Theories describing the important role of small particles for strengthening of metals have evolved since the pioneering work of Orowan. Here, this problem is analysed by a strain gradient plasticity (SGP) theory. The structure of the governing equations on non-dimensional form reveals that the plastic strain in the matrix material is to zeroth order approximation constant for a sufficiently small particle size $a$ in comparison to material length scale $\ell$. Based on this observation, a perturbation solution has been developed by expansions of all field variables in terms of $a/\ell$ and the volume fraction of particles $f$. The simple structure of the plastic strain field is also exploited to derive an upper bound solution from the principles of virtual work and maximum plastic dissipation. These analytical solutions are then used to derive expressions for the yield stress taking into account a random distribution of particles of various size and shape with elastic constants that differ from the matrix. The accuracy and range of validity of these solutions are demonstrated by comprehensive 2D and 3D finite element analyses of material volumes containing realistic distributions of particles of spherical and spheroidal shape of various elastic modulus. The results show that significant strengthening will arise provided that the representative particle size is smaller than the material length scale $\ell$ of the SGP material.

自 Orowan 的开创性工作以来，描述小颗粒对金属强化的重要作用的理论已经发展。在这里，这个问题通过应变梯度塑性 (SGP) 理论进行分析。无量纲形式的控制方程的结构表明，对于足够小的颗粒尺寸，基体材料中的塑性应变是零阶近似常数$\mathrm{一种}$ 与材料长度尺度相比 $\ell$. 基于这一观察，通过扩展所有场变量，开发了扰动解决方案$\mathrm{一种}/\ell$ 和粒子的体积分数 $F$. 还利用塑性应变场的简单结构从虚功和最大塑性耗散原理推导出上限解。然后将这些解析解用于推导出屈服应力的表达式，同时考虑到具有与基体不同的弹性常数的各种尺寸和形状的颗粒的随机分布。这些解决方案的准确性和有效性范围通过对材料体积进行全面的 2D 和 3D 有限元分析来证明，这些材料体积包含各种弹性模量的球形和类球体颗粒的真实分布。结果表明，如果代表性颗粒尺寸小于材料长度尺度，则会出现显着强化$\ell$ SGP 材料。