AI 中文总结
研究弹塑性遗传定律的最优历史编码，通过有限历史替代近似，证明绝对连续输入下的精确极小极大定理，得出最优编码器为等变差采样，还给出应力感知编码器及标量互补变量情况，确定累积变差为相关自然变量。
AI 中文摘要
我们直接以遗传形式制定与速率无关的弹性理想塑性响应，并研究其通过有限历史替代的近似。在材料点层面，本构定律是通过向应力空间中的闭凸弹性域进行度量投影生成的向量博弈算子。从阶跃输入的最接近点回映开始，我们转向绝对连续驱动历史，其本构定律具有微分形式：应力保持在弹性域内，速率差几乎处处属于法锥。在这种\(W^{1,1}\)设置下，遗传定律是因果的，收缩变差，并满足\(BV\)到\(L^\infty\)稳定性估计。然后我们用至多\(N\)个常数段的右连续阶跃替代来近似历史。对于绝对连续输入，我们证明了在\(L^\infty\)中由\(BV\)范数归一化的输入近似的精确极小极大定理：最优编码器由等变差采样给出。对于本构近似，通过允许编码器了解材料定律可得到正确的向量值极小极大陈述：它可以压缩精确应力历史\(\mathcal P(\pi)\)而非仅驱动历史\(\pi\)。在自然非退化假设\(0\in\operatorname{int}C\)下，所得的应力感知编码器与相同的离散遗传解码器相结合，给出精确值\((2N)^{-1}\)。还记录了标量互补变量情况：那里输入等变差编码器是精确的，因为标量停止/博弈算子在互补变量中是\(L^\infty\)非扩张的。结果确定累积变差是用于采样和压缩驱动及本构历史的自然变量。
英文摘要
We formulate rate-independent elastic-ideally-plastic response directly in hereditary form and study its approximation by finite history surrogates. At the material-point level, the constitutive law is the vector play operator generated by metric projection onto a closed convex elastic domain in stress space. Starting from the closest-point return mapping for step inputs, we pass to absolutely continuous driving histories, for which the constitutive law admits a differential form: the stress remains in the elastic domain and the difference of rates belongs almost everywhere to the normal cone. In this $W^{1,1}$ setting, the hereditary law is causal, contracts variation, and satisfies a $BV$-to-$L^\infty$ stability estimate. We then approximate histories by right-continuous step surrogates with at most $N$ constant pieces. For absolutely continuous inputs, we prove a sharp minimax theorem for input approximation in $L^\infty$, normalized by the $BV$ norm: the optimal encoder is given by equal-variation sampling. For constitutive approximation, the correct vector-valued minimax statement is obtained by allowing the encoder to be material-law aware: it may compress the exact stress history $\mathcal P(π)$ rather than only the driving history $π$. The resulting stress-aware encoder, followed by the same discrete hereditary decoder, gives the sharp value $(2N)^{-1}$ under the natural nondegeneracy assumption $0\in\operatorname{int}C$. The scalar complementary-variable case is also recorded: there the input equal-variation encoder is sharp because the scalar stop/play operator is $L^\infty$-nonexpansive in the complementary variable. The results identify cumulative variation as the natural variable for sampling and compressing both driving and constitutive histories.
