Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant–Kirchhoff Material

Abstract

This paper presents a detailed study on analytical solutions to a general nonlinear boundary-value problem in finite deformation theory. Based on canonical duality theory and the associated pure complementary energy principle in nonlinear elasticity proposed by Gao in (Mech Res Commun 26:31–37, 1999, [6], Wiley Encyclopedia of Electrical and Electronics Engineering, 1999, [7], Meccanica 34:169–198, 1999, [8]), we show that the general nonlinear partial differential equation for deformation is actually equivalent to an algebraic (tensor) equation in stress space. For St Venant–Kirchhoff materials, this coupled cubic algebraic equation can be solved principally to obtain all possible solutions. Our results show that for any given external source field such that the statically admissible first Piola–Kirchhoff stress field has nonzero eigenvalues, the problem has a unique global minimal solution, which is corresponding to a positive-definite second Piola–Kirchhoff stress \(\mathbf{T}\), and at most eight local solutions corresponding to negative-definite \(\mathbf{T}\). Additionally, the problem could have 15 unstable solutions corresponding to indefinite \(\mathbf{T}\). This paper demonstrates that the canonical duality theory and the pure complementary energy principle play fundamental roles in nonconvex analysis and finite deformation theory.