Stewart A. Silling

In this paper, we introduce a nonlocal extension of the Kim–Kim–Suzuki (KKS) phase-field corrosion model aimed at bridging local and nonlocal corrosion modeling approaches, such as phase-field and peridynamic frameworks. In this formulation, classical gradient operators are replaced with integral operators defined over a finite interaction horizon, naturally embedding an intrinsic length scale that aligns with nonlocal theories like peridynamics. Under precise assumptions on function spaces and kernel functions, we define a nonlocal free energy that integrates a standard bulk free energy density with a nonlocal interaction term. Through differentiation in an appropriate Hilbert space, we derive evolution equations, yielding a nonlocal Allen–Cahn equation for the phase-field and a nonlocal Cahn–Hilliard-type equation for the concentration. The latter is expressed as a gradient flow in a metric induced by the inverse of the nonlocal operator, mirroring the classical [Formula: see text] metric for conserved dynamics. We establish the well-posedness of these equations using Galerkin approximations, uniform energy estimates, and compactness arguments. Furthermore, we prove the convergence of the nonlocal model to its local KKS counterpart as the interaction horizon approaches zero, effectively unifying local and nonlocal perspectives. Numerical experiments, implemented via an explicit, matrix-free discretization, confirm the asymptotic compatibility of the scheme as the horizon vanishes and illustrate the transition from diffuse to sharp corrosion fronts. We further demonstrate the framework on a three-dimensional biodegradable magnesium implant screw, calibrated against experimental degradation data, and show that its explicit, matrix-free structure scales efficiently across compute nodes and onto GPUs, making engineering-scale simulations tractable.

Human tissues are highly organized structures with collagen fiber arrangements varying from point to point. Anisotropy of the tissue arises from the natural orientation of the fibers, resulting in location-dependent anisotropy. Heterogeneity also plays an important role in tissue function. It is therefore critical to discover and understand the distribution of fiber orientations from experimental mechanical measurements such as digital image correlation (DIC) data. To this end, we introduce the Heterogeneous Peridynamic Neural Operator (HeteroPNO) approach for data-driven constitutive modeling of heterogeneous anisotropic materials. Our goal is to learn a nonlocal constitutive law together with the material microstructure, in the form of a heterogeneous fiber orientation field, from load-displacement field measurements. We propose a two-phase learning approach. First, we learn a homogeneous constitutive law in the form of a neural network-based kernel function and a nonlocal bond force. This captures the complex homogeneous material responses from data. Second, we reinitialize the learnt bond force and train it together with the kernel and the fiber orientation field for each material point. Due to the inherent properties of the state-based peridynamic framework that we use, our HeteroPNO-learned material models are guaranteed to be objective and to satisfy the balances of linear and angular momenta. The effects of heterogeneity and the nonlinearity of the constitutive relationship are captured by the nonlocal kernel function, enabling physical interpretability. These features enable our HeteroPNO architecture to learn a constitutive model for biological tissue with anisotropic, heterogeneous response undergoing large deformation. To demonstrate the applicability of our approach, we apply the HeteroPNO in learning a material model and fiber orientation field from DIC measurements in biaxial testing. We evaluate the learnt fiber architecture, which is found to be consistent with observations from polarized spatial frequency domain imaging. The framework is capable of providing displacement and stress field predictions for new and unseen loading instances.

Nonlocal, integral operators have become an efficient surrogate for bottom-up homogenization, due to their ability to represent long-range dependence and multiscale effects. However, the nonlocal homogenized model has unavoidable discrepancy from the microscale model. Such errors accumulate and propagate in long-term simulations, making the resultant prediction unreliable. To develop a robust and reliable bottom-up homogenization framework, we propose a new framework, which we coin Embedded Nonlocal Operator Regression (ENOR), to learn a nonlocal homogenized surrogate model and its structural model error. This framework provides discrepancy-adaptive uncertainty quantification for homogenized material response predictions in long-term simulations. The method is built on Nonlocal Operator Regression (NOR), an optimization-based nonlocal kernel learning approach, together with an embedded model error term in the trainable kernel. Then, Bayesian inference is employed to infer the model error term parameters together with the kernel parameters. To make the problem computationally feasible, we use a multilevel delayed acceptance Markov chain Monte Carlo (MLDA-MCMC) method, enabling efficient Bayesian model calibration and model error estimation. We apply this technique to predict long-term wave propagation in a heterogeneous one-dimensional bar, and compare its performance with additive noise models. Owing to its ability to capture model error, the learned ENOR achieves improved estimation of posterior predictive uncertainty.

A technique is proposed for reproducing particle size distributions in three‐dimensional simulations of the crushing and comminution of solid materials. The method is designed to produce realistic distributions over a wide range of loading conditions, especially for small fragments. In contrast to most existing methods, the new model does not explicitly treat the small‐scale process of fracture. Instead, it uses measured fragment distributions from laboratory tests as the basic material property that is incorporated into the algorithm, providing a data‐driven approach. The algorithm is implemented within a nonlocal peridynamic solver, which simulates the underlying continuum mechanics and contact interactions between fragments after they are formed. The technique is illustrated in reproducing fragmentation data from drop weight testing on sandstone samples.