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An irreducible polynomial functional basis of two-dimensional Eshelby tensors

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Abstract

The two-dimensional (2D) Eshelby tensors are discussed. Based upon the complex variable method, an integrity basis of ten isotropic invariants of the 2D Eshelby tensors is obtained. Since an integrity basis is always a polynomial functional basis, these ten isotropic invariants are further proven to form an irreducible polynomial functional basis of the 2D Eshelby tensors.

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Abbreviations

M (2) :

two-dimensional (2D) Eshelby tensor

O 2 :

orthogonal group in the 2D physical space

R 2 :

rotation group in the 2D physical space

Q(θ):

a rotation of angle θ in the 2D physical space

Q̃ :

a special reflection in the 2D physical space

H m :

mth-order irreducible tensor space in the 2D physical space

e i :

orthonormal base in the 2D physical space

ℝ:

real number field

ℂn :

complex number field with the dimension n

Re(x):

real part of a complex number x

⊗:

tensor product

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Acknowledgements

The authors would like to express their gratitude to Prof. Wennan ZOU for his encouragement during the course of this work and for many useful discussions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Zhenyu Ming & Liping Zhang

  2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Yannan Chen

Authors
  1. Zhenyu Ming
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  2. Liping Zhang
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  3. Yannan Chen
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Corresponding author

Correspondence to Liping Zhang.

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Citation: MING, Z. Y., ZHANG, L. P., and CHEN, Y. N. An irreducible polynomial functional basis of two-dimensional Eshelby tensors. Applied Mathematics and Mechanics (English Edition) 40(8), 1169–1180 (2019) https://doi.org/10.1007/s10483-019-2502-6

Project supported by the National Natural Science Foundation of China (Nos. 11271221, 11771244, 11571178, and 11771405)

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Ming, Z., Zhang, L. & Chen, Y. An irreducible polynomial functional basis of two-dimensional Eshelby tensors. Appl. Math. Mech.-Engl. Ed. 40, 1169–1180 (2019). https://doi.org/10.1007/s10483-019-2502-6

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  • DOI: https://doi.org/10.1007/s10483-019-2502-6

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