Coupled dimensional energy balance and machine learning validation for ballistic response prediction of fiber composites

Official URL: https://dx.doi.org/10.1007/s12221-025-01273-9

In this study, we present a coupled, dimensional energy-balance model enhanced with machine-learning validation to predict residual-velocity curves and ballistic limits of fiber-reinforced composites. Projectile deceleration is described as a three-term balance involving strength-like, drag-like, and inertial effects, mapped to the nondimensional groups Π₀, Π₁, and Π₂; closed-form and RK4 solutions yield residual velocity and regime boundaries (Π₀ = Π₁, Π₁ = Π₂). Validation against six literature datasets (CFRP and aramid laminates; Vr–V0 curves) shows high accuracy: median R2 = 0.93–0.96 and typical RMSE = 10–30 m·s⁻1, with best case R2 = 0.976 and RMSE = 6.99 m·s⁻1 for thin CFRP. Ballistic-limit predictions accurately capture the nonlinear increase with thickness, with errors less than 1 m·s⁻1 in brittle CFRP and up to 10 m·s⁻1 in Kevlar laminates. A global master curve of wr = Vr/V0 versus ∥Π∥2 collapses all data and shows a consistent trend. Energy-budget analysis quantifies the contributions of the three terms: the strength term Π₀ dominates in about 90% of operational points, while drag-like effects are minimal and inertial effects only appear at thick or high-velocity limits; the dominance fractions and combined contributions support these shifts. The (V₀,h) regime map, derived by setting Π₀ = Π₁ and Π₁ = Π₂, separates design-relevant domains and aligns with observed transitions in Vr–V0 modes and slopes. An independent machine-learning check using Random Forests achieves R2 = 0.992, RMSE = 17.5 m·s⁻1, and MAE = 12.4 m·s⁻1 (fivefold cross-validation: R2 = 0.835 ± 0.145), supporting the mechanistic hierarchy through feature importance. The integrated physics-based model and machine-learning analysis provide traceable parameters (α, β, γ), uncertainty bounds, and practical screening maps for composite and geometric options under high-velocity impact.