Extended Studies On Eshelby’s Problems Of Non-elliptical Inclusions

Studies on the Eshelby’s inclusion problem have continued for over half a century.During this time,the equivalent inclusion method(EIM)based on uniformity of the interior Eshelby tensor of an ellipsoidal inclusion has proved to be the cornerstone of composite micromechanics,and delivered many standard schemes to estimate the efficient properties of heterogeneous materials,such as the self-consistent method,the Mori-Tanaka method,the IDD method,etc.However,until recently,a fundamental problem has been up in the air,that is to verify or falsify the Eshelby’s conjecture,which claims that an ellipsoidal inclusion is the only one to possess uniform interior fields.In dedications about this issue,inclusions with non-ellipsoidal/elliptical shapes attract many researchers’ interests.On the other hand,real inclusions in application are usually non-ellipsoidal/elliptical,which is another promotion of studying non-ellipsoidal/elliptical inclusion problems.In three-dimensional(3D)problems,except polyhedrons,analytical solutions of non-ellipsoidal inclusions are rarely reported for the complexity of descripting inclusions’ shapes.In two-dimensional(2D)cases,two kinds of typical non-elliptical inclusions,polygonal ones and those characterized by Laurent polynomials,are touched widely.However,some associated topics are still open,such as the integrity of the Eshelby tensor fields of smooth inclusions characterized by Laurent polynomials,the general analytical solutions of non-uniform eigenstrain problems in the aforementioned two kinds of 2D inclusions.In this thesis,we dedicated to do some extended research about the non-elliptical inclusion problem from two aspects:(1)the exterior elastic fields of smooth inclusions of Laurent polynomial type;(2)the polynomial eigenstrain problems of polygonal inclusions.The contributions of this dissertation mainly include the following three parts:(1)Starting from the boundary integral formulation of the Eshelby tensor,we developed a general method to explicitly derive the exterior elastic fields of smooth inclusions of Laurent polynomial type,and achieved the close-form solutions of the(N+1)-gonal hypocycloidal and the quasi-parallelogram shaped inclusions.By comparison with those of some classical and benchmarking models,the circle,the ellipse,etc.,the influence ranges of the exterior fields from different inclusion’s geometries are analyzed quantitatively.(2)Based on the complex variable method of isotropic elasticity,by constructing the prescribed eigenstrains in polynomial form of arbitrary order through the associated eigendisplacements,the elastic fields of arbitrary inclusions induced by polynomial eigenstrains are attributed to calculate the boundary integrals involved in a set of basic functions.For arbitrary polygonal inclusions,the involved boundary integrals are explicitly carried out,and the stress and displacement fields of some specific shaped inclusions,the triangle,the square,and the ellipse approximated by N-sided polygons,are numerically achieved.The effects of the shapes of inclusions and the forms of eigenstrains are analyzed by figures.(3)For the polynomial eigenstrain problem in anisotropic magneto-electro-elastic(MEE)materials,the prescribed extended eigenstains are also constructed through the associated extended eigendisplacements.From the extended Stroh formulism,some tedious derivations finally lead that the induced physical fields depend on two sets of basic functions,which contain boundary integrals over the transformed inclusion domain.For arbitrary polygonal inclusions,the involved boundary integrals are worked out explicitly.Numerical calculations conforms that the Eshelby’s polynomial conservation theorem is valid for anisotropic MEE materials.Field concentrations and singularities at vertexes of polygons are analyzed thoroughly,and demonstrated by the explicit forms of basic functions.