Analytical And Numerical Studies On The Eshebly Inclusion In An Elastic Semi-infinite Plane With Applications

The work is supported by the National Science Foundation of China titled "A Micromechanical Study of the Friction and Wear Behavior of Inhomogeneous materials containing Inclusions and Cracks(No.51875059)" and Fundamental Research Funds for the Central Universities entitled “The Theoretical and Experimental Study of Micromechanical Mechanisms of the Contact Fatigue for the Inhomogeneous Materials(No.106112017CDJQJ328839)”.The existence of inclusions and inhomogeneities in materials have a malignant impact on the mechanical properties,such as hardness,strength,ductility,fracture toughness.On the other hand,degraded reliability and service life of mechanical parts due to material inhomogeneity may directly affect the operation of the whole mechanical system.Thus,it is indispensable to quantitively evaluate the deteriorate effects on the mechanical behavior caused by inclusions and inhomogeneities.This work performs a systematical study on the problem of thermal inclusion in elastic half plane by means of analytical and numerical method,as well as finite element simulation.By extending the classical Eshelby inclusion theory in micromechanics,an effective algorithm is developed.In addition,the verification and applicability are explored extensively by using the finite element method.The results of the current work have practical significance for improving the mechanical properties of materials and enhance the resistability of fatigue.The main contents are as follows.First,the theoretical solution of a thermal inclusion in a half plane is derived,mainly for the elliptic cylindrical or planar elliptical inclusion subjected to uniform thermal expansion.From the basic theory of plane elasiticity,formation of plane problem with uniform thermal expansion inclusions is studied.Thereupon,based on the analytical solution of an ellipsoidal thermal inclusion in a half space,the closed-form solution of the full elastic fields,including the displacement,strain and stress,of the elliptic cylindrical thermal inclusion under plane strain is derived.In the process of derivation,several techniques such as tensorial indices,auxiliary functions,and the external unit normal vector of the imagary ellipse are introduced,making the expression simpler and more compact.According to the convertion rule between the plane problems,the plane stress solution is derived analytically.Consequently,the discontinuity across the interface of interior and exterior fields is discussed.In addition,the response field of a semi-infinite plane with a circular thermal inclusion in cylindrical coordinates is studied theoretically to achieve more extensive applications.Secondly,the correctness of the analytical solution and the validity of the numerical solution are verified by the results of the numerical solution of an elliptic cylindrical thermal inclusion in a half-plane.We have developed algorithms to calculate the displacement,strain and stress field caused by the elliptic cylindrical or planar elliptical inclusion originated from uniform expansion eigenstrain in half plane.Besides,the numerical solution is mainly based on the semi-analytical method.By discretizing the elliptical inclusion into a system of uniform rectangular element and superposition of the elementary solutions,the numerical solution of the elliptic inclusion are obtained.The fast Fourier algorithm(FFT)is incorporated into the numerical algorithms to improve the computational efficiency.Parametric studies are performed for an elliptic cylinder thermal inclusion in a half plane on the depth location and other geometric parameters,and the influence of free surface is discussed.Lastly,the finite element method(FEM)is used to verify the results of theoretical solution for the half-plane inclusion and contact problems.First,the finite element model of an elliptical inclusion in a half plane with uniform thermal expansion inclusion is established,and contour plots of the displacement,strain and stress in response field are reported.The correctness of the analytical solution is verified by the FEM.Then,the FEM studies are also implemented for the general two-dimensional conforming contact problems.Several benchmark examples are studied to examine the scope of application.The contact-inclusion model is analyzed by FEM.