Fundamental Study on Nonlinear Finite Element Method to Achieve High Accuracy

有限元法广泛应用于工程产品的结构分析中，以进行合理的设计和开发。为了使非线性有限元分析达到较高的精度，从1990年到1992年的三年中，本文研究了本构方程和变分原理及其对数值分析的影响。低弹性定义为速率型本构方程，用于描述金属合金的塑性行为；超弹性定义为势型本构方程，用于描述橡胶材料的塑性行为。若在简单剪切问题中采用线性运动硬化规律，则广泛应用于jaaumann应力导数的次弹性会导致应力-变形曲线振荡。在简单剪切问题中，基于未变形形态测量的格林-拉格朗日应变的超弹性理论不能准确预测剪切变形后剪应力响应的实验事实。对于超弹性分析，采用的混合变分原理不包括初始应力项，这对分岔分析有很大影响。为了克服本构方程和混合变分原理的这些缺点，本文对以下几点进行了研究。(1)将无旋转应变视为广义对数应变，建立了超弹性本构方程。(2)从与中间构型相关的共转方程和非共轴性的角度研究了低弹性的理论问题。(3)基于内时间理论建立了大塑性应变本构方程，并与实验进行了对比。(4)从理论上研究了考虑初始应力项和解稳定性的混合变分原理。(5)采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.