ε′ ′= =

Milano_2012 3434

0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

Gurson

Rousselier

0RΣ

h 0RΣ

Comparison

Seite 18

Milano_2012 3535

Extensions

Tvergaard & Needleman

damage function

( )c*

c c c

forfor

f ff

ff

ff f fκ≤⎧

= ⎨ + − ≥⎩

pgrowth nucl n p(1 ) kkf f f f AΕ ε= + = − +

nn

n

2p

nn

1exp22

fs

As

εεπ

⎛ ⎞−⎛ ⎞⎜ ⎟= − ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Chu & Needleman [1980]

void nucleation

Milano_2012 3636

0

c

n

n

n

1

22

3 1

00.120.050.050.151.51.0

2.25

fff

sqq

q q

ε

=======

= =

Effect of Triaxiality

Seite 19

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Tensile Test: GTN model

Simulation of deformation and damage in a round tensile bar

Milano_2012 38

SE(B): GTN model

Seite 20

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Simulation with the GTN model

0 50 100 1500

200

400

600

u3 [mm]

F [kN] Experiment

Simulation

1

23

axial force

punch force

2

1

x-axis 3

Punch Test

Milano_2012 40

Punch Test

Seite 21

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 J or CTOD

The models require determination of respective parameters:(1) Microstructural characteristics: volume fraction, shape, distance of

particles, ...(2) Initiation: f0, fn, εn, sn, coalescence: fc, final fracture: ff, ....(3) Shape of TSL, cohesive strength σc , separation energy Γc

(4) J(Δa) or δ(Δa)

Milano_2012 42

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,

Helmholtzzentrum Geesthacht