HCP materials are exceedingly being used as alloys and composites in several high strength light weight applications such as aerospace and aeronautical structures, deep sea maritime applications, and as biocompatible materials. To understand the deformation of HCP materials, reliable tools and techniques are required. One such technique is the Elasto-Plastic Self Consistency (EPSC) model. ESPC models use Eshelby’s Inclusion Theory as their basic formulation to model the strain experienced by a grain within a strained material sample. One of the oldest approximations (or models) used to model the grain’s strain within a strained sample is the Taylor’s Assumption (TA). TA assumes that each grain is strained to the same average value. EPSC models are different from the TA model since each grain modelled by the EPSC model would be strained to a different value. This is possible and obtained by solving an infinite domain boundary value problem. This key advantage of the EPSC model can therefore predict localized weak spots within material samples.EPSC models use the concept of eigen strain where the inhomogeneous grain is replaced with an equivalent inclusion. The technique proposed in this research is used to simulate uniaxial tension of rolled textured Magnesium. The number of deformation modes used in this research is seven. Both slipping systems and twinning systems are included in the simulation. The hardening phenomenon is described as a function of self-hardening as well as latent-hardening. As stated in (S. Kweon, 2020), modelling the interactive hardening requires a more robust numerical iterative technique. An improved robust iterative numerical technique is explained in (Daniel Raja, 2021) and (Soondo Kweon D. S., 2021). This research implements the equivalent inclusion theory in combination with the numerical iterative technique developed in the aforementioned papers.The report begins with the need for this research and advocates for the same. Then, the conceptional theories and the imaginary thought experiment performed by John D. Eshelby is presented. The concept of “Eigen Strain” which serves as the base work needed to understand and formulate the Equivalent Inclusion Theory is described in detail. The Equivalent Inclusion is then presented and developed. The concept of Green’s Function is presented and explained. These concepts serve as the building block for the derivation and calculation of the Eshelby Tensor which relates the concepts of eigen strain and constrained strain. The report concludes the theory section with the amalgamation of the ideas of the Green’s Function and Eigen Strain to develop the Eshelby Tensor for an Isotropic material as well as Anisotropic materials. In the following section, the unit cell accompanied with the deformation modes within the unit cell of an HCP material that are used in these simulations are presented. Following unit cell model, the crystal plasticity model which includes plastic deformation, hardening laws, and elastic deformation is elaborated. The results obtained from the simulation are presented and salient features are highlighted that are observed in the results. Lastly, the report concludes by pointing out key “take aways” from this research and identifies possible avenues for future research.Additionally, ten appendices are included towards the end of this report to enhance understanding of complicated derivations and solutions. Lastly, the author’s vita is included at the end of the report.

Eshelby-type inclusion problems of an infinite or a finite homogeneous isotropic elastic body containing an arbitrary-shape inclusion prescribed with an eigenstrain and an eigenstrain gradient are analytically solved. The solutions are based on a simplified strain gradient elasticity theory (SSGET) that includes one material length scale parameter in addition to two classical elastic constants. For the infinite-domain inclusion problem, the Eshelby tensor is derived in a general form by using the Green’s function in the SSGET. This Eshelby tensor captures the inclusion size effect and recovers the classical Eshelby tensor when the strain gradient effect is ignored. By applying the general form, the explicit expressions of the Eshelby tensor for the special cases of a spherical inclusion, a cylindrical inclusion of infinite length and an ellipsoidal inclusion are obtained. Also, the volume average of the new Eshelby tensor over the inclusion in each case is analytically derived. It is quantitatively shown that the new Eshelby tensor and its average can explain the inclusion size effect, unlike its counterpart based on classical elasticity. To solve the finite-domain inclusion problem, an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET are proposed and utilized. The solution for the disturbed displacement field incorporates the boundary effect and recovers that for the infinite-domain inclusion problem. The problem of a spherical inclusion embedded concentrically in a finite spherical body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. It is demonstrated through numerical results that the newly obtained Eshelby tensor can capture the inclusion size and boundary effects, unlike existing ones. Finally, a homogenization method is developed to predict the effective elastic properties of a heterogeneous material using the SSGET. An effective elastic stiffness tensor is analytically derived for the heterogeneous material by applying the Mori-Tanaka and Eshelby’s equivalent inclusion methods. This tensor depends on the inhomogeneity size, unlike what is predicted by existing homogenization methods based on classical elasticity. Numerical results for a two-phase composite reveal that the composite becomes stiffer when the inhomogeneities get smaller.

The Eshelby-type problems of an arbitrary-shape polygonal or polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material are analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensors for a plane strain inclusion with an arbitrary polygonal cross section and for an arbitrary-shape polyhedral inclusion are analytically derived in general forms in terms of three potential functions. These potential functions, as area integrals over the polygonal cross section and volume integrals over the polyhedral inclusion, are evaluated. For the polygonal inclusion problem, the three area integrals are first transformed to three line integrals using the Green's theorem, which are then evaluated analytically by direct integration. In the polyhedral inclusion case, each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach. In addition, the Eshelby tensor for an anti-plane strain inclusion with an arbitrary polygonal cross section embedded in an infinite homogeneous isotropic elastic material is analytically solved. Each of the newly derived Eshelby tensors is separated into a classical part and a gradient part. The latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. For homogenization applications, the area or volume average of each newly derived Eshelby tensor over the polygonal cross section or the polyhedral inclusion domain is also provided in a general form. To illustrate the newly obtained Eshelby tensors and their area or volume averages, different types of polygonal and polyhedral inclusions are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the each SSGET-based Eshelby tensor for all inclusion shapes considered vary with both the position and the inclusion size. It is also observed that the components of each averaged Eshelby tensor based on the SSGET change with the inclusion size.