Flexural waves in incompressible pre-stressed elastic composites

Abstract

In this paper some dynamic properties of symmetric, 4-ply (3 layer), pre-stressed, incompressible elastic laminated plates are discussed. The dispersion relation arising from flexural wave propagation is derived in respect of arbitrary strain energy functions. An asymptotic high wave number analysis is carried out which establishes that there are six possible phase speed limits of the dispersion relation, these being a surface wave speed, an interfacial wave speed or one of four associated shear wave speeds. All harmonics, with the possible exception of the first, are shown to tend to the least of all four shear wave speeds. These four distinct cases are all fully analysed with asymptotic expansions for the phase speed obtained up to and including third order. In two of the four cases sinusoidal terms are found to occur in these expansions at third order. If surface and interfacial waves exist then the corresponding wave speeds are shown to be the high wave number limit of the first two branches of the dispersion relation, provided they are lower in magnitude than all four shear wave speeds. In the case of interfacial waves this will always be the case; however, it is quite possible to have a surface wave speed greater than one, or even two, of these shear wave speeds. In such cases a distinct flattening of dispersion curve branches in the neighbourhood of the surface wave speed is observed, together with extremely flat maxima in neighbouring wave number regions on the associated group velocity curves. The implication is that in such cases a surface wave front will be formed from the combination of the various harmonics, rather than the high wave number limit of a single branch. The paper is concluded with a discussion of quasi-static solutions, neutral curves and stability. It is shown that possible regions exist in the low wave number regime within which bifurcation from the underlying homogeneous deformation is not possible.