Displacement differential equations of equilibrium in three-dimensional geometrically and physically nonlinear theory of elasticity for discontinuous closing equations in the cartesian coordinate system

We consider the construction of coefficients of equilibrium differential equations in displacements for a three-dimensional physically and geometrically nonlinear theory of elasticity. Differential equations of equilibrium are done in a rectangular Cartesian coordinate system. Closing equations are variable modules of volumetric and shear deformation; physical relations are approximated by bilinear functions. Physical dependencies are recorded for four possible rules of continuum deformation in accordance with bilinear graphs of volume and shear deformation. The rules of deformation on each linear section of bilinear diagrams of volumetric and shear deformation are determined by the secant modules of bilinear graphs of volumetric and shear deformation diagrams. The analysis shows that the coefficients of differential equations of equilibrium in displacements, which are second-order partial differential equations of displacements along spatial coordinates, are quadratic functions of the first derivatives of displacements along spatial coordinates. The constructed three-dimensional differential equations of equilibrium in displacements can be applied in the calculation of structures using three-dimensional equilibrium equations of physically and geometrically nonlinear theory of elasticity in displacements, the closing equations of physical relations for which are approximated by bilinear functions.