The principle of Saint-Venant in linear and non-linear plane elasticity

A two-dimensional elastic isotropic body {Mathematical expression} is considered. The body has self-equilbrated loadings on its short ends and is unstressed on its long sides. If it is assumed that the first four derivatives of the displacement vector are uniformly bounded and the bounds are sufficiently small, it can be shown from standard equations of non-linear elasticity that {Mathematical expression} where (i) e(x, y) is the strain tensor at any point (x, y). (ii) e{open}=sup | e(x,y)| (x, y) ε R (iii) k and a are positive constants which depend only upon the uniform bounds for the derivatives of the displacement vector. If the equations of linear elasticity are used, it need only be assumed that the norm of the strain tensor is uniformly bounded by e{open} (not necessarily small) to obtain {Mathematical expression} where a is an arbitrary positive parameter, k is a positive function of a. The quantity a may be chosen so as to give an optimal value for k. A discussion is also given as to how the above results can be extended to more general domains.