Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models

Samuel Forest (1) , Karam Sab (2)

1 CDM - Centre des Matériaux
2 NAVIER UMR 8205 - Laboratoire Navier

Germain's general micromorphic theory of order n is extended to fully non-symmetric higher-order tensor degrees of freedom. An interpretation of the microdeformation kinematic variables as relaxed higher-order gradients of the displacement field is proposed. Dynamical balance laws and hyperelastic constitutive equations are derived within the finite deformation framework. Internal constraints are enforced to recover strain gradient theories of grade n. An extension to finite deformations of a recently developed stress gradient continuum theory is then presented, together with its relation to the second-order micromorphic model. The linearization of the combination of stress and strain gradient models is then shown to deliver formulations related to Eringen's and Aifantis's well-known gradient models involving the Laplacians of stress and strain tensors. Finally, the structures of the dynamical equations are given for strain and stress gradient media, showing fundamental differences in the dynamical behaviour of these two classes of generalized continua.

Mots clés: generalized continua; dispersion relations; micromorphic continuum; strain gradient theory; stress gradient theory; finite deformation; method of virtual power

Domaines: Mécanique des matériaux [physics.class-ph]; Physique mathématique [math-ph]

Samuel Forest, Karam Sab. Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models. Mathematics and Mechanics of Solids, 2020, 25 (7), pp.1429-1449. ⟨10.1177/1081286517720844⟩. ⟨hal-02919094⟩