This paper is concerned with the characterization of the macroscopic behavior and statistics for the distribution of the stress and strain-rate fields in composites consisting of random and isotropic suspensions of rigid spherical particles in power-law viscoplastic materials. For this purpose, use is made of the Fully Optimized Second-Order (FOSO) homogenization method (Ponte Castañeda, 2016) in combination with recently developed estimates (Kammer and Ponte Castañeda, 2022) for the macroscopic properties of the associated ‘linear comparison composite’ (LCC). Special attention is devoted to the method’s ability to account for the dependence of the homogenized properties of the nonlinear composite on the Lode angle (third invariant) of the applied loading. It is found that, while, for large particle volume fractions , the effective flow stress is only weakly dependent on the Lode angle, for dilute volume fractions, the dependence on the Lode angle becomes more pronounced. In the ideally plastic limit, as tends to zero, the effective yield stress is shown to depend linearly on for axisymmetric shear, while this dependence becomes weaker with a non-analytic leading-order correction of for pure shear loading. This strong dependence on the Lode angle at dilute concentrations is shown to be due to significant differences in the local deformation patterns, which become strongly anisotropic and localize for pure shear conditions, but do not for axisymmetric shear. In turn, the FOSO homogenization method is able to capture the statistical features of these different deformation patterns by providing consistent estimates for the covariance tensor of the strain-rate field fluctuations in the matrix phase, which tend to become more strongly anisotropic for the pure shear case. As increases, the shear bands are deflected by the randomly dispersed spheres leading to a more isotropic distribution of the stress and strain-rate fields, which is consistent with a weaker Lode angle effect. The estimates can also capture the effect of strong particle interactions, including the existence of a rigidity threshold where the macroscopic flow stress and field fluctuations blow up.

中文翻译：

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颗粒增强粘塑性复合材料宏观行为和场统计的完全优化二阶估计

本文关注的是由幂律粘塑性材料中刚性球形颗粒的随机和各向同性悬浮液组成的复合材料中应力场和应变率场分布的宏观行为和统计特征的表征。为此，使用完全优化二阶 (FOSO) 均质化方法（Ponte Castañeda，2016）结合最近开发的估计（Kammer 和 Ponte Castañeda，2022）来评估相关“线性比较复合材料的宏观特性” ”（LCC）。特别注意该方法能够解释非线性复合材料的均质特性对所施加载荷的洛德角（第三个不变量）的依赖性。研究发现，对于大颗粒体积分数，有效流动应力仅较弱地依赖于洛德角，而对于稀体积分数，对洛德角的依赖性变得更加明显。在理想塑性极限下，当趋于零时，有效屈服应力与轴对称剪切呈线性关系，而对于纯剪切载荷，这种依赖性随着非解析主阶校正而变弱。这种对稀浓度下洛德角的强烈依赖性被证明是由于局部变形模式的显着差异造成的，局部变形模式变得强各向异性并在纯剪切条件下局部化，但在轴对称剪切条件下则不然。反过来，FOSO 均质化方法能够通过为基体相中应变率场波动的协方差张量提供一致的估计来捕获这些不同变形模式的统计特征，对于纯剪切，这些波动往往会变得更强的各向异性。案件。随着增加，剪切带被随机分散的球体偏转，导致应力和应变率场更加各向同性分布，这与较弱的洛德角效应一致。这些估计还可以捕捉强粒子相互作用的影响，包括宏观流应力和场波动爆发的刚性阈值的存在。