Explicit Analytical Solutions For A Complete Set Of The Eshelby Tensors Of An Ellipsoidal Inclusion & Numerical Solutions Based On FFT

The present work is supported by the National Natural Science Foundation of China(Grant No.51475057,“Numerical and experimental investigations of microstructural evolution in bearing steels under rolling contact fatigue”),Chongqing City Science and Technology Program(No.cstc2013jcyjA70013,“The fatigue life study of bear steels containing non-metallic inhomogeneities based on Monte Carlo simulation”)and Fundamental Research Funds for the Central University(No.CDJZR14285501).In practical engineering applications,the components in aerospace aircrafts,artificial satellites and many other machineries,are usually irreplaceable and the failure of transmission parts,such as bearings and gears,may lead to paralysis of the entire mechanical system.Consequently,the reliability and fatigue life of materials are crucial for the long-term operations of fundamental equipment.Since the macroscopic performance is dominated by the microstructure of material,it is meaningful to investigate the effect of material constituents at the microscopic scale from the perspectives of materials science and micromechanics.According to the Eshelby-Mura inclusion theory,the disturbed elastic field caused by material heterogeneity plays an important role to the localized failure mechanism of a material.The present work will focus on the microscopic mechanistic analysis and expects to provide an effective guidance for improving the mechanical or physical properties of the material.The main contents in this thesis include the following four parts:a)Based on the method of Green's function,a complete set of the Eshelby tensors corresponding to the displacement,displacement gradient,strain and stress are derived in explicit analytical expression.The matrix formation of the Eshelby tensor is also given.By taking advantage of the unit normal vectors of a confocal imaginary ellipsoidal,the explicit expressions for the exterior field are represented in a more compact,concise and geometrically meaningful form.The symmetry property of the Eshelby tensors and the interior elastic field of an ellipsoidal inclusion with uniform eigenstrain are discussed in detail.The results show that the stress and strain Eshelby tensors are constant depending on the elastic modulus and shape of the ellipsoid.In addition,the jump conditions for the strain and stress Eshelby tensors are discussed.The displacement Eshelby tensor is linearly related to its coordinates for a interior point and is continuous across the boundary of the ellipsoid.A benchmark example is also given to validate the present solution.b)The numerical method of an arbitrarily shaped inclusion in full space containing any distributed eigenstrain is proposed.The numerical approach makes use of the excitation-response notation.The computational domain may be divided into uniform cuboidal elements,where the analytical solution for a cuboidal inclusion is employed through superposition to achieve the numerical discretization.The present semi-analytical method combining analytical solution and numerical evaluation exhibits computational efficiency and is ease of numerical management.c)A key feature of the present semi-analytical method is that the displacement and stress formulas can be represented in the form of three-dimensional discrete convolution.In order to improve the computational efficiency,the fast Fourier transform(FFT)algorithm is utilized to accelerate the calculation.Moreover,the accuracy of the present numerical method is verified through comparison with the derived analytical solutions.d)The related half-space ellipsoidal inclusion is discussed.The method of images,potential theory as well as the FFT numerical approach,are discussed in this work.Furthermore,the explicit analytical solution of an ellipsoidal inclusion subjected to uniform thermal eigenstrain in a semi-infinite space is derived.The results exhibit good agreement with those obtained by the method of images.In addition,miscellaneous topics,including the equivalent inclusion method,the Euler angles for arbitrarily orientated inclusions,and the corresponding examples are presented.