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DOE OSTI2025

Liquid Sorption-Enhanced Haber–Bosch Process

The use of a liquid sorbent in a traditional Haber-Bosch process enables significant improvements in energy efficiency and potential cost savings for arguably the most important chemical process on the planet. The approach presented in this report employs an incompressible liquid sorbent that absorbs and releases ammonia (NH 3 ) under specific conditions. To achieve this, we investigate reactions of ammonia and pure phosphoric acid (H 3 PO 4 , PA), which rapidly neutralize to form an equilibrated solution of monoammonium phosphate (MAP) and diammonium phosphate (DAP) that functions as a reversible and regenerable sorbent. Through intimate contact of the gas-phase Haber-Bosch reaction mixture with this liquid absorbent, complete equilibrium uptake may be achieved in an appropriately sized separator, and facile separation occurs through the use of independent liquid and gas phases. Following depressurization and release of the ammonia product, only the incompressible fluid needs to be repressurized and returned to the reactor. This study documents proof-of-concept absorption and desorption experiments carried out in 75 mL batch reactors, predominantly charged with precise MAP and DAP mixtures that equilibrate at process-relevant temperatures and pressures. We then assemble the first thermodynamic relationships that underlie this advantaged separation strategy, validated by reactive force field (ReaxFF) interatomic potential simulations, and benchmarked with traditional separation routes via process modeling and technoeconomic analysis. The scale of energy consumption in the century-old Haber-Bosch process is massive, and the elegant liquid sorption approach reported here offers opportunities to enhance its energy efficiency for the next frontier of ammonia synthesis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Nature of transonic sub-Alfvénic turbulence and density fluctuations in the near-Sun solar wind

Context. Recent Parker Solar Probe (PSP) measurements have revealed that solar wind (SW) turbulence transits from a subsonic to a transonic regime near the Sun, while remaining sub-Alfvénic. These observations call for a revision of the existing SW models, where turbulence is considered to be both subsonic and sub-Alfvénic. Aims. In this work, we introduce a new magnetohydrodynamic (MHD) model of transonic sub-Alfvénic turbulence (TsAT). Methods. We used 3D MHD simulations initialized with parameters measured by PSP to investigate the properties of the new near-Sun SW transonic turbulent regime. We then derived a reduced set of MHD equations in the transonic sub-Alfvénic limit to interpret our numerical results. Results. Our TsAT model shows that turbulence is effectively nearly incompressible (NI) and has a 2D + slab (quasi-2D) geometry not only in the subsonic limit, but also in the transonic regime, as long as it remains sub-Alfvénic, a condition essentially enforced everywhere in the heliosphere by the strong local magnetic field. These predictions are consistent with 3D MHD simulations, showing that transonic turbulence is dominated by low-frequency quasi-2D incompressible structures, while compressible fluctuations are a minor component corresponding to low-frequency slow modes and high-frequency fast modes. Conclusions. Our new TsAT model extends existing NI theories of turbulence, and is potentially relevant for the theoretical and numerical modeling of space and astrophysical plasmas, including the near-Sun SW, the solar corona, and the interstellar medium.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Curious cross-field transport effects in multi-ion, magnetized plasma

In contrast to single-ion plasma, multiple-ion-species plasma exhibits new, curious, and large transport effects. On short timescales, where ions exchange momentum, magnetized multi-ion plasma behaves as a most unusual substance, compressible across field lines in number density but incompressible in charge density. It takes 40 times longer for electrons to participate. In this ion–ion cross-field transport regime, we identified the charge-incompressibility heat pump effect, transferring heat both spatially and between species. Curiously, the direction of impurity transport strongly depends on plasma magnetization, characterized by the ratio of light ion gyrofrequency to the collision frequency between light and heavy ion species. The expulsion of heavy ion impurities from a hotspot occurs sufficiently quickly to be observable on MagLIF, so long as plasma becomes sufficiently collisionally magnetized under implosion. Even more curious, multi-ion transport changes its nature in partially ionized plasma, where ions occupy different charge states. In this regime, we identify a partial-ionization deconfinement effect. The combination of cross-field transport, ionization, and recombination leads to a net ion charge moving across magnetic field lines on the ion–ion transport timescale as opposed to the electron–ion transport timescale. Cross-field transport effects in multi-ion plasma are important in a number of applications, including nuclear fusion and plasma mass filters.

Mlodik, M. E. (ORCID:0000000343003941)↗

DOE OSTI2024

On the accuracy of compressibility transformations

This study highlights the importance of satisfying the eddy viscosity equivalence below the logarithmic layer, to deriving accurate compressibility transformations. First, we analyze the ability of known transformations to satisfy the eddy viscosity equivalence and show that the accuracy of these transformations is strongly dependent on this ability. Second, in a step-by-step manner, we devise new transformations that satisfy this hypothesis. An approach based on curve fitting of the incompressible Direct Numerical Simulation data for eddy viscosity profiles below the logarithmic layer provides an extremely accurate transformation, which motivates self-contained methods, making use of mixing length formulas in the inner region. It is shown that the accuracy of existing transformations can be significantly improved by applying these ideas, below the logarithmic layer. Motivated by the effectiveness of the formulations derived from eddy viscosity equivalence, we introduce a new integral transformation based on Reynolds number equivalence between compressible and incompressible flows. This approach is based on defining a new compressible velocity scale, which affects the accuracy of transformations. Several choices for the velocity scale are tested, and in each attempt, it is shown that the eddy viscosity equivalence plays a very important role for the accuracy of compressibility transformations.

42 ENGINEERING↗

DOE OSTI2025

Taking control of compressible modes: bulk viscosity and the turbulent dynamo

Many polyatomic astrophysical plasmas are compressible and out of chemical and thermal equilibrium, introducing a bulk viscosity into the plasma via the internal degrees of freedom of the molecular composition, directly impacting the decay of compressible modes, $\mathrm{{\boldsymbol {\mathit {v}}}}_{\parallel }(\boldsymbol {k})$. This is especially important for small-scale, turbulent dynamo processes in the interstellar medium (ISM), which are known to be sensitive to the effects of compression. To control the viscous properties of $\mathrm{{\boldsymbol {\mathit {v}}}}_{\parallel }(\boldsymbol {k})$, we perform trans-sonic, visco-resistive dynamo simulations with additional bulk viscosity $\nu _{\text{bulk}}$, deriving a new $\nu _{\text{bulk}}$ Reynolds number $\text{Re}_{\text{bulk}}$, and viscous Prandtl number $\text{P}\nu \equiv \text{Re}_{\text{bulk}}/ \text{Re}_{\text{shear}}$, where $\text{Re}_{\text{shear}}$ is the shear viscosity Reynolds number. We derive a framework for decomposing $E_{\rm mag}$ growth rates into incompressible and compressible terms via orthogonal tensor decompositions of $\boldsymbol {\nabla }\otimes \mathrm{{\boldsymbol {\mathit {v}}}}$, where $\mathrm{{\boldsymbol {\mathit {v}}}}$ is the fluid velocity. We find that $\mathrm{{\boldsymbol {\mathit {v}}}}_{\parallel }(\boldsymbol {k})$ play a dual role, growing and decaying $E_{\rm mag}$, and that field-line stretching is the main driver of growth, even in compressible dynamos. In the absence of $\nu _{\text{bulk}}$ ($\text{P}\nu \rightarrow \infty$), $\mathrm{{\boldsymbol {\mathit {v}}}}_{\parallel }(\boldsymbol {k})$ pile up on small-scales, creating a spectral bottleneck, which disappears for $\text{P}\nu \approx 1$. As $\text{P}\nu$ decreases, $\mathrm{{\boldsymbol {\mathit {v}}}}_{\parallel }(\boldsymbol {k})$ are dissipated at increasingly larger scales, in turn suppressing incompressible modes through a coupling between high-k modes. We emphasize the importance of further understanding the role of $\nu _{\text{bulk}}$ in compressible astrophysical plasmas, which we estimate could be as strong as the shear viscosity in the cold ISM, and highlight that compressible direct numerical simulations without bulk viscosity have unresolved compressible mode dissipation scales.

MHD↗

DOE OSTI2026

Ab initio calculations of monopole sum rules: From finite nuclei to infinite nuclear matter

We compute moments of the isoscalar monopole response of 𝑁 = 𝑍 closed-shell nuclei based on chiral nucleon-nucleon plus three-nucleon interactions. We employ the random-phase approximation (RPA) and two ab initio many-body approaches, the in-medium similarity renormalization group (IMSRG) and coupled-cluster theory (CC). In the IMSRG framework, the moments are obtained as ground-state expectation values, whereas in the CC approach, they are evaluated through excited-state calculations. We find good agreement between the IMSRG and CC results across all nuclei studied. RPA provides a reasonable approximation to the correlated methods if the interaction is soft. From the calculated moments, we extract average energies of the monopole response, compute finite-nucleus incompressibilities, and estimate the incompressibility of symmetric nuclear matter by a fit to a leptodermous expansion. Our extrapolated values are lower than those obtained in nuclear-matter calculations with the same interactions, but the values are consistent with phenomenological ranges.

Bonaiti, Francesca [Michigan State Univ., East Lan↗

DOE OSTI2025

Assessment and Improvement of the SST-Gamma Transition Model in Nalu-Wind

We conduct laminar–turbulent boundary-layer transition simulations using a local correlation-based transition model for two-dimensional incompressible flow and present enhancements to improve the accuracy of transition predictions. Menter’s Galilean-invariant 𝛾 transition model is implemented in the incompressible, unstructured-grid flow solver Nalu-Wind and is validated against experimental data and results from NASA’s flow solvers. The test cases of the AIAA Transition Prediction and Modeling Workshop are investigated, namely, the T3A/T3B flat plates and the NLF(1)-0416 and S809 airfoils. Based on the results, best practices for transition simulations, particularly for an unstructured-grid flow solver, are identified. Additional airfoil simulations are conducted for two wind turbine airfoils, S822 at Reynolds numbers of 𝑂⁡(10 5 ) and DU00-W-212 at Reynolds numbers of 𝑂⁡(10 7 ), to assess the model at low and high Reynolds numbers. Furthermore, through this work, we propose several approaches to enhance transition simulations, including 1) enforcing positivity of the implicit operator for the source terms of the transition model, 2) employing a constant turbulence intensity in stationary external flow simulations, and 3) recommending meshing for unstructured-grid flow solvers. Finally, we provide detailed documentation of the validation and data for the canonical cases to the transition modeling community.

17 WIND ENERGY↗

DOE OSTI2025

Tensile Modeling PVC Gels for Electrohydraulic Actuators

Polyvinyl chloride (PVC)-dibutyl adipate (DBA) gels are a fascinating dielectric elastomer actuator showing promise in soft robotics. When actuated with high voltages, the gel deforms towards the anode. A recent application of PVC gels in electrohydraulic actuators motivates elastic and hyperelastic constitutive relationships for tensile loading modes. PVC gels with plasticizer-to-polymer weight ratios of 2:1, 4:1, 6:1, and 8:1 w/w were evaluated. PVC gels exhibit a linear elastic region up to 25% strain. The elastic modulus decreased with increasing plasticizer content from 288.8 kPa, 56.1 kPa, 24.7 kPa, to 11 kPa. Poisson’s ratio also decreased with increasing plasticizer content from 0.42, 0.43, 0.39, to 0.35. We suggest that the decrease in polymer concentration facilitates a weakly interconnected polymer network susceptible to chain slippage that hinders the network response, thus lowering Poisson’s ratio. Our work suggests that PVC gels can be treated as isotropic and incompressible for large strains and hyperelastic modeling; however, highly plasticized gels tend to act less incompressible at small strains. The power scaling law between the elastic modulus and plasticizer weight ratio showed high agreement, making the elastic modulus deterministic for any plasticizer content. The Neo–Hookean, Mooney–Rivlin, Yeoh, Gent, Ogden, and extended tube hyperelastic constitutive models are investigated. The Yeoh model shows the highest feasibility when evaluated up to 3.5 stretch, showing a maximum normalized root-mean-square-error of 6.85%. Together, these findings establish a constitutive basis for PVC-DBA gels, incorporating small strain elasticity, large strain non-linear behavior, and network analysis while providing suggestive insight into the network structure required for accurately modeling the EPIC.

Polymer Science↗

DOE OSTI2024

Scale Separation Effects on Simulations of Plasma Turbulence

Abstract Understanding plasma turbulence requires a synthesis of experiments, observations, theory, and simulations. In the case of kinetic plasmas such as the solar wind, the lack of collisions renders the fluid closures such as viscosity meaningless and one needs to resort to higher-order fluid models or kinetic models. Typically, the computational expense in such models is managed by simulating artificial values of certain parameters such as the ratio of the Alfvén speed to the speed of light ( v A / c ) or the relative mass ratio of ions and electrons ( m i / m e ). Although, typically care is taken to use values as close as possible to realistic values within the computational constraints, these artificial values could potentially introduce unphysical effects. These unphysical effects could be significant at sub-ion scales, where kinetic effects are the most important. In this paper, we use the 10-moment fluid model in the Gkeyll framework to perform controlled numerical experiments, systematically varying the ion–electron mass ratio from a small value down to the realistic proton–electron mass ratio. We show that the unphysical mass ratio has a significant effect on the kinetic range dynamics as well as the heating of both plasma species. The dissipative process for both ions and electrons becomes more compressive in nature, although the ions remain nearly incompressible in all cases. The electrons move from being dominated by incompressive viscous-like heating/dissipation to very compressive heating/dissipation dominated by compressions/rarefactions. While the heating change is significant for the electrons, a mass ratio of m i / m e ∼ 250 captures the asymptotic behavior of electron heating.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Kelvin–Helmholtz instability under stabilizing parallel magnetic field in nonhomogeneous compressible MHD flows

We study the Kelvin–Helmholtz instability (KHI) for the general case of a compressible, nonhomogeneous, magnetized plasma flow. The study is limited to a vortex sheet interface with an imposed parallel magnetic field. We introduce a new formalism based on a convective Mach number M c , a convective Alfvénic Mach number M Ac , and a total convective Mach number that combines the two. We derive an analytic expression of the KHI growth rate for a homogeneous flow (i.e., zero Atwood number, A=0) that converges toward both the expression for unmagnetized compressible flow and Chandrasekhar's expression for magnetized incompressible flow. Otherwise, the dispersion relation is solved numerically and allows deriving general stability diagrams of magnetized KHI for the triplet (A, M c , β −plasma) parameters. We show these parameters uniquely define all configurations for a parallel magnetic field. We also construct diagrams with respect to the convective Alfvénic Mach number, the β − plasma parameter, or the magnetic field showing which magnetic field strength is required for stabilizing a given shear flow. The theoretical growth rates are compared with 18 simulations made with the GAMERA code, currently used for 3D magnetospheric simulations. Finally, we apply our results to the analysis of a past KHI experiment performed at the OMEGA laser facility, showing linear theory succeeds to provide accurate estimates of the growth rate at early times. We further discuss how our results can inform future experiments in the high-Mach magnetized regime at the National Ignition Facility. Possible limitations of the study due to resistive, mixing, or turbulence effects are discussed.

compressible flows↗

DOE OSTI2025

Concurrent two-way coupling of global and local models across internal boundaries with non-matching discretizations

Coupling local and global models enables efficient simulation of multiscale systems, where global models capture large-scale behavior and local models, with enhanced physics, resolve finer details over a smaller region. Here, this paper presents a mathematically consistent method for coupling physics-based models of varying fidelity across adjacent, non-overlapping subdomains, even when discretizations do not match at the immersed interdomain interfaces. Incompressible Navier-Stokes equations (NSE) constitute the global model while residual-based turbulence model serves as the local high-fidelity model. In addition, a scalar advection-diffusion equation that models the convection of an active scalar field is appended to the turbulence model in the local domain. This scalar field does not have its complement in the global model, giving rise to unequal number of equations at the immersed boundary between local and global models. Interdomain coupling terms are derived via the Variational Multiscale Discontinuous Galerkin (VMDG) method with new developments in scale representation and efficient fine-scale estimation. While transient laminar flows modeled with NSE in the global domain can be resolved with relatively coarse mesh, turbulent flow calculations in the local model require much finer spatial discretizations as well as smaller time-step for appropriately resolving the turbulent flow physics. The proposed framework also accommodates non-matching meshes at the immersed boundaries. Test problems in 2D and 3D numerically showcase the concurrent two-way coupling of unknown fields across the immersed boundaries. The 3D test presents a case with an unequal number of equations, where the scalar field represents the convection of contaminant concentration. This provides more detailed physics in the local region and highlights its application in climate modeling and atmospheric sciences.

Variational Multiscale Discontinuous Galerkin (VMD↗

DOE OSTI2025

Optimal Polynomial Smoothers and One‐Sided V‐Cycles for Poisson Problems

The solution to the Poisson equation arising from the spectral element discretization of the incompressible Navier‐Stokes equations needs robust preconditioning strategies. One such strategy is multigrid. To realize the potential of multigrid methods, effective smoothing strategies are needed. Chebyshev polynomial smoothers, in conjunction with pointwise Jacobi or additive Schwarz methods (ASMs), prove to be an effective smoother. Other polynomial smoothers, however, may provide superior convergence to the multigrid preconditioner. The authors compare the standard Chebyshev polynomial smoothers to both the novel fourth‐kind Chebyshev polynomial smoothers proposed by Lottes as well as smoothers based on the polynomial of best uniform approximation to as proposed by Kraus, Vassilevski, and Zikatanov. At the cost of symmetry, further improvements may be made. For example, a order polynomial smoother on both sides of the V‐cycle may be substituted with an order polynomial smoother on one side at no additional cost. The choice of omitting the postsmoother in favor of higher‐order polynomial presmoothing is advantageous in cases where the multigrid approximation property constant is large. The authors consider a 2D model problem based on finite differences to motivate the choice of polynomial smoother, order, and whether to apply postsmoothing for the target application of high‐order ‐geometric multigrid methods for GPU architectures. Results from both domains demonstrate the substantial improvement of these approaches over the standard Chebyshev polynomial smoother with a symmetric V‐cycle.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

AMR-Wind: A Performance-Portable, High-Fidelity Flow Solver for Wind Farm Simulations

We present AMR-Wind, a verified and validated high-fidelity computational-fluid-dynamics code for wind farm flows. AMR-Wind is a block-structured, adaptive-mesh, incompressible-flow solver that enables predictive simulations of the atmospheric boundary layer and wind plants. It is a highly scalable code designed for parallel high-performance computing with a specific focus on performance portability for current and future computing architectures, including graphical processing units (GPUs). In this paper, we detail the governing equations, the numerical methods, and the turbine models. Establishing a foundation for the correctness of the code, we present the results of formal verification and validation. The verification studies, which include a novel actuator line test case, indicate that AMR-Wind is spatially and temporally second-order accurate. The validation studies demonstrate that the key physics capabilities implemented in the code, including actuator disk models, actuator line models, turbulence models, and large eddy simulation (LES) models for atmospheric boundary layers, perform well in comparison to reference data from established computational tools and theory. We conclude with a demonstration simulation of a 12-turbine wind farm operating in a turbulent atmospheric boundary layer, detailing computational performance and realistic wake interactions.

17 WIND ENERGY↗

DOE OSTI2025

10-th order of accuracy for numerical solution of 3-D elasticity equations for heterogeneous materials on unfitted Cartesian meshes

We have developed the Optimal Local Truncation Error Method (OLTEM) with 10-th order of accuracy on unfitted Cartesian meshes for a system of 3-D elasticity equations with smooth irregular interfaces. 5 x 5 x 5 = 125-point stencils (similar to those for quadratic finite elements) for elastic heterogeneous materials are used for OLTEM. There are no unknowns at the interface points between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal 10-th order of accuracy for OLTEM on unfitted Cartesian meshes, i.e., the increase by 7 orders in accuracy compared to quadratic finite elements on conformal meshes. A new post-processing procedure provides the 9-th order of accuracy for stresses in the 3-D case. Similar to basic computations it uses OLTEM with the 125-point stencils, the interface conditions and the elasticity equations. It was shown that the use of the elasticity equations for post-processing improves the accuracy of 0.1% stresses by 6 orders compared to post-processing without the use of PDEs. At an accuracy of for stresses, OLTEM with the new post-processing procedure reduces the number of degrees of freedom by 360 - 8000 times compared to quadratic finite elements with similar stencils. OLTEM with the 125-point stencils yields even more accurate results than high-order finite elements with much wider stencils. OLTEM provides accurate numerical results for compressible and nearly incompressible materials.

elasticity equations↗

DOE OSTI2025

Preserving nonlinear constraints in variational flow filtering data assimilation

Data assimilation aims to estimate the states of a dynamical system by optimally combining sparse and noisy observations of the physical system with uncertain forecasts produced by a computational model. The states of many dynamical systems of interest obey nonlinear physical constraints, and the corresponding dynamics is confined to a certain sub-manifold of the state space. Standard data assimilation techniques applied to such systems yield posterior states lying outside the manifold, violating the physical constraints. This work focuses on particle flow filters which use stochastic differential equations to evolve state samples from a prior distribution to samples from an observation-informed posterior distribution. The variational Fokker-Planck (VFP)—a generic particle flow filtering framework—is extended to incorporate non-linear, equality state constraints in the analysis. To this end, two algorithmic approaches that modify the VFP stochastic differential equation are discussed: (i) VFPSTAB, to inexactly preserve constraints with the addition of a stabilizing drift term, and (ii) VFPDAE, to exactly preserve constraints by treating the VFP dynamics as a stochastic differential-algebraic equation (SDAE). Additionally, an implicit-explicit time integrator is developed to evolve the VFPDAE dynamics. The strength of the proposed approach for constraint preservation in data assimilation is demonstrated on three test problems: the double pendulum, Korteweg-de-Vries, and the incompressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Benchmark study of the DTU OWC chamber with both two-way and one-way absorption

This paper reports on a benchmark study based on small-scale (1:50) measurements of a single, oscillating water column chamber mounted sideways in a long flume. The geometry of the OWC chamber is extracted from a barge-like, attenuator-type floating concept “KNSwing” with 40 chambers targeted for deployment in the Danish part of the North Sea. In addition to traditional two-way energy extraction we also consider one-way energy extraction with passive venting and compare chamber response, pressures and total absorbed energy between the two methods. A blind study was established for the numerical modeling, with participants applying several implementations of weakly nonlinear potential flow theory and commercial Navier–Stokes solvers (CFD). Both compressible and incompressible models were used for the air phase. Potential flow calculations predict more energy absorption near the chamber resonance for one-way absorption than for two-way absorption, but the opposite is found from the experimental measurements. This outcome is mainly attributed to energy losses in the experimental passive valve system, but this conclusion must be confirmed by better experimental measurements. Modeling the one-way valve in CFD proved to be very challenging and only one team was able to provide results which were generally closer to the experiments. The study illustrates the challenges associated with both numerical and experimental analysis of OWC chambers. Air compressibility effects were not found to be important at this scale, even with the large volume of additional air used for the one-way case.

16 TIDAL AND WAVE POWER↗

DOE OSTI2025

Butterfly valve performance factors using the multiphysics object oriented simulation environment

Butterfly valves are typically used in nuclear reactors to control incompressible fluid flow with high inlet velocities. Performance factors for butterfly valves include the pressure drop across the valve and the loss coefficient from which hydrodynamic torque and flow coefficients can be computed. This work explores a computational fluid dynamics approach for butterfly valve performance factors using the open-source Multiphysics Object Oriented Simulation Environment (MOOSE) framework. While MOOSE is often used in the nuclear energy modeling and simulation community for simulations ranging from fuel characterization to heat pipe simulation, this work employs the MOOSE open-source Navier–Stokes solver capability for simulating butterfly valve performance factors and compares those to experimentally measured results from the Advanced Test Reactor at Idaho National Laboratory at Reynolds numbers in the order of 10 6 for the partially opened configuration. The MOOSE framework results are compared against experimentally measured butterfly valve performance factors across five valve opening angles using meshes with order 10 4 – 10 5 elements. This validation serves to enable MOOSE-based multiphysics simulations incorporating the open-source Navier–Stokes module.

97 - MATHEMATICS AND COMPUTING↗

DOE OSTI2025

A chain stretch-based gradient-enhanced model for damage and fracture in elastomers

Similar to quasi-brittle materials, it has been recently shown that elastomers can exhibit a macroscopically diffuse damage zone that accompanies the fracture process. In this study, we introduce a stretch-based gradient-enhanced damage (GED) model that allows the fracture to localize and also captures the development of a physically diffuse damage zone. This capability contrasts with the paradigm of the phase field method for fracture, where a sharp crack is numerically approximated in a diffuse manner. Capturing fracture localization and diffuse damage in our approach is achieved by considering nonlocal effects that encompass network topology, heterogeneity, and imperfections. These considerations motivate the use of a statistical damage function dependent upon the nonlocal deformation state. From this model, fracture toughness is realized as an output. While GED models have been classically utilized for damage modeling of structural engineering materials (e.g., concrete), they face challenges when trying to capture the cascade from damage to fracture, often leading to damage zone broadening (de Borst and Verhoosel, 2016). This deficiency contributed to the popularity of the phase-field method over the GED model for elastomers and other quasi-brittle materials. Other groups have proceeded with damage-based GED formulations that prove identical to the phase-field method (Lorentz et al., 2012), but these inherit the aforementioned limitations. To address this issue in a thermodynamically consistent framework, we implement two modeling features (a nonlocal driving force bound and a simple relaxation function) specifically designed to capture the evolution of a physically meaningful damage field and the simultaneous localization of fracture, thereby overcoming a longstanding obstacle in the development of these nonlocal strain- or stretch-based approaches. Here, we discuss several numerical examples to understand the features of the approach at the limit of incompressibility, and compare them to the phase-field method as a benchmark for the macroscopic response and fracture energy predictions.

Elastomers↗