Theoretical Modeling of Plasticity - BSSM

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


1. Introduction 2. Plasticity modeling3. Isotropic hardening4. Anisotropic hardening5. Final remarks

International Seminar on Metal Plasticity June 19, 2017

Outline


1. Dislocation glide (slip)

Rauch et al., 2011

1. Introduction – Deformation mechanisms

Hull, 1983

2. Other mechanisms

Plastic behavior in metals


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


• 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)


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)


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


• 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


• 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


• 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)


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


2090-T3 Hill 1948 (stress-based)

r-value-based


Hill (1948) plane stress


0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 20 40 60 80

r-va

lue


2090-T32090-T3 Hill 1948 (stress-based)

ExperimentsStress-based

Nor

mal

ized

flow

str

ess


2090-T3 Hill 1948 (stress-based)

r-value-based




-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


r-value-based

TD n

orm

aliz

ed s

tres

s




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)


Hill (1948) limitations


• 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



• 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




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






0.00

0.50

1.00

1.50

2.00

0 20 40 60 80

ExperimentsYld2000-2dYld2004-18p

Nor

mal

ized

flow

str

ess






-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


2090-T3

Hill (r-based)

Hill (σ-based)

Yld2000-2d

Yld2004-18p




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


Strength differential (SD) effect

• Constant coefficient 𝐾


-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


Cazacu et al., 2006

Twinning yield surfaces


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


-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


Biaxial compression testing

𝑟 = YZY[


ISO 16842: 2014 Metallic materials −Sheet and strip −Biaxial tensile testing method using a cruciform test

piece


Biaxial tension testing

Tubular specimens

Cruciform specimensKuwabara and Sugawara, 2013


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


Hakoyama andKuwabara, 2015

Contour of plastic work for DP 600 steel


• Kinematic hardening

• Differential hardening

• Combined kinematic hardening and distortional plasticity

• Distortional plasticity only

4. Anisotropic hardening

• Combined kinematic - isotropic hardening

Approaches


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



Example based on experimental data

RD uniaxial tension

Balanced biaxial tension

AISI 409 stainless steel

Solid line: ExperimentDash line: Crystal plasticity


-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


Example based on crystal plasticity of Zr

Plunkett et al., 2006


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


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


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



Translating surfaces

Two surfaces (Dafalias and Popov, 1975)


Multiple nested surfaces(Mroz, 1967)


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)


• Bounding surface evolution


4. Anisotropic hardening – Validation

Biaxial compression testing – Stacked sheet specimens

Tozawa, 1978


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


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


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


• 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


• 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)



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)


Coefficient identification• Sequential


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


Choi et al., 2017

Forward and reverse simple shear


Fu et al. (2017)


Forward and reverse simple shear (TRIP 780 steel)



Fu et al. (2017)

Forward and reverse simple shear cycles (DP 600 & TWIP 980 steels)


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

Documents

Theoretical Modeling of Plasticity - BSSM