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DOE OSTI2025

Towards robust surrogate models: Benchmarking machine learning approaches to expediting phase field simulations of brittle fracture

Data-driven approaches have the potential to make modeling complex, nonlinear physical phenomena significantly more computationally tractable. For example, computational modeling of fracture is a core challenge where machine learning techniques have the potential to provide a much needed speedup that would enable progress in areas such as multi-scale modeling and uncertainty quantification. Currently, phase field modeling (PFM) of fracture is one such approach that offers a convenient variational formulation to model crack nucleation, branching and propagation. To date, machine learning techniques have shown promise in approximating PFM simulations. While standard fracture benchmarks represent realistic scenarios frequently observed in practice, they typically do not provide sufficiently challenging tests for data-driven methods. Here, to address this gap, we introduce a challenging dataset based on PFM simulations designed to benchmark and advance ML methods for fracture modeling. This dataset includes three energy decomposition methods, two boundary conditions, and 1000 random initial crack configurations for a total of 6000 simulations. Each sample contains 100 time steps capturing the temporal evolution of the crack field. Alongside this dataset, we also implement and evaluate Physics Informed Neural Networks (PINN), Fourier Neural Operators (FNO), and UNet models as baselines, and explore the impact of ensembling strategies on prediction accuracy. With this combination of our dataset and baseline models drawn from the literature we aim to provide a standardized and challenging benchmark for evaluating machine learning approaches to solid mechanics. Our results highlight both the promise and limitations of popular current models, and demonstrate the utility of this dataset as a testbed for advancing machine learning in fracture mechanics research.

Benchmark dataset↗

DOE OSTI2015

Multi-scale modeling of microstructure dependent intergranular brittle fracture using a quantitative phase-field based method

In this study, the fracture behavior of brittle materials is strongly influenced by their underlying microstructure that needs explicit consideration for accurate prediction of fracture properties and the associated scatter. In this work, a hierarchical multi-scale approach is pursued to model microstructure sensitive brittle fracture. A quantitative phase-field based fracture model is utilized to capture the complex crack growth behavior in the microstructure and the related parameters are calibrated from lower length scale atomistic simulations instead of engineering scale experimental data. The workability of this approach is demonstrated by performing porosity dependent intergranular fracture simulations in UO 2 and comparing the predictions with experiments.

36 MATERIALS SCIENCE↗

DOE OSTI2020

A numerical-homogenization based phase-field fracture modeling of linear elastic heterogeneous porous media

Most porous media, such as geomaterials and biomaterials are highly heterogeneous in nature, and they contain large variations of microscopic pore structures, such as pore sizes, pore distribution, and pore shapes. The oscillation of microscopic structures is a substantial challenge in theoretical characterization and is usually ignored in continuous modeling. However, mechanical behavior of porous media such as deformation and failure, are essentially impacted by the microscopic heterogeneity which needs to be considered in modeling a porous media. Here, this research proposes a numerical modeling framework with a capability to investigate the effect of microscopic heterogeneity on the macroscopic fracture behavior in porous media by using a numerical homogenization technique, combined with the phase-field fracture modeling method. This numerical modeling strategy computes a homogenized elasticity tensor based on microscopic heterogeneous pore structures heterogenous porous domain by solving boundary value problems at microscopic domain. The strain energy and subsequent propagation of macroscopic fractures will be updated using homogenized stiffness information. Using this numerical scheme, the microscopic pore structure’s impact on the fracture behavior through the homogenized elastic tensor will be taken into account. This multiscale technique is benchmarked against classical problems. Finally, the results highlight the importance of the underlying pore structure and reveal that both fracture strength and propagation path can be influenced by the microscopic heterogeneity.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Modeling brittle fracture due to anisotropic thermal expansion in polycrystalline materials

Here, this work investigated brittle fracture of polycrystalline materials due to thermal stresses arising from anisotropic thermal expansion. We used phase-field fracture simulations with the properties of alpha-uranium (α-U) and assumed a linear elastic mechanical response. Three-dimensional simulations were used to predict fracture for various conditions and crystallographic textures. We found that fracture was more pronounced during cooling than during heating because the anisotropy increased with temperature. We also found that the total crack surface area increased with increasing average misorientation, while the net shape change of the material decreased with increasing misorientation. Two-dimensional simulations in which one crystallographic coefficient of thermal expansion (CTE) was set to zero indicated that the expansion behavior in the crystallographic direction with the smallest CTE was the primary cause of fracture.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2026

On the roles of welding residual stresses in determination of fracture toughness in austenitic stainless steel SUS 304 pipeline girth welds

Welding residual stresses especially the high tensile stresses are proved to have negative impacts on the fatigue and fracture behaviors of welded structures. In this study, a virtual fabrication of test specimens from welding process to specimen preparation was carried out by numerical simulation. An austenitic stainless steel multi-pass pipe welding was simulated by transient thermal–mechanical finite element analysis, the residual stresses were then mapped into the test specimen to evaluate fracture toughness. The findings in this study confirmed that, residual stress can be high in a sub-sized compact tensile specimen, which may accelerate or hinder the crack propagation during actual fatigue and fracture tests as reported in recent years. The influence of the cutting location and orientation of the specimen on fracture performance was investigated systematically to provide a fundamental understanding of welding residual stress and necessary insights into the specimen preparation procedure. Considering the limitation of measuring techniques and the complexity of the stress distribution, the developed numerical model can be a very useful tool to elucidate the stress evolution and quantify the effect of remaining welding stress on fracture toughness.

Fracture behavior↗

DOE OSTI2026

3D seismic imaging of a fracture damage zone controlling reservoir compartmentalization at the Raft River EGS using multi-azimuth walkaway VSP

Accurate imaging of steeply dipping fracture zones in crystalline enhanced geothermal systems (EGS) is critical for constraining permeability architecture and guiding stimulation design. However, such structures remain poorly resolved by conventional surface seismic methods. We present a fully three-dimensional (3D) elastic-waveform inversion-migration workflow applied to multi-azimuth walkaway vertical seismic profiling (VSP) data acquired in a deviated borehole at the Raft River EGS. The workflow integrates first-arrival traveltime tomography, multi-scale elastic waveform inversion (EWI), and elastic least-squares reverse-time migration (ELSRTM) to recover high-resolution compressional-and shear-wave velocity models and to image structural discontinuities in the crystalline basement. The results reveal a laterally continuous low-velocity anomaly, with shear-wave velocity reductions of 25-30%, consistent with fractureinduced mechanical weakening. Two steeply dipping discontinuities bound a 50-80 m wide fracture damage zone. Independent constraints from microseismic clustering and geochemical compartmentalization corroborate the geometry and structural significance of this feature. Synthetic modeling further confirms that structures of this scale are resolvable (~30 m). These findings indicate that the Narrows structure is a distributed fracture damage zone rather than a discrete fault plane. By resolving fracture-zone geometry at the tens-of-meters scale using a single borehole, this workflow provides a practical and transferable approach for improving structural characterization, reducing uncertainty in permeability architecture, and supporting reservoir modeling and stimulation design in fractured crystalline EGS reservoirs.

58 GEOSCIENCES↗

DOE OSTI2015

Laser notching ceramics for reliable fracture toughness testing

A new method for notching ceramics was developed using a picosecond laser for fracture toughness testing of alumina samples. The test geometry incorporated a single-edge-V-notch that was notched using picosecond laser micromachining. This method has been used in the past for cutting ceramics, and is known to remove material with little to no thermal effect on the surrounding material matrix. This study showed that laser-assisted-machining for fracture toughness testing of ceramics was reliable, quick, and cost effective. In order to assess the laser notched single-edge-V-notch beam method, fracture toughness results were compared to results from other more traditional methods, specifically surface-crack in flexure and the chevron notch bend tests. Lastly, the results showed that picosecond laser notching produced precise notches in post-failure measurements, and that the measured fracture toughness results showed improved consistency compared to traditional fracture toughness methods.

36 MATERIALS SCIENCE↗

DOE OSTI2019

Hyperelastic phase-field fracture mechanics modeling of the toughening induced by Bouligand structures in natural materials

Bouligand structures are widely observed in natural materials; elasmoid fish scales and the exoskeleton of arthropods, such as lobsters, crabs, mantis shrimp and insects, are prime examples. In fish scales, such as those of the Arapaima gigas, the tough inner core beneath the harder surface of the scale displays a Bouligand structure comprising a layered arrangement of collagen fibrils with an orthogonal or twisted staircase (or plywood) architecture. A much rarer variation of this structure, the double-twisted Bouligand structure, has been discovered in the primitive elasmoid scales of the coelacanth fish; this architecture is quite distinct from “modern” elasmoid fish scales yet provides extraordinary resistance to deformation and fracture. Here we examine the toughening mechanisms created by the double-twisted Bouligand structure in comparison to those generated by the more common single Bouligand structures. Specifically, we have developed an orientation-dependent, hyperelastic, phase-field fracture mechanics method to computationally examine the relative fracture toughness of elasmoid fish scales comprising single vs. double-twisted Bouligand structures of fibrils. The model demonstrates the critical role played by the extra inter-bundle fibrils found in coelacanth fish scales in enhancing the toughness of Bouligand-type structures. Synthesis and fracture tests of 3-D printed Bouligand-type materials are presented to support the modeling and complement our understanding of the fracture mechanisms in Bouligand-type structures.

3-D printing↗

DOE OSTI2022

Grain-boundary fracture mechanisms in Li 7 La 3 Zr 2 O 12 (LLZO) solid electrolytes: When phase transformation acts as a temperature-dependent toughening mechanism

Garnet-type, solid electrolytes, such as Li 7 La 3 Zr 2 O 12 (LLZO), are a promising alternative to liquid electrolytes for lithium-metal batteries. However, such solid-electrolyte materials frequently exhibit undesirable lithium (Li) metal plating and fracture along grain boundaries. In this work, we employ atomistic simulations to investigate the mechanisms and key fracture properties associated with intergranular fracture along one such boundary. Our results show that, in the case of a Σ5 (310) grain boundary, this boundary exhibits brittle fracture behavior, i.e. the absence of dislocation activity ahead of the propagating crack tip, accompanied with a decrease in work of separation, peak stress, and maximum stress intensity factor as the temperature increases from 300 K to 1500 K. As the crack propagates, we predict two temperature-dependent Li clustering regimes. For temperatures at or below 900 K, Li tends to cluster in the bulk region away from the crack plane driven by a void-coalescence mechanism concomitant a simultaneous cubic-to-tetragonal phase transition. The tetragonalization of LLZO in this temperature regime acts as an emerging toughening mechanism. At higher temperatures, this phase transition mechanism is suppressed leading to a more uniform distribution of Li throughout the grain-boundary system and lower fracture properties as compared to lower temperatures.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Method to account for natural fracture induced elastic anisotropy in geomechanical characterization of shale gas reservoirs

Shale has been usually recognized as a transverse isotropic (TI) medium in conventional geomechanical log interpretation due to its laminated nature. However, when natural fractures exist in the shale rock, additional elastic anisotropy is introduced, converting laminated Shale to an orthorhombic (OB) medium. Previous studies illustrate that neglecting the natural fracture induced anisotropy in shale geomechanical log interpretation could lead to inaccurate evaluations of elastic moduli and in-situ stresses. In this paper, a new method is developed to account for the natural fracture induced anisotropy in geomechanical log interpretation based upon the TI acoustic model developed by the author and a characterization technique of elastic wave anisotropy (Sayers, 1991). The new OB model incorporates the four acoustic log data inputs and five modeling constraints in a nonlinear optimization algorithm to solve for the nine independent stiffness coefficients of an OB rock, and further to solve for the geomechanical properties and in-situ stress profiles in an OB formation. The new method was validated with a Marcellus Gas Shale field case. Both the new OB model and the conventional TI model were applied to interpret the minimum horizontal stress profile for the same formation. By comparing the results, the OB model is more robust than the TI from two aspects. First, the average stress magnitude predicted by the OB model is closer to the one measured by the Diagnostic Fracture Injection Test (DFIT). Second, the OB model predicts a more obvious stress barrier between the lower Marcellus and upper Onondaga Limestone than the TI model does. Finally, the predicted stress barrier is consistent with the observation of the microseismic events of a horizontal well drilled and completed nearby, which reveals that no hydraulic fracture propagates downward through the bottom boundary of Marcellus Shale into the underlying Onondaga Limestone.

03 NATURAL GAS↗

DOE OSTI2018

Dynamic damage and fracture of a conductive glass under high-rate compression: A synchrotron based study

Dynamic damage and fracture of conductive glass are investigated using a split Hopkinson pressure bar, implemented with in situ X-ray phase contrast imaging (XPCI) and optical imaging for comparison. Quantitative comparison between X-ray and optical images demonstrates that XPCI exhibits much higher resolution in resolving micro cracks and dynamic fracture modes. Multiple predamage and fracture modes of glass samples under dynamic loading are revealed with XPCI to depend on competing nucleation of initial flaws across scales, which give rise to a scattered fracture strength distribution. The fracture strengths increase with increasing strain rates due to accelerated crack propagation and damage growth. Quantitative gray-scale statistical analysis of XPCI and optical images yields spatial and temporal evolutions of damage. Unexpected plateaus essentially without damage growth are observed on the damage curves of glass, deviating from conventional theoretical predictions. The damage plateaus are attributed to growth, closure and re-expansion of inclined main cracks, due to interactions of stress waves with crack tips. In conclusion, the current results also demonstrate a reliable experimental technique for dynamic damage characterization of brittle materials.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Thermal-shock experiments for separate-effects validation of UO 2 fuel fracture models

Because of the important role that fracture plays in the behavior of ceramic UO2 fuel in a nuclear reactor environment, fracture models are a major component of fuel performance codes. As with any aspect of fuel performance, it is crucial to validate these fracture models against experimental data; however, obtaining well-controlled data for conditions representative of a reactor environment is difficult. Quenching is proposed here as a relatively simple approach for using a laboratory environment to achieve conditions that approximate those of a reactor environment. In this paper, an experimental apparatus containing a single instrumented fuel pellet in a sealed section of copper tubing is developed. It is then applied to a series of seven experiments in which the apparatus is first heated to a high temperature (580-680°C) by immersion in a molten salt bath, then quenched in a cold bath (-10-4 °C). Development of these experiments was guided by numerical simulations, and post-test simulations were performed to aid in understanding the experimental behavior and assessing the accuracy of the predictions of fracture initiation and propagation. In addition, similar experiments were performed on solid copper rods to provide temperature-dependent heat transfer coefficients for use in simulations of these experiments. Here, this study demonstrates that quenching is a viable approach for generating thermal gradients representative of those in prototypical light-water reactor conditions at powers of about 5–10 kW/m. Fracture is expected to begin at these power levels, and moderate amounts of radial and axial cracking was observed in these quenching tests.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2023

Mitigation of spall fracture by evolving porosity

Ductile materials subject to impact loading conditions can undergo spall fracture when an incoming compressive stress wave reflects off interfaces and free surfaces as a tensile stress wave. Experimental observations suggest that in ductile materials, spall fracture is driven by the evolution of porosity. However, the presence of initial porosity in ductile materials also introduces plastic compressibility, which can attenuate the incoming compressive stress wave and, as a result, reduce the amplitude of the reflected tensile stress wave. This, in turn, can mitigate or delay spall fracture. In this work, we report on finite deformation finite element calculations that analyze the response of porous ductile materials subjected to impact loading conditions. Two sets of calculations are carried out, in the first set the material contains initial porosity values ranging from 0% to 5% while in the second set the material also undergoes stress-controlled porosity nucleation. Both sets of calculations are carried out for a wide range of imposed impact velocities. Here, our results show that porosity in ductile materials can, under certain circumstances, mitigate spall fracture by attenuating stress waves. Results correlating the effects of impact velocity, initial porosity, and porosity nucleation on spall fracture are presented and the underlying mechanisms are discussed.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Plasticity and fracture behavior of Inconel 625 manufactured by laser powder bed fusion: Comparison between as-built and stress relieved conditions

In this work, the influence of stress relief on the plasticity and fracture behavior of Inconel 625 fabricated through laser powder bed fusion additive manufacturing (AM) was investigated. The as-built versus stress relieved microstructures were compared, showing similar grain structures but the presence of ~10 vol % δ phase in the stress relieved condition, and no δ phase in the as-built condition. Mechanical tests under plane strain tension were performed on the stress relieved samples, and an anisotropic plasticity model was calibrated and validated using finite element simulations. Uniaxial and notched tension tests were performed on both as-built and stress relieved samples to probe the effect of stress relief on stress state- and direction-dependent fracture behavior. It was found that on average, the fracture strain of the stress relieved samples along the build direction was 30% higher than that along the perpendicular build direction in the stress state range studied, and the stress relief heat treatment resulted in a 45% decrease in fracture strain. The fracture strain in stress relieved samples was more strongly dependent on stress state than in as-built samples.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Temperature-Dependent Fracture Resistance of Silicon Nanopillars during Electrochemical Lithiation

During the lithation of silicon anodes, the solid-state diffusion of lithium into Li x Si follows the Arrhenius law, the resulting morphology and fracture behavior are determined by the silicon anode operation temperature. Here, we reveal the temperature dependence of the lithiation mechanics of crystalline silicon nanopillars (SiNPs) via microscopic observations of the anisotropic growth and fracture behavior. In this work, we fabricated 1D SiNP structures with various orientations ($\langle100\rangle, \langle110\rangle$, and $\langle111\rangle$) as working electrodes and operated them at temperatures ranging from –20 to 40 °C. The lithiation of crystalline silicon at low temperatures exhibited preferential volume expansion along $\langle110\rangle$ and decreased fracture resistance. Furthermore, low temperatures caused the catastrophic fracture of amorphous silicon after the second lithiation. Our findings demonstrate the importance of silicon anode temperature control to prevent mechanical fracture during the cycle of lithium-ion batteries in harsh environments (e.g., electric vehicles in winter).

25 ENERGY STORAGE↗

DOE OSTI2019

Impact of Confining Stress on Capillary Pressure Behavior During Drainage Through Rough Fractures

In this work we study, numerically, the behavior of capillary pressure $(P_c)$ during slow immiscible displacement in a rough fracture as a function of the degree of fracture aperture heterogeneity that results from two distinct mechanisms: normal confining stress and fracture surface correlation. We generate synthetic self-affine rough fractures at different correlation scales, solve the elastic contact problem to model the effect of confining stress, and simulate slow immiscible displacement of a wetting fluid by a nonwetting one using a modified invasion percolation model that accounts for in-plane curvature of the fluid-fluid interface. Our modeling results indicate that the power spectral density, $S(f)$, of $P_c$, can be used to qualitatively characterize fracture aperture heterogeneity. We show that the distribution of forward avalanche sizes follows a power law $N_f (S_f)$ ∝ $S^{-a}_f$ , with exponent $α$ = 2, in agreement with previously reported values for porous media and equal to the expected theoretical exponent for a self-organized criticality process.

58 GEOSCIENCES↗

DOE OSTI2024

Sensitivity Analysis in the Presence of Intrinsic Stochasticity for Discrete Fracture Network Simulations

Abstract Large‐scale discrete fracture network (DFN) simulators are standard fare for studies involving the sub‐surface transport of particles since direct observation of real world underground fracture networks is generally infeasible. While these simulators have successfully been used in several engineering applications, estimates of output quantities of interest (QoI) — such as breakthrough time of particles reaching the edge of the system — suffer from two distinct types of uncertainty. A run of a DFN simulator requires several parameters to be set that dictate the placement and size of fractures, the density of fractures, and the overall permeability of the system; uncertainty on the proper parameters will lead to uncertainty in the QoI, called epistemic uncertainty. Furthermore, since these input settings to DFN simulators control the stochastic processes which place fractures and govern flow, understanding how this randomness affects the QoI requires several runs of the simulator at distinct random seeds. The uncertainty in the QoI attributed to different realizations (i.e., different seeds) of the same random process (i.e., identical input parameters) leads to a second type of uncertainty, called aleatoric uncertainty. In this paper, we perform a Sensitivity Analysis, which directly attributes the uncertainty observed in the QoI to the epistemic uncertainty from each input parameter and to the aleatoric uncertainty. Beyond the specific takeaways on which input variables influence uncertainty in the QoI the most, a major contribution of this paper is the introduction of a statistically rigorous workflow for characterizing the uncertainty in DFN flow simulations that exhibit heteroskedasticity.

58 GEOSCIENCES↗

DOE OSTI2026

Geologic stress modulates fluid mixing at fracture intersections

Fracture intersections are critical links that enable flow and transport in subsurface fracture networks, and their behavior strongly influences fluid mixing in a network. Although all subsurface fractures are subjected to geological stress, we lack a fundamental understanding of how fracture intersection geometry evolves under stress and how these changes influence fluid mixing. Here, we combine 3D printing, 3D X-ray tomographic imaging, and 3D pore-scale numerical simulations to reveal stress-induced changes in intersection geometry and their impact on mixing. Mixing is found to be strongly affected by partial closure of an intersection under stress. As an intersection closes, the void area for fluid flow and diffusion decreases leading to substantial deviations between conventional mixing models and full pore-scale modeling. To address this, we propose a modified mixing model that accounts for intersection deformation, which is essential for accurate modeling of solute transport and mixing through fracture networks.

15 GEOTHERMAL ENERGY↗