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Elasticity in three dimensions

  • A New Mathematical Formulation
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],

  • Computational Modeling of 3D Mixed-Boundary-Value Elastic Problems
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,

Further reading  
M. Zubaer Hossain, Master's
Thesis, Department of Mechanical Engineering, BUET, Dhaka-1000, Bangladesh