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Applied Mathematics and Mechanics >

2022 , Vol. 43 >Issue 5: 667 - 688

DOI: https://doi.org/10.1007/s10483-022-2845-5

Articles

Elliptical inclusion in an anisotropic plane: non-uniform interface effects

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  • 1. School of Electrical and Mechanical Engineering, Jinling Institute of Technology, Nanjing 211169, China;
    2. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received date: 2022-01-20

  Revised date: 2022-03-09

  Online published: 2022-05-05

Supported by

the National Natural Science Foundation of China (No.11902147) and the Natural Science Foundation of Jiangsu Province of China (No.BK20190393)

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Abstract

We study the plane deformation of an elastic composite system made up of an anisotropic elliptical inclusion and an anisotropic foreign matrix surrounding the inclusion. In order to capture the influence of interface energy on the local elastic field as the size of the inclusion approaches the nanoscale, we refer to the Gurtin-Murdoch model of interface elasticity to describe the inclusion-matrix interface as an imaginary and extremely stiff but zero-thickness layer of a finite stretching modulus. As opposed to isotropic cases in which the effects of interface elasticity are usually assumed to be uniform (described by a constant interface stretching modulus for the entire interface), the anisotropic case considered here necessitates non-uniform effects of interface elasticity (described by a non-constant interface stretching modulus), because the bulk surrounding the interface is anisotropic. To this end, we treat the interface stretching modulus of the anisotropic composite system as a variable on the interface curve depending on the specific tangential direction of the interface. We then devise a unified analytic procedure to determine the full stress field in the inclusion and matrix, which is applicable to the arbitrary orientation and aspect ratio of the inclusion, an arbitrarily variable interface modulus, and an arbitrary uniform external loading applied remotely. The non-uniform interface effects on the external loading-induced stress distribution near the interface are explored via a group of numerical examples. It is demonstrated that whether the nonuniformity of the interface effects has a significant effect on the stress field around the inclusion mainly depends on the direction of the external loading and the aspect ratio of the inclusion.

Key words： elliptical inclusion; anisotropic material; interface stress; Gurtin-Murdoch model; membrane-type interface

Cite this article

Pengyu PEI, Ming DAI . Elliptical inclusion in an anisotropic plane: non-uniform interface effects[J]. Applied Mathematics and Mechanics, 2022 , 43(5) : 667 -688 . DOI: 10.1007/s10483-022-2845-5

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