Anti-symmetric motion of a pre-stressed incompressible elastic layer near shear resonance

A two-dimensional model is derived for anti-symmetric motion in the vicinity of the shear resonance frequencies in a pre-stressed incompressible elastic plate. The method of asymptotic integration is used and a second-order solution, for infinitesimal displacement components and incremental pressure, is obtained in terms of the long-wave amplitude. The leading-order hyperbolic governing equation for the long-wave amplitude is observed to be not wave-like for certain pre-stressed states, with time and one of the in-plane spatial variables swapping roles. This phenomenon is shown to be intimately related to the possible existence of negative group velocity at low wave number, i.e. in the vicinity of shear resonance frequencies

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

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near its cut-off frequencies is derived. Leading-order solutions for displacement and pressure are obtained in terms of the long wave amplitude by direct asymptotic integration. A governing equation, together with corrections for displacement and pressure, is derived from the second-order problem. A novel feature of this (two-dimensional) hyperbolic governing equation is that, for certain pre-stressed states, time and one of the two (in-plane) spatial variables can change roles. Although whenever this phenomenon occurs the equation still remains hyperbolic, it is clearly not wave-like. The second-order solution is completed by deriving a refined governing equation from the third-order problem. Asymptotic consistency, in the sense that the dispersion relation associated with the two-dimensional model concurs with the appropriate order expansion of the three-dimensional relation at each order, is verified. The model has particular application to stationary thickness vibration of, or transient response to high frequency shock loading in, thin walled bodies

On localised vibrations in incompressible pre-stressed transversely isotropic elastic solids

This paper is concerned with 2D localised vibration in incompressible pre-stressed fibre-reinforced elastic solids and the closely related problem of surface wave propagation in such materials. An appropriate constitutive model is derived and its stability discussed within the context of the strong ellipticity condition. Surface wave propagation in an associated half-space is considered, with the particular cases of propagation along a principal direction of primary deformation and that of almost inextensible fibres also investigated. The problems of free and forced edge vibration of a semi-infinite strip are analysed, revealing a link between the natural edge frequencies and the associated Rayleigh surface wave speed

On a Lamb-type problem for a bi-axially pre-stressed incompressible elastic plate

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The definitive publisher authenticated version J. D. KAPLUNOV AND A. V. PICHUGIN (2006). On a Lamb-type problem for a bi-axially pre-stressed incompressible, IMA Journal of Applied Mathematics. 71, 171−185. OUP, is available online at: http://dx.doi.org/10.1093/imamat/hxh097The far-field response of a bi-axially pre-stressed incompressible elastic plate, subjected to an instantaneous edge impulse loading, is studied using a refined long-wave low-frequency theory. The second-order correction introduced by the refined theory is demonstrated to smooth the discontinuity associated with one of the wave-fronts predicted by the leading order hyperbolic theory. The character of the so-called quasi-front is shown to depend greatly on both the material parameters and the pre-stress and may be either classical receding or advancing. Additionally, and in contrast to the analogous problem in linear isotropic elasticity, in a pre-stressed plate the dilatational quasi-front may propagate slower than the shear wave-front. This situation is demonstrated to lead to the formation of a head-wave quasi-front

On the problem of a thin rigid inclusion embedded in a Maxwell material

We consider a plane viscoelastic body, composed of Maxwell material, with a crack and a thin rigid inclusion. The statement of the problem includes boundary conditions in the form of inequalities, together with an integral condition describing the equilibrium conditions of the inclusion. An equivalent variational statement is provided and used to prove the uniqueness of the problem’s solution. The analysis is carried out in respect of perfect and non-perfect bonding of the rigid inclusion. Additional smoothness properties of the solutions, namely the existence of the time derivative, are also established

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

We address the problem of wave propagation in a pre-stressed elastic layer with mixed boundary conditions, the layer having one fixed and one free face. Numerical analysis provides a good initial insight into the influence of these boundary conditions on dispersion characteristics. In the long wave regime, there is clearly no evidence of low-frequency motion and thus an absence of any long wave fundamental mode-like features. In the short wave regime, however, the dispersion relations does show evidence of low-frequency dispersion phenomena. The first harmonic’s short wave phase speed limit is shown to be distinct from that of all other harmonics; this coincides with the associated Rayleigh surface wave speed. The short wave analysis is completed with the derivation of approximate solutions for the higher harmonics. Asymptotic long wave approximations of the dispersion relation are then obtained for motion within the vicinity of the thickness stretch and thickness shear resonance frequencies. These approximations are required to obtain the relative asymptotic orders of the displacement components for frequencies within the vicinity of either the shear or stretch resonance frequencies. This enables an analogue of the asymptotic stress-strain state to be established through asymptotic integration

An Ill-posed Cauchy Type Problem for an Elastic Strip

This study deals with an incorrectly posed, plane elasticity, boundary value problem for a strip. The strip is loaded by a concentrated load of known intensity applied to one side and the displacements on this side are also known. The problem is therefore over-determined on one side of the boundary; in contrast no boundary conditions are specified on the other side of the strip. Therefore, the problem is ill-posed with the specified boundary conditions. The problem can be reduced to a system of integral equations derived from basic properties of holomorphic functions, which are used to prove uniqueness of the considered boundary value problem. An analytical solution of the problem is obtained by applying Fourier transforms. The inversion of the Fourier transform is performed with the use of the Stieltjes integral. This is a non-stable operation, which necessitates the application of a regularisation technique in order to build stable solutions. For numerical implementation we discuss the regularisation procedure based on the SVD truncation method