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 well-known gradient models involving the Laplacians of stress and strain tensors. Finally, the structure of the dynamical equations is given for strain and stress gradient media, showing fundamental differences in the dynamical behavior of these two classes of generalized continua.
Reference: Forest and K. Sab, Finite deformation second order micromorphic theory and its relations to strain and stress gradient models, Mathematics and Mechanics of Solids, 2017.