Stress- and Strain-Based Boundary Value Problems of Elasticity

Boundary value problems in the theory of elasticity are traditionally formulated in terms of displacements, while stresses and strains are calculated from the displacements. Usually, the calculation of stresses and strains is accompanied by approximation errors. In this paper, alternative formulations of elasticity problems in stresses and strains are developed. It is shown that, to formulate boundary value problems (BVP) in stresses, it suffices to consider the three off-diagonal Beltrami–Michell equations together with the three equilibrium equations. Similarly, the off-diagonal strain compatibility equations are considered in conjunction with the three equilibrium equations expressed in terms of strains. In addition, the Beltrami–Michell equations and the strain compatibility equations are reduced to the Poisson equations for the stress and strain tensors, respectively. For the numerical solution of the formulated BVP in stresses and strains, the finite difference method and the finite element method within the FreeFEM++ framework were applied. A comparison of numerical results for the problems of stretching a rectangular plate under a parabolic load and of a dam subjected to hydrostatic pressure and self-weight with known solutions demonstrates good agreement, thereby confirming the validity of the proposed boundary value problems regarding stresses and strains.