Regularization of Divergent Integrals in Boundary Integral Equations for Elastostatics

Let consider a homogeneous, linearly elastic body, which in three-dimensional (3-D) Euclidean space ℝ3 occupies volume V with smooth boundary ∂V The region V is an open bounded subset of the 3-D Euclidean space ℝ3 with a C0,1 Lipschitzian regular boundary ∂V The boundary contains two parts $$\partial V_u$$ and $$\partial V_p$$ such that $$\partial V_u \cap \partial V_p = \emptyset \mbox{and} \partial V_u \cup \partial V_p = \partial V$$ On the part $$\partial V_u$$ are prescribed displacements u i (x) of the body points and on the part $$\partial V_p$$ are prescribed tractions p i (x), respectively. The body may be affected by volume forces b i (x). We assume that displacements of the body points and their gradients are small, so its stress-strain state is described by the small strain deformation tensor ε ij (x) Then differential equations of equilibrium in the form of displacements may be presented in the form $$A_{ij} u_j + b_i = 0, \quad A_{ij} = \mu\delta_{ij}\partial_k \partial_k + (\Lambda + \mu) \partial_i \partial_j \quad \forall{\rm x} \in V,$$ where λ and μ are Lamé constants,μ > 0 and λ > –μ, and δ ij is the Kronecker symbol.