Group theoretic analysis and similarity solution for a nonlinear viscous rod subjected to time dependent velocity impact

Unidirectional wave motion in a nonlinear viscous rod obeying Norton's law in creep, subjected to time dependent velocity impact is considered. From the basic equations of the problem and the four parameter dimensional group of transformations, absolute invariants of the group are constructed to obtain similarity transformations. Similarity representation of the original system of partial differentiation equations is formulated as a system of nonlinear ordinary differential equations with auxiliary conditions. Closed form solutions are obtained for a linear rod, for a nonlinear rod subjected to constant velocity impact and a weekly nonlinear rod. Nonlinear case is solved by a numerical approach based on the quasilinearization method.