Scaling crossover in droplet impact force on elastic substrates

Droplet impacts are fundamental to fluid-structure interactions, shaping processes from erosion to bioprinting. While previous scaling laws have provided insights into droplet dynamics, force scaling laws remain insufficiently understood, particularly for soft substrates where both the droplet and substrate deform significantly. Here, we show that droplet impacts on elastic substrates exhibit a scaling crossover in maximum impact force, transitioning from inertial force scaling, typical for rigid substrates under high inertia, to Hertzian impact scaling, characteristic of rigid spheres on elastic substrates. Using high-speed photoelastic tomography, we captured high-resolution dynamic stress fields and identified a similarity parameter governing the interplay between droplet inertia, substrate elasticity, and deformation time scales. Our findings redefine how substrate properties influence impact forces, demonstrating that droplets under high inertia-long thought to follow inertial force scaling-can instead follow Hertzian impact scaling on soft substrates. This framework provides practical insights for designing soft, impact-resistant materials.