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The second Eshelby problem and its solvability

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Abstract

It is still a challenge to clarify the dependence of overall elastic properties of heterogeneous materials on the microstructures of non-elliposodal inhomogeneities (cracks, pores, foreign particles). From the theory of elasticity, the formulation of the perturbance elastic fields, coming from a non-ellipsoidal inhomogeneity embedded in an infinitely extended material with remote constant loading, inevitably involve one or more integral equations. Up to now, due to the mathematical difficulty, there is almost no explicit analytical solution obtained except for the ellipsoidal inhomogeneity. In this paper, we point out the impossibility to transform this inhomogeneity problem into a conventional Eshelby problem by the equivalent inclusion method even if the eigenstrain is chosen to be non-uniform. We also build up an equivalent model, called the second Eshelby problem, to investigate the perturbance stress. It is probably a better template to make use of the profound methods and results of conventional Eshelby problems of non-ellipsoidal inclusions.

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Authors and Affiliations

  1. Institute for Advanced Study, Nanchang University, 330031, Nanchang, China

    Wen-Nan Zou & Quan-Shui Zheng

  2. Department of Engineering Mechanics, Tsinghua University, 100084, Beijing, China

    Quan-Shui Zheng

Authors
  1. Wen-Nan Zou
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  2. Quan-Shui Zheng
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Correspondence to Wen-Nan Zou.

Additional information

The project was supported by the National Natural Science Foundation of China (10872086 and 11072105).

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Zou, WN., Zheng, QS. The second Eshelby problem and its solvability. Acta Mech Sin 28, 1331–1333 (2012). https://doi.org/10.1007/s10409-012-0139-y

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  • DOI: https://doi.org/10.1007/s10409-012-0139-y

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