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 1999, 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 no-zero eigenvalues, the problem has a unique global minimal solution, which is corresponding to a positive-definite second Piola-Kirchhoff stress S, and at most eight local solutions corresponding to negative-definite S. Additionally, the problem could have 15 unstable solutions corresponding to indefinite S. This paper demonstrates that the canonical duality theory and the pure complementary energy principle play fundamental roles in nonconvex analysis and finite deformation theory.

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“…For a given statically admissible stress field τ ∈ T c , this tensor equation could have at most 27 solutions T(x) at each material point x ∈ , but only one T(x) 0, which leads to a global minimal solution [122]. For many real-world problems, the statically admissible stress τ ∈ T c can be uniquely obtained and the canonical dual algebraic equation (80) can be solved to obtain all possible stress solutions.…”

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“…By the fact that W (F) must be an objective function [37], there exists a real-valued function Ψ(C) such that W (F) = Ψ(F T F) (see [5]). For most reasonable materials (say the St. Venant-Kirchhoff material [22]), the function Ψ(C) is a usually convex function of the Cauchy strain measure C = F T F such that its complementary energy density can be uniquely defined by the Legendre transformation…”

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“…For a given statically admissible stress field τ ∈ T c , this tensor equation could have at most 27 solutions T(x) at each material point x ∈ Ω, but only one T(x) 0, which leads to a global minimal solution [68]. For many real-world problems, the statically admissible stress τ ∈ T c can be uniquely obtained and the canonical dual algebraic equation ( 79) can be solved to obtain all possible stress solutions.…”