Theory and numerics of geometrically non-linear gradient plasticity

This work presents the theory and the numerics of a thermodynamically consistent formulation of geometrically non-linear gradient plasticity. Due to the lack of the classical local continuum formulation to produce physically meaningful and numerically converging results within localization computations, a thermodynamically motivated gradient plasticity formulation is envisioned. Especially within the framework of crystal plasticity we resort to physically motivated arguments in terms of geometrically necessary dislocations densities that imply the incorporation of higher gradients. In a first simplified approach presented here we adopt the gradient of the internal hardening variable as a provision for geometrically necessary dislocations. We start from a thermodynamic formulation within a geometrically non-linear setting including the additional contribution of the gradient of the internal history variable. This introduces e.g. the vectorial hardening flux and the quasi-nonlocal drag stress. At the numerical side, besides the balance of linear momentum, the algorithmic consistency condition has to be solved in weak form. Thereby, the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non-linear programming. Finally, some demonstrative numerical examples complement the presentation.

[1]  Lars Pilgaard Mikkelsen.  Post-necking behaviour modelled by a gradient dependent plasticity theory , 1997 .

[2]  P. Steinmann.  FORMULATION AND COMPUTATION OF GEOMETRICALLY NON-LINEAR GRADIENT DAMAGE , 1999 .

[3]  Gianfranco Capriz,et al.  Continua with Microstructure , 1989 .

[4]  E. Cosserat,et al.  Théorie des Corps déformables , 1909, Nature.

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

[6]  Rhj Ron Peerlings,et al.  Gradient enhanced damage for quasi-brittle materials , 1996 .

[7]  E. Aifantis.  On the role of gradients in the localization of deformation and fracture , 1992 .

[8]  R. Billardon,et al.  A two-field finite element formulation for elasticity coupled to damage , 1994 .

[9]  Claudia Comi,et al.  Computational modelling of gradient‐enhanced damage in quasi‐brittle materials , 1999 .

[10]  J. Pamin,et al.  Gradient-dependent plasticity in numerical simulation of localization phenomena , 1994 .

[11]  Castrenze Polizzotto,et al.  A thermodynamics-based formulation of gradient-dependent plasticity , 1998 .

[12]  Dominic G.B. Edelen,et al.  On the thermodynamics of systems with nonlocality , 1971 .

[13]  Jerzy Pamin,et al.  Some novel developments in finite element procedures for gradient-dependent plasticity , 1996 .

[14]  Robert L. Taylor,et al.  Complementary mixed finite element formulations for elastoplasticity , 1989 .

[15]  U. F. Kocks.  Polyslip in single crystals , 1960 .

[16]  Rhj Ron Peerlings,et al.  Some observations on localisation in non-local and gradient damage models , 1996 .

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

[18]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[19]  Thomas Svedberg,et al.  An algorithm for gradient-regularized plasticity coupled to damage based on a dual mixed FE-formulation , 1998 .

[20]  Ronald W. Armstrong,et al.  The (cleavage) strength of pre-cracked polycrystals , 1987 .

[21]  P. Pinsky.  A finite element formulation for elastoplasticity based on a three-field mixed variational equation , 1987 .

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

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

[24]  P. Steinmann,et al.  Theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity , 2001 .

[25]  A. Acharya,et al.  Incompatibility and crystal plasticity , 1995 .

[26]  de R René Borst,et al.  Wave propagation, localization and dispersion in a gradient-dependent medium , 1993 .

[27]  Paul Steinmann,et al.  On the numerical treatment and analysis of finite deformation ductile single crystal plasticity , 1996 .

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

[29]  Z. Bažant,et al.  Nonlocal Continuum Damage, Localization Instability and Convergence , 1988 .

[30]  E. Aifantis.  On the Microstructural Origin of Certain Inelastic Models , 1984 .

[31]  C. Truesdell,et al.  The Non-Linear Field Theories Of Mechanics , 1992 .

[32]  H. Zbib,et al.  A thermodynamical theory of gradient elastoplasticity with dislocation density tensor. I: Fundamentals , 1999 .

[33]  T. Svedberg.  On the Modelling and Numerics of Gradient-Regularized Plasticity Coupled to Damage , 1999 .

[34]  H. Stumpf,et al.  STRAIN MEASURES, INTEGRABILITY CONDITION AND FRAME INDIFFERENCE IN THE THEORY OF ORIENTED MEDIA* , 1998 .

[35]  Paolo Maria Mariano,et al.  Multifield theories in mechanics of solids , 2002 .

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

[37]  D. Caillerie,et al.  One-dimensional localisation studied with a second grade model , 1998 .

[38]  B. Nedjar,et al.  Elastoplastic-damage modelling including the gradient of damage: formulation and computational aspects , 2001 .

[39]  Zdenek P. Bazant,et al.  Why Continuum Damage is Nonlocal: Micromechanics Arguments , 1991 .

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

[41]  A. Menzel,et al.  Erratum to “On the continuum formulation of higher gradient plasticity for single and polycrystals”: [Journal of the Mechanics and Physics of Solids 48 (2000) 1777–1796] , 2001 .

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

[43]  J. C. Simo,et al.  Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory , 1992 .

[44]  Gérard A. Maugin,et al.  The thermomechanics of nonlinear irreversible behaviors : an introduction , 1999 .