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.
