Exponential Decay for the Saint-Venant Principle

Abstract

We consider a semi-infinite beam B = {(z,t)(VBAR)z(ELEM)(OMEGA), t (GREATERTHEQ) 0} where (OMEGA) is a bounded domain in (//R)('2) with the cone property and a C('3)-smooth boundary. By applying semigroup theory and spectral theory, we show that in our formulation Saint-Venant's principle is true for a class of stored energy functions of the type W = 1/2u('2) + Q(u(,,1),u(,,2)), where u is the displacement along the t-axis. By using the theory of stable manifolds, we also prove the existence of a mild solution for the associated traction boundary value problem in elastostatics.