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This paper presents a detailed study on analytical solutions to a generalnonlinear boundary-value problem in finite deformation theory. Based oncanonical duality theory and the associated pure complementary energy principlein nonlinear elasticity proposed by Gao in 1999, we show that the generalnonlinear partial differential equation for deformation is actually equivalentto an algebraic (tensor) equation in stress space. For St Venant-Kirchhoffmaterials, this coupled cubic algebraic equation can be solved principally toobtain all possible solutions. Our results show that for any given externalsource field such that the statically admissible first Piola-Kirchhoff stressfield has no-zero eigenvalues, the problem has a unique global minimalsolution, which is corresponding to a positive-definite second Piola-Kirchhoffstress S, and at most eight local solutions corresponding to negative-definiteS. Additionally, the problem could have 15 unstable solutions corresponding toindefinite S. This paper demonstrates that the canonical duality theory and thepure complementary energy principle play fundamental roles in nonconvexanalysis and finite deformation theory.

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