Analytic displacement solutions for linearly varying eigenstrain in a thermal inclusion enclosed by line segments and circular arcs

Abstract

Planar Eshelby’s inclusion problem has laid the cornerstone for many fundamental aspects of micromechanics. Previous research has primarily focused on the stress/strain solutions, but analyses of the displacements are less documented, particularly for inclusions of complex shapes. This paper presents a closed-form displacement solution for inclusions whose cross sections are made up of any combination of line segments and circular arcs, under the circumstance of linearly distributed eigenstrains. In light of the method of Green’s function, the displacements are formulated through area integrals, which are subsequently converted into contour integrals along the boundary. The proposed method offers a direct, efficient, and analytical approach for determining the displacements. Additionally, the accuracy and reliability of the present algorithms are confirmed through comparisons with results obtained from the isoparametric inclusion model and the traditional finite element method (FEM).

Keywords:

• Linearly distributed eigenstrains
• closed-form solutions
• contour integral
• Green’s function

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Science Foundation of China (Grant Nos. 11932004, 51875059 and 52205192). The author would like to thank the Graduate Research and Innovation Foundation of Chongqing, China (Grant No. CYS21011). X.J. would also like to acknowledge the support from Chongqing City Science and Technology Program (Grant No. cstc2020jcyj-msxmX0850).