Solution of the stress-equilibrium and compatability equations in the presence of body forces and arbitrary temperature fields

We have solved the stress-equilibrium and compatibility equations in three dimensions in terms of a scalar stress function which satisfies a fourth order partial differential equation. The solution also includes operations on two integrals which exhibit the longitudinal and shear components explicitly. All possible relationships involving equilibrium and compatibility conditions result in the same fourth order differential equation involving time and space coordinates. For the static case this equation reduces to the biharmonic equation ${\nabla ^4}\chi = 0$.