Acceleration waves and shock waves in transversely isotropic elastic non-conductors

Abstract

We consider the propagation of plane acceleration waves and weak nonlinear shock waves in an unstrained transversely isotropic elastic non-conductor which is incompressible but extensible along the preferred direction denoted by the unit vector e. It is shown that such a material may transmit two different shock waves in the direction of any unit vector n which is neither perpendicular nor parallel to the preferred direction e. One of these shock waves is polarised in the plane Π spanned by e and n and carries an entropy jump of third order in the shock amplitude, whilst the other is polarised perpendicularly to Π and carries merely a fourth order entropy jump. In the degenerate cases when the angle θ between n and e is either π/2 or 0, the two different shock waves which can propagate both carry fourth order entropy jumps. In all of these cases, an asymptotic evolution law describing the decay of the shock amplitude is obtained from a reduced first order governing partial differential equation which is itself obtained by differentiating a certain Riemann invariant that is approximately constant under the isentropic assumption adopted here (in common with all studies of non-conductors). We consider also the limiting case in which the material is inextensible along the preferred direction whilst remaining incompressible. It is shown that when θ = π 2 , two shock waves can propagate as before, but when θ = 0, no shock wave can propagate, and when θ ≠ π 2, 0 , only one shock wave can propagate. Shock waves which can propagate in such a constrained material are all of the second type. The shock wave travelling in the direction of inextensibility is allowed by the purely linear theory of such constrained materials but is forbidden by the weak nonlinear theory.