ON THE CAVITIES AND RIGID INCLUSIONS CORRESPONDENCE AND THE COSSERAT SPECTRUM

The correspondence in two-dimensional elasticity between the stress fi elds of cavities and rigid inclusions has been obtained by DUNDURs [1] and MARKENSCOFF [3]. It was shown that if the limit of the stress of the inclusion boundary-value problem, which depends on the elastic con stants, exists when the Poisson's ratio nu tends to 1, then this solve s the traction boundary-value problem for the cavity problem since it satisfies equilibrium and boundary conditions, and, by the uniqueness theorem, exists and is unique. In three dimensions the solution of the traction boundary-value problem of elasticity does depend on Poisson' s ratio since the Beltrami-Mitchell compatability conditions for the s tress depend on Poisson's ratio. So the similar argument for the corre spondence between cavities and rigid inclusions cannot in principle be made. However, the Beltrami-Mitchell compatability conditions are ind ependent of nu if the dilatation is a constant or a Linear function of the position. In this case we can show that the same result goes thro ugh for the correspondence. In order to investigate the behavior of th e solutions in the vicinity of nu = 1, we use some results obtained fo r the Cosserat spectrum by MIKHLIN [4], MAZ'YA and MIKHLIN [3], see al so [6]. The existence of the limit for 20 and 3D when nu tends to 1 is proved on the basis of the fact that the eigenvalue omega = - 1 of th e Cosserat spectrum is isolated.