Dispersion of Small Amplitude Waves in a Pre-Stressed, Compressible Elastic Plate

Small amplitude waves in a pre-stressed compressible elastic layer with one fixed and one free face

“…For the parameters used in Figure 1(a), this corresponds to a shear wave with speedv S = √ γ 1 ≈ 0.62. Following a labelling system introduced in [10], we refer to a wave front formed this way as a type 1 wave front. We also notice from Figure 1(a) a flattening of the harmonics in a narrow wave speed region, forming a longitudinal wave front, travelling with the speedv L = √ α 11 ≈ 1.16.…”

“…The short wave limit of all harmonics, apart from the first, defined byv =v O ≈ 0.76, demonstrate oscillatory behaviour in the region below v =v S . We remark that for a pre-stressed compressible elastic material, a shear wave can travel faster, with the same speed or even slower than a longitudinal wave, a fact seemingly first noted in [10]. For investigation of the short wave limiting behaviour we shall use the classification, adopted in [10], namely:…”

“…In passing, we remark that if γ 2 = α 22 there is a coincidence of both the speeds of the longitudinal and shear waves propagating along Ox 2 ; and the thickness stretch and thickness shear resonance frequencies. For a discussion of the associated dispersion phenomena the reader is referred to [10]. Dividing both sides of the dispersion relation (29) by cos(2q 1 η) cos(2q 2 η), we obtain two asymptotic balances, namely tan(…”

“…After the numerical investigations in the plane strain problem were carried out in [7] and [8], an asymptotic long and short wave analysis was carried out for a variety of layered pre-stressed structures, see [9] and [10]. Three dimensional motion in the form of a plane wave travelling parallel to the plane of the layer was also examined in [11].…”

Abnormal long wave dispersion phenomena in a slightly compressible elastic plate with non-classical boundary conditions

“…In fairly recent times this problem has been extended to elucidate both the influences of pre-stress and/or anisotropy, see for example [1] and [2] in the incompressible case and [3] and [4] for the compressible case. There have also been a small number of articles investigating the effect of different conditions on the faces, and in particular so-called fixed face conditions whereby the boundary conditions are taken to be those of zero dispalcement, see [5] and [6].…”