introduction to fracture and damage mechanics · 2012-04-04 · introduction to fracture and damage...
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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)
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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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Milano_2012 15
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)
Seite 2
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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
Seite 4
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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
Seite 5
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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)
Seite 2
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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
Seite 4
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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
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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
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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
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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 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
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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 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
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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
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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
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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 !
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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
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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
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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
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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 extension in an Al alloy
Schematic view of process zone with “unit cells”Broberg [1999]
Seite 12
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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
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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
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• 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
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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
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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
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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
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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