Surface integrals for domains with fractal boundaries and some applications to elasticity

Abstract

Some possible formulations of integration of differential forms over a non–smooth boundary are introduced. It is supposed that the differential of a form is integrable on the whole domain. In applications to continuum mechanics, the condition of integral convergence selects the physically interesting cases, when the energy over the domain is finite. Conditions providing continuity of surface integrals for domains in Rn with fractal boundary of non–integer Hausdorff dimension and non–integer box dimension for both continuous and special discontinuous differential forms are described. The results obtained are applied to elastic domains in R3 with fractal boundaries. Some discontinuous differential forms are considered when the forms are formed by the use of the elastic stress and displacement fields. The proof of the uniqueness theorem of solutions to problems of elastostatics for bodies with fractal boundaries is given.