Series Solution For The Problem Of An Inhomogeneous Inclusion With Arbitrary Shape In The Infinite Plane

As an extension of the classical Eshelby's problem,and due to its practical significance in composite materials and production,the problem of inhomogeneous inclusion with arbitrary shape has attracted the attention of so many scholars in recent years.Different from the conventional homogeneous problem,it is unlikely to obtain explicitly analytical solutions for heterogeneous inclusion problem because of its mathematical difficulties.That's why among many researchers who have tackled this issue,only few can really get results.Based on Riemann mapping theory and complex function theory,the remote uniform loading and thermoelastic problems of inhomogeneous are studied in this paper.By means of the knowledge of Cauchy type integral and Faber polynomials,the first order expression and higher order recurrence formula of perturbation potential functions for heterogeneous uniform loading problem are obtained,at the same time,the highest power of the corresponding independent variable is given.Then,the basic potential functions are calculated according to the homogeneous inclusion problem.So,the series solution of the heterogeneous inclusion problem is obtained by combining the two approaches.And in the process of solving thermoelastic problems,the derivation steps of perturbation potential function are given because it does not involve basic potential issues.The result of the above two kinds of problems are expressed by explicit series solutions,and the analysis and verification work including convergence and validity of the series solutions are carried out.The effectiveness analysis is to compare the series solution of stress with the numerical solution of ABAQUS finite element method,and the main conclusions are as follows:(1)The series solution has converged basically when calculating the eighth order,and reflects the superiority of using Faber polynomial in the research method;in the validity analysis,the fractional error between the series solution and the finite element numerical solution is very small,most of them are around 0.1,and the largest one is just about 0.95,which shows that the series solution has high reliability.(2)In the heterogeneous problem of far-field loading,stress in the hard inclusion is greater than the stress which is given in the far-field.This phenomenon can be reversed under soft inclusion.And on the boundary,they have opposite positions of the maximum stress point or the extreme stress point.The above results can provide some theoretical references for other scholars,with hoping to promote the development of researches on heterogeneous inclusion.