Large Deformation Plasticity of Amorphous Solids, with

Application and Implementation into Abaqus

Kristin M. MyersJanuary 11, 2007

Plasticity ES 246 - HarvardReferences:[1] Anand, L., Gurtin, M.E., 2003. “A theory of amorphous solids undergoing large deformations, with application to polymeric glasses.” International Journal of Solids and Structures 40, 1465-1487.

[2] Boyce, M. C., Arruda, E. M., 1990. “An experimental and analytical investigation of the large strain compressive and tensile response of glassy polymers.” Polymer Engineering and Science 30 (20), 1288-1298.

[3] Lubliner, J. Plasticity Theory. 1990. Macmillan Publishing Company. (Chapter 8)

[4] Abaqus 6.5-4 Documentation “Getting Started with ABAQUS/EXPLICIT.” Hibbitt, Karlsson & Sorensen,INC.

Motivation – Examples of Materials

• Amorphous Solids – – polymeric and metallic glasses (i.e. Polycarbonate)– Rubber degradation

• Biomaterials – Soft Collageneous Biological Tissue

(i.e. cartilage, cervical tissue, skin, tendon, etc.)– Engineering Collagen Scaffolds

(i.e. skin, nerve, tendon etc.)

Material Characteristics:1. Large stretches – elastic & inelastic2. Highly non-linear relationships between stress/strain3. Time-Dependent; viscoplasticity4. Strain hardening or softening after initial yield5. Non-linearity of tension & compression behavior (Bauschinger effect)

Experimental Results – Polycarbonate

From Boyce and Arruda

• Large deformation regime• Strain-softening after initial yield• Back stress evolution after yield drop to create strain-

hardening

TENSIONCOMPRESSION

Kinematics – Multiplicative Decomposition of the Deformation Gradient

),( tXyF

),( tXyv 1 FFvgradL

peFFF

XF p

XF e

XdXFdl p

ldXFxd e

Deformation GradientDecomposition of deformation gradient into its elastic and plastic components (Kroner-Lee)

Velocity tensor

Velocity Gradient

Segment of the “relaxed configuration”

Segment of the “current configuration”

“Relaxed Configuration”: Intermediate configuration created by elastically unloading the current configuration and relieving the part of all stresses.

Kinematics – Multiplicative Decomposition of the Deformation

Gradient II

• Conditions of Plastic Flow– Incompressible

– Irrotational

1

epee FLFLLeeeee WDFFL

1

ppppp WDFFL 1

ee LsymD ee LskwW

pp LsymD pp LskwW

0

1det

p

p

Ltr

F

0pW pp DL

pppFDF

eFFJ detdet

Principle of ObjectivityPrinciple of Material Frame

Indifference)(),()(),(),( * tqtXytQtXytXy

)(tQ

TQGQG *

Smooth time-dependent rigid transformations of the Eulerian Space:

Principle of Relativity: relation of the two motions is equivalentRelative motion of two observers Eulerian bases

To be objective (in general):

gQg *

FQF ee FQF

TTee QQQLQL

Tee QDQD TTee QQQWQW

The relaxed and reference configurations are invariant to the transformations of the Eulerian Space PF PD

Principal of Virtual Power• External expenditure of power = internal

energy

Macroscopic Force Balance

• Internal energy Wint is invariant under all changes in frame

Microforce Balance

0 bfTdiv

P

tbdP

ext dVvfdAvntW ~~)(

Pt

PPedVDTJLTW ~~ 1

int

int*int WW

Pt

PPe dVDTJDTW 1int

Dissipation Inequality and Constitutive Framework

• 2nd Law of Thermodynamics: The temporal increase in free energy ψ of any part P be less than or equal to the power expended on P 0 PPe DTDTJ

),,,(

),,,(ˆ

),(ˆ),(ˆ

PPei

PPePP

Pe

Pe

DFFh

DFFTT

FFTT

FF

• Constitutive framework: Free energy, stress, and internal variables are a function of deformation.

Constitutive Theory – Framework• Frame Indifference

– Euclidean Space– Amorphous Solids: material are invariant

under all rotations of the Relaxed and Reference Configuration

),,,(

),,,(

),(

),(

PPei

PPePP

ePee

Pe

DBCh

DBCTT

FBCTFT

BCT

),,,(

),,,(ˆ

),(ˆ),(ˆ

PPei

PPePP

Pe

Pe

DFFh

DFFTT

FFTT

FF

Constitutive Theory – Thermodynamic Restrictions

and Flow RuleP

Pe

e BB

CC

Plug into dissipation inequality

ee

Pee F

CBCFJT

),(2 1

PP

Pe

oP Y

BBCsymT

),(2 0),,,( PPPeP DDBCY

Energy dissipated per unit volume (in the relaxed configuration) must be purely dissipatative. Dissipative FLOW STRESS:

FLOW RULE:Define:

P

Pe

oback BBCsymS ),(2

backoPPeP SSDBCY ),,,(

e

Pee

oee

o CBCCsymTCsymS ),(220

• Free Energy

• Equations for Stress

Constitutive Equations

Pe 22

2/1 eeo

e EtrKEG )( PP

12 eeo

e EtrKEGT eee RTRT

T

Teee RTRT

Te Stress conjugate to Ee

= Cauchy Stress

material parameters

Constitutive prescription

)(1)(~ cvcv bss

• FLOW RULE for Plastic STRETCHING

• Evolution of Internal Variables

2

Po

eoPP BTD

PPPFDF 10, XF P

moP

ps

/1

DP=(magnitude)(DIRECTION)

Po

eo BT 2

1

Effective Stress:

Constitutive Equationsmaterial parameters

Constitutive prescription

P

cvo

Po

ssg

sshs

1

)(~1

= evolution of shear resistance (captures strain softening)= change in free-volume from initial state

Saturation value:

L

P

PL

R L

1

3

L0

L

Amorphous polymeric materials:• Wavy kinked fibrous network structure• Resistance of the network in tension• Have finite distensibility (maximum stretch )• Once material overcomes the resistance to intermolecular chain motion

chains will align w/principle plastic stretch (Bp,λp)•Alignment decreases the configurational entropy

creates an internal network back stress Sback

L

undeformed

deformed

Micrograph by Roeder et al, 2001

0LL

Stretch

Force

LLimiting extensibility

Force-stretch relationship: - Initially compliant behavior followed by increase in stiffness as the limiting stretch is approached

Parameters:

•Rubbery Modulus•Limiting stretch

Evolution of the Back Stress: Langevin Statistics

P

P

P

)(

31

1)()coth()( L

....2 x

L

P

LRP

State Variables in Summary: In VUMAT

C**********************************************************************C STATE VARIABLES - Variables that need to be evolved with TIMEC STATEV(1) = Fp(1,1) -- PLASTIC DEFORMATION GRADIENT, (1,1) COMP.C STATEV(2) = Fp(1,2) -- PLASTIC DEFORMATION GRADIENT, (1,2) COMP.C STATEV(3) = Fp(1,3) -- PLASTIC DEFORMATION GRADIENT, (1,3) COMP.C STATEV(4) = Fp(2,1) -- PLASTIC DEFORMATION GRADIENT, (2,1) COMP.C STATEV(5) = Fp(2,2) -- PLASTIC DEFORMATION GRADIENT, (2,2) COMP.C STATEV(6) = Fp(2,3) -- PLASTIC DEFORMATION GRADIENT, (2,3) COMP.C STATEV(7) = Fp(3,1) -- PLASTIC DEFORMATION GRADIENT, (3,1) COMP.C STATEV(8) = Fp(3,2) -- PLASTIC DEFORMATION GRADIENT, (3,2) COMP.C STATEV(9) = Fp(3,3) -- PLASTIC DEFORMATION GRADIENT, (3,3) COMP.CC STATEV(10)= Internal variable S - shear resistanceCC STATEV(11)= dFp(1,1) -- incre in PLASTIC DEFORMATION GRADIENT, (1,1) COMP.C STATEV(12)= dFp(1,2) -- incre in PLASTIC DEFORMATION GRADIENT, (1,2) COMP.C STATEV(13)= dFp(1,3) -- incre in PLASTIC DEFORMATION GRADIENT, (1,3) COMP. C STATEV(14)= dFp(2,1) -- incre in PLASTIC DEFORMATION GRADIENT, (2,1) COMP.C STATEV(15)= dFp(2,2) -- incre in PLASTIC DEFORMATION GRADIENT, (2,2) COMP. C STATEV(16)= dFp(2,3) -- incre in PLASTIC DEFORMATION GRADIENT, (2,3) COMP. C STATEV(17)= dFp(3,1) -- incre in PLASTIC DEFORMATION GRADIENT, (3,1) COMP. C STATEV(18)= dFp(3,2) -- incre in PLASTIC DEFORMATION GRADIENT, (3,2) COMP. C STATEV(19)= dFp(3,3) -- incre in PLASTIC DEFORMATION GRADIENT, (3,3) COMP.CC STATEV(20)= Internal variable eta: eta=0 at virgin state of the material,C and change in free volume with time evolutionCC**********************************************************************

Material Parameters in Summary: In VUMAT

C----------------------------------------------------------------------C MATERIAL PARAMETERSCC Elastic PropertiesC EG = elastic shear modulusC EK = elastic bulk modulusC Langevin Properties (Statistical Mechanics)C MU_R = rubbery modulusC LAMBDA_L = network locking stretchCC D0 = reference (initial) plastic shear-strain rateC m = plastic strain rate dependency (m=0; rate independent)C ALPHA = coefficent of pressure dependencyC Internal Variable S coefficients (s monitors the isotropic resistance to deformationC H0 = initial hardening rateC SCV = equilibrium hardening strengthC SO = initial resistance to flow (yield point)C Coefficients for ETA - free volumeC G0 = coefficent of plastic dilantancyC b = coefficient for evolving etaC NCV = equilibrium value for free volumeC----------------------------------------------------------------------

VUMAT Program• F_t = F at start of step• F_tau = F at end of step• U_tau = U at end of step• For the first time step

– Initialize state variables– Fp_tau = 1– Fe_tau=F_tau– Calculate Ce_tau– Calculate Ee_tau– Calculate Te_tau– Calculate T_tau– Rotate Cauchy stress to Abaqus

Stress and update Abaqus stress variables

• For other time steps– Get state variables from last step– Calculate Fp_tau– Normalize Fp– Calculate Fp_tau_inv– Calculate Fe_tau– Calculate Ce_tau– Calculate Ee_tau– Calculate Te_tau– Calculate pressure

– Calculate Tmendel; Mendel stress– Calculate μ Bp_tau_dev; Back

Stress (USE LANGEVIN)– Calculate Tflow; Flow Stress– Calculate tau: Equivalent Shear

Stress– IF tau is not ZERO THEN

• EVOLVE DP; calculate ANUp;• EVOLVE dFp• EVOLVE S• EVOLVE eta

– IF tau is ZERO• Do not evolve state variables

– Update Fp, F, C, U, T– Update state variables– Update Abaqus stresses