A finite volume based approach is employed in the solution of unit cell problems at different orders of the asymptotic field expansion to construct a homogenization theory for anti-plane shear loading of unidirectional fiber-reinforced periodic structures. This new construction complements and further extends our recent contribution to asymptotic homogenization based on locally-exact elasticity unit cell solutions, He and Pindera (2020), to unit cells with multiple inclusions of arbitrary shapes. The present approach builds upon the previously developed finite-volume direct averaging micromechanics theory applicable under uniform strain fields, and extends it to account for strain gradients and non-vanishing microstructural scale relative to structural dimensions. The unit cell problems at different orders of the asymptotic field expansion are solved by satisfying local equilibrium equations in each subvolume of the discretized microstructure in a surface-averaged sense. This facilitates construction of local stiffness matrices at the subvolume level and subsequent assembly into the global stiffness matrix for the unit cell response under uniform and gradient strain fields. Comparison of the calculated microfluctuation functions and associated stress fields under uniform and gradient strain fields with those reported in the literature verifies the finite volume asymptotic solutions. The new theory's ability to accurately recover local fields is further illustrated through comparison with the direct numerical solution of a periodic structure with varying number of inclusions under gradient loading. The proposed homogenization approach is an efficient and accurate alternative to current numerical techniques for the analysis of periodic materials experiencing strain gradients regardless of microstructural scale, inclusion shape and number, demonstrated by analyzing arrays characterized by elliptical and square inclusions and multi-inclusion unit cells.

基于有限体积的方法用于解决渐进场扩展不同阶数的晶胞问题，以构建单向纤维增强周期性结构的反平面剪切载荷的均质化理论。这种新的构造补充并进一步扩展了我们最近对基于局部精确弹性单位细胞解决方案He和Pindera（2020）的渐近均质化的贡献，扩展到具有多个任意形状包含物的单位细胞。本方法建立在先前开发的可在均匀应变场下应用的有限体积直接平均微力学理论的基础上，并将其扩展为考虑了应变梯度和相对于结构尺寸不变的微观结构尺度。通过满足离散平均微观结构的每个子体积中的局部平衡方程，可以解决渐近场扩展不同阶次的晶胞问题。这有助于在子体积水平上构造局部刚度矩阵，并随后组装到整体刚度矩阵中，以便在均匀应变场和渐变应变场下对单位单元进行响应。在均匀和梯度应变场下计算的微涨落函数和相关应力场与文献报道的比较证明了有限体积渐近解。通过与梯度载荷下包含不同数量夹杂物的周期结构的直接数值解进行比较，进一步说明了新理论精确恢复局部场的能力。