Resolving differential equations of physically nonlinear theory of elasticity in terms of stresses for a plane strain

The paper is aimed to obtain resolving differential equations of physically nonlinear theory of elasticity in terms of stresses for a plane strain. These equations represent a mathematical model of the continuum whose variable coefficient of the volume expansion (compressibility) is a function of average stress only, and the variable coefficient of the shear is a function of tangential stress intensity only. The resolving differential equations are obtained by inserting the physical relations, in which the strains are expressed in terms of stresses, into Saint-Venant's compatibility condition written for a plane problem. As a result, a physically nonlinear analogue of the Levy equation for linear theory of elasticity is derived. When balance equations are satisfied irrespective of volume forces, stress function introducing yields a physically nonlinear analogue of the Levy equation represented as a physically nonlinear analogue of the biharmonic equation for a plane strain. As opposed to physically linear theory of elasticity, where biharmonic equations are homogeneous, the analogue to the biharmonic equation of physically nonlinear theory of elasticity is inhomogeneous. The form of the right side of the biharmonic equation is governed by the analyzed mathematical model of continuum. The obtained results can be used when solving the problems of physically nonlinear theory of elasticity in terms of stresses.