A new mathematical formulation, specifically suitable for
finite-difference analysis of stresses and displacements of
three-dimensional mixed-boundary-value elastic problems. Earlier,
mathematical models of elasticity were very deficient in handling
three-dimensional practical stress problems. In the present model, a new
scheme of reduction of unknowns is used to formulate the
three-dimensional problem in terms of a single potential function,
defined in terms of the three displacement components. Compared to the
conventional models, the present model provides numerical solution of
higher accuracy in a shorter period of computational time. The
application of the potential function formulation is demonstrated here
through a number of classical problems of solid mechanics, and the
results are compared with the available solutions in the literature. The
comparison of the results establishes the rationality of the present
approach. The governing differential equation is derived as [1-2],
The governing equation transforms into the well-known biharmonic
form if one the spatial variation is neglected. The boundary conditions
are expressed in terms of the displacement potential function [1-2],
Using finite-difference
modeling techniques, the general equation derived by Hossain et al. is
discretized onto a 3D rectangular mesh [1-2] . The detail
finite-difference modeling of a classical 3D boundary value elastic
problem is demonstrated in references [1-2]. Here, the central
difference form and the corresponding stencil of the governing
differential equation is shown below,
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