Waves and uniqueness in multi-porosity elasticity

Abstract

ABSTRACTIn this article, we present and review equations which describe the evolutionary behavior of double and triple porosity elastic materials. Uniqueness and continuous dependence results are described for a double porosity anisotropic elastic body when the elastic coefficients are positive, in a precise sense. We then review work on uniqueness and continuous dependence when the elastic coefficients are not required to be sign definite. With the advent of auxetic materials, and other materials which have negative Poisson ratios, sign indefinite elastic coefficients are important. We further establish uniqueness for a double porosity elastic body which occupies an unbounded spatial domain. This is not a trivial extension from the bounded domain case since the pressures and deformation and their derivatives are allowed to have substantial growth in space at infinity and so classical techniques like the energy method fail. After this we review work on acceleration wave evolution in a linear elastic double porosity body and then in a nonlinear elastic one. For the linear elastic double porosity problem, it is seen that the amplitude of the acceleration wave will decay exponentially in time. In the nonlinear case, however, we see that shocks may develop in a finite time. Finally, we present models for the evolutionary behavior of a triple porosity linearly elastic body.