Uncoupled dual formulations of the variational boundary element method in problems of the theory of elasticity

Abstract

Uncoupled dual formulations (UDFs), different from those considered previously [1–3], are proposed for the boundary functionals of the linear theory of elasticity, in the sense that the displacements and stresses are varied independently, and the equations of state on the boundary are treated as constraints involving Lagrange multipliers. The idea of this device—using Lagrange multipliers to get rid of restrictions in the variational problem, represented by the equations of state—was used previously [4]to formulate dual variational problems of the linear theory of elasticity based on the Lagrange-Castigliano principle. A finite element approximation of the solutions of these problems yields mixed formulations of the finite element method [5]. Thus, the boundary element approximations (BEAs) proposed below for the UDF may be regarded as a special mixed finite element method. Simultaneous BEA of the displacements and stresses extends the applicability of UDFs to cases in which allowance must be made for singularities of the solution, e.g. in contact problems of the theory of elasticity and in fracture mechanics (crack problems).