Finite element analysis and algorithms for single‐crystal strain‐gradient plasticity

SUMMARY We provide optimal a priori estimates for finite element approximations of a model of rate-independent single-crystal strain-gradient plasticity. The weak formulation of the problem takes the form of a variational inequality in which the primary unknowns are the displacement and slips on the prescribed slip systems, as well as the back-stress associated with the vectorial microstress. It is shown that the return mapping algorithm for local plasticity can be applied element-wise to this non-local setting. Some numerical examples illustrate characteristic features of the non-local model. Copyright © 2012 John Wiley & Sons, Ltd.

[1]  N. A. Flecka,et al.  A reformulation of strain gradient plasticity , 2001 .

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

[3]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[4]  Christian Miehe,et al.  A comparative study of stress update algorithms for rate‐independent and rate‐dependent crystal plasticity , 2001 .

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

[6]  W. Han,et al.  Plasticity: Mathematical Theory and Numerical Analysis , 1999 .

[7]  Morton E. Gurtin,et al.  Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector , 2005 .

[8]  Christian Wieners,et al.  A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing , 2010, Comput. Vis. Sci..

[9]  Michael Ortiz,et al.  Concurrent Multiscale Computing of Deformation Microstructure by Relaxation and Local Enrichment with Application to Single-Crystal Plasticity , 2007, Multiscale Model. Simul..

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

[11]  Swantje Bargmann,et al.  Modeling of polycrystals using a gradient crystal plasticity theory that includes dissipative micro-stresses , 2011 .

[12]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

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

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

[15]  W. Dörfler,et al.  Convergence of an adaptive hp finite element strategy in higher space-dimensions , 2010 .

[16]  Andreas Rieder,et al.  Local inversion of the sonar transform regularized by the approximate inverse , 2011 .

[17]  Norman A. Fleck,et al.  A reformulation of strain gradient plasticity , 2001 .

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

[19]  Christian Wieners,et al.  Distributed Point Objects. A New Concept for Parallel Finite Elements , 2005 .

[20]  Barbara I. Wohlmuth,et al.  A Primal-Dual Finite Element Approximation for a Nonlocal Model in Plasticity , 2011, SIAM J. Numer. Anal..

[21]  Christian Wieners,et al.  On the superlinear convergence in computational elasto-plasticity , 2011 .

[22]  A. Needlemana,et al.  A comparison of nonlocal continuum and discrete dislocation plasticity predictions , 2002 .

[23]  Andreas Rieder,et al.  Towards a general convergence theory for inexact Newton regularizations , 2009, Numerische Mathematik.

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

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

[26]  H. Alber.  Materials with Memory: Initial-Boundary Value Problems for Constitutive Equations with Internal Variables , 1998 .

[27]  M. Gurtin,et al.  Gradient single-crystal plasticity with free energy dependent on dislocation densities , 2007 .

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

[29]  M. Gurtin,et al.  The Mechanics and Thermodynamics of Continua , 2010 .

[30]  H. Alber.  Materials with Memory , 1998 .

[31]  R. Borja,et al.  Discrete micromechanics of elastoplastic crystals , 1993 .

[32]  Morton E. Gurtin,et al.  A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations , 2005 .

[33]  W. Brekelmans,et al.  A three-dimensional dislocation field crystal plasticity approach applied to miniaturized structures , 2007 .

[34]  Christian Wieners,et al.  A parallel block LU decomposition method for distributed finite element matrices , 2011, Parallel Comput..

[35]  Mgd Marc Geers,et al.  A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity , 2006 .

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

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

[38]  V. Tvergaard,et al.  On the formulations of higher-order strain gradient crystal plasticity models , 2008 .

[39]  Peter Gudmundson,et al.  A unified treatment of strain gradient plasticity , 2004 .

[40]  B. D. Reddy,et al.  The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 2: single-crystal plasticity , 2011 .

[41]  Alexander Mielke,et al.  Chapter 6 – Evolution of Rate-Independent Systems , 2005 .