[论文解读] Discovering the mechanics of ultra-low density elastomeric foams in elite-level racing shoes

Jeremy A. McCulloch, Scott L. Delp+1 more|arXiv (Cornell University)|Feb 13, 2026

Prosthetics and Rehabilitation Robotics被引用 0

一句话总结

该论文将两种超低密度弹性体泡沫在顶尖跑鞋中的大规模力学测试与本构神经网络相结合，发现紧张、压缩和剪切行为的紧凑、可解释模型，并实现 gait 级别仿真。

ABSTRACT

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.

研究动机与目标

• 表征用于顶尖竞速鞋的两种超低密度弹性泡沫在多种加载模式下的力学响应。
• 开发数据驱动、物理信息本构模型，具备高准确性、可解释性，能够在不同加载情景下具备泛化能力。
• 识别关键材料性质（拉伸、压缩和剪切的刚度；能量回弹）及其对鞋性能与耐久性的影响。
• 实现有限元与 gait 级别仿真，以量化个体运动员层面的跑步经济性、性能提升和受伤风险。

提出的方法

• 实验对每种泡沫各取五个样品，在单轴拉伸、无约束与有限约束压缩以及简单剪切条件下进行测试。
• 处理数据以获得每种加载模式和泡沫的平均应力-拉伸（stretch）曲线。
• 开发一种基于不变量和主拉伸输入的本构神经网络，从中学习自由能函数，再由此推导应力。
• 在自由能结构中加入单一不变量、混合不变量和主拉伸项，以捕捉张力-压缩不对称性。
• 通过网络设计和损失函数将物理约束（参考构型零应力、当 J->0 或 ∞ 时的适当极限）强制纳入。
• 使用加权最小二乘损失结合 L0.5 正则化进行训练，以促进模型稀疏性与可解释性。

Figure 1: Sample preparation. We prepare samples from the ASICS Metaspeed Sky and Edge racing shoes by removing the rubber outsole and separating the carbon-fiber plate from the underlying foam. We section the extracted foam into slabs, and cut rectangular samples for uniaxial tension testing and cy

实验结果

研究问题

• RQ1 研究这两种超低密度泡沫在拉伸、压缩和剪切上的力学响应如何？
• RQ2 是否存在一个具有不变量和主拉伸输入的本构神经网络，能够发现紧凑、可解释的模型，准确捕捉泡沫在各加载模式下的行为？
• RQ3 在拉伸、压缩和剪切中，这两种泡沫在刚度和能量回弹方面有何差异，这对中底的稳定性和能量效率意味着什么？
• RQ4 所发现的模型是否能够支撑有限元和 gait 级别的仿真，以量化个体运动员的跑步经济性和受伤风险？

主要发现

• 两种泡沫都表现出张力-压缩不对称性和近似为零的有效泊松比（横向应变可忽略）。
• 压缩刚度在两者之间相似（268 ± 16 kPa 与 299 ± 29 kPa）。
• 其中一种泡沫的拉伸刚度较高（884 ± 69 kPa 对 623 ± 96 kPa），剪切刚度几乎翻倍（219 ± 20 kPa 对 117 ± 24 kPa）。
• 在各自加载模式下，两种泡沫的能量回弹都较高，一种泡沫为 83.3 ± 1.5%，另一种为 88.9 ± 1.8%。
• 本构神经网络在所有加载模式下都实现了高拟合度（R2 接近于 1），且使用由不变量和主拉伸项构成的紧凑、可解释模型。
• 该方法使基于物理的仿真成为可能，可用于评估跑步性能、受伤风险以及耐用设计洞察，适用于可穿戴弹性泡沫材料。

Figure 2: Mechanical testing. We test five samples of each foam in uniaxial tension, left, unconfined compression, middle, and confined compression, right.