This paper is devoted to the standard theory of gradient plasticity and visco-plasticity, cf. Gudmundson (2004), Fleck and Willis (2009) and the quoted references. Within the framework of the generalized standard materials and the generalized principle of virtual work, our attention is focussed on the mathematical basis of the theory in order to underline the basic assumptions, general results and principal difficulties compared to the classical theory of plasticity. Gradient terms can be introduced in the expression of the energy potential or the dissipation potential. Various general results of the literature on the constitutive modeling and on the response of a solid under a given loading are revisited in this unified description. The quasi-static response is governed by a variational inequality which is an alternative form of Biot equation. A full discussion of this evolution equation is given in terms of the energy and dissipation potentials of the solid. The rate problem and the rate variational principle are considered and compared to the incremental problem and the incremental variational principle which result from an implicit discretization of the evolution equations. In particular, rate and incremental minimum principles are related to a stability condition.

本文致力于梯度可塑性和粘塑性的标准理论，参见。Gudmundson（2004），Fleck和Willis（2009）以及引用的参考文献。在通用标准材料和虚拟工作通用原理的框架内，我们的注意力集中在该理论的数学基础上，以强调与经典可塑性理论相比的基本假设，一般结果和主要困难。可以在能量势或耗散势的表达中引入梯度项。在这个统一的描述中，重新讨论了有关本构模型和固体在给定载荷下的响应的文献的各种一般结果。准静态响应由变分不等式控制，这是比奥方程的另一种形式。根据固体的能量和耗散势，对该演化方程进行了全面讨论。考虑了速率问题和速率变分原理，并将其与由演化方程的隐式离散化产生的增量问题和增量变分原理进行了比较。特别是，速率和增量最小原理与稳定性条件有关。