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DOE OSTI2023

A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

Here we discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by, we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use C 0 and C 1 virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.

42 ENGINEERING↗

DOE OSTI2024

A variational phase-field framework for thermal softening and dynamic ductile fracture

A variational phase field model for dynamic ductile fracture is presented. Here, the model is designed for elasto-viscoplastic materials subjected to rapid deformations in which the effects of heat generation and material softening are dominant. The variational framework allows for the consistent inclusion of plastic dissipation in the heat equation as well as thermal softening. It employs a coalescence function to degrade fracture energy during regimes of high plastic flow. A variationally consistent form of the Johnson–Cook model is developed for use with the framework. Results from various benchmark problems in dynamic ductile fracture are presented to demonstrate capabilities. In particular, the ability of the model to regularize shear band formation and subsequent damage evolution in two- and three-dimensional problems is demonstrated. Importantly, these phenomena are naturally captured through the underlying physics without the need for phenomenological criteria such as stability thresholds for the onset of shear band formation.

42 ENGINEERING↗

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↗

DOE OSTI2026

Predicting fracture behavior in single crystal nickel using a coupled crystal plasticity phase field damage approach

Understanding the fracture behavior of single crystal metals is critical for predicting material performance under mechanical loading. Here, in this study, we investigate the fracture characteristics of single crystal nickel tensile bars using a crystal plasticity coupled phase field damage (CP-PFD) model. Experimental tensile tests were conducted on 15 specimens spanning five crystallographic orientations and three thickness variants per orientation. The results revealed two distinct fracture modes: brittle fractures with 45-degree angled surfaces and ductile fractures characterized by significant necking. The CP-PFD model successfully replicated these fracture behaviors, demonstrating strong agreement with experimental observations. The model effectively predicted the strain at which necking and fracture occurred, as well as the orientation-dependent fracture mechanisms. By comparing experimental and simulated fracture surfaces, we establish the CP-PFD model as a robust tool for predicting single crystal behavior and damage evolution. This work provides insight into the microstructural dependence of fracture behavior and establishes a predictive framework for modeling orientation-dependent damage evolution in single-crystal nickel.

Crystal plasticity↗

DOE OSTI2025

GEOS-DEV/FORGE

This dataset is a repository that provides the input deck for numerical models associated with two Utah FORGE research projects: "Closing the Loop Between In situ Stress Complexity and EGS Fracture Compexity" (Project Number: 2-2446) and "Coupled Investigation of Fracture Permeability Impact on Reservoir Stress and Seismic Slip Behavior" (Project Number: 5-2428). Three kinds of simulation decks are included: (i) hydraulic fracturing simulation (HydroFrac folder), (ii) phase-field simulation (PhaseField folder), and (iii) thermo-hydro-mechanical simulation (ThermoHydroMech folder).

Cusini, Matteo [Lawrence Livermore National Labora↗

DOE OSTI2024

Intermediate Principal Stress Effects on the 3D Cracking Behavior of Flawed Rocks Under True Triaxial Compression

Crack initiation, growth, and coalescence in flawed rocks have been extensively studied for 2D (planar, penetrating) flaws under uniaxial/biaxial compression. However, little is known as to the mechanisms and processes of cracking from 3D flaws under true triaxial compression, where the intermediate principal stress (σ 2 ) is distinguished from the major and minor principal stresses. In this work, we systematically investigate the effects of σ 2 on the 3D cracking behavior of rock specimens with preexisting flaws, through the use of mechanistic simulations of mixed-mode fracture in rocks. We explore how two characteristics of σ 2 , namely, (i) its orientation with respect to the flaw and (ii) its magnitude, affect two aspects of the cracking behavior, namely, (i) the cracking pattern and (ii) the peak stress. Results show that the orientation of σ 2 exerts more control over the cracking pattern than the flaw inclination angle. The peak stress becomes highest when σ 1 is parallel to the flaw, whereas it becomes lowest when σ 2 is parallel to the flaw. Also, the effects of σ 2 magnitude are more significant when σ 2 becomes more oblique to the flaw plane. On the basis of our observations, we propose mechanisms underlying the cracking behavior of 3D flawed rocks under true triaxial compression.

58 GEOSCIENCES↗

DOE OSTI2025

Coupled phase field damage and crystal plasticity analysis of intragranular fracture: The role of crystallographic orientation and voids

Damage evolution in engineering metal alloys at the grain scale exhibits significant microstructural heterogeneity and anisotropy. These heterogeneities create local hotspots for stress and strain localization, leading to void nucleation. Crystal orientation influences the active slip systems around voids, affecting lattice rotation and potentially forming discontinuities. At low triaxiality, voids may change shape due to lower stress, rotation, elongation, and coalescence. At high triaxiality, the correlation between crystal orientation and void growth rate becomes stronger, resembling the behavior observed in isolated single crystals. Therefore, understanding the effects of crystal orientation, heterogeneous strain, and defect evolution is crucial for single crystal fracture characterization. Here, in this work, a coupled phase-field damage (PFD) and crystal plasticity (CP) model is implemented within a finite element framework to analyze crystal deformation and failure. The CP method employs a dislocation density-based constitutive model, while intragranular failure is modeled using an anisotropic PFD method. The PFD model considers both the stored energy due to elastic stretching and the energy release due to defect formation and crack formation. A single crystal Al2219 with an intracrystalline spherical void is chosen to analyze fracture. The study finds that fracture propagation is strongly correlated with crystal orientations. This coupled CP-PFD model provides accurate failure prediction in crystalline materials by incorporating the effects of crystal orientations and existing voids. This study demonstrates how the local microstructure and defects influence plastic deformation and failure mechanisms in metal alloys.

Aluminum alloy↗

DOE OSTI2024

Quarterly Research Performance Progress Report (Q8)

As part of Task 1, we have started by testing our modeling capabilities by reproducing isothermal DFIT simulations presented in the literature. Once satisfied with the results we have started by targeting the modeling of the DFITs at conducted at well 58-32. We have a identified a specific test (cycle 4 in zone 2) as the most interesting to be model with GEOS hydraulic fracturing module. Thus, we have first produced results with an isothermal model and adjusted model parameters to get a satisfying match with field pressure data. The, we have added thermal effects and compared the modeling results with and without thermal effects to estimate how thermal effects may influence test interpretation. Models seem to suggest that, for small volumes of fluid, thermal effects are moderate. In Task 2, we have adapted GEOS phase-field formulation to be able to simulate near-wellbore hydraulic fracture nucleation and propagation. We have devised a novel formulation that, compared to other existing ones, incorporates rock strengths. We have submitted a journal publication about our work. We are currently employing this phase-field formulation to model the experiments taking place at U Pitt and help us understand the effect of various parameters. In Task 3, we have built a model of the region surround well 16A and have started modeling stage 3 stimulation because of its simpler planar geometry. After calibrating simulation parameters using known analytical solutions, we have simulated the stage 3 stimulation using our isothermal hydraulic fracturing module, varying the permeability field, the stress conditions including different physics to get a better understanding of the numerical challenges and of the effects of varying these parameters on the simulation results. In Task 4 laboratory experimentation, a set of specialized drilling and injection tools has been customized and constructed to accommodate an inclined well with an orientation of up to 30 degrees relative to material anisotropy or principal stress axes. These inclined samples have also undergone thermal stress and hydraulic fracturing at a temperature of 190 degrees Celsius. Furthermore, both vertical and deviated sampling testing setups enable an extended analysis of post-peak pressure behaviors, facilitating post-test pressure analyses such as the G-function, step rate, and fracture reopening measurements. Thus, the key components of in-situ stress estimation can be extracted and validated through our experiment, providing a solid foundation for validating existing in-situ stress estimation theories or proposing new ones. Simultaneously, we are integrating computer vision techniques with traditional experimental fracture observation methods such as multi-overcore/slicing and water-penetration fracture observation. This combination will prove beneficial in populating the hydraulic fracture patterns database, generated under challenging EGS conditions. This approach aims to deepen our understanding of the complexities in EGS reservoirs and pave the way for future data-driven investigations. Additionally, PITT has also equipped the ELE International compression machine, which is now prepared for conducting indirect tensile and fracture toughness tests. These tests will aid in characterizing how rock fabrics influence the resulting fracture patterns. Additionally, we have completed the required personnel training and gained access to Scanning Electron Microscopy (SEM) and Energy Dispersive Spectroscopy (EDS) for conducting more detailed characterization and analysis of rock fabrics, as well as the examination of thermal and hydraulically induced fracture patterns. Thus, the PITT team has effectively demonstrated the capabilities of our experimental apparatuses in exploring the thermal effects, well deviation angles, material anisotropy, and operational choices (such as circulation rate, injection fluid viscosity, and injection rate) and their impact on pressure responses and fracture trajectories under the Utah FORGE conditions.

15 GEOTHERMAL ENERGY↗

DOE OSTI2025

Material Fracturing and Failure Simulation Datasets

Fracturing is a fundamental physics phenomena with broad relevance across multiple domains, ranging from infrastructure integrity, aerospace durability, reservoir production, and seismic events. We present a diverse dataset of simulated fracture evolution and material failure generated from two numerical solvers: the phase-field method and the combined finite-discrete element method (FDEM). These solvers differ in formulation, physical fidelity, and computational efficiency. The dataset includes five materials: PBX, anisotropic shale, tungsten, aluminum, and steel. For each, phase-field simulations span 400,000 cases: 200,000 under uniaxial tension and 200,000 under biaxial tension. The computationally expensive FDEM simulations include 90,000 split evenly among PBX, shale, and tungsten under uniaxial loading. All simulations begin with randomized initial fracture patterns. Each entry includes temporal data capturing fracture propagation dynamics. This comprehensive dataset is designed to support the development of foundational or surrogate machine learning approaches for predicting material failure. While no such models are introduced here, the dataset lays a robust foundation for advancing future research and innovation in these areas.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Arbitrary Order Virtual Element Methods for High‐Order Phase‐Field Modeling of Dynamic Fracture

ABSTRACT Accurate modeling of fracture nucleation and propagation in brittle and ductile materials subjected to dynamic loading is important in predicting material damage and failure under extreme conditions. Phase‐field fracture models have garnered a lot of attention in recent years due to their success in representing damage and fracture processes in a wide class of materials and under a variety of loading conditions. Second‐order phase‐field fracture models are by far the most popular among researchers (and increasingly, among practitioners), but fourth‐order models have started to gain broader acceptance since their more recent introduction. The exact solution corresponding to these high‐order phase‐field fracture models has higher regularity. Thus, numerical solutions of the model equations can achieve improved accuracy and higher spatial convergence rates. In this work, we develop a virtual element framework for the high‐order phase‐field model of dynamic fracture. The virtual element method (VEM) can be regarded as a generalization of the classical finite element method. In addition to many other desirable characteristics, the VEM allows computing on polytopal meshes. Here, we use ‐conforming virtual elements and the generalized‐ time integration method for the momentum balance equation, and adopt ‐conforming virtual elements for the high‐order phase‐field equation. We verify our virtual element framework using classical quasi‐static benchmark problems and demonstrate its capabilities with the aid of numerical simulations of dynamic fracture in brittle materials.

42 ENGINEERING↗

DOE OSTI2024

Phase-field modeling of thermally-grown oxide and damage evolution in environmental barrier coatings

Silicon carbide-based ceramic matrix composites protected by environmental barrier coatings (EBCs) present a promising materials solution for next-generation gas turbines. Developing more robust and efficient EBCs is therefore of significant technological importance. During the service in high-temperature oxidative environments, there is a thermally grown oxide (TGO) layer, spontaneously formed in the EBC system. TGO is recognized as a critical factor for the degradation and failure of EBCs, yet the detailed mechanisms of TGO growth and its effect on EBC failure remain unclear. In this study we develop a comprehensive chemo-mechano-phase-field model to simulate growth of the TGO in EBCs, factoring in creep and deformation, and especially the cracking behaviors. The volume expansion due to TGO growth and the resulting large inelastic deformation are addressed by using our recently developed, so-called incremental realization of inelastic deformation (IRID) algorithm, in combination with an adapted Hu-Chen spectral solver for elasticity. Simulations of TGO growth are performed considering different growth modes of TGOs determined mainly by the ratio of oxidant permeability in the topcoat to that in the TGO itself. Large-scale three-dimensional (3D) simulations are performed to model the formation of interconnecting vertical/channel cracks (often called ‘mud cracks’). The simulated crack morphology are in excellent agreement with the experimental observations from the literature. The simulations also provide insights into the cracking of EBCs and its dependence on the structure and constituent properties of the coating system. Furthermore, these results demonstrate the developed damage model can be a useful tool for design of more durable EBCs.

Cracking↗

DOE OSTI2023

A probabilistic-phase field model for the fracture of brittle materials

Abstract We develop a computational method to determine the failure probability of brittle materials under general mechanical loading conditions. The method is a combination of two parts: (1) numerical simulations of materials with multiple cracks using phase field theory, where the complete fracture process is viewed as ‘damage percolation’ along critical paths or clusters of cracks, rather than the traditional weak-link failure mechanism of Weibull, and (2) an extension of the Batdorf statistical theory of fracture to finite domains, where it is implemented within the finite element framework. The results of phase-field simulations at the ‘percolation threshold’ are used as failure data in the Batdorf theory to determine the overall probability of failure. The input to this approach is the size distribution of cracks in a pristine material. An example is shown, where alumina samples that were previously tested by Abe and coworkers (Abeet al2003J. Am. Ceram. Soc.861019–21) in four-point loading are compared to the results of our numerical simulations. The approach developed here has the advantage of being extendable to more complex thermomechanical loading.

Materials Science↗

DOE OSTI2023

Multiscale modeling for high burnup structure formation and associated pulverization

This report summarizes the lower length scale modeling work performed in the fiscal year 2023 under the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program to capture the microstrutural evolution and associated pulverization criteria in the high burnup regions. This year, the resolution mechanisms within the cluster dynamics code has been updated and it influence on the bubble growth in HBS regions at the mesoscale is studied. To improve the accuracy of phase-field models of fission gas bubble growth, a new Helmholtz free energy for high-density Xe gas is derived from a virial equation of state. Two previously used strategies for representing net vacancy production in phase-field models of fission gas bubble growth are compared with each other and with analytical models of bubble growth. The vacancy source only model is found to be more convenient to parameterize realistically compared with the vacancy source+sink model. Furthermore, the phase-field-fracture simulation have been performed using MD- informed failure stress values and realistic HBS structure obtained from the phase-field simulations. We also present the uncertainty bands on prediction of the critical stress to account for the effect of the lower length scale variabilities on the failure criteria at the mesoscale. It is observed that pulverization may occur in partially restructured regions with restructuring fraction as low as 17%. In light of this, an update to the BISON’s pulverization criteria is recommended.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2024

Fracture Dynamics in Silicon Anode Solid-State Batteries

Solid-state batteries (SSBs) with silicon anodes could enable improved safety and energy density compared to lithium-ion batteries. However, degradation arising from the massive volumetric changes of silicon anodes during cycling is not well understood in solid-state systems. Here, we use operando X-ray computed microtomography to reveal micro- to macro-scale chemo-mechanical degradation processes of silicon anodes in SSBs. Mud-type channel cracks driven by biaxial tensile stress form across the electrode during delithiation. We also find detrimental cracks at the silicon/ solid electrolyte interface that form due to local reaction competition between neighboring domains of different sizes. Continuum phase-field damage modeling quantifies stress-driven channel cracking and shows that the lithiated silicon stress state is critical for determining the extent of interfacial fracture. This work reveals the mechanisms that govern SSBs compared to conventional lithium-ion batteries and provides guidelines for engineering chemomechanically resilient electrodes for high-energy batteries.

25 ENERGY STORAGE↗

DOE OSTI2025

Coupled momentum balance and phase-field solver with fenicsx module

Code solves momentum balance and phase-field equations simultaneously. The differential equations are solved on a discretized domain with appropriate boundary and initial conditions using finite element method. Primary purpose of the code is to simulate brittle fracture under dynamic loading. Constitutive equations are that of linear elasticity with degradation of stress due to fracture. Small strain formulation is used.

Zecevic, Milovan↗

DOE OSTI2026

Capturing the fractocohesive length scale in elastomers through a statistical mechanics-based gradient enhanced damage model

Here, this study aims to examine modeling flaw sensitivity in elastomers. The direct incorporation of polymer chain statistical mechanics considerations into a continuum stretch-based gradient-enhanced damage formulation, in turn, allows a representation of diffuse chain damage and fracture events that align with known micromechanical mechanisms. Through a series of numerical experiments, we simulate crack propagation and extract the fracture energy as an output of the model, while keeping track of the micromechanical signatures of diffuse chain damage that accommodate fracture propagation and eventually influence flaw sensitivity. Finally, by combining the fracture toughness and the work to rupture, we identify a fractocohesive length of the material, corresponding to the full width of the damage process zone. As the damage-to-fracture cascade in the proposed GED model is influenced by the introduction of a length scale associated with network imperfection and long-range load transfer, the emerging relationship of the two length scales is discussed, providing a potential link between microscopic damage mechanisms and the observed macroscopic fracture response.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Surrogate Model Integration with MOOSE XFEM for Creep Crack Growth

Ferritic-martensitic steels are key structural materials for advanced reactors but experience time-dependent deformation and damage under prolonged high temperature and irradiation, leading to creep-driven crack initiation and growth. High-fidelity models—crystal plasticity with irradiation mechanisms, phase-field for microstructural evolution, and continuum-damage viscoplasticity—capture the underlying physics but are too computationally intensive for broad design-space exploration and uncertainty quantification. This milestone advances a scalable alternative by integrating a microstructure-sensitive surrogate creep model into the Multiphysics Object-Oriented Simulation Environment (MOOSE) finite element framework and extending it to fracture via the extended finite element method (XFEM). The surrogate model, developed with collaborators at Sandia and Los Alamos National Laboratories, maps relevant microstructural descriptors to the viscoplastic response of HT9. We embed this surrogate within a coupled deformation-damage workflow in MOOSE/XFEM to simulate creep-driven crack initiation and propagation. Implementation enhancements include updates to the material interface, a plastic correction phase involving microstructure evolution, and fracture criteria to ensure numerical robustness and compatibility with the surrogate structure. Demonstrations on canonical creep benchmarks spanning uniaxial and multiaxial states show that the surrogate reproduces key trends of high-fidelity models while substantially reducing computational cost. The resulting capability bridges physics fidelity and performance, providing a practical path to a predictive, microstructure-aware assessment of creep and fracture in reactor materials.

36 - MATERIALS SCIENCE↗

DOE OSTI2024

Phase-field modeling of crack growth under coupled creep-fatigue

Crack growth under coupled creep-fatigue strongly influences service life of metallic components operating at high temperatures. However, there lack numerical models that enable direct simulations of crack growth and plasticity development under coupled creep-fatigue. Here, in this study, a unified phase-field model is developed for simulating crack growth under fatigue, creep, and coupled creep-fatigue. This model is able to reproduce the Paris laws for both brittle and ductile materials in pure fatigue mode, which shows that the degradation rate due to cyclic stress in ductile materials is generally larger than that in brittle materials. Fatigue simulations with different yield strengths demonstrate that plasticity facilitates crack growth in typical ductile materials. High-throughput simulations are performed under coupled creep-fatigue conditions, with varying hold-stress levels and hold-time. By analyzing crack growth rate under different hold-time, a creep-fatigue interaction term is obtained. Simulation results demonstrate that the creep-fatigue interaction is caused by smaller stress gradient and enhanced degradation rate ahead of the crack tips under coupled creep-fatigue, compared to the situation of pure fatigue. This work reveals the origin of different Paris-law exponents in brittle and ductile materials and demonstrates how the interplay between creep and fatigue affects crack growth.

36 MATERIALS SCIENCE↗