有限塑性率问题的虚功方程及自我校正方法的推证和应用

DERIVATION AND APPLICATION FOR VIRTUAL WORK EQUATION OF RATE PROBLEM IN FINITE PLASTICITY AND SELF-CORRECTING METHOD

本文首先从数学上证明了在有限塑性问题的虚功方程或变分原理中,Cauchy应力是与Euler应变,Kirchhoff应力是与Lagrange应变,名义应力是与位移梯度配对的。接着,在此基础上导出了率形式的,以Kirchhoff应力率表示的虚功方程。本文还以不同于Tvergaard的方法导出了他于1984年提出的自我校正方法,使方法中的Lagrange乘子显示出物理意义。最后作为方法的应用,本文根据所导出的率形式的虚功方程建立的有限元公式和自我校正方法对囿棒拉伸时缩颈过程和缩颈传播作了有限元分析。并比较了采用或不采用自我校正方法两种情况下的有限元计算精度。分析时考虑了材料的硬化。

In this paper, firstly, it is proved by mathmatics that Cauchy Stress is conjugate to Euler Strain, Kirchhoff stress is conjuate to Lagrange strain, and nominal stress is conjugate to displcement gradient. Then, virtual work equation in rate form and expressed by Kirchhoff stress is derived. The self-correcting method proposed by Tvergaard in 1984 is also derived by a different way,thus givinS lagrange multipliers λan obvious physical interpretation. Lastly, as the application of the methods, a FEM analysis is done for necking and neck propagation of a round tensile bar, and the computational solutions based on using or not using the self-correcting method are compared for accuacy. The strain hardening is considered.