Abstract
The positive definiteness of elasticity tensors plays an important role in the elasticity theory. In this paper, we consider the bi-block symmetric tensors, which contain elasticity tensors as a subclass. First, we define the bi-block M-eigenvalue of a bi-block symmetric tensor, and show that a bi-block symmetric tensor is bi-block positive (semi)definite if and only if its smallest bi-block M-eigenvalue is (nonnegative) positive. Then, we discuss the distribution of bi-block M-eigenvalues, by which we get a sufficient condition for judging bi-block positive (semi)definiteness of the bi-block symmetric tensor involved. Particularly, we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite, including bi-block (strictly) diagonally dominant symmetric tensors and bi-block symmetric (B)B0-tensors. These give easily checkable sufficient conditions for judging bi-block positive (semi)definiteness of a bi-block symmetric tensor. As a byproduct, we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.
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Acknowledgements
The first author’s work was supported by the National Natural Science Foundation of China (Grant No. 11871051).
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Huang, ZH., Li, X. & Wang, Y. Bi-block positive semidefiniteness of bi-block symmetric tensors. Front. Math. China 16, 141–169 (2021). https://doi.org/10.1007/s11464-021-0874-0
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DOI: https://doi.org/10.1007/s11464-021-0874-0
Keywords
- Bi-block symmetric tensor
- bi-block symmetric Z-tensor
- bi-block symmetric B 0-tensor
- diagonally dominant bi-block symmetric tensor
- bi-block M-eigenvalue