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DOE OSTI2023

Thiol–ene Click Chemistry Incorporates Hydroxyl Functionality on Polycyclooctene to Tune Properties

Polyolefins compose the majority of plastic waste, but conventional mechanical recycling degrades their properties, thereby reducing their value. We report the functionalization of a model for dehydrogenated polyethylene, polycyclooctene (PCOE), with thiol–ene click chemistry to install pendant hydroxyl ethyl thioethers. Functionalization of PCOE using mercaptoethanol via thiol–ene click chemistry yielded functionalization between 1.4 and 22.9% based on ethylene monomeric units. Reactions were well-controlled by varying the reagent stoichiometry and reaction time. Crystallinity and melting temperature decreased, and glass transition temperature increased with greater functionalization. Contact angle measurements reveal an increase in surface polarity with functionalization. Here, comparisons with poly(ethylene-co-vinyl alcohol) (EVOH) show comparable surface polarity at similar levels of alcohol functionalization. At 12% functionalization, the ultimate shear stress (USS) of functionalized PCOE in an adhesive configuration is 4.10 ± 0.48 MPa, comparable to EVOH. At >12% functionalization, the failure mode changed from adhesive to mixed adhesive–cohesive, and the USS decreased.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Finite-volume pionless effective field theory for few-nucleon systems with differentiable programming

Finite-volume pionless effective field theory provides an efficient framework for the extrapolation of nuclear spectra and matrix elements calculated at finite volume in lattice QCD to infinite volume, and to nuclei with larger atomic number. In this work, it is demonstrated how this framework may be implemented via a set of correlated Gaussian wave functions optimized using differentiable programming and via solution of a generalized eigenvalue problem. This approach is shown to be significantly more efficient than a stochastic implementation of the variational method based on the same form of correlated Gaussian wave functions, yielding comparably accurate representations of the ground-state wave functions with an order of magnitude fewer terms. The efficiency of representation allows such calculations to be extended to larger systems than in previous work. Further, the method is demonstrated through calculations of the binding energies of nuclei with atomic number A ϵ {2,3,4} in finite volume, matched to lattice QCD calculations at quark masses corresponding to m π = 806 MeV, and infinite-volume effective field theory calculations of A ϵ {2,3,4,5,6} systems based on this matching.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2019

Mobile sensing of point-source gas emissions using Bayesian inference: An empirical examination of the likelihood function

This paper evaluates likelihood function forms for Bayesian inference of point-source gas emissions using a mobile sensor. Whereas Bayesian inference has been successfully used to estimate emission rates from time-averaged concentration data measured by stationary sensors, data collected by mobile sensors do not represent ensemble or time-averaged conditions. To examine the potential impact of this contrast, controlled release experiments were conducted with a mobile sensor measuring concentrations repeatedly along transverse cross sections of the downwind plumes. Experiments were conducted with measurements made at different downwind distances, different sensor heights, and with different obstacle states. An examination is made between two commonly-used likelihood functions, the Gaussian and the log-normal. For experiments conducted in the absence of obstacles, the Bayesian estimates using the log-normal likelihood function yield a much smaller bias than those based on the Gaussian likelihood function. This finding is consistent with the non-Gaussian nature of concentration fluctuations near a point-source. For experiments conducted in the presence of obstacles, the Bayesian inference based on the Gaussian likelihood function exhibits a better performance. This can be explained by the enhanced turbulent mixing due to the obstacle-introduced wake eddies. Overall, we find that the selection of the likelihood function can be physically related to the underlying conditions, and the proper selection is critical to ensure the performance of the Bayesian inference for source characterization using mobile sensing data.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Dynamic thermal behavior of polycrystalline LaB 6 hollow cathodes

Lanthanum hexaboride (LaB6) hollow cathodes have demonstrated a capability for long life operation, which is critical to many space exploration missions. Thermal characterization of LaB6 hollow cathodes has revealed lower than expected electron emitter temperatures when the cathode reaches a steady state. This phenomenon is observed at discharge currents ranging from 5 to 35 A and xenon mass flow rates of 5–25 SCCM in cathodes with three different orifice diameters. Thus, the currently accepted value of the work function for polycrystalline LaB 6 , 2.67 eV, does not describe well the emission characteristics of LaB6 hollow cathodes operating with internal gas discharges at a steady state. We use empirically measured temperatures combined with a model of the hollow cathode emitter and xenon discharge to estimate the value of the work function, yielding a value ranging from 2.1 to 2.44 eV. This lower work function value implies that LaB6 hollow cathodes are expected to have even longer lifetimes than previously anticipated, further establishing them as a more suited alternative to other conventional cathode technologies for the task of long duration travel. Direct measurements of the work function as a function of depth on a hollow cathode emitter using x-ray photoelectron spectroscopy and ion beam milling indicate that the work function decreases with depth. We postulate several mechanisms that could explain the observed work function enhancement. Altogether, our results have important implications to the design, study approach, and operation of LaB 6 cathodes and potentially other cathodes with hollow configuration. Finally, our work opens the question of why the work function is reduced upon interaction with Xe plasma.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Effect of surface defects on damage and failure of dynamically driven cylindrical shells

Here, in this work, expanding cylinder experiments are performed using a gas gun on the aluminum 6061 alloy to understand the dynamic deformation and failure behavior of this material in the presence of surface defects in the form of scratches. The experiments are then modeled with the modified Tepla (for tensile plasticity) model, which is equipped with the damage yield function of Gurson [“Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules for porous ductile media,” (1977)] and Tvergaard and Needleman [Acta Metall. 32(1), 157–169 (1984)], over-stress viscosity [F. Addessio and J. Johnson, J. Appl. Phys. 74(3), 1640–1648 (1993)], and the underlying viscoplastic model of Preston et al. [J. Appl. Phys. 93(1), 211–220 (2003)] (Preston–Tonks–Wallace model) for the dense material. The Tepla model parameters were calibrated using the Bayesian approach against experimentally measured free-surface velocity data obtained from plate impact experiments on the aluminum 6061 alloy. The evolution of radial expansion and shapes of the deformed cylinder from the simulations on a pristine cylinder wall and a cylinder wall with a longitudinal surface defect shows excellent agreement with the corresponding experimental results. Finally, the role of surface defect size and cylinder wall geometry on the dynamic strength of the cylinder wall was also investigated through a series of simulations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2019

Selective Photo‐Oxygenation of Light Alkanes Using Iodine Oxides and Chloride

Abstract Partial oxidation of light alkanes to generate alkyl esters has been achieved under photochemical conditions using mixtures of iodine oxides and chloride salts in trifluoroacetic acid (HTFA). The reactions are catalytic in chloride and are successful using compact fluorescent light, but higher yields are obtained using a mercury lamp. In this photo‐initiated oxyesterification process, the robust alkyl ester products are resistant to over‐oxidation, and under optimized conditions yields for alkyl ester production of ∼50 % based on methane, ∼60 % based on ethane (with a total functionalized yield of EtX (X=TFA or Cl) of 80 %) and ∼30 % based on propane have been demonstrated. The reaction also proceeds in aqueous HTFA and dichloroacetic acid with lower yields. Mechanistic studies indicate that the process likely operates by a chlorine hydrogen atom abstraction pathway wherein alkyl radicals are generated, trapped by iodine, and converted to alkyl trifluoroacetates in situ .

Liebov, Nichole S.↗

DOE OSTI2021

Too big, too small, or just right? A benchmark assessment of density functional theory for predicting the spatial extent of the electron density of small chemical systems

Multipole moments are the first-order responses of the energy to spatial derivatives of the electric field strength. The quality of density functional theory prediction of molecular multipole moments thus characterizes errors in modeling the electron density itself, as well as the performance in describing molecules interacting with external electric fields. However, only the lowest non-zero moment is translationally invariant, making the higher-order moments origin-dependent. Therefore, instead of using the 3 × 3 quadrupole moment matrix, we utilize the translationally invariant 3 × 3 matrix of second cumulants (or spatial variances) of the electron density as the quantity of interest (denoted by K). The principal components of K are the square of the spatial extent of the electron density along each axis. A benchmark dataset of the principal components of K for 100 small molecules at the coupled cluster singles and doubles with perturbative triples at the complete basis set limit is developed, resulting in 213 independent K components. The performance of 47 popular and recent density functionals is assessed against this Var213 dataset. Several functionals, especially double hybrids, and also SCAN and SCAN0 predict reliable second cumulants, although some modern, empirically parameterized functionals yield more disappointing performance. The H, Li, and Be atoms, in particular, are challenging for nearly all methods, indicating that future functional development could benefit from the inclusion of their density information in training or testing protocols.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Geodyn Material Library: Pseudocap models for dry porous tocks

This report describes the second edition of the Pseudocap Strength models for porous rocks implemented in GEODYN material library. The first model was developed in 2007 and calibrated for concrete. Then, the model parameters were calibrated based on triaxial tests reported for limestones and sandstones of various porosities. In these models some key parameters were chosen as functions of the reference porosity,Φ. Since then multiple modifications were implemented in the model, therefore, it has been recalibrated for some common porous materials such as limestones, sandstones, alluvium, tuffs and granite. Two types of models are described in this repot. The first type (called Pseudocap Model or PM) is for rocks from a specific location. Parameters were calibrated for several specific geologic materials. The second type (called Generic Pseudocap Model or GPM) is useful for the sites where only basic information (rock type, porosity) is available. Generic models include built-in correlations between porosities and other mechanical properties observed for certain rock types. The models of both types were validated by comparing not only to quasi-static triaxial tests for these materials but also to shock Hugoniot data and spherical explosion data for some materials. All models were derived in the frame of isotropic plasticity. They are designed to be used in explicit finite element/difference codes. Tangent stiffness tensor is not provided but can be calculated numerically for the model to be used in implicit finite element codes. For an isotropic material the stress can be decomposed into volumetric and deviatoric parts. The volumetric part is modeled using an Equation of state (EOS) which calculates the pressure and the bulk sound speed as functions of the internal specific energy and density. Here a simple, Mie-Gruneisen EOS is presented, but tabulated EOS (LEOS) provided by the library can be used as well. The stress is limited by the yield surface which depends on three invariants of the stress tensor and specific internal energy. In addition, to capture the strain-rate dependence a simple multiplier is used for the yield surface which depends on the equivalent plastic strain rate. The failure surface (the ultimate yield, Y f , defined later) is chosen in the Hoek-Brown form, commonly used in rock mechanics. It includes measurable parameters such as Unconfined Compressive Strength (UCS) as well as scale parameters characterizing the quality of the rock such as GSI (Geologic Strength Index). Thus, even though the model is calibrated for small samples it offers a way to extrapolate the strength to the field scale using geological characterization of the rock mass. The model captures effects of brittle-ductile transition in rocks by introducing a cap multiplier to the yield function. The rate of dilatancy (bulking) is proportional to the slope of the yield surface affected by the cap. Therefore, it takes place only at low confinements when the pressure is less than the brittle-ductile transition pressure, P BD . On the contrary, the porous compaction takes place at pressures higher than P BD . The cap moves as the porosity is compacted or new porosity is generated due to dilatancy. The porous compaction is modeled using an evolution equation which includes deviatoric stress so that the onset of compaction corresponds to the cap surface. The model captures effects of shear-enhanced compaction which is an important for porous rocks. Section 2 describes the modeling framework and Section 3 presents the model calibration procedure. Section 4 compares experimental data for various rocks versus model predictions. The model parameters used for this comparison are given in Appendix. The files with material constants are available with the latest GEODYN material library distribution.

58 GEOSCIENCES↗

DOE OSTI2020

The nature of the chemical bond and the role of non-dynamical and dynamical correlation in Be2

In the spin-coupled generalized valence bond (SCGVB) description of Be2, there is a pair of electrons in highly overlapping “inner” orbitals corresponding to a traditional σ bond, but this bond is compromised by Pauli repulsion arising from its overlap with a second “outer” pair. The presence of this outer pair of electrons leads to a repulsive potential energy curve at long range and a bound, but metastable molecule at short range. To obtain further insights into the nature of the bond in Be2, we determined the non-dynamical and dynamical correlation contributions to the potential energy curve of Be2 using four different choices for the zero-order wave function: Restricted Hartree–Fock (RHF), SCGVB, valence-CASSCF(4,4), and valence-CASSCF(4,8). The SCGVB and valence-CASSCF(4,4) wave functions yield similar breakdowns of the total correlation energy, with non-dynamical correlation being the more important contribution. For the RHF and valence-CASSCF(4,8) wave functions, dynamical correlation is critical, without which the potential energy curve is purely repulsive. High accuracy calculations on the HBen−1Be–BeBen−1H molecule as a function of n (n = 1–6) suggest that the intrinsic strength of a Be–Be σ bond uncompromised by Pauli repulsion is on the order of 62–63 kcal/mol, and its length is 2.13–2.14 Å, ∼60 kcal/mol stronger and ∼0.35 Å shorter than in Be2.

Xu, Lu. T. (ORCID:0000000182064902)↗

DOE OSTI2025

Depolymerization as a Design Strategy: Depolymerization Etching of Polymerization-Induced Microphase Separations

Thermally triggered depolymerization has traditionally been viewed through the lens of sustainability and recycling, not as a constructive tool for materials design. Herein, we show that selective, thermally triggered depolymerization to gaseous monomer serves as a solvent-free strategy for generating porosity in nanostructured polymer materials, offering a means to bypass the mass transport limitations inherent in conventional solution-based etching. As a demonstration platform, we employed polymerization-induced microphase separation (PIMS) to generate disordered bicontinuous block copolymer structures with embedded depolymerizable domains. By incorporating a methacrylate block susceptible to thermal depolymerization within a cross-linked, depolymerization-resistant styrenic matrix, we developed a process we term depolymerization etching of polymerization-induced microphase separations (DEPIMS). This approach enables highly selective and efficient domain removal via reversion to monomer to produce mesoporous materials with high surface areas (>200 m 2 /g). Subsequent surface functionalization yielded mesoporous adsorbents with tunable uptake kinetics and among the highest dye adsorption capacities reported for PIMS-derived materials, demonstrating the adaptability of the DEPIMS platform for chemical separations. DEPIMS can also be extended to a gram-scale, one-pot approach to yield mesoporous materials with recoverable monomer in under 12 h. These findings reposition thermal depolymerization from a sustainability tool to a broadly enabling strategy for scalable, on-demand fabrication of functional nanostructured materials.

Depolymerization↗

DOE OSTI2021

Observation of a chiral wave function in the twofold-degenerate quadruple Weyl system BaPtGe

Topological states in quantum materials are defined by bulk wave functions that possess nontrivial topological invariants. While edge modes are widely presented as signatures of nontrivial topology, how bulk wave functions can manifest explicitly topological properties remains unresolved. Here, using high-resolution inelastic x-ray spectroscopy (IXS) combined with first principles calculations, we report experimental signatures of chiral wave functions in the bulk phonon spectrum of BaPtGe, which we show to host a previously undiscovered twofold-degenerate quadruple Weyl node. The chirality of the degenerate phononic wave function yields a nontrivial phonon dynamical structure factor, S(Q,ω), along high-symmetry directions, that is in excellent agreement with numerical and model calculations. Furthermore, our results establish IXS as a powerful tool to uncover topological wave functions, providing a key missing ingredient in the study of topological quantum matter.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Novel theranostic agent for PET imaging and targeted radiopharmaceutical therapy of tumour-infiltrating immune cells in glioma

Background: Malignant gliomas are deadly tumours with few therapeutic options. Although immunotherapy may be a promising therapeutic strategy for treating gliomas, a significant barrier is the CD11b + tumour-associated myeloid cells (TAMCs), a heterogeneous glioma infiltrate comprising up to 40% of a glioma's cellular mass that inhibits anti-tumour T-cell function and promotes tumour progression. A theranostic approach uses a single molecule for targeted radiopharmaceutical therapy (TRT) and diagnostic imaging; however, there are few reports of theranostics targeting the tumour microenvironment. Methods: Utilizing a newly developed bifunctional chelator, Lumi804, an anti-CD11b antibody (αCD11b) was readily labelled with either Zr-89 or Lu-177, yielding functional radiolabelled conjugates for PET, SPECT, and TRT. Findings: 89 Zr/ 177 Lu-labeled Lumi804-αCD11b enabled non-invasive imaging of TAMCs in murine gliomas. Additionally, 177 Lu-Lumi804-αCD11b treatment reduced TAMC populations in the spleen and tumour and improved the efficacy of checkpoint immunotherapy. Interpretation: 89 Zr- and 177 Lu-labeled Lumi804-αCD11b may be a promising theranostic pair for monitoring and reducing TAMCs in gliomas to improve immunotherapy responses.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2026

Self‐Limiting Polymerization‐Induced Crystallization‐Driven Self‐Assembly (SL‐PI‐CDSA) Enables Templated Synthesis of Chiral Plasmonic, Hybrid 2D Hexagonal Assemblies

Examples of self-regulating synthetic self-assembly are relatively few, with most known chemical systems relying on kinetic rather than thermodynamic control. Herein, we demonstrate the rapid generation (t ≤ 5 min) of size-tunable ultralow dispersity (Ð ≤ 1.01) 2D hexagonal nanosheets governed by self-limiting self-assembly (SLSA). Self-assembly in natural systems occurs with exquisite control of structure, function, and dimension. We demonstrate that key aspects of biological assembly can be rationally applied toward the development of bottom-up approaches for the construction of chiral nanomaterials. To this end, self-limiting polymerization-induced crystallization-driven self-assembly (SL-PI-CDSA) of modular and templating aryl isocyanide (AIC) monomers yields functional 2D assemblies permitting post-polymerization/assembly modifications. Detailed study of the internal and external structure of the hexagonal nanosheets reveals topological defects which offer mechanistic insights into both their assembly and subsequent utilization. Specifically, these features enable fabrication of chiral hybrid organic–inorganic nanomaterials incorporating chiral plasmonic metal nanoparticles (MNPs). Our results suggest that the synergistic interplay of template-driven confinement and hierarchical chirality induce symmetry breaking of in situ-generated gold MNPs. We anticipate that the platform presented will facilitate fabrication of new hybrid, chiral organic–inorganic nanostructures.

36 MATERIALS SCIENCE↗

DOE OSTI2020

nPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integro-differential equations with sparse, noisy, unstructured, and multifidelity data. PINNs incorporate all available information, including governing equations (reflecting physical laws), initial-boundary conditions, and observations of quantities of interest, into a loss function to be minimized, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal Laplace operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius $\delta$ goes to zero, and to the fractional Laplacian as $\delta$ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We also provide theoretical convergence rates with respect to $\delta$ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, $\delta$ and $\alpha$, characterizing the kernel of the unified operator. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon, that is, different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order $\alpha(y)$, which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as $\delta$. More importantly, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integro-differential equations with sparse, noisy, unstructured, and multifidelity data. PINNs incorporate all available information, including governing equations (reflecting physical laws), initial-boundary conditions, and observations of quantities of interest, into a loss function to be minimized, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal Laplace operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius δ goes to zero, and to the fractional Laplacian as δ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We provide theoretical convergence rates with respect to δ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, δ and α, characterizing the kernel of the unified operator. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon, that is, different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order α(γ), which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as δ. More importantly, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Nonequilibrium information entropy approach to ternary fission of actinides

Ternary fission of actinides probes the state of the nucleus at scission. Light clusters are produced in space and time very close to the scission point. Within the nonequilibrium statistical operator method, a generalized Gibbs distribution is constructed from the information given by the observed yields of isotopes. Using this relevant statistical operator, yields are calculated taking excited states and continuum correlations into account, in accordance with the virial expansion of the equation of state. Furthermore, clusters with mass number A≤10 are well described using the nonequilibrium generalizations of temperature and chemical potentials. Improving the virial expansion, in-medium effects may become of importance in determining the contribution of weakly bound states and continuum correlations to the intrinsic partition function. Yields of larger clusters, which fail to reach this quasiequilibrium form of the relevant distribution, are described by nucleation kinetics, and a saddle-to-scission relaxation time of about 7000 fm/c is inferred. Light-charged particle emission, described by reaction kinetics and virial expansions, may therefore be regarded as a very important tool to probe the nonequilibrium time evolution of actinide nuclei during fission.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2025

Uncertainty propagation in feed-forward neural network models

We develop new uncertainty propagation methods for feed-forward neural network architectures with leaky ReLU activation functions subject to random perturbations in the input vectors. In particular, we derive analytical expressions for the probability density function (PDF) of the neural network output and its statistical moments as a function of the input uncertainty and the parameters of the network, i.e., weights and biases. A key finding is that an appropriate linearization of the leaky ReLU activation function yields accurate statistical results even for large perturbations in the input vectors. This can be attributed to the way information propagates through the network. We also propose new analytically tractable Gaussian copula surrogate models to approximate the full joint PDF of the neural network output. To validate our theoretical results, we conduct Monte Carlo simulations and a thorough error analysis on a multi-layer neural network representing a nonlinear integro-differential operator between two polynomial function spaces. Our findings demonstrate excellent agreement between the theoretical predictions and Monte Carlo simulations.

MLP networks↗

DOE OSTI2024

Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanics

Data-driven constitutive modeling with neural networks has received increased interest in recent years due to its ability to easily incorporate physical and mechanistic constraints and to overcome the challenging and time-consuming task of formulating phenomenological constitutive laws that can accurately capture the observed material response. However, even though neural network-based constitutive laws have been shown to generalize proficiently, the generated representations are not easily interpretable due to their high number of trainable parameters. Sparse regression approaches exist that allow for obtaining interpretable expressions, but the user is tasked with creating a library of model forms which by construction limits their expressiveness to the functional forms provided in the libraries. Here, in this work, we propose to train regularized physics-augmented neural network-based constitutive models utilizing a smoothed version of $L^0$-regularization. This aims to maintain the trustworthiness inherited by the physical constraints, but also enables interpretability which has not been possible thus far on any type of machine learning-based constitutive model where model forms were not assumed a priori but were actually discovered. During the training process, the network simultaneously fits the training data and penalizes the number of active parameters, while also ensuring constitutive constraints such as thermodynamic consistency. We show that the method can reliably obtain interpretable and trustworthy constitutive models for compressible and incompressible hyperelasticity, yield functions, and hardening models for elastoplasticity, using synthetic and experimental data. This work aims to set a new paradigm for interpretable machine learning models in the broad area of solid mechanics where low and limited data is available along with prior knowledge of physical constraints that the learned maps need to obey. This paradigm can potentially be extended to a broader spectrum of scientific exploration.

Data-driven constitutive models↗