Fundamental Study on Nonlinear Finite Element Method to Achieve High Accuracy

有限元方法被广泛应用于工程产品的结构分析中，以便进行合理的设计和开发。为了提高非线性有限元分析的精度，本文对1990~1992年三年间的本构方程和变分原理及其对数值分析的影响进行了研究。亚弹性被定义为速率型本构方程，用于描述金属合金的塑性行为，而超弹性被定义为势型本构方程，用于橡胶材料。在简单剪切问题中，如果采用线性运动硬化定律，亚弹性中广泛使用的Jaumann应力导数会使应力-变形曲线出现振荡。基于未变形构型的格林-拉格朗日应变的超弹性不能准确地预测简单剪切问题中剪切变形后的剪切应力响应这一实验事实。对于超弹性分析，所采用的混合变分原理没有考虑初始应力项，这对分叉分析有很大影响。为了克服本构方程和混合变分原理中的这些不足，本文对以下几点进行了连续的研究：(1)将无旋转应变作为广义对数应变，提出了超弹性本构方程。(2)从共转方程和与中间构型相关的非共轴性的角度研究了亚弹性的理论问题。(3)基于内时理论建立了塑性大应变的本构方程，并与实验结果进行了比较。(4)从理论上研究了考虑初应力项和解的稳定性的混合变分原理。(5)利用无旋转应变进行了亚弹性和超弹性的数值分析。

The finite element method is widely used in the structural analysis of engineering products for the reasonable design and developments. The present study is concerned with the constitutive equation and variational principles and their effects on numerical analysis in order to achieve high accuracy in the non-linear finite element analysis during three years from 1990 to 1992. Hypoelasticity, defined as the rate type constitutive equation, is used to describe plastic behaviors of metal alloys, on the other hand, hyperelasticity, defined as the potential type constitutive equation, is used for rubber materials. Hypoelasticity widely used with Jaumann stress derivative gives oscillation in stress-deformation curve if linear kinematic hardening law is adopted in simple shear problem. Hyperelasticity, based on Green-Lagrange strain measured from the undeformed configuration, cannot predict accurately the experimental fact that shear stress respond behind shear deformation in simple shear problem. For the analysis of hyperelasticity, the employed mixed variational principle does not include the initial stress term, which significantly affects bifurcation analysis. In order to overcome such shortcomings in constitutive equation and mixed variational principles, the following points are studied consecutively.(1) By using rotationless strain regarded as the generalized logarithmic strain, hyperelastic constitutive equation is proposed.(2) Theoretical aspects of hypoelasticity is studied from the viewpoint of corotational equation and non-coaxiality relating to intermediate configurations.(3) Constitutive equation for large plastic strain is formulated based on internal time theory and compared to the experiments.(4) Mixed variational principles taking account of the initial stress term and stability of solutions are studied theoretically.(5) Numerical analysis of hypo and hyperelasticity is carried out by using rotationless strain.