Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity

TL;DR

This paper develops a pure complementary energy variational method based on canonical duality-triality theory to analytically solve anti-plane shear problems in finite elasticity, revealing multiple solutions and the importance of nonconvex analysis.

Contribution

It introduces a novel analytical solution framework for large deformation anti-plane shear problems using duality theory, addressing nonconvexities and solution multiplicity.

Findings

Multiple solutions can exist at each material point.

Complementary gap function and triality theory identify extremal solutions.

Convexity conditions only provide local minimality, not uniqueness.

Abstract

This paper presents a pure complementary energy variational method for solving anti-plane shear problem in finite elasticity. Based on the canonical duality-triality theory developed by the author, the nonlinear/nonconex partial differential equation for the large deformation problem is converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular (poly-, quasi-, and rank-one) convexities provide only local minimal criteria, the Legendre-Hadamard condition does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis.