On Saint-Venant's principle in finite anti-plane shear: An energy approach

Abstract

Some time ago a version of Saint-Venant’s principle was formulated and established for finite elastostatics [1]. As was discussed in [1], the issues of concern in connection with Saint-Venant’s principle in the nonlinear theory of elasticity are considerably more involved then those arising in the linear theory. (For a survey of results on Saint-Venant’s principle, primarily for linear theories, see e.g. [2–5].) One difficulty is that the appropriate Saint-Venant solutions need to be carefully characterized (see e.g. [6–13] and the references cited therein). Secondly, in the absence of superposition, consideration of self-equilibrated end loads is no longer sufficient. Furthermore, instabilities may have to be taken into account. Also the decay rate for end effects, even if exponential, might depend on the overall loading as well as on geometry and material characteristics. Several of these issues have been considered in recent studies in the nonlinear elasticity context [1, 14–23] as well as in investigations of spatial decay of solutions of nonlinear elliptic partial differential equations [24–32].