Bifurcation and Instability of Non-Associative Elastoplastic Solids

Global and local uniqueness and stability criteria for elastoplastic solids with non-associative flow rules are presented. Hill’s general theory is developed in the form generalized by Raniecki to non-associativity. Local stability criteria are presented and systematically discussed in a critical way. These are: positive definiteness and non-singularity of the constitutive operator, and positive definiteness (strong ellipticity) and non-singularity (ellipticity) of the acoustic tensor. The former criteria are particularly relevant for homogeneous deformation of solids subject to all-round controlled nominal surface tractions. Dually, the latter criteria are particularly relevant for homogeneous deformation of solids subject to displacements prescribed on the entire boundary. Flutter instability as related to complex conjugate eigenvalues of the acoustic tensor is also briefly discussed.

[1]  H. Petryk.  General conditions for uniqueness in materials with multiple mechanisms of inelastic deformation , 2000 .

[2]  Rodney Hill,et al.  Aspects of Invariance in Solid Mechanics , 1979 .

[3]  H. Petryk.  Stability and Constitutive Inequalities in Plasticity , 1993 .

[4]  D. Bigoni.  On smooth bifurcations in non-associative elastoplasticity , 1996 .

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

[6]  Bounding approach to the bifurcation point of annular plates with nonassociated flow law subjected to uniform tension at their outer edges , 1988 .

[7]  M. Kleiber.  On plastic localization and failure in plane strain and round void containing tensile bars , 1986 .

[8]  Rodney Hill,et al.  Elastic potentials and the structure of inelastic constitutive laws , 1973 .

[9]  K. Chau.  Buckling, barrelling, and surface instabilities of a finite, transversely isotropic circular cylinder , 1995 .

[10]  Rodney Hill,et al.  Some basic principles in the mechanics of solids without a natural time , 1959 .

[11]  Otto T. Bruhns,et al.  Bounds to bifurcation stresses in solids with non-associated plastic flow law at finite strain , 1981 .

[12]  D. Bigoni,et al.  Eigenvalues of the Elastoplastic Constitutive Operator , 1994 .

[13]  Kenneth Runesson,et al.  A note on nonassociated plastic flow rules , 1989 .

[14]  I. Vardoulakis.  Bifurcation analysis of the plane rectilinear deformation on dry sand samples , 1981 .

[15]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[16]  Michael Ortiz,et al.  Effect of boundaries and interfaces on shear-band localization , 1991 .

[17]  H. Petryk,et al.  Theory of Bifurcation and Instability in Time-Independent Plasticity , 1993 .

[18]  M. Kleiber.  Numerical study on necking-type bifurcations in void-containing elastic-plastic material , 1984 .

[19]  M. Gurtin,et al.  An introduction to continuum mechanics , 1981 .

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

[21]  Shear band formulations in finite strain elastoplasticity , 1994 .

[22]  J. Rudnicki,et al.  Bifurcations of compressible pressure-sensitive materials in plane strain tension and compression , 1990 .

[23]  Ioannis Vardoulakis,et al.  Rigid granular plasticity model and bifurcation in the triaxial test , 1983 .

[24]  R. Hill,et al.  On discontinuous plastic states, with special reference to localized necking in thin sheets , 1952 .

[25]  Uniqueness and bifurcation in elastic-plastic solids , 1993 .

[26]  B. Loret,et al.  Localization of deformation in plane elastic–plastic solids with anisotropic elasticity , 2000 .

[27]  N. S. Ottosen,et al.  Acceleration Waves in Elasto-Plasticity , 1991 .

[28]  Tomasz Hueckel,et al.  Coupling of elastic and plastic deformations of bulk solids , 1976 .

[29]  John W. Hutchinson,et al.  Post-bifurcation behavior in the plastic range , 1973 .

[30]  P. Chadwick,et al.  Singular surfaces in linear thermoelasticity , 1965 .

[31]  D. Bigoni,et al.  Loss of strong ellipticity in non-associative elastoplasticity , 1992 .

[32]  R. Hill.  The mathematical theory of plasticity , 1950 .

[33]  K. Thermann,et al.  Second-order bifurcation in elastic-plastic solids , 1985 .

[34]  Plastic bifurcation and postbifurcation analysis for generalized standard continua , 1989 .

[35]  J. Miles.  Fluid-pressure eigenstates and bifurcation in tension specimens under lateral pressure , 1973 .

[36]  H. Petryk.  Material instability and strain-rate discontinuities in incrementally nonlinear continua , 1992 .

[37]  The energy criteria of instability in time-dependent inelastic solids , 1991 .

[38]  M. Hayes.  ON THE DISPLACEMENT BOUNDARY-VALUE PROBLEM IN LINEAR ELASTOSTATICS , 1966 .

[39]  J. Rice,et al.  Current topics in non-elastic deformation of geological materials , 1979 .

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

[41]  G. Maier,et al.  NONASSOCIATED AND COUPLED FLOW RULES OF ELASTOPLASTICITY FOR ROCK-LIKE MATERIALS , 1979 .

[42]  R. Hill.  A general theory of uniqueness and stability in elastic-plastic solids , 1958 .

[43]  Viggo Tvergaard,et al.  Influence of void nucleation on ductile shear fracture at a free surface , 1982 .

[44]  R. Brannon,et al.  Influence of non-classical elastic-plastic constitutive features on shock wave existence and spectral solutions , 1993 .

[45]  T. Hueckel,et al.  A note on strain localization for a class of non-associative plasticity rules , 1990, Ingenieur-Archiv.

[46]  J. Mandel.  Conditions de Stabilité et Postulat de Drucker , 1966 .

[47]  E. Ryzhak.  On stable deformation of “unstable” materials in a rigid triaxial testing machine , 1993 .

[48]  D. Bigoni,et al.  Strong ellipticity of comparison solids in elastoplasticity with volumetric non-associativity , 1992 .

[49]  Louis Van Hove,et al.  Sur l'extension de la condition de Legendre du calcul de variations aux intégrales multiples à plusieurs fonctions inconnues , 1947 .

[50]  Davide Bigoni,et al.  On the eigenvalues of the acoustic tensor in elastoplasticity , 1994 .

[51]  D. Bigoni,et al.  On uniqueness for frictional contact rate problems , 1999 .

[52]  Viggo Tvergaard,et al.  Cavitation instabilities in elastic-plastic solids , 1991 .

[53]  J. Rice,et al.  A note on some features of the theory of localization of deformation , 1980 .

[54]  R. Hill.  Acceleration waves in solids , 1962 .

[55]  R. Hill.  Eigenmodal deformations in elastic/plastic continua , 1967 .

[56]  John W. Hutchinson,et al.  Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension , 1974 .

[57]  Davide Bigoni,et al.  The quasi-static finite cavity expansion in a non-standard elasto-plastic medium , 1989 .

[58]  B. Loret.  Does deviation from deviatoric associativity lead to the onset of flutter instability , 1992 .

[59]  R. Ogden.  LOCAL AND GLOBAL BIFURCATION PHENOMENA IN PLANE STRAIN FINITE ELASTICITY , 1985 .

[60]  M J Keedwell,et al.  Rheology and soil mechanics , 1984 .

[61]  T. Hueckel,et al.  Uniqueness and localization. II, Coupled elastoplasticity , 1991 .

[62]  William Prager.  Discontinuous Fields of Plastic Stress and Flow , 1954 .

[63]  N. Young.  Bifurcation phenomena in the plane compression test , 1976 .

[64]  G. Piero,et al.  Some properties of the set of fourth-order tensors, with application to elasticity , 1979 .

[65]  D. Bigoni.  On flutter instability in elastoplastic constitutive models , 1995 .

[66]  C. Horgan,et al.  Cavitation in Nonlinearly Elastic Solids: A Review , 1995 .

[67]  D. Schaeffer,et al.  The flutter instability in granular flow , 1992 .

[68]  J. Prévost,et al.  Loss of hyperbolicity in elastic-plastic solids with deviatoric associativity , 1990 .

[69]  Rodney Hill,et al.  Progress in solid mechanics , 1963 .

[70]  M. Gurtin.  The Linear Theory of Elasticity , 1973 .

[71]  R. Hill,et al.  On constitutive inequalities for simple materials—I , 1968 .

[72]  J. Rice,et al.  CONDITIONS FOR THE LOCALIZATION OF DEFORMATION IN PRESSURE-SENSITIVE DILATANT MATERIALS , 1975 .

[73]  J. Hadamard,et al.  Leçons sur la propagation des ondes et les equations de l'hydrodynamique , 2015 .

[74]  Alan Needleman,et al.  Non-normality and bifurcation in plane strain tension and compression , 1979 .

[75]  Surface boundary conditions trigger flutter instability in non-associative elastic-plastic solids , 1995 .

[76]  B. Loret,et al.  Effects of elastic anisotropy on strain localization andflutter instability in plastic solids , 1999 .

[77]  Millard F. Beatty,et al.  Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues—With Examples , 1987 .

[78]  John W. Hutchinson,et al.  Bifurcation phenomena in the plane tension test , 1975 .

[79]  K. Chau.  Non-normality and bifurcation in a compressible pressure-sensitive circular cylinder under axisymmetric tesion and compression , 1992 .

[80]  Davide Bigoni,et al.  Uniqueness and localization—I. Associative and non-associative elastoplasticity , 1991 .