Large Deformation Plasticity of Amorphous Solids, with Application and Implementation into Abaqus
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Large Deformation Plasticity of Amorphous Solids, with Application and Implementation into AbaqusKristin M. MyersJanuary 11, 2007Plasticity ES 246 - HarvardReferences: 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.
 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.
 Lubliner, J. Plasticity Theory. 1990. Macmillan Publishing Company. (Chapter 8)
 Abaqus 6.5-4 Documentation Getting Started with ABAQUS/EXPLICIT. Hibbitt, Karlsson & Sorensen,INC.
Motivation Examples of MaterialsAmorphous Solids polymeric and metallic glasses (i.e. Polycarbonate)Rubber degradationBiomaterials Soft Collageneous Biological Tissue(i.e. cartilage, cervical tissue, skin, tendon, etc.)Engineering Collagen Scaffolds(i.e. skin, nerve, tendon etc.)Material Characteristics:Large stretches elastic & inelasticHighly non-linear relationships between stress/strainTime-Dependent; viscoplasticityStrain hardening or softening after initial yieldNon-linearity of tension & compression behavior (Bauschinger effect)
Experimental Results PolycarbonateFrom Boyce and ArrudaLarge deformation regimeStrain-softening after initial yieldBack stress evolution after yield drop to create strain-hardeningTENSIONCOMPRESSION
Kinematics Multiplicative Decomposition of the Deformation GradientDeformation GradientDecomposition of deformation gradient into its elastic and plastic components (Kroner-Lee)Velocity tensorVelocity GradientSegment of the relaxed configurationSegment of the current configurationRelaxed Configuration: Intermediate configuration created by elastically unloading the current configuration and relieving the part of all stresses.
Kinematics Multiplicative Decomposition of the Deformation Gradient IIConditions of Plastic FlowIncompressible
Principle of ObjectivityPrinciple of Material Frame IndifferenceSmooth time-dependent rigid transformations of the Eulerian Space:Principle of Relativity: relation of the two motions is equivalentRelative motion of two observers Eulerian basesTo be objective (in general):
The relaxed and reference configurations are invariant to the transformations of the Eulerian Space
Principal of Virtual PowerExternal expenditure of power = internal energy
Macroscopic Force Balance
Internal energy Wint is invariant under all changes in frame
Dissipation Inequality and Constitutive Framework2nd Law of Thermodynamics: The temporal increase in free energy of any part P be less than or equal to the power expended on PConstitutive framework: Free energy, stress, and internal variables are a function of deformation.
Constitutive Theory FrameworkFrame IndifferenceEuclidean SpaceAmorphous Solids: material are invariant under all rotations of the Relaxed and Reference Configuration
Constitutive Theory Thermodynamic Restrictions and Flow RulePlug into dissipation inequalityEnergy dissipated per unit volume (in the relaxed configuration) must be purely dissipatative. Dissipative FLOW STRESS:FLOW RULE:
Equations for Stress
Constitutive EquationsTe Stress conjugate to Ee= Cauchy Stressmaterial parametersConstitutive prescription
FLOW RULE for Plastic STRETCHING
Evolution of Internal Variables
DP=(magnitude)(DIRECTION)Effective Stress:Constitutive Equationsmaterial parametersConstitutive prescription= evolution of shear resistance (captures strain softening)= change in free-volume from initial stateSaturation value:
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 motionchains will align w/principle plastic stretch (Bp,p)Alignment decreases the configurational entropy creates an internal network back stress Sback
undeformeddeformedMicrograph by Roeder et al, 2001StretchForce-stretch relationship: - Initially compliant behavior followed by increase in stiffness as the limiting stretch is approachedParameters: Rubbery ModulusLimiting stretchEvolution of the Back Stress: Langevin Statistics
State Variables in Summary: In VUMATC**********************************************************************C STATE VARIABLES - Variables that need to be evolved with TIMECSTATEV(1) = Fp(1,1) -- PLASTIC DEFORMATION GRADIENT, (1,1) COMP.CSTATEV(2) = Fp(1,2) -- PLASTIC DEFORMATION GRADIENT, (1,2) COMP.CSTATEV(3) = Fp(1,3) -- PLASTIC DEFORMATION GRADIENT, (1,3) COMP.CSTATEV(4) = Fp(2,1) -- PLASTIC DEFORMATION GRADIENT, (2,1) COMP.CSTATEV(5) = Fp(2,2) -- PLASTIC DEFORMATION GRADIENT, (2,2) COMP.CSTATEV(6) = Fp(2,3) -- PLASTIC DEFORMATION GRADIENT, (2,3) COMP.CSTATEV(7) = Fp(3,1) -- PLASTIC DEFORMATION GRADIENT, (3,1) COMP.CSTATEV(8) = Fp(3,2) -- PLASTIC DEFORMATION GRADIENT, (3,2) COMP.CSTATEV(9) = Fp(3,3) -- PLASTIC DEFORMATION GRADIENT, (3,3) COMP.CCSTATEV(10)= Internal variable S - shear resistanceCCSTATEV(11)= dFp(1,1) -- incre in PLASTIC DEFORMATION GRADIENT, (1,1) COMP.CSTATEV(12)= dFp(1,2) -- incre in PLASTIC DEFORMATION GRADIENT, (1,2) COMP.CSTATEV(13)= dFp(1,3) -- incre in PLASTIC DEFORMATION GRADIENT, (1,3) COMP. CSTATEV(14)= dFp(2,1) -- incre in PLASTIC DEFORMATION GRADIENT, (2,1) COMP.CSTATEV(15)= dFp(2,2) -- incre in PLASTIC DEFORMATION GRADIENT, (2,2) COMP. CSTATEV(16)= dFp(2,3) -- incre in PLASTIC DEFORMATION GRADIENT, (2,3) COMP. CSTATEV(17)= dFp(3,1) -- incre in PLASTIC DEFORMATION GRADIENT, (3,1) COMP. CSTATEV(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,Cand change in free volume with time evolutionCC**********************************************************************
Material Parameters in Summary: In VUMATC----------------------------------------------------------------------C MATERIAL PARAMETERSCCElastic PropertiesCEG = elastic shear modulusCEK = elastic bulk modulusCLangevin Properties (Statistical Mechanics)CMU_R = rubbery modulusCLAMBDA_L = network locking stretchCCD0 = reference (initial) plastic shear-strain rateCm = plastic strain rate dependency (m=0; rate independent)CALPHA = coefficent of pressure dependencyCInternal Variable S coefficients (s monitors the isotropic resistance to deformationCH0 = initial hardening rateCSCV = equilibrium hardening strengthCSO = initial resistance to flow (yield point)CCoefficients for ETA - free volumeCG0 = coefficent of plastic dilantancyCb = coefficient for evolving etaCNCV = equilibrium value for free volumeC----------------------------------------------------------------------
VUMAT ProgramF_t = F at start of stepF_tau = F at end of stepU_tau = U at end of stepFor the first time stepInitialize state variablesFp_tau = 1Fe_tau=F_tauCalculate Ce_tauCalculate Ee_tauCalculate Te_tauCalculate T_tauRotate Cauchy stress to Abaqus Stress and update Abaqus stress variablesFor other time stepsGet state variables from last stepCalculate Fp_tauNormalize FpCalculate Fp_tau_invCalculate Fe_tauCalculate Ce_tauCalculate Ee_tauCalculate Te_tauCalculate pressure
Calculate Tmendel; Mendel stressCalculate Bp_tau_dev; Back Stress (USE LANGEVIN)Calculate Tflow; Flow StressCalculate tau: Equivalent Shear StressIF tau is not ZERO THENEVOLVE DP; calculate ANUp;EVOLVE dFpEVOLVE SEVOLVE eta IF tau is ZERODo not evolve state variablesUpdate Fp, F, C, U, TUpdate state variablesUpdate Abaqus stresses
The assertion that if a motion y generates a stress T then the transformed motion y_star generates of stress T_star = QTQ_transform. (Lubliner)Main constituents of the fibrous network are type I and type III collagen, and collagen content in terms of the dry weight of the tissue amounts to 70%, elastin only accounts for 2% dry weight of the fibrous networkMacroscopic stress levels developed depends on both the amount of stretch of the individual fibril and as well as the deformation of the network
Force stretch behavior of the individual fibril dominated by its wavy crimped nature. First it unbends and the dominant changes due to changes in the bending energy, later on stretching of the individual covalent bonds->much stiffer responseThe behavior is similar to the force extension relationship of a freely jointed molecular chain. (phenomenological similarities)Origin of the stress is different, nevertheless the statistical chain model was adopted to represent the individual collagen fibers Then have to establish link between the individual fibril response and the network as a whole . Propose 8 chain model, in which the unit cell deforms with the principal stretches of the deformation.