Stress-gradient materials are generalized continua with two generalized stress variables: the Cauchy stress field and its gradient. For homogenization purposes, we introduce an extension to stress-gradient materials of the principle of Hashin and Shtrikman. The variational principle is first stated within the framework of periodic homogenization, then extended to random homogenization. Contrary to the usual derivation of the classical principle, we adopt here a stress-based approach, much better suited to stress-gradient materials. We show that, in many cases of interest, the third-order trial eigenstrain may be discarded, leaving only one (second-order) trial eigenstrain in the functional to optimize. For N-phase material, the bounds are very similar in structure to their classical counterpart. One notable difference is the fact that, even in the case of isotropy, the bounds depend on some additionnal microstructural parameters (besides the usual volume fractions).

应力梯度材料是具有两个广义应力变量的广义连续体：柯西应力场及其梯度。出于均质化的目的，我们引入了Hashin和Shtrikman原理对应力梯度材料的扩展。变分原理首先在周期性均化的框架内陈述，然后扩展到随机均化。与经典原理的通常推导相反，我们在这里采用基于应力的方法，该方法更适合于应力梯度材料。我们表明，在许多感兴趣的情况下，三阶试验本征可能会被丢弃，而在功能优化中只剩下一个（二阶）试验本征。对于N相材料的边界在结构上与经典材料非常相似。一个显着的差异是以下事实：即使在各向同性的情况下，边界也取决于某些附加的微观结构参数（除了通常的体积分数）。