Theory of mixtures for micromorphic materials-II. Elastic constitutive equations

Abstract

In Part I of this work[1], the balance equations were derived for micromorphic and micropolar mixtures. In this part, the general form of the non-linear, anisotropic, elastic, constitutive equations for micromorphic and micropolar mixtures are developed. Such equations should have application in describing the behavior of polyatomic or polymolecular crystal lattices as well as that of such materials as polycrystalline mixtures and granular composites. As an illustration, the general micropolar equations are given explicitly for a linear elastic, isotropic, two constituent mixture. The field equations are developed for the case of restricted coupling, and with these the propagation of a plane wave is studied. The longitudinal and transverse displacement waves of classical theory arise as well as longitudinal and transverse microrotation waves. Dispersion of the two displacement waves arises from the intrinsic structure of the material and is expected to be of importance at high frequencies. The two transverse waves are complexly coupled. Simplification by assuming no microrotation shows that the propagation velocity of the classical transverse displacement wave is dependent on the microrotation material parameter.