Ultra-low-density elastomeric foams enable lightweight systems that combine high compliance with efficient energy return. In high-performance racing shoes, these foams are critical for low weight, high cushioning, and efficient energy return; yet, their constitutive behavior remains difficult to model and poorly understood. Here we integrate mechanical testing and machine learning to discover the mechanics of two ultra-low density elastomeric polymeric foams used in elite-level racing shoes. Across uniaxial tension, confined and unconfined compression, and simple shear, both foams exhibit pronounced tension-compression asymmetry, negligible lateral strains consistent with an effective Poisson's ratio close to zero, and low hysteresis indicative of an efficient energy return. Both foams provide a similar compressive stiffness (268kPa vs. 299kPa), while one foam exhibits nearly double the shear stiffness (219kPa vs. 117kPa), implying a substantially greater lateral stability at a comparable vertical energy return (83% vs. 89%). By integrating these data into constitutive neural networks, paired with sparse regression, we discover compact, interpretable single-invariant models, supplemented by mixed-invariant or principal-stretch based terms, that capture the unique signature of the foams with R2 values close to one. From a human performance perspective, these models enable finite-element and gait-level simulations of high-performance racing shoes to quantify running economy, performance enhancements, and injury risks on an individual athlete level. More broadly, this work establishes a scalable and interpretable approach for constitutive modeling of highly compressible, ultra-light elastomeric foams with applications to wearable technologies, soft robotics, and energy-efficient mobility systems.

翻译：超低密度弹性体泡沫能够实现兼具高柔顺性与高效能量回馈的轻量化系统。在高性能竞速跑鞋中，这类泡沫对于实现低重量、高缓震和高效能量回馈至关重要；然而，其本构行为仍难以建模且缺乏深入理解。本研究通过整合力学测试与机器学习方法，揭示了两种应用于精英级竞速跑鞋的超低密度弹性体聚合物泡沫的力学特性。在单轴拉伸、受限与非受限压缩以及简单剪切等多种加载模式下，两种泡沫均表现出显著的拉压不对称性、可忽略的横向应变（对应有效泊松比接近零），以及低滞后特性（表征高效能量回馈）。两种泡沫具有相近的压缩刚度（268kPa vs. 299kPa），而其中一种泡沫的剪切刚度近乎翻倍（219kPa vs. 117kPa），这意味着在垂直能量回馈率相近（83% vs. 89%）的情况下，该泡沫能提供显著增强的横向稳定性。通过将这些数据输入本构神经网络并结合稀疏回归方法，我们建立了紧凑且可解释的单不变量模型，辅以混合不变量或基于主拉伸的修正项，该模型以接近1的R²值精确捕捉了泡沫的独特力学特征。从人体运动表现角度，这些模型支持通过有限元与步态级仿真对高性能竞速跑鞋进行分析，从而在个体运动员层面量化跑步经济性、性能提升潜力及损伤风险。更广泛而言，本研究为高可压缩超轻弹性体泡沫的本构建模建立了可扩展且可解释的研究框架，该框架可应用于可穿戴技术、软体机器人及节能移动系统等领域。