Homogenization of Composite and Cellular Materials Incorporating Microstructure and Surface Energy Effects

In the last few decades, the popularity of composite and cellular materials has rapidly increased through their widespread applications in multiple engineering fields including aerospace, automotive, civil and biomedical. However, to a large extent, the success of their practical applications depends on our ability to predict their mechanical behavior by using high-fidelity mechanics models. Micromechanical modeling of composite and cellular materials is a challenging task due to the heterogeneous nature of such materials and the interactions among various constituent phases at the microscopic level, which result in non-homogeneous deformation, strain and stress fields. Therefore, it is necessary to develop simple but accurate models that allow replacing heterogeneous materials with equivalent homogeneous comparison solids, whose macroscopic properties can be easily predicted. A plethora of homogenization methods have been developed for predicting effective properties of heterogeneous composite materials based on their microstructures. Hill’s lemma is a milestone in the development of micromechanics that provides the foundation for homogenization analyses. In this work, Hill’s lemma is extended to non-Cauchy continua by using three higher-order elasticity theories and a surface elasticity theory. The extended versions of Hill’s lemma are then applied to homogenize 2-D and 3-D composites and cellular materials. In Chapter 2, two extended versions of Hill’s lemma are provided for non-Cauchy continua satisfying a modified couple stress theory. The first version is used to obtain the classical elasticity tensor, while the second version is employed to determine the couple stress elasticity tensor. The kinematic boundary conditions used in each version of the extended Hill’s lemma are then modified to accommodate composites with periodic microstructures by introducing periodic parts to the displacement and micro-rotation fields and reconstructing the Hill–Mandel condition. In Chapter 3, two versions of the extended Hill’s lemma for non-Cauchy continua satisfying the couple stress theory are proposed. Four sets of boundary conditions (BCs) are identified using each version. For each BC set obtained, admissibility and average field requirements are checked. Furthermore, the equilibrium is examined for the cases with the kinetic BCs, and the compatibility is checked for the cases with the kinematic BCs. To illustrate the two newly proposed versions of the extended Hill’s lemma, a homogenization analysis is conducted for a two-phase composite using a meshfree radial point interpolation method. In Chapter 4, a modified strain energy-based method for homogenization of 2-D and 3-D cellular materials is developed using an extended version of Hill’s Lemma built on the micropolar elasticity theory. The newly proposed method requires that the nodal equilibrium equations to be explicitly satisfied at all nodes, unlike the existing approach which does not enforce the equilibrium conditions at interior nodes. Two sample cases of homogenization of cellular materials are studied by applying the new method – one for a 2-D lattice structure and the other for a 3-D pentamode material. In Chapter 5, an extended version of Hill’s lemma for non-Cauchy continua satisfying the modified couple stress theory in the bulk and the surface elasticity theory in the surface layer is proposed. Four sets of boundary conditions are identified using this extended Hill’s lemma. The four boundary condition sets proposed here are all checked for admissibility. The kinetic boundary condition set is then applied to homogenize a two-phase composite. The numerical results are compared with those obtained using a finite element model constructed using the COMSOL Multiphysics through its weak form PDE interface