Complete system of polynomial solutions of differential equations of elasticity (Q2753462)

A new representation of general solution of the Lamé elastostatic equations is derived. The following theorem is proved: For any solution of the equation \((1-2\nu)\nabla^2 {\mathbf u} + \nabla\nabla\cdot {\mathbf u} = 0\) there exist three harmonic functions \(H_1\), \(H_2\),and \(H_3\) such that \({\mathbf u} = (3-4\nu) H_2 {\mathbf i}_3 - x_3\nabla H_2 +\nabla \times (H_1{\mathbf i}_3) + \nabla H_0\). Substitution of harmonic polynomials instead of these harmonic functions gives a system of polynomial solutions of differential equations. The constructed system of vector-functions is complete, and a sequence of approximate solutions, constructed on its basis, converges to the exact solution.