Analytical Study For Problem Of An Infinite 2D Elastic Body With A Non-elliptical Inhomogeneous Inclusion Having The Same Shear Module As The Matrix

The inhomogeneity problem takes an important position in the inclusion position,and its achievements have very important applications in the development and research of composite materials.In engineering practices,most inclusions are heterogeneous and non-ellipsoidal/elliptical,so the research on the problem of nonellipsoidal/elliptical inhomogeneity is of more practical significance.However,due to the difficulty in the mathematical treatment,it is still difficult to work out its explicit analytical solution.Most of the existing literatures have to be numerical or approximate,and the solutions of non-ellipsoid /elliptical inhomogeneity undergoing polynomial eigenstrain is even less concerned.Based on the theory of complex function,such as Riemann map and the KolosovMuskhelishvili(K-M)potential method,this paper studies the problem of non-elliptical inhomogeneity which has distinct properties to the matrix but shares the same shear modulus.In this research,the general non-elliptical shape is characterized by the Laurent polynomial.Two issues are mainly considered: 1)one is the elastic disturbance field of an infinite body containing an inhomogeneity under the actions of remote uniform stress and uniform eigenstrain;2)another is the elastic disturbance field of material containing inhomogeneity under the action of polynomial eigenstrain.During the derivation,the rigid-body displacement of the inhomogeneity relative to the matrix is specially considered.The main results and conclusions are listed as follows:1)The equivalent method is applied to convert the problem of perturbation field due to the remote uniform loadings into that of an equivalent uniform eigenstrain,and the explicit analytical solution of the K-M potentials of the elastic perturbation field is obtained.The analytical results of some examples under volumetric eigenstrain are verified to be consistent with the numerical solution(ANSYS),and the analytical results reduced to the ellipse case is completely consistent with the existing solution of the inhomogeneous and homogeneous ellipse inclusion.At the same time,the analytical solution degenerated to the case of homogeneity shaped by the hypocycloidal curve is also in complete agreement with the results in the existing literature.2)With construction of the eigen displacement corresponding to the polynomial eigenstrain,the analytical solution of the elastic perturbation potentials of inhomogeneity sharing the same shear modulus of the matrix and undergoing polynomial eigenstrain of any order is obtained.The obtained analytical solution degenerated to the case of elliptical inclusion under the action of linear eigenstrain is completely consistent with the results in the existing literature.3)The rigid-body displacement of the inhomogeneity relative to the matrix under the action of uniform load and polynomial eigenstrain is obtained and the explicit solution is achieved.The explicit results of non-elliptic inhomogeneity problem,first obtained in this paper,may have certain applications in practice and would be a prelude in tackling the more general non-elliptic inhomogeneity problems.