Decomposition of displacement gradient and strain definition

Abstract

For continuum, the displacement gradient can be divided into symmetry part and asymmetry part. Based on Chen Zhi-da’s research, the asymmetry part is related with an orthogonal rotation. But the definition of rotation angle in Chen’s theory is not correct for large deformation or rotation. The definition of rotation angle was improved to be suitable for large deformation or rotation. The results show that the displacement gradient can be divided into symmetry part and orthogonal rotation part. This result improves Chen Zhi-da’s S-R decomposition theoretic research and Finger-Truesdell’s polar decomposition theorem. Based on the new formulation of displacement gradient decomposition, it shows that S-R decomposition can be extended to large deformation and large rotation cases. The symmetry part contains a coupling tensor directly related with rotation tensor, this coupling tensor causes that the strain defined significantly differ from actual value calculated based on measured stress field. The additional stress related with the coupling tensor is the function of square of elements of asymmetry part of displacement gradient. The strain defined on Finger-Truesdell’s polar decomposition theorem loses a factor related with expanding of rotation. As the rotation contains asymmetry stress, the traditional equilibrium equation only meets moment conservation, and the angular-moment conservation is derived to accept the asymmetry stress related with rotation. As the velocity gradient can be gotten by taking time differentiation of displacement, hence the results are also true for fluids.

Keywords

displacement gradient / strain definition / decomposition / equilibrium equation / rational mechanics

References

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