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

Abstract

A two-dimensional elastic isotropic body $$R:{\text{ }}[0 < x{\text{ < 2}}l;{\text{0 < }}y < h]{\text{ with }}\frac{h}{l} \ll {\text{1}}$$ 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 $$\left| {e{\text{(}}x,y{\text{)}}} \right| \leqq \varepsilon {\text{exp}}\left[ { - k\frac{{{\text{(}}x - a{\text{)}}}}{h}} \right],{\text{ }}x < l$$ where (i) e(x, y) is the strain tensor at any point (x, y). (ii) ɛ=sup ¦ e(x,y)¦ (x, y) e R (iii) k and a are positive constants which depend only upon the uniform bounds for the derivatives of the displacement vector.