n the fourth order tensor valued function of the stress in return map algorithm |
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Authors: | Mingxiang Chen |
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Institution: | Civil Engineering School of Wuhan University, Wuhan 430072, China |
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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. |
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Keywords: | Integration of the constitutive equations stress update return map algorithm isotropy representation theorem |
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