Embracing nonlinearity and geometry: a dimensional analysis guided design of shock absorbing materials.

Design of shock absorbers requires a delicate balance between mechanical properties and geometric design, allowing them to be compressible yet strong enough to withstand crushing loads. Here, we present a unified framework for designing compact and lightweight shock absorbers by employing a streamlined kinematic model and dimensional analysis. We derive geometric constraints on the thickness and cross-sectional area of a protective foam with a given stress-strain response to ensure that acceleration and compressive strain remain within critical limits. Additionally, we identify the optimal mechanical properties that yield the most compact and lightweight protective foam pads for absorbing impact energy. Contrary to common belief, we demonstrate that foams with a nonlinear stress-strain response can effectively achieve thin and lightweight protective pads, particularly when a large cross-sectional area is required. Guided by this design framework, we introduce optimal architected designs of vertically aligned carbon nanotube (VACNT) foams-a low-density hierarchical material system.