Inequalities for the Equilibrium of Slightly Deformable Elastic Bodies

Abstract

In general, the boundary-value problem of static elasticity is difficult to solve even in the linear theory. I shall take up the matter in the next chapter, but now I call attention to the fact that it certainly may be useful, and often it is sufficient, to establish certain inequalities for the components of stress and displacement. After all, a method of integration generally gives the values of the unknown functions only to within some error. For engineering use it is often sufficient to make sure that the magnitude of a given component of stress is less than a certain critical value. Therefore, lower bounds for the stress components may be useful for indicating a dangerous state of stress. To this end certain interesting results were found by Signorini [1, 6, 7], while several other inequalities have been established recently (Grioli [3, 4, 9, 10]; Bressan [3]). The results given in this chapter concern bodies only slightly deformed, so that the classical linear theory is sufficient. However, some of these results remain valid for arbitrary continuous media, but most are valid only for homogeneous, anisotropic, elastic bodies whose elastic potential energy is a positive-definite quadratic form in the stress components.