By Gowri Srinivasan, Daniel O’Malley, and M. Giselle Fernández-Godino
Many engineering applications utilize brittle materials—such as glass, ceramics, graphite, concrete, and some metals like beryllium—for their stiffness, lightweight properties, and ability to maintain their shapes at extreme temperatures. For example, beryllium is particularly useful in defense and aerospace applications for building lightweight instruments with high precision and controlling fission reactions. Concrete, on the other hand, is widely used in civil applications as the primary substance for buildings and bridges. Though brittle materials have many desirable properties, they are also prone to catastrophic failures that arise because the materials can handle only some elastic deformation and almost no plastic deformation. Instead of bouncing back (elastic behavior) or permanently deforming (plastic behavior), brittle materials fail with little warning. Understanding the mechanisms that cause them to fail is imperative for avoiding accidents that can jeopardize the safety and security of people and systems.
Figure 1. Beryllium disc fragments that result from high-velocity impact. Figure courtesy of [1].
Brittle materials fail through the nucleation, evolution, and coalescence of microcracks. Figure 1 displays the aftermath of several experiments wherein one beryllium disc crashed into another at an extreme speed [1], revealing a handful of large, readily-visible cracks. The finite-discrete element method (FDEM) is ideally suited for modeling crack dynamics at the mesoscale, since the finite elements describe intact cells in the material and the discrete elements characterize the cracks (void space). Researchers at Los Alamos National Laboratory (LANL) have developed an FDEM solver—the Hybrid Optimization Software Suite (HOSS)—that models the evolution of many cracks and generates an accurate prediction of the failure path and failure time. Unfortunately, the cost associated with these simulations can be enormous. For example, performing just a single simulation may be unmanageable at large spatial or temporal scales, even with high-performance computing resources. And performing many simulations for optimization or uncertainty quantification can be problematic at more modest scales. Continuum model simulations provide a relatively inexpensive alternative. However, these simulations sacrifice the ability to track each crack for computational efficiency by homogenizing microcrack information through upscaled parameters. This upscaling process is scientifically challenging, as it tends to lose crucial information that is related to the structure and evolution of cracks.
Machine learning (ML) is well-suited for modeling the dynamics of systems where phenomena at smaller length scales significantly impact the larger scales of interest [6]. Access to sufficient ground truth data for validation—in terms of experimental evidence and mechanistic, lower-length scale models—renders ML a powerful tool for scale-bridging applications. Here, ML offers the tantalizing possibility of learning about crack evolution and coalescence via time series data from high-fidelity FDEM simulations. This data can inform a variety of ML models that predict damage characteristics in brittle materials—such as where and when materials will fail—and deliver additional information about stress, strain, and individual cracks.
Recent research has utilized ML to model brittle materials that undergo low strain rate dynamic tensile loading [3, 5]. Tensile loading refers to a load that pulls an object apart (like the rope in a tug-of-war game), and low strain rate conditions result from the slow application of tensile loading. We simulated a concrete plate 2m \(\times\) 3m whose upper edge moves upward at a speed of 0.3 m/s, resulting in a low strain rate of 0.1 s-1. The simulation, which we performed in two dimensions using the FDEM code, has 20 initial cracks with identical length (30 cm), random uniform location, and three different possible orientations. We then explored multiple conceptual approaches to predict the path of failure when compared with the expensive FDEM code (HOSS). The best of these approaches, called Microcrack Pair Informed Coalescence (McPIC), treats each crack as a node in a dynamic graph where the presence of edges indicates the coalescence of two cracks. The model predicts two things: (i) whether two cracks will coalesce, and (ii) the timing of the coalescence if they do. The development of McPIC involved an investigation of several supervised ML algorithms, including decision trees, random forests, and neural networks. The neural networks provided a marginally better performance than the random forest approach, and decision trees performed the worst.
Moving beyond these initial models, we developed a workflow in which ML-based upscaling methods feed into a continuum-scale Lagrangian simulator called FLAG [2, 4, 7]. While the upfront cost of generating the training data for this type of ML model can be expensive, every continuum-scale simulation that utilizes the ML upscaling experiences computational savings afterwards. This workflow has produced a continuum-scale simulator that is both accurate and faster by three to four orders of magnitude [2].
At LANL, our workflow is of particular interest for high strain rate scenarios that occur during explosions. High-impact velocities (sometimes on the order of 1 km/s) produce a high strain rate. Figure 2a depicts the setup for a flyer plate experiment wherein the flyer plate crashes into the target plate with an impact velocity of 721 m/s.
Figure 2. High strain rate simulations. 2a. Setup for the flyer plate test simulations. The flyer plate has a starting velocity of 0.721 km/s and initially has contact with the target plate. 2b. Evolution of the shock wave velocity at the middle rear of the target plate for a second experiment with an impact velocity of 1.246 km/s. FLAG-ML improves the predictions obtained by FLAG without an imposed damage model. Figure courtesy of [2].
A critical insight during the development of the upscaling ML model was the observation that a small set of features could accurately inform the continuum-scale model. In our case, the two key features were the maximum stress and length of the longest crack. We took this into account while also acknowledging that the full crack length distribution, crack orientations, and stress fields at the microscale do not seem to play a substantial role in this setting. The combination of ML upscaling and the continuum scale model FLAG, referred to as FLAG-ML, was able to make predictions as accurately as HOSS. FLAG-ML also showed good agreement with experiments that utilized an impact velocity that was approximately 73 percent faster than the HOSS simulations that we used for training. Figure 2b shows the ML prediction for the shock wave velocity at the bottom of the target plate as a function of time. The ML emulator was trained with the HOSS simulation data of a 0.721 km/s flyer plate impact, and FLAG-ML was validated against experiments for which the flyer plate impact velocity was 1.2 km/s. This powerful result could enable transfer learning — the application of an ML model that is trained for a particular problem to a similar but somewhat different problem.
The relentless growth of computational power over the past several decades has enabled the use of detailed, high-fidelity simulations. Now it employs a large amount of data to drive the training of fast ML models. Scientists often exploit data from high-fidelity simulations to train these ML models; this dynamic can influence the study of material behavior under extreme loading conditions, as the fast ML models are able to bridge scales. While significant progress has occurred in this field, the work is representative of a new beginning in upscaling and scale-bridging. Researchers traditionally had to perform years of in-depth studies to upscale complex phenomena, sacrificing accuracy for efficiency through approximations that neglect known physics. ML provides an alternative that can exchange people-years for central-processing-unit-years. Although this assertion does not downplay the importance of the traditional tactics, there are grounds for optimism that a data-centric ML approach will offer solutions to challenging problems.
References
[1] Cady, C., Adams, C., Prime, M., Hull, L., Addessio, F., Bronkhorst, C., …, Brown, D. (2011). Characterization of S200-F beryllium using shock loading and quasi-static experiments. (Technical Report LA-UR-11-06976). Los Alamos, NM: Los Alamos National Laboratory.
[2] Fernández-Godino, M.G., Panda, N., O’Malley, D., Larkin, K., Hunter, A., Haftka, R.T., & Srinivasan, G., (2021). Accelerating high-strain continuum-scale brittle fracture simulations with machine learning. Comput. Mater. Sci., 186, 109959.
[3] Hunter, A., Moore, B.A., Mudunuru, M., Chau, V., Tchoua, R., …, Srinivasan, G. (2019). Reduced-order modeling through machine learning and graph-theoretic approaches for brittle fracture applications. Comput. Mater. Sci., 157, 87-98.
[4] Larkin, K., Rougier, E., Chau, V., Srinivasan, G., Abdelkefi, A., & Hunter, A. (2020). Scale bridging damage model for quasi-brittle metals informed with crack evolution statistics. J. Mechan. Phys. Solids, 138, 03921.
[5] Moore, B.A., Rougier, E., O’Malley, D., Srinivasan, G., Hunter, A., & Viswanathan, H. (2018). Predictive modeling of dynamic fracture growth in brittle materials with machine learning. Comput. Mater. Sci., 148, 46-53.
[6] Srinivasan, G., Hyman, J.D., Osthus, D.A., Moore, B.A., O’Malley, D., Karra, S., …, Viswanathan, H.S. (2018). Quantifying topological uncertainty in fractured systems using graph theory and machine learning. Sci. Rep., 8, 11665.
[7] Vaughn, N., Kononov, A., Moore, B., Rougier, E., Viswanathan, H., & Hunter, A. (2019). Statistically informed upscaling of damage evolution in brittle materials. Theor. Appl. Fracture Mechan., 102, 210-221.
Gowri Srinivasan is the Verification and Analysis group leader in the X Computational Physics Division at Los Alamos National Laboratory (LANL). Her research interests include uncertainty quantification and machine learning. Daniel O’Malley is a staff scientist in the Earth and Environmental Sciences division at LANL. M. Giselle Fernández-Godino is a simulation data scientist for above and below-ground physics in the Atmospheric Science Research and Applications group in Lawrence Livermore National Laboratory's Atmospheric, Earth, & Energy Science Division. Her current work focuses on machine learning and uncertainty quantification approaches in the context of atmospheric flow, transport, and hazard assessment.