Gradient plasticity theories are of utmost importance for accounting for size effects in metals, especially on the grain scale. Today, there are several methods used to derive the governing equations for the additional degrees of freedom in gradient plasticity theories. Here, the equivalence between an extended principle of virtual power and an extended energy balance is shown. The energy balance of a Boltzmann continuum is supplemented by contributions based on a scalar-valued degree of freedom. It is considered to be invariant with respect to a change of observer. This yields unambiguously the existence of a corresponding micro-stress vector, which is presumed from the outset in the context of an extended principle of virtual power. A thermodynamically consistent nonlocal evolution equation for the additional, scalar-valued degree of freedom is obtained by evaluation of the dissipation inequality in terms of the Clausius–Duhem inequality. Partitioning the nonlocal flow rule yields a partial differential equation, often referred to as micro-force balance. The approach presented is applied to derive a slip gradient crystal plasticity theory regarding single slip. Finally, the distribution of the plastic slip is exemplified with respect to a laminate material consisting of an elastic and an elastoplastic phase.

中文翻译：

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扩展能量平衡的不变性

梯度可塑性理论对于解释金属的尺寸效应（尤其是晶粒尺寸）至关重要。如今，有几种方法可用于为梯度可塑性理论中的附加自由度推导控制方程。在此，示出了虚拟功率的扩展原理与扩展的能量平衡之间的等价关系。玻尔兹曼连续体的能量平衡由基于标量值自由度的贡献补充。它被认为相对于观察者的变化是不变的。这就明确地产生了相应的微应力矢量，该矢量在虚拟功率扩展原理的背景下从一开始就被推测出来。附加的热力学一致的非局部演化方程，标量值的自由度是通过根据克劳修斯-杜海姆不等式评估耗散不等式获得的。对非局部流动规则进行划分会产生一个偏微分方程，通常称为微力平衡。提出的方法可用于推导有关单滑移的滑移梯度晶体可塑性理论。最后，对于由弹性相和弹塑性相组成的层压材料举例说明了塑料滑移的分布。