A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints

A generalized strain energy-based homogenization method for 2-D and 3-D cellular materials with and without periodicity constraints is proposed using Hill’s Lemma and the matrix method for spatial frames. In this new approach, the equilibrium equations are enforced at all boundary and interior nodes and each interior node is allowed to translate and rotate freely, which differ from existing methods where the equilibrium conditions are imposed only at the boundary nodes. The newly formulated homogenization method can be applied to cellular materials with or without symmetry. To illustrate the new method, four examples are studied: two for a 2-D cellular material and two for a 3-D pentamode metamaterial, with and without periodic constraints in each group. For the 2-D cellular material, an asymmetric microstructure with or without periodicity constraints is analyzed, and closed-form expressions of the effective stiffness components are obtained in both cases. For the 3-D pentamode metamaterial, a primitive diamond-shaped unit cell with or without periodicity constraints is considered. In each of these 3-D cases, two different representative cells in two orientations are examined. The homogenization analysis reveals that the pentamode metamaterial exhibits the cubic symmetry based on one representative cell, with the effective Poisson’s ratio v¯ being nearly 0.5. Moreover, it is revealed that the pentamode metamaterial with the cubic symmetry can be tailored to be a rubber-like material (with v¯ ≅0.5) or an auxetic material (with v¯< 0).

Two Versions of the Extended Hill’s Lemma for Non-Cauchy Continua Based on the Couple Stress Theory

Two versions of the extended Hill’s lemma for non-Cauchy continua satisfying the couple stress theory are proposed. Each version can be used to determine two effective elasticity (stiffness) tensors: one classical and the other higher order. The classical elasticity tensor relates the symmetric part of the force stress to the symmetric strain, whereas the higher-order elasticity tensor links the deviatoric part of the couple stress to the non-symmetric curvature. Four sets of boundary conditions (BCs) are identified using Version I, and three sets of BCs are obtained from Version II of the extended Hill’s lemma, which can all satisfy the Hill–Mandel condition. For each BC set selected, 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. The effective elastic constants obtained in this analysis are compared with those predicted by the Voigt and Reuss bounds and computed through a finite element model constructed using COMSOL, which verifies and supports the current method.

Effective Elastic Behavior of Irregular Closed-Cell Foams

This paper focuses on the computational modeling of the effective elastic properties of irregular closed-cell foams. The recent Hill’s lemma periodic computational homogenization approach is used to predict the effective elastic properties. Three-dimensional (3D) rendering is reconstructed with the tomography slices of the real irregular closed-cell foam. Its morphological description is analysed to generate realistic numerical closed-cell structures by the Voronoi-based approach. The influences of the Representative Volume Element (RVE) parameters (i.e., the number of realizations and the volume of RVE) and the relative density on the effective elastic properties are studied. Special emphasis is placed on the appropriate choice of boundary conditions. Satisfying agreements between the homogenized results and the experimental results are observed.

Extended Hill’s lemma for non-Cauchy continua based on the simplified strain gradient elasticity theory

Hill's lemma for the Cauchy continuum has been playing an important role in micromechanics. An extended version of Hill's lemma for non-Cauchy continua is formulated using the simplified strain gradient elasticity theory (SSGET), which contains only one material length scale parameter and can account for the microstructure-dependent strain gradient effect. As a corollary of the extended Hill's lemma, the Hill–Mandel macro-homogeneity condition for non-Cauchy continua is obtained along with the general forms of kinematically and statically admissible boundary conditions that are required for constructing an energetically equivalent homogeneous comparison material. Based on these general forms, four sets of uniform boundary conditions are identified, which are implementable in material tests and can be directly used in homogenization analyses of heterogeneous materials. It is shown that when the strain gradient effect is suppressed, the extended Hill's lemma recovers the classical Hill's lemma for the Cauchy continuum and the extended Hill–Mandel condition reduces to its classical counterpart.