Material force for dynamic crack propagation in multiphase micromorphic materials

Micromorphic theory, which considers material body as a continuous collection of deformable particles of finite size and inner structure; each has nine independent degrees of freedom describing the stretches and rotations of the particle in addition to the three classical translational degrees of freedom of its center, is briefly introduced in this work. The concept of material forces, which may also be referred as Eshelbian mechanics, is extended to micromorphic theory. The balance law of pseudo-momentum is formulated. The detailed expressions of Eshelby stress tensor, pseudo-momentum, and material forces are derived for thermoelastic micromorphic solid. It is found that the material forces are due to (1) body force and body moment, (2) temperature gradient and (3) material inhomogeneities in density, microinertia, and elastic coefficients. The general expression of material forces due to the presence of dynamically propagating crack front has also been derived. It is found that, at the crack front, material force is reduced to the J-integral in a very special and restrictive case.