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Mingxiang Chen. n the fourth order tensor valued function of the stress in return map algorithm[J]. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(2): 228-238. DOI: 10.6052/0459-1879-2010-2-2009-016
 Citation: Mingxiang Chen. n the fourth order tensor valued function of the stress in return map algorithm[J]. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(2): 228-238. DOI: 10.6052/0459-1879-2010-2-2009-016

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

• The inversion of a fourth order tensor valued functionof the stress and its transformation to the second order tensor are requiredin the return map algorithm for implicit integration of the constitutiveequation. Based on a set of the base tensors which are mutually orthogonal,this paper presents an effective methodology to perform those tensoroperations for the isotropic constitutive equations. In the scheme, two ofthe base tensors are the second order identity tensor and the deviatoricstress tensor, respectively. Another base tensor is constructed using anisotropic second order tensor valued function of the stress. The three basetensors are coaxial. By making use of the representation theorem forisotropic tensorial functions, all the second order, the fourth order tensorvalued functions of the stress involved can be represented in terms of thebase tensors. It shows that the operations between the tensors are specifiedby the simple relations between the corresponding matrices. The inversion ofa fourth order tensor is reduced to the inversion of corresponding 3\times3 matrix, and its transformation to the second tensor is equivalent totransformation of 3\times 3 matrix to 3\times 1 column matrix. Finally,some discussions are given to the application of those transformationrelationships to the iteration algorithm for the integration of theconstitutive equations.

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