A new strain energy-based method for homogenization of 2-D and 3-D cellular materials is developed using an extended version of Hill’s Lemma for the non-Cauchy continuum satisfying the micropolar elasticity theory. Both kinematic and mixed boundary conditions (BCs) that obey the Hill-Mandel condition are identified. The newly proposed method requires that the nodal equilibrium equations be explicitly satisfied at all boundary and interior nodes, unlike the existing approach which does not enforce the equilibrium conditions at interior nodes. For 2-D cellular materials, it is shown that purely kinematic BCs (with prescribed displacement and micro-rotation vectors) and mixed BCs (with prescribed couple stress traction and displacement vectors) can both be used for homogenization, and periodic constraints can be readily accommodated in the former. For 3-D cellular materials, it is demonstrated that mixed BCs (with prescribed couple stress traction and displacement vectors) enable the homogenization satisfying the Hill-Mandel condition. 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. For the 2-D lattice material, the effective classical and micropolar stiffness tensors are obtained using purely kinematic BCs (with and without periodic constraints) and mixed BCs. The closed-form expressions of the effective stiffness tensor components derived here are compared to those given by the existing strain energy-based homogenization method and are seen to be more accurate. For the 3-D pentamode material, mixed BCs are employed in the homogenization, and the effective stiffness tensor components are obtained in closed-form expressions for the first time. The numerical results for the pentamode material given by these analytical formulas are found to agree well with those provided by a finite element model constructed using COMSOL.

使用希尔的引理的扩展版本开发了一种基于应变能的，用于2D和3D细胞材料均质化的新方法，适用于满足微极性弹性理论的非Cauchy连续体。识别出符合Hill-Mandel条件的运动边界条件（BCs）和混合边界条件（BCs）。新提出的方法要求节点平衡方程在所有边界节点和内部节点处均得到明确满足，这与现有方法不要求在内节点处达到平衡条件不同。对于二维细胞材料，已证明纯运动学BC（具有指定的位移和微旋转矢量）和混合BC（具有指定的耦合应力牵引和位移矢量）均可用于均质化，并且周期性约束很容易容纳在前。对于3-D多孔材料，已证明混合BC（具有指定的耦合应力牵引和位移矢量）可以实现满足Hill-Mandel条件的均质化。通过应用这种新方法，研究了两种均质化细胞材料的示例情况–一种用于2-D晶格结构，另一种用于3-D五模材料。对于二维晶格材料，使用纯运动学BC（有和没有周期约束）和混合BC可获得有效的经典和微极刚度张量。将此处得出的有效刚度张量分量的闭式表达式与现有的基于应变能的均质化方法给出的闭式表达式进行比较，可以发现它们更精确。对于3-D五模态材料，在均质化中使用混合BC，并以封闭形式首次获得有效刚度张量分量。发现这些分析公式给出的五模式材料的数值结果与使用COMSOL构建的有限元模型提供的数值结果非常吻合。