Small vibrations of prestrained constrained elastic materials: The idealized fibre-reinforced material

We derive the general equations satisfied by small vibrations of arbitrary form superimposed upon a finite, static deformation (not necessarily homogeneous) of an elastic body of arbitrary anisotropy suffering an unspecified number of constraints of fully general form. Specialization is then made to a fibre-reinforced material, modelled here as an incompressible material that is inextensible in the fibre direction. The slowness surface is a one-sheeted, centro-symmetric surface except that two slownesses are possible for waves travelling along, or normal to, the fibredirection. In many of these exceptional directions the slowness surface exhibits singular behaviour which is fully discussed. Numerical illustrations are presented.