用张量分析方法推导含偶应力弹性力学有限元理论

Derivation of the Finite Element Theory of Elasticity with Couple Stress by Tensor Analysis

经典弹性力学理论用位移梯度表示无限小变形,不考虑旋转变形,把微元体的旋转视为刚体旋转。含偶应力弹性力学理论将旋转变形以旋转张量表示,微元体旋转和微元体平动位移同量级,而旋转张量和应变张量同量级,旋转张量与旋转矢量一一对应,用旋转矢量的梯度表示旋转变形。含偶应力弹性力学理论本构关系包括应力-应变关系和偶应力-曲率张量关系,用等参变换方法离散单元位移到节点上,从虚功原理出发,增加罚函数项以降低有限元方程对高阶单元的需求,推导了拟解决三维及二维问题的含偶应力弹性线力学有限元理论,可得三维及二维问题中位移、应力、应变等分布情况,对结构进行力学评价。

The classical theory of elasticity expresses infinitesimal deformation by displacement gradient.The rotation of the infinitesimal body is regarded as the rotation of the rigid body without considering the rotation deformation.In the theory of elasticity with couple stress,rotation deformation is represented by rotation tensor.The rotation and translational displacement of the micro element are the same order,while the rotation and strain tensors are the same order.The rotation tensor corresponds to the rotation vector one by one,and the gradient of the rotation vector is used to represent the rotation deformation.The constitutive relation of elasticity with couple stress includes stress-strain relation and couple stress curvature tensor relation.Discrete the displacement of element to nodes by using isoparametric transformation method.Based on the principle of virtual work,a penalty function is added to reduce the demand of higher-order elements in the finite element equation.The finite element theory of elastic lines with coupled stresses to solve the plane problem is derived.The distribution of displacement,stress and strain in the problem of plane can be obtained,and the mechanical evaluation of the structure can be carried out.