The boundary integral equation method, and the boundary element method offer important advantages in approximate analysis of continuum mechanics over the so-called domain type numerical methods. This paper deals with a derivation of new integral equation formulations when establishing numerical solutions of many kinds of problems in continuum mechanics. The principal objective is to present a general method for developing such formulations, and to illustrate its application to a number of different areas of solid and fluid mechanics. Usually the fundamental equation in continuum mechanics is given in terms of the configuration variable u by Au=(T^*ET)u=f In the boundary element method, it is customary to derive the corresponding integral equation from the above equation by using the fundamental solution for the adjoint operator A^* of A. In this paper, we start with the following so-called primal set of canonical equation associated with the operator A instead of the primal equation : Kinematic relation Tu=v Balance equation T^*σ=f Constitutive equation Ev=σ From the above primal set, the new integral equation set in terms of not only the configuration variable u but also the variable σ is easily derived by using the general methodology presented in this paper. The presented method is directly applicable to numerous problems encountered in continuum mechanics. New integral equation sets and related fundamental solutions for problems of elastostatics and incompressible viscous fluid flow problems are presented illustratively. The simplicity and generality of the methodology proposed in this paper allow for the formulation of very general class of problems, non-linear, and time-dependent problems.
