• Research paper •

### n the fourth order tensor valued function of the stress in return map algorithm

Mingxiang Chen

1. Civil Engineering School of Wuhan University, Wuhan 430072, China
• Received:2009-01-09 Revised:2009-03-26 Online:2010-03-18 Published:2010-03-18

Abstract: 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\times 3 matrix, and its transformation to the second tensor is equivalent to transformation of 3\times 3 matrix to 3\times 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.