Optimal history encoding for elastic-plastic hereditary laws: Sharp input and constitutive approximation

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 $\mathcal P(\pi)$ rather than only the driving history $\pi$. 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.