An Explicit Formula for the Dry Packing Density of Aggregate Gradations Derived from the de Larrard Compressible Packing Model by Symbolic Regression

The dry packing density φ of an aggregate mixture governs binder demand in concrete and the achievable density of unbound graded-aggregate bases. The Compressible Packing Model (CPM) of de Larrard gives φ only as the root of an implicit equation. Factoring out the analytical single-class solution and applying symbolic regression only to the multi-class correction gives an explicit, differentiable approximation for continuous Andreasen–Andersen gradations, assuming one specific packing density β for all size classes; it vanishes as β→0 and K→0, though its asymptotic behavior differs from the CPM’s. On 1000 independent gradations, the four-variable formula reproduces the CPM with R2 = 0.990 and root-mean-square error 0.0064 (0.0108 at the highest compaction index, at most 0.053 on the domain boundary). The surrogate can be smooth because the compaction relation sums over all candidates, filtering out the kink that the virtual packing density develops at each governing-class change. Its closed-form gradients are approximate and cannot locate interior optima in the distribution modulus. For TCVN 8859:2023 base gradations, the formula error stays below 0.020. The formula is validated against the CPM only; the CPM itself, checked separately against 64 glass-bead measurements, gives errors of 0.014–0.016. Crushed-aggregate validation remains future work.