Study On The Analytical Solution Of Eshelby Problem For Planar Anisotropic Material With A Smooth Inclusions

Eshelby problem study findings have been widely used in engineering practice over the past few years.On the other hand,research on the Eshelby problem for anisotropic materials is comparatively underdeveloped due to the quick growth and expanding uses of semiconductor materials.The pertinent analytical solution for the Eshelby problem with planar anisotropic general smooth form inclusions is still in its infancy.In this essay,we investigate how to tackle problems of this nature based on the Riemann mapping theory and our understanding of the theory of functions of complex variables.Following the pre-determination of the intrinsic strain using the extended Stroh formula and the complex variable function,the analytic functional representation of the intrinsic displacement based on the Cauchy-type integral and Faber polynomials is obtained.Thereafter,the fundamental equations of the homogeneous intercalation problem for anisotropic materials under uniform intrinsic strain are listed.Through the use of orthogonality relations,the issue is converted into a degenerate RiemannHilbert problem,after which the theory of singular integrals of analytical functions is used to provide an analytical solution for the perturbed physical field caused by arbitrary smooth-shaped(expressed by Laurent polynomials)inclusions under uniform eigenstrain.In order to verify the validity and accuracy of the analytical solution,the results of this paper’s calculations are compared and analyzed with the previous results and the results of an ANSYS finite element analysis.The analytical solution to the aforementioned problem is reduced to the calculation of a set of elementary functions in this paper.Finally,the major conclusions listed below are reached.(1)At the smooth inclusions inflection point,stress concentration will also happen.The less smooth the inclusions shape,the more likely it is to cause the stress concentration phenomenon,and the absolute maximum stress value will be higher.(2)The disturbance field is also affected by the symmetry of the inclusions’ geometry;the positive stress field is symmetric about the axis of the inclusions,but the shear stress is antisymmetric.For other academics researching the inclusion’s problem,the aforementioned conclusions can serve as some theoretical references.The in-depth analysis of this work can aid in improving knowledge of the intercalation problem’s structure and process as well as offering some theoretical backing for dealing with similar issues.