Large strain finite element analysis of a local second gradient model: application to localization

SUMMARY A large strainnite element formulation based on a local second gradient plasticity model is presented. The corresponding constitutive equations were developed as a direct extension of microstructured (SIAM J. Appl. Math. 1973; 25(3):556 - 575; Arch. Rational Mech. Anal. 1964; 16:51 - 78) or micromorphic (In Mechanics of Generalized Continua, IUTAM Symposium, Kroner (ed.), Springer: Berlin, 1968; 18-35; J. Math. Mech. 1966; 15(6):909 - 924.) continua in which a mathematical constraint between the micro kinematics description and the usual macrodeformation gradienteld has been introduced. This constraint is enforced in a weak sense by the use of Lagrange multipliers in order to avoid diculties with the C 1 continuity, for thenite element method. Correspondingnite elements are then constructed involving the Lagrange multiplierseld. A geometrically non-linear 2-Dnite element code is developed within a framework of an incremental method. For every step, a full Newton-Raphson involving a numerical consistent tangent stiness operator for the complete model (i.e., the second gradient terms as well as the classical ones) is done and some numerical tests allow to validate the method and to discuss the inuence of the geometrical non-linearity. Copyright ? 2002 John Wiley & Sons, Ltd.

[1]  Norman A. Fleck,et al.  A phenomenological theory for strain gradient effects in plasticity , 1993 .

[2]  René Chambon,et al.  Plastic continuum with microstructure, Local Second Gradient Theories for Geomaterials : Localization Studies , 2001 .

[3]  C. Truesdell,et al.  The Non-Linear Field Theories of Mechanics , 1965 .

[4]  I. Kunin,et al.  Elastic Media with Microstructure II , 1982 .

[5]  Hussein M. Zbib,et al.  On the localization and postlocalization behavior of plastic deformation. II: On the evolution and thickness of shear bands , 1988 .

[6]  P. Steinmann.  A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity , 1994 .

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

[8]  Z. Bažant,et al.  Nonlocal damage theory , 1987 .

[9]  R. Toupin.  Elastic materials with couple-stresses , 1962 .

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

[11]  N. Fleck,et al.  FINITE ELEMENTS FOR MATERIALS WITH STRAIN GRADIENT EFFECTS , 1999 .

[12]  René de Borst,et al.  Gradient-dependent plasticity: formulation and algorithmic aspects , 1992 .

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

[14]  R. Toupin.  ELASTIC MATERIALS WITH COUPLE STRESSES, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS , 1962 .

[15]  I. Vardoulakis,et al.  The thickness of shear bands in granular materials , 1987 .

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

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

[18]  R. D. Mindlin.  Micro-structure in linear elasticity , 1964 .

[19]  Zdeněk P. Bažant,et al.  Non-local yield limit degradation , 1988 .

[20]  Xikui Li,et al.  FINITE ELEMENT METHOD FOR GRADIENT PLASTICITY AT LARGE STRAINS , 1996 .

[21]  Ioannis Vardoulakis,et al.  A gradient flow theory of plasticity for granular materials , 1991 .

[22]  Hussein M. Zbib,et al.  A Gradient-Dependent Flow Theory of Plasticity: Application to Metal and Soil Instabilities , 1989 .

[23]  P. Germain,et al.  The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure , 1973 .

[24]  D. C. Drucker,et al.  Mechanics of Incremental Deformation , 1965 .

[25]  R. Borst.  SIMULATION OF STRAIN LOCALIZATION: A REAPPRAISAL OF THE COSSERAT CONTINUUM , 1991 .

[26]  Xikui Li,et al.  A mixed strain element method for pressure‐dependent elastoplasticity at moderate finite strain , 1998 .

[27]  Hans Muhlhaus,et al.  A variational principle for gradient plasticity , 1991 .

[28]  Zdenek P. Bazant,et al.  Imbricate continuum and its variational derivation , 1984 .

[29]  A. Eringen,et al.  LINEAR THEORY OF MICROPOLAR ELASTICITY , 1965 .

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

[31]  Cv Clemens Verhoosel,et al.  Non-Linear Finite Element Analysis of Solids and Structures , 1991 .

[32]  Ioannis Vardoulakis,et al.  A finite element displacement formulation for gradient elastoplasticity , 2001, Bifurcation and Localisation Theory in Geomechanics.

[33]  R. D. Mindlin.  Second gradient of strain and surface-tension in linear elasticity , 1965 .

[34]  René Chambon,et al.  Second gradient models as a particular case of microstructured models: a large strain finite elements analysis , 2000 .

[35]  Z. Bažant.  Mechanics of Distributed Cracking , 1986 .

[36]  Robert Charlier,et al.  An algorithm and a method to search bifurcation points in non‐linear problems , 2001 .

[37]  J. C. Simo,et al.  Consistent tangent operators for rate-independent elastoplasticity☆ , 1985 .

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

[39]  A. C. Eringen,et al.  Mechanics of Micromorphic Continua , 1968 .

[40]  R. Borst,et al.  CONTINUUM MODELS FOR DISCONTINUOUS MEDIA , 1991 .

[41]  Z. C. Xia,et al.  Plane-Stress Deformation in Strain Gradient Plasticity , 2000 .

[42]  E. Kröner,et al.  Elasticity theory of materials with long range cohesive forces , 1967 .

[43]  Hussein M. Zbib,et al.  On the localization and postlocalization behavior of plastic deformation. I: On the initiation of shear bands , 1988 .

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

[45]  J. Rice.  Localization of plastic deformation , 1976 .

[46]  R. D. Mindlin.  MICROSTRUCTURE IN LINEAR ELASTICITY , 1999 .

[47]  R. Rivlin,et al.  Simple force and stress multipoles , 1964 .

[48]  I. Kunin.  The Theory of Elastic Media with Microstructure and the Theory of Dislocations , 1968 .