New approach to numerical solution to three-dimensional mixed-boundary-value elastic problems

Abstract

This thesis develops a new mathematical formulation, specially suitable for numerical analysis of stresses and displacements of three-dimensional mixed-boundary-value elastic problems. Earlier, mathematical models of elasticity were very deficient in handling the practical stress problems of solid mechanics, especially the three-dimensional ones. Analytical methods of solution have not gained that much popularity in the field of stress analysis, mainly because of the inability of dealing with complex boundary conditions and shapes. The existing mathematical models for the solution of threedimensional problems involve finding either six stress components or three displacement components simultaneously. But, solving for even three functions satisfying three simultaneous second order partial differential equations and variously mixed conditions on the bounding surfaces is well-neigh impossible. As a result, no serious attempt has been reported so far in the literature even to solve a model problem with uniform boundary conditions using these approaches. Stress analysis of structural problems is mainly handled by numerical methods. The major numerical methods in use are (a) the method of finite-difference and (b) the method of finite-element. Finite-difference method is an ideal numerical approach for solving partial differential equations. Despite its ideal characteristics, finite-difference method has been supplanted in most solid mechanics applications in engineering by the more popular finite-element method. In the present research, the supremacy is again brought back to the finite-difference technique form the finite-element method through a novel formulation of three-dimensional stress analysis of solid structures and a scheme of boundary management. The development of a novel formulation for the solution of three-dimensional mixedboundary value elastic problems is presented here in details. A new scheme of reduction of variables is used to develop the formulation. In this approach the threedimensional problem is reduced to the determination of a single potential function, defined in terms of the three displacement components, satisfying a single differential equation of equilibrium. Finite-difference technique is used to discretize the governing fourth-order differential equation as well as the second and third order differential equation associated with the boundary conditions. Ultimately, the problem is solved by finding the solution for the single discretized variable from a system of linear algebraic equations resulting from the discretization of the domain into mesh points. Compared to the conventional approaches, the present method provides the solution of higher accuracy in a shorter period of time. Finally, the application of the present numerical approach has been demonstrated through the solution of a classical three-dimensional problem of solid mechanics, and the results are compared with those obtained by the standard finite-element method. The comparison of solutions firmly establishes the reliability as well as rationality of the new mathematical model.