Asymptotic Behaviour of Energy Integrals for Particular Problems of Mathematical Physics
Vladimir Gilelevich Maz'ya, С. А. Назаров, Boris A. Plamenevskij · Birkhäuser Basel eBooks · 2000
This chapter specifies the asymptotic formulas obtained in Chapter 7. The first section deals with Dirichlet’s problem for Laplace’s operator in domains that are disturbed near a corner or conic point and in domains with one or more small holes. The asymptotic behaviour of the energy integral for Neumann’s problem in domain with a small hole is given in 8.2, and Dirichlet’s problem for the biharmonic operator in such a domain is considered in 8.3. Chapter 8.4 derives Griffith-Irvin’s formula, mentioned in the beginning of Chapter 7, for the change of energy depending on the length of crack. In the final sixth section of this chapter we describe the asymptotic behaviour of potential energy for the stress and deformation state of a plane domain perturbed in the neighborhood of a corner. The necessary facts concerning behaviour of the solutions of problems of the theory of elasticity in a neighborhood of the sector vertex are put together in 8.5.