Analytical And Numerical Studies Of Axisymmetric Eshelby’s Inclusion In A Half Space

The work is supported by the National Natural Science Foundation of China(Grant Nos.51475057 and 51875059),and Fundamental Research Funds for the Central Universities(Project No.106112017CDJQJ328839).The titled research can be traced back to Eshelby’s pioneering work in 1957.Ever since then,the inclusion model has been providing a cutting-edge theory for exploring the mechanical properties and thermal failure mechanism of materials.Particularly,a broad range of applications have been evidenced in various machine components including gears,bearings and clutches,and also in semiconductor and electronic devices,in addition to many technical problems originated from localized failure on the scale of microns.Therefore,studying the comprehensive performance and life degradation of mechanical parts via the inclusion model has significant theoretical and practical values.The main contents of this thesis are as follows.First,the application of equivalent inclusion method is expounded,and the numerical equivalent inclusion method is introduced to solve the inhomogeneity of arbitrary shape in a half space.Based on the fundamental governing equations of the inclusion/eigenstrain theory,the complete solution of Eshelby’s ellipsoidal inclusion is formulated in explicit and geometrically meaningful form.The current work highlights an analytical study of the axisymmetric inclusions subjected to uniform thermal expansion in a half space,where the formulae are represented in closed-form for convenience of practical using.The derivations are deeply rooted in the potential theory with assistance of the Galerkin vector method.Through constructing the unit outer normal vectors in conjunction with some auxiliary functions,we derived closed-form expression for the displacements,strains and stresses both inside and outside the inclusion.Subsequently,the finite element software is used to solve the inclusion problem with uniform thermal expansion in a half space,and the contour plots are provided for stresses,strains and displacements.Through comparison with the numerical results,the present solution of an axisymmetric inclusion subjected to uniform thermal eigenstrains is validated.Subsequently,comprehensive parametric studies on the influences of the shape and the depth location in the half space are conducted.The current work evidently shows that all these geometric parameter combinations play an important role on the effects of the free surface.Next,axisymmetric problem considering inclusions with non-uniform thermal expansion in a half space is examined.In this case,the normal eigenstrain component along the depth direction differs from the other two normal components.In view of the superposition principle,the problem is decomposed into two sub-problems,where the former considers a uniform thermal expansion and the latter is concerned with the normal eigenstrain component only existing along the depth direction.The latter problem is carefully analyzed using the potential theory with the assistance of the Galerkin vector,where closed-form expressions of the elastic field are derived.Then,the numerical analyses are carried out for the purpose of validation,with main results given in contour plots.A benchmark example reports the elastic fields along the target line utilizing the analytical solution,showing almost identical results with those of the numerical solution.Lastly,a computational scheme is proposed for inclusions with multiple-layer in a half space,where the multi-layer inclusion may be numerically discretized into a group of corresponding single-layer inclusions.It is demonstrated that through the unit normal vectors and auxiliary functions,the analytical solution to the double-layer inclusion problem can be constructed,and the strategy is extended to any multi-layer inclusions.The numerical simulation of the double-layer inclusion problem by the finite element software is carried out,and the results are compared with the analytical solution to verify the effectiveness and validity of the present theoretical derivation.