The relationship between yield stress and the distribution of microscopic plastic deformation was numerically investigated by using a crystal plasticity finite element method (CP-FEM) in the model where particles were randomly distributed. It was in order to reveal which particle spacing. i.e., the maximum, minimum or average particle spacing, can be taken as the representative length which controls yielding. The critical resolved shear stress for the onset of the slip deformation in any element was defined under the extended equation in the Bailey-Hirsch type model. The model includes the term of the Orowan stress obtained from the local values of the representative length. Each particle spacing was distributed with a standard deviation of approximately 2 to 3 times larger than the average particle spacing. The macroscopic mechanical properties obtained with CP-FEM were in good agreement with those experimentally obtained. The onset of microscopic slip deformation depended on the particle distribution. Plastic deformations started first in the area where the particle size is larger, then the plastic region grows in the areas where the particle spacing is smaller. Slip deformation had occurred in 90% of the matrix phase by the macroscopic yield point. The length factor in the Orowan equation was the average spacing of the particles in the model, which is in good agreement with Foreman and Makin. The CP-FEM indicated that in dispersed hardened alloys, microscopic load transfer occurred between the areas where the large particles spacing and the small one at the yielding.

在颗粒随机分布的模型中，使用晶体塑性有限元方法（CP-FEM）数值研究了屈服应力与微观塑性变形分布之间的关系。这是为了揭示哪个粒子间距。即，最大，最小或平均粒子间距可以用作控制屈服的代表性长度。在Bailey-Hirsch型模型的扩展方程式下，定义了任何元素中滑动变形发生的临界解析剪切应力。该模型包括从代表长度的局部值获得的Orowan应力项。每个粒子间距的分布标准偏差大约是平均粒子间距的2到3倍。CP-FEM获得的宏观力学性能与实验获得的力学性能非常一致。微观滑动变形的开始取决于颗粒的分布。塑性变形首先从粒径较大的区域开始，然后，塑料区域在粒子间距较小的区域中生长。在宏观屈服点处，基体相的90％发生了滑移变形。Orowan方程中的长度因子是模型中粒子的平均间距，与Foreman和Makin很好地吻合。CP-FEM表明，在分散的硬化合金中，在屈服时大颗粒间距与小颗粒间距之间发生了微观载荷传递。