We analyse the mixed energetic–dissipative potential (MP) recently proposed by our group to predict, within higher-order strain gradient plasticity (SGP), reliable size-dependent responses under general loading histories. Such an MP follows former proposals by Chaboche, Ohno and co-workers for nonlinear kinematic hardening in the context of size-independent metal plasticity. The MP is given by M quadratic addends that each transitions, at a different threshold value, into a linear dissipative contribution. Hence, the MP involves 2M positive material parameters, given by the M threshold values and the M moduli weighing each quadratic recoverable term. We analytically demonstrate that, under proportional loading, the MP limit for M → ∞ converges to a less-than-quadratic potential with well-defined properties. This result is of crucial importance for identifying the material parameters of any model adopting the MP. Moreover, our analysis provides a formula for the characterization of the energetic and dissipative parts of any possible MP limit, showing that, regarding the capability to describe the effect of diminishing size within SGP, the MP can be selected such that its contribution to the strengthening (i.e. an increase in yield point) is mostly dissipative, whereas its contribution to the increase in strain hardening is mostly recoverable.

我们分析了我们小组最近提出的混合能量耗散势（MP），以预测在高阶应变梯度可塑性（SGP）范围内在一般载荷历史下可靠的尺寸依赖性响应。这种MP遵循Chaboche，Ohno及其同事先前提出的在尺寸无关的金属可塑性范围内进行非线性运动硬化的建议。MP由M个二次加法给出，每个二次加法以不同的阈值转换为线性耗散贡献。因此，MP包含2 M个积极的材料参数，由M个阈值和M个模量加权每个二次可恢复项给出。我们通过分析证明，在比例载荷下，MP极限为M  →∞收敛到具有明确定义的性质的小于二次方的势。该结果对于确定采用MP的任何模型的材料参数至关重要。此外，我们的分析提供了一个公式，用于表征任何可能的MP限制的高能和耗散部分，表明对于描述SGP内尺寸减小的影响的能力，可以选择MP以使其对强化的贡献（即屈服点的增加）大部分是耗散的，而其对应变硬化增加的贡献大部分是可恢复的。