Variational gradient plasticity at finite strains. Part III: Local–global updates and regularization techniques in multiplicative plasticity for single crystals

The third part of our work on variational inelasticity with long-range effects outlines a formulation and finite element implementation of micromechanically-motivated multiplicative gradient plasticity for single crystals. In order to partially overcome the complexity of full multislip scenarios, we suggest a new viscous regularized formulation of rate-independent crystal plasticity, that exploits in a systematic manner the long- and short-range nature of the involved variables. To this end, we outline a multifield scenario, where the macro-deformation and the plastic slips on crystallographic systems are the primary fields. Related to these primary fields, we define as the long-range state the deformation gradient, the plastic slips and their gradients. We then introduce as the short-range plastic state the plastic deformation map, the dislocation density tensor and scalar hardening parameters associated with the slip systems. It is then shown that the evolution of the short range state is fully determined by the evolution of the long-range state. This separation into long- and short-range states is systematically exploited in the algorithmic treatment by a new update structure, where the short-range variables play the role of a local history base. The model problem under consideration accounts in a canonical format for basic effects related to statistically stored and geometrically necessary dislocation flow, yielding micro-force balances including non-convex cross-hardening, kinematic hardening and size effects. Further key ingredients of the proposed algorithmic formulation are geometrically exact updates of the short-range state and a distinct regularization of the rate-independent dissipation function that preserves the range of the elastic domain. The formulation is shown to be fully variational in nature, governed by rate-type continuous and incremental algorithmic variational principles. We demonstrate the modeling capabilities and algorithmic performance by means of representative numerical examples for multislip scenarios in fcc single crystals.

[1]  E. Kröner,et al.  Nicht-lineare Elastizitätstheorie der Versetzungen und Eigenspannungen , 1959 .

[2]  Huajian Gao,et al.  Indentation size effects in crystalline materials: A law for strain gradient plasticity , 1998 .

[3]  Christian Miehe,et al.  Exponential Map Algorithm for Stress Updates in Anisotropic Multiplicative Elastoplasticity for Single Crystals , 1996 .

[4]  K. Runesson,et al.  Gradient crystal plasticity as part of the computational modelling of polycrystals , 2007 .

[5]  A. Seeger.  Theorie der Gitterfehlstellen , 1955 .

[6]  Stefan Müller,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Lower Semi-continuity and Existence of Minimizers in Incremental Finite-strain Elastoplasticity , 2022 .

[7]  John L. Bassani,et al.  Plastic flow of crystals , 1993 .

[8]  Christian Miehe,et al.  Variational gradient plasticity at finite strains. Part I: Mixed potentials for the evolution and update problems of gradient-extended dissipative solids , 2014 .

[9]  R. Hill.  Generalized constitutive relations for incremental deformation of metal crystals by multislip , 1966 .

[10]  E. Kröner,et al.  Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen , 1959 .

[11]  Morton E. Gurtin,et al.  A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations , 2002 .

[12]  J. Nye.  Some geometrical relations in dislocated crystals , 1953 .

[13]  Frank Reginald Nunes Nabarro,et al.  Theory of crystal dislocations , 1967 .

[14]  R. Bullough,et al.  Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[15]  M. Becker.  Incompatibility and instability based size effects in crystals and composites at finite elastoplastic strains , 2006 .

[16]  Vasily V. Bulatov,et al.  On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals , 2004 .

[17]  Christian Miehe,et al.  A multi-field incremental variational framework for gradient-extended standard dissipative solids , 2011 .

[18]  Morton E. Gurtin,et al.  A comparison of nonlocal continuum and discrete dislocation plasticity predictions , 2003 .

[19]  M. Ortiz,et al.  The variational formulation of viscoplastic constitutive updates , 1999 .

[20]  K. Runesson,et al.  Modeling of polycrystals with gradient crystal plasticity: A comparison of strategies , 2010 .

[21]  W. Brekelmans,et al.  Scale dependent crystal plasticity framework with dislocation density and grain boundary effects , 2004 .

[22]  E. Hall,et al.  The Deformation and Ageing of Mild Steel: III Discussion of Results , 1951 .

[23]  Bob Svendsen,et al.  Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations , 2002 .

[24]  R. Asaro,et al.  Micromechanics of Crystals and Polycrystals , 1983 .

[25]  T. Böhlke,et al.  Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[26]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[27]  J. Rice,et al.  Constitutive analysis of elastic-plastic crystals at arbitrary strain , 1972 .

[28]  Alan Needleman,et al.  An analysis of nonuniform and localized deformation in ductile single crystals , 1982 .

[29]  U. F. Kocks.  A statistical theory of flow stress and work-hardening , 1966 .

[30]  C. Teodosiu,et al.  Lattice defect approach to plasticity and viscoplasticity , 1974 .

[31]  M. Gurtin,et al.  Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations , 2002 .

[32]  T. Lin,et al.  Physical Theory of Plasticity , 1971 .

[33]  Michael Ortiz,et al.  Computational modelling of single crystals , 1993 .

[34]  E. Arzt.  Size effects in materials due to microstructural and dimensional constraints: a comparative review , 1998 .

[35]  Lallit Anand,et al.  Elasto-viscoplastic constitutive equations for polycrystalline metals: Application to tantalum , 1998 .

[36]  Paul Steinmann,et al.  Views on multiplicative elastoplasticity and the continuum theory of dislocations , 1996 .

[37]  L. Anand,et al.  A computational procedure for rate-independent crystal plasticity , 1996 .

[38]  M. Ashby.  The deformation of plastically non-homogeneous materials , 1970 .

[39]  G. Taylor.  The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical , 1934 .

[40]  J. Schröder,et al.  Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials , 1999 .

[41]  C. Miehe,et al.  Anisotropic finite elastoplastic analysis of shells: simulation of earing in deep-drawing of single- and polycrystalline sheets by Taylor-type micro-to-macro transitions , 2004 .

[42]  D. Parks,et al.  Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density , 1999 .

[43]  S. Bargmann,et al.  On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation , 2010 .

[44]  R. Asaro,et al.  Finite element analysis of crystalline solids , 1985 .

[45]  V. Tvergaard,et al.  A finite deformation theory of higher-order gradient crystal plasticity , 2008 .

[46]  Morton E. Gurtin,et al.  A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on the accumulation of geometrically necessary dislocations , 2008 .

[47]  Morton E. Gurtin,et al.  On the plasticity of single crystals: free energy, microforces, plastic-strain gradients , 2000 .

[48]  P. Perzyna.  Temperature and rate dependent theory of plasticity of crystalline solids , 1988 .

[49]  Norman A. Fleck,et al.  Boundary layers in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity , 2001 .

[50]  N. Fleck,et al.  Strain gradient plasticity , 1997 .

[51]  Anthony G. Evans,et al.  A microbend test method for measuring the plasticity length scale , 1998 .

[52]  N. Petch,et al.  The Cleavage Strength of Polycrystals , 1953 .

[53]  U. F. Kocks.  Thermodynamics and kinetics of slip , 1975 .

[54]  S. Reese,et al.  Continuum Thermodynamic Modeling and Simulation of Additional Hardening due to Deformation Incompatibility , 2003 .

[55]  Mgd Marc Geers,et al.  Non-local crystal plasticity model with intrinsic SSD and GND effects , 2004 .

[56]  J. Hutchinson,et al.  Bounds and self-consistent estimates for creep of polycrystalline materials , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[57]  Jens Lothe John Price Hirth,et al.  Theory of Dislocations , 1968 .

[58]  D. Hull,et al.  Introduction to Dislocations , 1968 .

[59]  M. Geers,et al.  Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories , 2009 .

[60]  M. Gurtin,et al.  On the characterization of geometrically necessary dislocations in finite plasticity , 2001 .

[61]  Paul Steinmann,et al.  On the continuum formulation of higher gradient plasticity for single and polycrystals , 2000 .

[62]  J. Rice.  Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity , 1971 .

[63]  M. Ashby,et al.  Strain gradient plasticity: Theory and experiment , 1994 .

[64]  M. Lambrecht,et al.  Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals , 2002 .

[65]  K. S. Havner,et al.  Finite Plastic Deformation of Crystalline Solids , 1992 .