基于余能原理计算简单剪切变形

Calculation of Simple Shear Deformation Based on the Complementary Energy Principle

以高玉臣提出的基于基面力的弹性大变形余能原理为基础,利用Lagrange乘子,放松平衡条件和力边界条件对余能泛函的约束,给出了基于基面力的广义余能泛函。利用极分解定理,将表示单位质量的余应变能分为刚性转动和纯变形两部分,考虑相邻单元间边界面力的平衡条件,推导出了平面四边形单元的有限元公式,并利用其进行了简单剪切变形的计算。数值计算结果与理论结果的对比表明,当剪切角不超过35°时,二者是非常接近的。

Using the complementary energy principle for large elasticity proposed by Gao Yuchen based on the base forces and Lagrange multipliers,the generalized complementary energy functional based on base forces was obtained by releasing constraint conditions of equilibrium equations and force boundary conditions.According to polar decomposition,the complementary energy of unit mass could be decomposed into two parts：rigid rotation and pure deformation parts.The finite element formulae for plane quadrilateral element were gained with the consideration of equilibrium conditions between adjacent elements,which was used to the computation for simple shear deformation.Comparison shows that the numerical results are very closed to theoretical solutions when the shear angle is not exceed 35°.