Bi-Analytic Functions and their Applications in Elasticity

This paper is a review of some of our relevant work on bi-analytic functions and their applications in elasticity. The main basis lies in a mechanical interpretation of bi-analytical functions. First we provide a brief introduction of (λ, k) bi-analytical functions and the relation between the basic problems of plane elasticity, and (λ, k) bi-analytical functions. A mixed-contact problem solved by using (λ, k) bi-analytical functions and singular integral operators and some basic problems of elasticity reduced to BVP on some Riemann surfaces are demonstrated. Moreover, we restrict our attention to quasistatic linear thermoelasticity. We discuss the problems in general simply-connected domains. We first use the Bergman kernel function and the Green function of the domain to obtain integral representations for the BVPs and then decouple the thermoelastic systems on the bounded simply-connected domain. By utilizing the contractive mapping principle the decoupled temperature equation is studied. The representations of solutions of the field equations are obtained and their solvability results are proved. Finally their numerical solutions are given by means of the Galerkin finite element method and wavelet method respectively.