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.