A Micromechanical Study Of Heterogenous Materials Containing Inclusions Or Dislocations

The mechanical failure at the surface or subsurface relies heavily on the microstructural evolution in machine components.The study of the interaction between inhomogeneities and dislocations is a classical topic in solid mechanics and may provide a better understanding of the strengthening and hardening mechanisms in engineering materials.In the framework of micromechanics,the present dissertation intends to use a combination of theoretical and numerical analyses to explore the effects of inclusions and dislocations on the material properties.The main contributions are listed as follows:1)Based on the method of Green's function and potential theory,a complete set of the Eshelby tensors corresponding to the displacement,strain,and stress are derived for a general case of three-dimensional(3D)inclusions subjected to uniformly distributed eigenstrains.For a cuboidal inclusion,a complete elastic solution is analytically obtained in closed-form with the help of an effective notation.Through a limit process for plane strain conditions,analytical solutions of displacements,strains,and stresses are derived for a rectangular inclusion,and are further deduced for the case of plane stress.It is demonstrated that both interior and exterior elastic fields for all the elementary solutions can be represented in a unified algebraic form.In light of the properties of discrete convolutions,the present elementary solutions can be seamlessly combined with the fast Fourier transform(FFT)technique to enable an efficient numerical computation.The results demonstrated that the present work provides an effective tool to analyze the effects of the morphology,size,distribution form,and content level of inclusions on material properties.2)Under plane strain condition,the 2D Green's function for displacement is derived from a point eigenstrain excitation.By Green's theorem,the area integral can be converted to a contour integral along the boundary of the inclusion.For a typical line element forming a part of the inclusion boundary,the resultant displacements for both the interior and exterior points may be written in a unified algebraic form using the coordinates of the endpoints,which is convenient for geometric visualization and numerical implementation.Subsequently,the corresponding strain components may be derived from the displacement solution by differentiation.It is demonstrated that the path-integral formulation can be adapted to deal with a special case of linear eigenstrains,and the corresponding solutions are obtained analytically for either a horizontal or vertical line element.Furthermore,the present work shows that the displacement at the vertices or boundaries of a polygonal inclusion is bounded,while the strain Eshelby tensors may encounter a jump across the interface between the polygonal inclusion and the surrounding matrix.3)A complete set of Eshelby tensors are derived for the displacement,strain and stress in a 2D inclusion subjected to uniformly distributed eigenstrains.In the case of linear eigenstrains,the corresponding solution can be represented through an area integral over the inclusion domain.The applicability of Green's theorem is examined for both interior and exterior fields of the inclusion.It is shown that Green's theorem can be directly applied for the interior and exterior displacement fields,while a compensation term may be needed when evaluating the interior stress components.Accordingly,the line element solutions are derived in closed-form for the displacements,strains and stresses.It should be noted that the stress components in both interior and exterior fields can be expressed in a unified algebraic form by appropriate choice of the primitive functions.Benchmark of uniform and linear eigenstrains are provided to validate the present work,and parametric studies on different inclusion shapes are further conducted to demonstrate the effectiveness of the current solutions.4)The general formulation of the equivalent inclusion method(EIM)for solving the inhomogeneous inclusion problems is presented and is then applied to the 2D anti-plane case for two sub-problems: one is a screw dislocation interacting with a circular inhomogeneity,and the other is a circular inhomogeneity subjected to initial eigenstrains.Based on the energy principle,the governing equations of the interaction energy between a screw dislocation and an inhomogeneous inclusion are established.With the assistance of equivalent eigenstrains,analytical solutions of the interaction energy and force on dislocations are respectively obtained for a circular inhomogeneous inclusion interacting with a screw dislocation.For the numerical studies,the elementary solutions for the stress and the interaction energy are derived first,and an effective iterative method based on the numerical equivalent inclusion method(NEIM)is proposed for treating an arbitrarily shaped inhomogeneous inclusion.By virtue of the FFT techniques,the elastic energy for inhomogeneities and the interaction energy may be obtained with tremendous computational savings.It is worthwhile to note that the present anti-plane analysis furnishes one of the rare analytical works involving non-uniform equivalent eigenstrains.5)Using the energy principle and Gauss' s theorem,an integral representation for the interaction energy between an edge dislocation and an inhomogeneity is presented in this work.Since the integration is over a finite domain of the inclusion/inhomogeneity or along a dislocation trajectory,the assessment of the interaction energy becomes more convenient.With the assistance of the NEIM,this work also provides a novel numerical scheme,where the interaction energy can be numerically solved based on the line or area elements.Both the 2-norm and the maximum norm are examined for the relative errors,and the results demonstrate that the stress disturbance field caused by the inhomogeneity has a significant influence on the final solutions,especially when the dislocation is located in the neighborhood of the inhomogeneity.Compared with the0-th iterative scheme,the present computational method shows excellent numerical convergence and stability.Furthermore,the effects of different shapes of the inhomogeneity on the interaction energy are also considered,and parametric studies illustrate that the present method appears to be convenient and efficient for handling arbitrarily shaped inhomogeneities interacting with an edge dislocation.