A gradient approach to localization of deformation 227 materials

By utilizing methods recently developed in the theory of fluid interfaces, we provide a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials. The approach overcomes one of the major shortcomings in constitutive equations for solids admitting localization of deformation at finite strains, i.e. their inability to provide physically acceptable solutions to boundary value problems in the post-localization range due to loss of ellipticity of the governing equations. Specifically, strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density. The modified strain energy function leads to equilibrium equations which remain always elliptic. Explicit solutions of these equations can be found for certain classes of deformations. They suggest not only the direction but also the width of the deformation bands providing for the first time a predictive unifying method for the study of preand post-localization behavior. The results derived here are a three-dimensional extension of certain one-dimensional findings reported earlier by the second author for the problem of simple shear.

[1]  C. Truesdell,et al.  The Classical Field Theories , 1960 .

[2]  William L. Ko,et al.  Application of Finite Elastic Theory to the Deformation of Rubbery Materials , 1962 .

[3]  Richard Bellman,et al.  Plastic Flow and Fracture in Solids , 1962 .

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

[5]  G. Fichera,et al.  Existence Theorems in Linear and Semi‐Linear Elasticity , 1974 .

[6]  E. Aifantis,et al.  On the thermodynamic theory of fluid interfaces: infinite intervals, equilibrium solutions, and minimizers , 1986 .

[7]  J. Serrin,et al.  Equilibrium solutions in the mechanical theory of fluid microstructures , 1983 .

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

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

[10]  E. Aifantis Dislocation Kinetics and the Formation of Deformation Bands , 1983 .

[11]  R. Toupin,et al.  Theories of elasticity with couple-stress , 1964 .

[12]  J. K. Knowles,et al.  On the failure of ellipticity of the equations for finite elastostatic plane strain , 1976 .

[13]  R. S. Rivlin,et al.  Multipolar continuum mechanics , 1964 .

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

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

[16]  J. Serrin,et al.  The mechanical theory of fluid interfaces and Maxwell's rule , 1983 .

[17]  Viggo Tvergaard,et al.  Flow Localization in the Plane Strain Tensile Test , 1981 .

[18]  James K. Knowles,et al.  On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics , 1978 .

[19]  J. K. Knowles,et al.  On the ellipticity of the equations of nonlinear elastostatics for a special material , 1975 .

[20]  Z. Marciniak,et al.  Limit strains in the processes of stretch-forming sheet metal , 1967 .

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

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

[23]  J. E. Dunn,et al.  On the Thermodynamics of Interstitial Working , 1983 .

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

[25]  A. l. Leçons sur la Propagation des Ondes et les Équations de l'Hydrodynamique , 1904, Nature.