Generalized Hooke's law for isotropic second gradient materials

In the spirit of Germain the most general objective stored elastic energy for a second gradient material is deduced using a literature result of Fortuné & Vallée. Linear isotropic constitutive relations for stress and hyperstress in terms of strain and strain-gradient are then obtained proving that these materials are characterized by seven elastic moduli and generalizing previous studies by Toupin, Mindlin and Sokolowski. Using a suitable decomposition of the strain-gradient, it is found a necessary and sufficient condition, to be verified by the elastic moduli, assuring positive definiteness of the stored elastic energy. The problem of warping in linear torsion of a prismatic second gradient cylinder is formulated, thus obtaining a possible measurement procedure for one of the second gradient elastic moduli.

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

[2]  Pierre Seppecher.  Etude des conditions aux limites en théorie du second gradient: cas de la capillarité , 1989 .

[3]  Wolfgang Ehlers,et al.  On theoretical and numerical methods in the theory of porous media based on polar and non-polar elasto-plastic solid materials , 1998 .

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

[5]  M. Sokołowski.  Theory of couple-stresses in bodies with constrained rotations , 1970 .

[6]  W. Drugan,et al.  A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites , 1996 .

[7]  L. Schwartz.  Théorie des distributions , 1966 .

[8]  Bianchi identities in the case of large deformations , 2001 .

[9]  R. Borst,et al.  Localisation in a Cosserat continuum under static and dynamic loading conditions , 1991 .

[10]  A. Goldman.  Sur une conjecture de D.G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne , 1998 .

[11]  P. Seppecher,et al.  Edge Contact Forces and Quasi-Balanced Power , 1997, 1007.1450.

[12]  E. Aifantis,et al.  Strain gradient elasticity theory for antiplane shear cracks. Part I: Oscillatory displacements , 2000 .

[13]  H. Georgiadis,et al.  Problems of the Flamant–Boussinesq and Kelvin Type in Dipolar Gradient Elasticity , 2007 .

[14]  Ching S. Chang,et al.  Application of higher-order tensor theory for formulating enhanced continuum models , 2000 .