Oscillatory Structured Shock Waves in a Nonlinear Elastic Rod With Weak Viscoelasticity

Abstract

The propagation of longitudinal shock waves in a thin circular viscoelastic rod is investigated theoretically as the counterpart of the torsional shock waves previously considered in [1, 2]. Assuming a “nearly elastic” rod, the approximate equation is first derived by taking account of not only the finite deformation but also the lateral contraction or dilatation of rod. The latter gives rise to the geometrical dispersion, which is taken in the form of Love’s theory for an elastic rod. Taking two typical relaxation functions, the structures of the steady shock waves are investigated in detail, one being the exponential function type and the other the power function type. The effect of geometrical dispersion is emphasized. Finally a brief discussion is included on the simplified evolution equations for a far field behavior.