arXiv subjects

Jeremy A. McCulloch

Publications and source records attributed to Jeremy A. McCulloch.

4 recordsLinked to original sources

arXiv2026

Mechanical resilience of ultra-low-density racing-shoe foams

High-performance racing shoes rely on ultra-low-density elastomeric foams that undergo large, repeated deformations during running. Yet little is known about how their mechanical properties vary throughout the shoe or change with repeated use. Here, we characterize the midsole foam in an elite-level racing shoe from the heel, midfoot, and toe of a new shoe and a shoe worn for 300 miles. Microscopy reveals a characteristic pore length scale of 128+/-18 um and supports an approximately isotropic continuum description. We then quantify the mechanical response under tension, compression, and shear. Remarkably, despite 300 miles of real-world use, the foam retains its mechanical response across all three loading modes and shoe regions. Energy return remains largely unchanged, with values of 85-93% in tension and compression and 64-71% in shear. At the same time, we observe strong regional variations, with tensile and compressive stiffnesses 38-51% lower in the toe than in the heel. The foam also exhibits a pronounced tension-compression asymmetry in Poisson's ratio. Together, these findings reveal a spatially structured and mode-dependent mechanical response that remains largely preserved after 300 miles of use. This mechanical resilience may extend the functional lifetime of racing shoes, with implications for runners, replacement recommendations, and sustainability.

physics.comp-ph↗

arXiv2026

Discovering the mechanics of ultra-low density elastomeric foams in elite-level racing shoes

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.

cs.CE↗

arXiv2025

Discovering uncertainty: Gaussian constitutive neural networks with correlated weights

When characterizing materials, it can be important to not only predict their mechanical properties, but also to estimate the probability distribution of these properties across a set of samples. Constitutive neural networks allow for the automated discovery of constitutive models that exactly satisfy physical laws given experimental testing data, but are only capable of predicting the mean stress response. Stochastic methods treat each weight as a random variable and are capable of learning their probability distributions. Bayesian constitutive neural networks combine both methods, but their weights lack physical interpretability and we must sample each weight from a probability distribution to train or evaluate the model. Here we introduce a more interpretable network with fewer parameters, simpler training, and the potential to discover correlated weights: Gaussian constitutive neural networks. We demonstrate the performance of our new Gaussian network on biaxial testing data, and discover a sparse and interpretable four-term model with correlated weights. Importantly, the discovered distributions of material parameters across a set of samples can serve as priors to discover better constitutive models for new samples with limited data. We anticipate that Gaussian constitutive neural networks are a natural first step towards generative constitutive models informed by physical laws and parameter uncertainty.

cs.CE↗

arXiv2024

On sparse regression, Lp-regularization, and automated model discovery

Sparse regression and feature extraction are the cornerstones of knowledge discovery from massive data. Their goal is to discover interpretable and predictive models that provide simple relationships among scientific variables. While the statistical tools for model discovery are well established in the context of linear regression, their generalization to nonlinear regression in material modeling is highly problem-specific and insufficiently understood. Here we explore the potential of neural networks for automatic model discovery and induce sparsity by a hybrid approach that combines two strategies: regularization and physical constraints. We integrate the concept of Lp regularization for subset selection with constitutive neural networks that leverage our domain knowledge in kinematics and thermodynamics. We train our networks with both, synthetic and real data, and perform several thousand discovery runs to infer common guidelines and trends: L2 regularization or ridge regression is unsuitable for model discovery; L1 regularization or lasso promotes sparsity, but induces strong bias; only L0 regularization allows us to transparently fine-tune the trade-off between interpretability and predictability, simplicity and accuracy, and bias and variance. With these insights, we demonstrate that Lp regularized constitutive neural networks can simultaneously discover both, interpretable models and physically meaningful parameters. We anticipate that our findings will generalize to alternative discovery techniques such as sparse and symbolic regression, and to other domains such as biology, chemistry, or medicine. Our ability to automatically discover material models from data could have tremendous applications in generative material design and open new opportunities to manipulate matter, alter properties of existing materials, and discover new materials with user-defined properties.

cs.LG↗