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Graduate Institute of Ferrous Technology Pohang University of Science and Technology - 1 -
International Seminar on Metal PlasticityRome, Italy, June19, 2017
Organizers: Marco Rossi, Sam Coppieters
Frédéric Barlat
Graduate Institute of Ferrous TechnologyPohang University of Science and Technology, Republic of Korea
THEORETICAL MODELING OF PLASTICITY
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1. Introduction 2. Plasticity modeling3. Isotropic hardening4. Anisotropic hardening5. Final remarks
International Seminar on Metal Plasticity June 19, 2017
Outline
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1. Dislocation glide (slip)
Rauch et al., 2011
1. Introduction – Deformation mechanisms
Hull, 1983
2. Other mechanisms
Plastic behavior in metals
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Continuum scale plasticity
Multiphase material modeling (unit cells)
Crystal plasticity (slip, twinning and homogenization schemes)
1. Introduction – Approaches
Dislocation dynamics (dislocation network with interaction rules)
Atomistic (lattice with interatomic potentials) and ab-initio
This presentation is about continuum scale plasticity
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• External variables (elastic strain, plastic strain, strain rate, temperature)
• State variables assumed to represent deformation mechanisms (explicitly or implicitly)
• Thermodynamically conjugate variables through the expression of the free energy (stress, entropy)
Variables
2. Plasticity modeling – Approaches
Context of this presentation
• Rate and temperature-independent behavior (mostly)
• Isotropic hardening (applicable for monotonic loading)
• Anisotropic hardening (applicable for non-monotonic loading)
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Plasticity concepts
• Hardening model (stress-strain curve in uniaxial tension)
• Flow rule 𝑑𝛆 (𝑑𝜀%&'() = − ,-⁄ 𝑑𝜀/0(1 for isotropic material in uniaxial
tension)
• Yield condition (applied stress equal to yield stress in uniaxial tension)
2. Material modeling – Approaches
• Elastic–plastic decomposition
Total strain increment
𝑑𝛆%0% = 𝑑𝛆2/' + 𝑑𝛆4/' (𝑑𝛆4/' = 𝑑𝛆 in this presentation)
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Hardening rule
• 𝑥 represents (scalar or tensorial) state variables
• Same yield condition but with evolving state variables (microstructure)
Yield condition
𝛷 𝛔 = 0 for instance 𝜎; 𝜎<= − 𝜎> = 0
𝛷 𝛔, 𝛩, 𝑥 = 0 for instance 𝜎; 𝛔 − 𝜎A 𝛩, 𝑥 = 0
2. Material modeling – Plasticity concepts
• Effective stress and yield stress
• Yield condition defines yield surface
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• Argument based on crystal plasticity by Bishop and Hill (1951) general approach (not restricted to specific boundary conditions)
Flow rule• Associated or non-associated. For metal, associated flow rule is
consistent with plastic deformation mechanisms
𝑑𝛆 = 𝑑𝜆 CDC𝛔
• Work-equivalent effective strain 𝜎;𝑑𝜀 = 𝝈 ∶ 𝑑𝛆 defines a possible state variable (accumulated deformation or accumulated dislocations)
2. Material modeling – Plasticity concepts
Choice of the yield condition fully defines the material behavior
• Associated flow rule reduces to 𝑑𝛆 = 𝑑𝜀 CHIC𝛔
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• Principal stresses are invariants
• The effective stress is based on invariants such as von-Mises, Tresca, Hershey, etc.
𝜎; = HJKHL MN HLKHO MN HOKHJ M
-
, '⁄− 𝜎A 𝜀 = 0 (Hershey, 1954)
• Reduces to Tresca or von-Mises for specific values of 𝑎
• Non-quadratic and convex yield function
• Identification of 𝜎A 𝜀 , e.g., using least square approximation. Issue for extrapolation
𝜎; 𝛔 − 𝜎A 𝜀 = 0
3. Isotropic hardening – Isotropic material
Isotropic yield conditions
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• Reduces to von Mises for specific values of F, G, H and N
• Plane stress case
𝜎; 𝛔 − 𝜎A 𝜀 = 0
𝜎; = 𝐹 𝜎>> − 𝜎RR- + 𝐺 𝜎RR − 𝜎TT - + 𝐻 𝜎TT − 𝜎>>
- + 2𝑁𝜎T>-, -⁄
= 𝜎A 𝜀
• Issue for identification F, G, H and N: Based on flow stresses or 𝑟-value in uniaxial tension?
• Same yield condition as for isotropic case but stress components must be expressed in material symmetry axes (eg., RD, TD, ND)
𝑟 = YZY[
3. Isotropic hardening – Anisotropic material
Anisotropic yield conditions
Effective stress based on Hill (1948)
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0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
0 20 40 60 80
Nor
mal
ized
flow
str
ess
Angle from RD (deg.)
2090-T3 2090-T3 Hill 1948 (stress-based)
Experiments
stress-based
Nor
mal
ized
flow
str
ess
Angle from RD (deg.)
2090-T3 Hill 1948 (stress-based)
r-value-based
3. Isotropic hardening – Anisotropic material
Hill (1948) plane stress
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0.00
0.50
1.00
1.50
2.00
2.50
3.00
0 20 40 60 80
r-va
lue
Angle from RD (deg.)
2090-T32090-T3 Hill 1948 (stress-based)
ExperimentsStress-based
Nor
mal
ized
flow
str
ess
Angle from RD (deg.)
2090-T3 Hill 1948 (stress-based)
r-value-based
3. Isotropic hardening – Anisotropic material
Hill (1948) plane stress
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-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
TD n
orm
aliz
ed s
tres
s
RD normalized stress
2090-T3
stress-based
3. Isotropic hardening – Anisotropic material
r-value-based
TD n
orm
aliz
ed s
tres
s
RD normalized stress
Hill (1948) plane stress
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𝜎; = 𝐹 𝜎>> − 𝜎RR' + 𝐺 𝜎RR − 𝜎TT ' + 𝐻 𝜎TT − 𝜎>>
' + 2𝑁𝜎T>', '⁄
= 𝜎A 𝜀
• Note that Hill (1948) cannot be generalized directly, i.e.,
• This formulation does not work because it is component-based, not invariant based
• Cannot, in general, model uniaxial tension properly
Non-quadratic yield functions and isotropic hardening
• Use average behavior (still inaccurate) or non-associated flow rule with strain potential (not based on the physics of slip)
3. Isotropic hardening – Anisotropic material
Hill (1948) limitations
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• For instance, with two transformations 𝛔′ % = 𝐂 % ∶ 𝛔′ (𝑡 = 1,2 )
Non-quadratic yield functions
𝜎; =HJa J KHL
a J MN -HL
a L NHJa L M
N -HJa L NHL
a L M
-
, '⁄
= 𝜎A 𝜀
• Plane stress case: Yld2000-2d
• Total of eight anisotropy coefficients in 𝐂 , and 𝐂 -
• Linear stress transformation approach
• Reduces to isotropic Hershey (1954) when 𝐂 , and 𝐂 - are the identity
3. Isotropic hardening – Anisotropic material
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• General stress state Yld2004-18p
𝜎; =14c 𝜎d4
e , − 𝜎dfe - '
,,g
4,f
, '⁄
= 𝜎A 𝜀
• Total of 16 independent anisotropy coefficients in 𝐂 , and 𝐂 -
• Reduces to isotropic Hershey (1954) when 𝐂 , and 𝐂 - are the identity
• Advantage of linear transformations compared to other approaches for plastic anisotropy: Preserve convexity of the isotropic function
3. Isotropic hardening – Anisotropic material
Non-quadratic yield functions
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0.80
0.85
0.90
0.95
1.00
1.05
0 20 40 60 80
ExperimentsYld2000-2dYld2004-18p
Nor
mal
ized
flow
str
ess
Angle from RD (deg.)
2090-T3 2090-T3 Hill 1948 (stress-based)
3. Isotropic hardening – Anisotropic material
Non-quadratic yield functions
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0.00
0.50
1.00
1.50
2.00
0 20 40 60 80
ExperimentsYld2000-2dYld2004-18p
Nor
mal
ized
flow
str
ess
Angle from RD (deg.)
2090-T3 2090-T3 Hill 1948 (stress-based)
3. Isotropic hardening – Anisotropic material
Non-quadratic yield functions
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-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
TD n
orm
aliz
ed s
tres
s
RD normalized stress
2090-T3
Hill (r-based)
Hill (σ-based)
Yld2000-2d
Yld2004-18p
3. Isotropic hardening – Anisotropic material
Non-quadratic yield functions
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𝜎; =HJa KhHJa
MN HLa KhHLa
MN HOa KhHOa
M
i
, '⁄
− 𝜎A 𝜀 = 0
• Isotropic yield function (Cazacu et al., 2006)
• Compression to tension ratio HjH[= -M ,Kh MN- ,Nh M
-M ,Nh MN- ,Kh M
, '⁄
• Anisotropic yield function using linear transformation
3. Isotropic hardening – Anisotropic material
Strength differential (SD) effect
• Constant coefficient 𝐾
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-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50
Xtal FCCXtal BCCYld FCCYld BCC
σ1
/ �σ
σ 2 / �
σ
TwinningFCC: {111} <11-2>BCC: {112) <-1-11>
Isotropic material
3. Isotropic hardening – Anisotropic material
Cazacu et al., 2006
Twinning yield surfaces
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Good agreement between experiments, crystal plasticity and- Plane stress Yld2000-2d- General stress Yld2004-18p • stacked sheet specimen
5052-H35
1100-O
3. Isotropic hardening – Validation
Biaxial compression testing
Tozawa, 1978
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-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Yld2004-18pTBH polycrystal
Exp. locusExp. shear
Shear
σyy
/ �σ
σxx
/ �σAl-2.5%Mg
0.63 (yld)
0.62 (exp.)
0.61 (TBH)
Good agreement between: - Experiments - Crystal plasticity- Plane stress Yld2000-2d- General stress Yld2004-18p
3. Isotropic hardening – Validation
Biaxial compression testing
𝑟 = YZY[
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ISO 16842: 2014 Metallic materials −Sheet and strip −Biaxial tensile testing method using a cruciform test
piece
3. Isotropic hardening – Validation
Biaxial tension testing
Tubular specimens
Cruciform specimensKuwabara and Sugawara, 2013
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0 300 600 9000
300
600
900 εp
0 = 0.002 0.03 0.06 0.09 0.12 0.16
True
stre
ss σ
y /MPa
True stress σx /MPa
6.826.45
6.205.985.90
a=7.68
Yld2000-2d
(a)DP600 steel
3. Isotropic hardening – Validation
Hakoyama andKuwabara, 2015
Contour of plastic work for DP 600 steel
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• Kinematic hardening
• Differential hardening
• Combined kinematic hardening and distortional plasticity
• Distortional plasticity only
4. Anisotropic hardening
• Combined kinematic - isotropic hardening
Approaches
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Differential hardening
• Hill and Hutchinson (1992)
• Can be modelled by varying the coefficients of an isotropic hardening model
• Relatively simple but based on one given strain path only
4. Anisotropic hardening – Differential hardening
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4. Anisotropic hardening – Differential hardening
Example based on experimental data
RD uniaxial tension
Balanced biaxial tension
AISI 409 stainless steel
Solid line: ExperimentDash line: Crystal plasticity
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-1200
-800
-400
0
400
800
-1200 -800 -400 0 400 800
-1200 -800 -400 0 400 800
σ yy
σ xx
60%
1%
45% 35%
5%
25%
(MPa)
(MPa
)
CPB06VPSC♦
Zr plate
4. Anisotropic hardening – Differential hardening
Example based on crystal plasticity of Zr
Plunkett et al., 2006
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0
5
10
15
20
25
30
35
40
0 0.15 0.3 0.45
Spec
imen
hei
ght (
mm
)
Strain (ln[r/r0])
Major profile
In red,model withouttexture
evolution
Exp.
Model
0
5
10
15
20
25
30
35
40
0 0.15 0.3 0.45
Spec
imen
hei
ght (
mm
)
Strain (ln[r/r0])
Minor profile
In red,model withouttexture
evolution
Exp.
Model
Major profile Minor profile Footprint
4. Anisotropic hardening – Differential hardeningExample based on crystal plasticity of Zr (Taylor impact test application)
Major profile Minor profile Footprint
Plunkett et al., 2006Maudlin et al., 1999
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𝛷 𝛔, 𝑥 = 𝜎; 𝛔− 𝐗 − 𝜎> = 0 (Prager, 1949)
• One state variable: Back-stress 𝐗 = 𝐱, �� = 𝐶𝐃
Yield surface translate
4. Anisotropic hardening – Kinematic hardening
Linear kinematic hardening
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σ s,X( ) = Y +R ε( )Yield stress
Back-stress Hardening
Evolution equations
�� = 𝐶𝐃 − 𝛾𝐗𝜀
�� =𝑑𝑅𝑑𝜀
𝜀
• Chaboche et al. (1979)
Non-linear kinematic hardening
• Hu and Teodosiu (1995)
σ s,X,M( ) = Y +R S,P( )Texture anisotropy
(constant)Strength of
dislocation structure
Polarization of dislocation
structureBack-stress
4. Anisotropic hardening – Kinematic hardening
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Translating surfaces
Two surfaces (Dafalias and Popov, 1975)
4. Anisotropic hardening – Kinematic hardening
Multiple nested surfaces(Mroz, 1967)
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Two surfaces
f = φ s − α( )− Y = 0
F = φ s − β( )− B +R( ) = 0
• Yield surface evolution
Initial size boundary surface
Yield stressBack-stress 1
Back-stress 2 Hardening
π-plane
Bounding surface
Yield surface
• Yoshida and Uemori (2002)
4. Anisotropic hardening – Kinematic hardening
• Bounding surface evolution
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4. Anisotropic hardening – Validation
Biaxial compression testing – Stacked sheet specimens
Tozawa, 1978
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3. Isotropic hardening – ValidationDistortional hardeningYield loci at ε = 0.2% for steel (S10C) pre-
stretched by various strains in tension
εpre = 0.05
εpre = 0.20εpre = 0.10
Tozawa, 1978
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εpre = 0.20
Yield loci at ε = 0.2% for steel (S10C) pre-
stretched by various strains in tension
εpre = 0.05εpre = 0.10
4. Anisotropic hardening – Validation
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Homogeneous anisotropic hardening (HAH)
ω F s, f1q , f2q ,h( )q + ⎧
⎨⎩
⎫⎬⎭
1q= σ R ε( )
Flow stressFluctuating component
(Bauschinger effect)
!Microstructure deviator• Tensorial state variable with evolution rule• Mimics delays in formation / rearrangement of dislocation structures• Provides a reference for distortion
• replaced by for cross-loading & latent effectsω s( ) ωCL s( )f1q , f2q• describe the amount of distortion
Homogenous function
No kinematic hardening
Effective stress
ω s( )q
Effective stress
Stable component
σ s( ) =
4. Anisotropic hardening – Distortional plasticity only
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• Effects captured− Distortional hardening− Bauschinger effect
• Loading sequences − (I) RD tension− (II) RD compression
Reverse loading
• Anisotropic material 1
• Note− Half and half
h
Yld2000-2d(real anisotropic
material)
4. Anisotropic hardening – HAH model
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• Effects captured− Distortional hardening− Bauschinger effect− Cross-loading contraction− Latent hardening
• Loading sequence− (I) RD tension− (II) Near TD plane strain
tension
Cross-loading
• Anisotropic material 2
• Note− Proportional loading (proof)
h
Yld2000-2d(real anisotropic
material)
4. Anisotropic hardening – HAH model
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Reverse loading• Independent identification of coefficients (Bauschingerand other effects)
Cross-loading• Independent identification of coefficients (latent hardening and other effects)
• Same identification procedure as that of isotropic hardening ( , )ω s( )σ R ε( )
Proportional loading• HAH reduces to isotropic hardening response (anisotropic yield function)
4. Anisotropic hardening – HAH model
Coefficient identification• Sequential
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0.00 0.02 0.04 0.06 0.080
200
400
600
800
1000
1200
1400
1600
Experiement HAH model YU model
Pre-strain: 1.8 % and 2.6 %
Tension
True
stre
ss (M
Pa)
True strain
Compression
4. Anisotropic hardening – HAH model
Choi et al., 2017
Forward and reverse simple shear
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Fu et al. (2017)
4. Anisotropic hardening – HAH model
Forward and reverse simple shear (TRIP 780 steel)
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4. Anisotropic hardening – HAH model
Fu et al. (2017)
Forward and reverse simple shear cycles (DP 600 & TWIP 980 steels)
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Plasticity remains a topic with many challenges
• Anisotropic material under isotropic and anisotropic hardening
• Numerical implementation
• Identification with complex evolution equations
• Applicability for industrial problems
5. Final remarks