On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models

Higher order gradient continuum theories have often been proposed as models for solids that exhibit localization of deformation (in the form of shear bands) at sufficiently high levels of strain. These models incorporate a length scale for the localized deformation zone and are either postulated or justified from micromechanical considerations. Of interest here is the consistent derivation of such models from a given microstructure and the subsequent comparison of the solution to a boundary value problem using both the exact microscopic model and the corresponding approximate higher order gradient macroscopic model.In the interest of simplicity the microscopic model is a discrete periodic nonlinear elastic structure. The corresponding macroscopic model derived from it is a continuum model involving higher order gradients in the displacements. Attention is focused on the simplest such model, namely the one whose energy density involves only the second order gradient of the displacement. The discrete to continuum comparisons are done for a boundary value problem involving two different types of macroscopic material behavior. In addition the issues of stability and imperfection sensitivity of the solutions are also investigated.

[1]  A. Cemal Eringen,et al.  Nonlinear theory of micro-elastic solids—II☆ , 1964 .

[2]  B. D. Coleman.  Necking and drawing in polymeric fibers under tension , 1983 .

[3]  J. K. Knowles,et al.  Non-elliptic elastic materials and the modeling of elastic-plastic behavior for finite deformation , 1987 .

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

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

[6]  J. K. Knowles,et al.  On the dissipative response due to discontinuous strains in bars of unstable elastic material , 1988 .

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

[8]  Elias C. Aifantis,et al.  The physics of plastic deformation , 1987 .

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

[10]  A. Molinari,et al.  Analytical Characterization of Shear Localization in Thermoviscoplastic Materials , 1987 .

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

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

[13]  M. Gurtin,et al.  Structured phase transitions on a finite interval , 1984 .

[14]  G. Geymonat,et al.  QUELQUES REMARQUES SUR L'HOMOGENEISATION DES MATERIAUX ELASTIQUES NONLINEAIRES , 1990 .

[15]  Attila Askar,et al.  Lattice dynamical foundations of continuum theories , 1986 .

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

[17]  T. Belytschko,et al.  Localization limiters in transient problems , 1988 .

[18]  Nicolas Triantafyllidis,et al.  A gradient approach to localization of deformation. I. Hyperelastic materials , 1986 .

[19]  D. Gazis,et al.  SURFACE EFFECTS AND INITIAL STRESS IN CONTINUUM AND LATTICE MODELS OF ELASTIC CRYSTALS , 1963 .

[20]  R. Clifton.  High Strain Rate Behavior of Metals , 1990 .

[21]  Stelios Kyriakides,et al.  The initiation and propagation of a localized instability in an inflated elastic tube , 1991 .

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

[23]  B. Maker,et al.  On the Comparison Between Microscopic and Macroscopic Instability Mechanisms in a Class of Fiber-Reinforced Composites , 1985 .

[24]  E. Aifantis,et al.  A note on the paper “on the thermodynamic theory of fluid interfaces: Infinite intervals, equilibrium solutions, and minimizers” , 1990 .

[25]  Bernard D. Coleman,et al.  On shear bands in ductile materials , 1985 .

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

[27]  A. Cemal Eringen,et al.  NONLINEAR THEORY OF SIMPLE MICRO-ELASTIC SOLIDS-I , 1964 .

[28]  Z. Bažant.  Softening Instability: Part I—Localization Into a Planar Band , 1988 .

[29]  Nicolas Triantafyllidis,et al.  An Investigation of Localization in a Porous Elastic Material Using Homogenization Theory , 1984 .

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

[31]  R. James.  Displacive phase transformations in solids , 1986 .

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

[33]  J. Waals.  The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density , 1979 .

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

[35]  I. Kunin,et al.  Elastic Media with Microstructure I: One-Dimensional Models , 1982 .

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

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

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