Newly Formulated General Solutions for the Navier Equation in Linear Elasticity

Abstract

The Navier equations are reformulated to be third-order partial differential equations. New anti-Cauchy-Riemann equations can express a general solution in 2D space for incompressible materials. Based on the third-order solutions in 3D space and the Boussinesq-Galerkin method, a third-order method of fundamental solutions (MFS) is developed. For the 3D Navier equation in linear elasticity, we present three new general solutions, which have appeared in the literature for the first time, to signify the theoretical contributions of the present paper. The first one is in terms of a biharmonic function and a harmonic function. The completeness of the proposed general solution is proven by using the solvability conditions of the equations obtained by equating the proposed general solution to the Boussinesq-Galerkin solution. The second general solution is expressed in terms of a harmonic vector, which is simpler than the Slobodianskii general solution, and the traditional MFS. The main achievement is that the general solution is complete, and the number of harmonic functions, three, is minimal. The third general solution is presented by a harmonic vector and a biharmonic vector, which are subjected to a constraint equation. We derive a specific solution by setting the two vectors in the third general solution as the vectorizations of a single harmonic potential. Hence, we have a simple approach to the Slobodianskii general solution. The applications of the new solutions are demonstrated. Owing to the minimality of the harmonic functions, the resulting bases generated from the new general solution are complete and linearly independent. Numerical instability can be avoided by using the new bases. To explore the efficiency and accuracy of the proposed MFS variant methods, some examples are tested.