On the existence and formulation of variational principles for systems of scalar partial differential equations

A generalization of the effective stiffness theory of Sun, Achenbach and Herrmann is used to derive a set of approximate equations describing the propagation of a stress wave in a non-linear, geometrically dispersive composite material. The equations which are derived are singular perturbation equations having two kinds of singularities, one caused by dispersion and one caused by the non-linearity. A non-linear dispersion equation which describes the propagation of a periodic wave is derived using an averaged Lagrangian technique developed by Whitham. It is shown that the resulting solution is a simple wave system. With this technique, the singularity associated with the non-linearity may be eliminated and a solution which is convergent beyond the time of shock formation found. Application of this solution to a stress pulse and the group velocity for this system are discussed.

P.J. Rausch

Papers

Propagation of periodic waves in a non-linear, geometrically dispersive solid