The Analytical And Closed-form Solutions For The Half-space Ellipsoidal Inclusion And Arbitrarily Shaped Plane Inclusion

The current dissertation is supported by the National Natural Science Foundations of China entitled “Numerical and Experimental Investigations of Microstructural Evolution in Bearing Steels under Rolling Contact Fatigue(No.51475057)” and “A Micromechanical Study of the Friction and Wear Behavior of Inhomogeneous materials containing Inclusions and Cracks(No.51875059)”;Fundamental Research Funds for the Central Universities titled “The Theoretical and Experimental Study of Micromechanical Mechanisms of the Contact Fatigue for the Inhomogeneous Materials(No.106112017CDJQJ328839)” and the Research Fund(No.2018CDYJSY0055);the Graduate Research and Innovation Foundation of Chongqing(No.CYB17025).Inhomogeneous materials are broadly used in mechanical transmission system,aerospace,wind energy,and defense technology owing to their excellent properties of light weight,high strength and corrosion resistances.This dissertation intends to use the method of micromechanics which is laid by Eshelby,to explore the inclusion problems.This subject contains the displacement solution of elliptic cylindrical shaped Eshelby inclusion,stress analysis of the polygonal inclusion with uniform eigenstrains,analytical solutions for the elastic field of half-space thermal ellipsoidal inclusion,closed-form solution for the horizontally aligned thermal-porous spheroidal inclusion in a semi-infinite space,and the interacting or competing problems of the full-space ellipsoidal inhomogeneities subject to the thermal-porous eigenstrain and the half-space dilatational inclusions.The main contents of this paper are listed as follows:Firstly,based on the three-dimensional displacement solution of Eshelby's ellipsoidal inclusion,the closed-form solution of a two-dimensional Eshelby inclusion of elliptic cylindrical shape is derived.The exterior field of the elliptic cylindrical inclusion tends to be more complex,but the obstacle is overcome by utilizing the outward unit normal vector of an imaginary confocal ellipse.The current work complements our previous study on stress and strain solutions,and hence concludes that a complete elasticity solution for a 2D Eshelby's inclusion may be obtained in explicit closed-form.The proposed formulation in matrix form is presented for ease of programming and engineering applications.Secondly,we present a contour integral method for deriving the closed-form solution for the stresses produced by an arbitrary shaped plane inclusion with uniform eigenstrain.On the basis of Green's functions,the stresses at field point can be solved through converting area integral to the contour integral along the boundary of inclusion.It is demonstrated that both the interior and exterior stress field can be analytically represented in a unified expression in terms of only elementary functions.The case of an irregular nonelliptical inclusion with thermo-porous eigenstrain is derived.The current approach is directly applicable to the non-uniform eigenstrain problem and is effective and straightforward way to determine the stress distributions.Furthermore,the stresses for the rectangular inclusion subjected to the linear eigenstrain and the influence of roughness degree for the weakly non-circular inclusions are also investigated.Thirdly,based on the explicit analytical solutions for the complete elastic field produced by an ellipsoidal thermal inclusion in a semi-infinite space,the current result demonstrates that the interior strains and stresses are no longer uniform,and appear to be much more complex.Nevertheless,the expressions can be represented in a more compact and geometrically meaningful form by introducing the unit outward normal vector.The degenerate case of a spherical thermal inclusion are derived in a closed form,and is verified by the well-known Mindlin solution.Furthermore,the current work analytically explores this asymmetric problem of thermo-porous spheroidal inclusion with the assistance of geometric interpretation.The complete solution to the displacement,strain and stress is formulated in Cartesian coordinates for ease of engineering applications.Lastly,Porous materials are known to be of practical importance to many areas of applied science and engineering,including material science,petroleum and construction engineering,etc.In this paper,the interactions between multiple ellipsoidal inhomogeneities with arbitrary orientation due to the pore pressure and temperature changes are investigated.Based on the well-known equivalent inclusion method,an approximate analytical approach is presented to evaluate the full fields with respect to the displacement,strain and stress of arbitrarily distributed inhomogeneities.This method avoids increasing the numerical complexity of the computational problem,and is an effective and straightforward way to study the elastic fields of multiple inhomogeneities.The illustrative benchmark examples are performed to validate the robustness and accuracy of the current method.In addition,the case of half-space multiple inclusions subjected to the poro-thermo eigenstrains are also investigated.