三维St-Venant-Kirchhoff材料有限变形问题的解析解
Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material
摘要：
本文详细研究了有限变形理论中一般非线性边值问题的解析解。基于正则对偶理论和Gao于1999年提出的非线性弹性力学中的纯余能原理，我们证明了一般的非线性变形偏微分方程实际上等价于应力空间中的代数（张量）方程.对于St Venant-Kirchhoff材料，这个耦合的三次代数方程可以原则上求解以获得所有可能的解。结果表明，对于任意给定的外部源场，若第一应力场的特征值不为零，则问题存在唯一的整体极小解，该解对应于一个正定的第二Piola-Kirchhoff应力S，最多存在8个对应于负定的第二Piola-Kirchhoff应力S的局部解.此外，该问题可能有15个不稳定的解决方案，对应于不确定的S。本文论证了正则对偶理论和纯余能原理在非凸分析和有限变形理论中的基础作用。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
