An Energy Estimate for the Biharmonic Equation and Its Application to Saint-venant’s Principle in Plane Elastostatics

Among the many relatively recent studies of the issue known in the theory of elasticity as Saint-Venant's principle, several have been devoted to the construction of certain kinds of energy inequalities for the biharmonic differential equation in plane domains*. Such inequalities are designed to estimate the spatial rate of decay of stored energy away from a portion of the boundary which carries a "selfequilibrated" load, the remainder of the boundary being traction-free. These energy estimates for two-dimensional, linear problems in plane strain-as well as generalizations of such estimates appropriate to anisotropic materials or to axisymmetric threedimensional problems-have all been based on one or the other of two basic arguments involving differential inequalities. The first of these was given in Knowles (1966) and produced an explicit estimate of the spatial decay rate valid for bounded, simply connected domains of general shape; subsequently, Flavin (1974) showed that a slight modification of the analysis given in Knowles (1966) leads to an improved estimate of the decay rate. The second kind of argument is that given more recently by Oleinik and Yosifian ( l978a, b )*"'; their results apply to bounded domains of general shape and account for the influence on the estimated decay rate of certain geometrical features of the underlying domain not reflected in the results of Knowles (1966) or Flavin (1974).