Grain boundary strengthening effect for polycrystalline copper is studied numerically using crystal plasticity in conjunction with cylindrical indentation simulations under the plane strain condition. In order to compare with an isotropic, heterogeneous continuum model a new constitutive relation is developed. This new nonlocal continuum model that encompasses the heterogeneity in yield strength based on the exponentiated Weibull function can predict the plastic properties of materials in the micron length scale. The spatial description of the deformation gradient two-point tensor is utilized to capture the intrinsic size effect in line with the subsequent deformation measures. Moreover, the total geometrically necessary dislocation density is obtained from the non-zero components of Nye dislocation density tensor. From the simulation, the relationship between the effective Green–Lagrange strain and effective stress measures is explained using the persistent long-range order and intermittent short-range order. The observation derived from the analogy between the cylindrical indentation and the progress in cylindrical voids describes how different average grain sizes of polycrystalline materials are compared with the behavior of isotropic materials. The trajectories of directions of both principal stretch and maximum shear strain explain that the internal stresses induced by cylindrical indentation either hinder or reinforce the dislocation flow, forming strain localization sporadically. The grain size dependence of polycrystalline modeling incorporates the Hall–Petch strengthening as well as the homogenization of anisotropic polycrystalline metal into the isotropic effective medium. This is a physically-based model that is used to model failure characterization in metals.

中文翻译：

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圆柱压痕中多晶塑性建模的晶粒尺寸依赖性

使用晶体塑性结合圆柱压痕模拟在平面应变条件下对多晶铜的晶界强化效应进行了数值研究。为了与各向同性、异质连续介质模型进行比较，开发了一种新的本构关系。这种新的非局部连续介质模型包含基于取幂威布尔函数的屈服强度异质性，可以预测微米长度尺度上材料的塑性特性。利用变形梯度两点张量的空间描述来捕捉与后续变形措施一致的内在尺寸效应。此外，总的几何必要位错密度是从 Nye 位错密度张量的非零分量中获得的。从模拟来看，有效格林-拉格朗日应变和有效应力测量之间的关系用持久的长程有序和间歇的短程序来解释。从圆柱压痕和圆柱空隙进展之间的类比得出的观察结果描述了如何将多晶材料的不同平均晶粒尺寸与各向同性材料的行为进行比较。主拉伸方向和最大剪切应变方向的轨迹解释了圆柱形压痕引起的内应力阻碍或加强了位错流动，偶尔形成应变局部化。多晶建模的晶粒尺寸依赖性将霍尔-佩奇强化以及各向异性多晶金属均匀化到各向同性有效介质中。这是一个基于物理的模型，用于模拟金属的失效特征。