On interfacial waves in pre-stressed layered incompressible elastic solids

Abstract

Interfacial waves along the plane boundary between two pre-stressed incompressible elastic solids are considered. One of the solids is a half-space while the other has arbitrary uniform thickness. The principal axes of the underlying pure homogeneous deformation in the two solids are aligned, with one axis normal to the interface. For propagation along an in-plane principal axis, the dispersion equation is derived in respect of a general strain-energy function. Conditions on the pre-strain, pre-stress and material parameters that ensure the existence of a unique interfacial wavespeed at low frequencies are obtained, and it is shown that, in special circumstances, non-dispersive waves can exist at the low-frequency limit. Asymptotic results at the high-frequency limit are also obtained. For the case of equibiaxial pre-strain, more specific conditions are derived for the existence of interfacial waves at the low- and high-frequency asymptotes, and these provide information on the existence of waves for the whole frequency range. A particular feature of the structure considered is that it may act as a mechanical filter in different frequency regimes depending on the pre-strain, pre-stress and material parameters. When the wavespeed vanishes, the dispersion equation reduces to a bifurcation equation, solutions of which define states of stress and deformation which form boundaries of the region of stability of the underlying state of stress and deformation in the two materials for given material properties. The bifurcation equation is examined separately and an explicit bifurcation criterion is given for equibiaxial deformations. The results are illustrated graphically by considering several numerical examples based on a certain class of strain-energy functions, which includes the neo-Hookean strain-energy function. The results highlight low- and high-frequency features and demonstrate the influence of pre-stress and deformation on the multiplicity of propagating modes.

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

“…Although pre-stress may well be introduced during the manufacturing process, it is the aforementioned load supporting scenario that mainly motivates this work. In view of the important applications of this particular material model within such a context, a number a recent studies have focused on various aspects of wave propagation and vibration in plates, half-spaces and layered structures composed of such material, see for example Dowaikh & Ogden (1991); Ogden & Roxburgh (1993); Ogden & Sotiropoulos (1995); Rogerson & Fu (1995);Rogerson (1997); Rogerson & Sandiford (1997).…”

The effect of initial stress on the propagation of surface waves in a layered half-space

“…The formula (80) is the same as one derived in Ogden and Sotiropoulos (1995) [equation First, the limiting case kh → 0 corresponds to a half-space for which the secular equation…”

“…Following the arguments in Dowaikh and Ogden (1990) and Ogden and Sotiropoulos (1995) we may deduce that there is an upper bound on the wave speed according to…”

“…For a layered half-space of incompressible isotropic elastic material subject to a pure homogeneous finite deformation and an accompanying stress (a so-called pre-stress) the propagation of Rayleigh-type surface waves in a principal plane of the underlying deformation was examined in detail in Ogden and Sotiropoulos (1995) on the basis of the linearized theory of incremental deformations superimposed on a finite deformation. In the special case of the Murnaghan theory of second-order elasticity Akbarov and Ozisik (2004) also examined the effect of pre-stress on the propagation of surface waves.…”

Resonance phenomena at the interface of two perfectly bonded, prestressed elastic strips

“…This introduces the effect of dispersion, which is again highly sensitive to changes in the principal stretches. Such dependency has been investigated by Ogden and Sotiropoulos [1995] and Sotiropoulos and Sifniotopoulos [1995].…”