关于返回映射算法中应力的四阶张量值函数  

ON THE FOURTH ORDER TENSOR VALUED FUNCTION OF THE STRESS IN RETURN MAP ALGORITHM

作　　者：陈明祥[1] 

机构地区：[1]武汉大学土木建筑工程学院,武汉430072

出　　处：《力学学报》2010年第2期228-237,共10页Chinese Journal of Theoretical and Applied Mechanics

摘　　要：针对各向同性材料,基于一组相互正交的基张量,建立了一套有效的相关运算方法.基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴.根据张量函数表示定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组基所表示.推演结果表明:这些张量之间的运算,表现为对应系数矩阵之间的简单关系.其中,四阶张量求逆归结为对应的3×3系数矩阵求逆,它对二阶张量的变换则表现为该矩阵对3×1列阵的变换.最后,对这些变换关系应用于返回映射算法的迭代格式进行了相关讨论.The inversion of a fourth order tensor valued function of the stress and its transformation to the second order tensor are required in the return map algorithm for implicit integration of the constitutive equation.Based on a set of the base tensors which are mutually orthogonal,this paper presents an effective methodology to perform those tensor operations for the isotropic constitutive equations.In the scheme,two of the base tensors are the second order identity tensor and the deviatoric stress tensor,respectively.Another base tensor is constructed using an isotropic second order tensor valued function of the stress.The three base tensors are coaxial.By making use of the representation theorem for isotropic tensorial functions,all the second order,the fourth order tensor valued functions of the stress involved can be represented in terms of the base tensors.It shows that the operations between the tensors are specified by the simple relations between the corresponding matrices.The inversion of a fourth order tensor is reduced to the inversion of corresponding 3×3 matrix,and its transformation to the second tensor is equivalent to transformation of 3×3 matrix to 3×1 column matrix.Finally,some discussions are given to the application of those transformation relationships to the iteration algorithm for the integration of the constitutive equations.

关 键 词：本构方程积分 应力更新 返回映射算法 各向同性 张量函数表示定理 

分 类 号：O344.1[理学—固体力学]