Linear and Nonlinear Wave Propagation in Coated or Uncoated Elastic Half-spaces

Abstract

In these lectures, we discuss the following three closely related topics.(i) Unification of different methods for deriving evolution equations for surface acoustic waves. Early studies on nonlinear surface acoustic waves were thwarted by very complicated derivation of evolution equations. Worse still, different methods seemed to have given different evolution equations. Later on, it became known that all these methods except one yield the same evolution equation, but even at the time when we started to prepare the current lecture notes, there still existed a method that does not agree with the other methods. Such a situation is unsatisfatory since each method has some following. The purpose of the lectures on this topic is three-fold. Firstly, we aim to show that derviation of the evolution eqution for nonlinear surface waves can be carried out in one A4 page even in the most general case. Secondly, we show that the odd method that used to give a different evolution equation can in fact be used to obtain the same evolution equation if it is properly executed. Thus, we set the record straight: all known methods should and do give the same evolution equation! Thirdly, we express our evolution equation in terms of results from the linear surface-wave theory built on the Stroh formulation, and we explain how the coefficients in the evolution equation can be evaluated efficiently.(ii) Linear wave propagation in a coated elastic half-space. This is partly in preparation for our discussion of the third topic, but the problem is of much interest in its own right. We show how the dispersion relation can be expressed elegantly in terms of the surface-impedance matrices associated with the layer and the half-space. We derive a two-term expression for the wave speed in the long-wavelength limit.(iii) Periodic and solitary waves in a coated elastic half-space. An uncoated elastic half-space cannot in general support solitary waves due to lack of dispersion although it has been argued previously that the nonlocal character of nonlinearity may give rise to the existence of steady travelling waves. We derive the nonlinear evolution equation for small-amplitude long-wavelength travellingThe original publication is available at www.springerlink.com http://www.springerlink.com/content/978-3-211-73571-8/#section=264495&page=2&locus=50 2 waves propagating in a coated elastic half-space where the thin coating induces weak dispersion. When this evolution equation is linearized, we recover the two-term dispersion relation obtained in (ii). We explain a simple method that can be used to compute periodic or solitary travelling-wave solutions.

Surface waves supported by thin-film/substrate interactions

“…For the case of vanishing residual stress and with the O(h 2 ) terms neglected, these equations are consistent with equation (4.6) of Fu (2007), which is valid for materials having general symmetry. Further papers relevant to this subject are those by Cai & Fu (2000) and Every & Shuvalov (2002).…”

A long-wave model for the surface elastic wave in a coated half-space

“…As examples of ongoing interest to the problem, we mention recent publications (Fu 2007;Steigmann & Ogden 2007;Qian et al 2009 and references therein).…”

“…The theory of elastic surface waves for coated bodies is highly important for many modern applications, for example, thin-film technology. As examples of ongoing interest to the problem, we mention recent publications (Fu 2007;Steigmann & Ogden 2007;Qian et al 2009 and references therein). *Author for correspondence (prikazchikovda@yandex.ru).…”

A refined model for the buckling of film/substrate bilayers

“…Having derived the relation between traction and displacement at the interface from the side of the coating layer, we now employ the Stroh formulation (Stroh, 1958) to derive its counterpart from the side of half-space. The following derivations are adapted from Fu (2007).…”

“…It was shown in Fu (2007) that the matrix G is related to the surface impedance matrix M (Ingebrigtsen & Tonning, 1969) by…”