This paper presents a general method to estimate the first and second moments of the stress and deformation fields in linearized elastic and viscous composites and polycrystals. The methodology can be seamlessly extended by means of the ‘linear comparison’ variational homogenization methods (Ponte Castañeda, 1991, 2016) to extract the corresponding field statistics in composites and polycrystals with nonlinear properties, including hyperelastic composites, as well as viscoplastic composites, undergoing finite deformations. Expressions are obtained for the overall response and statistics of the deformation gradient field in hyperelastic composites with linearized (incremental) response characterized by elasticity tensors exhibiting major symmetry, but not minor symmetry. These results are then specialized for composites with fully symmetric elasticity tensors, and use is made of a well-known analogy to convert these results into corresponding estimates for the overall viscosity tensor and statistics of the velocity gradient for linearly viscous composites. Subsequently, the velocity-gradient statistics are projected onto the appropriate symmetric and anti-symmetric subspaces to extract the strain-rate and spin tensor statistics, respectively. The formulation recovers known relations for the overall response and fields statistics, but also allows for the computation of hitherto unexplored quantities, such as the second moments of the spin fluctuations in viscous composites. To illustrate the results, applications are considered for various examples, including two-phase viscous composites, as well as viscous polycrystals, with special types of isotropic and anisotropic linearized response.

本文提出了一种通用方法，用于估计线性化的弹性和粘性复合材料及多晶中的应力和形变场的第一和第二矩。可以通过“线性比较”变分均质方法（PonteCastañeda，1991年，2016年）无缝扩展该方法，以提取具有非线性特性的复合材料和多晶体（包括超弹性复合材料以及粘塑性复合材料）中相应的场统计数据。有限变形。获得了具有线性（增量）响应的高弹复合材料的整体响应和形变梯度场的统计表达式，其特征在于弹性张量显示出主要对称性，但没有次要对称性。然后将这些结果专门用于具有完全对称弹性张量的复合材料，并使用众所周知的类比将这些结果转换为整体粘度张量的相应估计值以及线性粘性复合材料的速度梯度统计数据。随后，将速度梯度统计量投影到适当的对称和反对称子空间上，以分别提取应变率统计量和自旋张量统计量。该公式恢复了整体响应和场统计的已知关系，但也允许计算迄今尚未探索的数量，例如粘性复合材料中自旋涨落的第二阶矩。为了说明结果，我们考虑了各种示例的应用，包括两相粘性复合材料，