The Quasilinear Wave Equation for Antiplane Shearing of Nonlinearly Elastic Bodies

We formulate an efficient numerical algorithm based on finite-difference approximations and inspired by algorithms from gas dynamics to treat the quasilinear wave equationwtt=[α(wx2+wy2)wx]x+[α(wx2+wy2)wy]ygoverning antiplane motions of incompressible, isotropic nonlinearly elastic bodies in two-dimensions. In particular, we are concerned with the treatment of focusing and shocks for bodies whose material response differs markedly from that of linear elasticity. We carefully validate our method by comparing our results with those of the axisymmetric version of this equation in polar coordinates.