An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate

Abstract

An asymptotically consistent two-dimensional theory is developed to help elucidate dynamic response in finitely deformed layers. The layers are composed of incompressible elastic material, with the theory appropriate for long-wave motion associated with the fundamental mode and derived in respect of the most general appropriate strain energy function. Leading-order and refined higher-order equations for the mid-surface deflection are derived. In the case of zero normal initial static stress and in-plane tension, the leading-order equation reduces to the classical membrane equation, with its refined counterpart also being obtained. The theory is applied to a one-dimensional edge loading problem for a semi-infinite plate. In doing so, the leading- and higher-order governing equations are used as inner and outer asymptotic expansions, the latter valid within the vicinity of the associated quasi-front. A solution is derived by using the method of matched asymptotic expansions.