Analytic Solutions to Large Deformation Problems Governed by Generalized Neo-Hookean Model

TL;DR

This paper develops analytical solutions for large deformation problems in nonlinear elasticity using a unified algebraic approach, revealing the role of external forces in ellipticity and solution uniqueness.

Contribution

It introduces a novel pure complementary energy principle and a triality theory to obtain complete analytical solutions for 3-D finite deformations under the generalized neo-Hookean model.

Findings

Analytical solutions for 3-D finite deformation problems are derived.

Ellipticity depends on external forces in nonconvex systems.

Uniqueness is established through quasiconvexity and generalized ellipticity.

Abstract

This paper addresses some fundamental issues in nonconvex analysis. By using pure complementary energy principle proposed by the author, a class of fully nonlinear partial diforerential equations in nonlinear elasticity is able to converted a unified algebraic equation, a complete set of analytical solutions are obtained for 3-D finite deformation problems governed by generalized neo-Hookean model. Both global and local extremal solutions to the nonconvex variational problem are identifored by a triality theory. Connection between challenges in nonlinear analysis and NP-hard problems in computational science is revealed. Results show that Legendre-Hadamard condition can only guarantee ellipticity for convex problems. For nonconvex systems, the ellipticity depends not only on the stored energy, but also on the external force field. Uniqueness is proved based on a quasiconvexity and a generalized ellipticity condition. Application is illustrated for logarithm stored energy.