Neural operators [1–5] have recently become popular tools for designing solution maps between function spaces in the form of neural networks. Differently from classical scientific machine learning approaches that learn parameters of a known partial differential equation (PDE) for a single instance of the input parameters at a fixed resolution, neural operators approximate the solution map of a family of PDEs [6,7]. Despite their success, the uses of neural operators are so far restricted to relatively shallow neural networks and confined to learning hidden governing laws. In this work, we propose a novel nonlocal neural operator, which we refer to as nonlocal kernel network (NKN), that is resolution independent, characterized by deep neural networks, and capable of handling a variety of tasks such as learning governing equations and classifying images. Our NKN stems from the interpretation of the neural network as a discrete nonlocal diffusion reaction equation that, in the limit of infinite layers, is equivalent to a parabolic nonlocal equation, whose stability is analyzed via nonlocal vector calculus. The resemblance with integral forms of neural operators allows NKNs to capture long-range dependencies in the feature space, while the continuous treatment of node-to-node interactions makes NKNs resolution independent. The resemblance with neural ODEs, reinterpreted in a nonlocal sense, and the stable network dynamics between layers allow for generalization of NKN's optimal parameters from shallow to deep networks. This fact enables the use of shallow-to-deep initialization techniques [8]. Our tests show that NKNs outperform baseline methods in both learning governing equations and image classification tasks and generalize well to different resolutions and depths.
Additive manufactured Ti-5Al-5V-5Mo-3Cr (Ti-5553) is being considered as an AM repair material for engineering applications because of its superior strength properties compared to other titanium alloys. Here, we describe the failure mechanisms observed through computed tomography, electron backscatter diffraction (EBSD), and scanning electron microscopy (SEM) of spall damage as a result of tensile failure in as-built and annealed Ti-5553. We also investigate the phase stability in native powder, as-built and annealed Ti-5553 through diamond anvil cell (DAC) and ramp compression experiments. We then explore the effect of tensile loading on a sample containing an interface between a Ti-6Al-V4 (Ti-64) baseplate and additively manufactured Ti-5553 layer. Post-mortem materials characterization showed spallation occurred in regions of initial porosity and the interface provides a nucleation site for spall damage below the spall strength of Ti-5553. Preliminary peridynamics modeling of the dynamic experiments is described. Finally, we discuss further development of Stochastic Parallel PARticle Kinteic Simulator (SPPARKS) Monte Carlo (MC) capabilities to include the integration of alpha (α)-phase and microstructural simulations for this multiphase titanium alloy.
Nonlocal models provide a much-needed predictive capability for important Sandia mission applications, ranging from fracture mechanics for nuclear components to subsurface flow for nuclear waste disposal, where traditional partial differential equations (PDEs) models fail to capture effects due to long-range forces at the microscale and mesoscale. However, utilization of this capability is seriously compromised by the lack of a rigorous nonlocal interface theory, required for both application and efficient solution of nonlocal models. To unlock the full potential of nonlocal modeling we developed a mathematically rigorous and physically consistent interface theory and demonstrate its scope in mission-relevant exemplar problems.
A straight fiber with nonlocal forces that are independent of bond strain is considered. These internal loads can either stabilize or destabilize the straight configuration. Transverse waves with long wavelength have unstable dispersion properties for certain combinations of nonlocal kernels and internal loads. When these unstable waves occur, deformation of the straight fiber into a circular arc can lower its potential energy in equilibrium. The equilibrium value of the radius of curvature is computed explicitly.
Computer Methods in Applied Mechanics and Engineering
Shojaei, Arman; Hermann, Alexander; Cyron, Christian J.; Seleson, Pablo; Silling, Stewart A.
Efficient and accurate calculation of spatial integrals is of major interest in the numerical implementation of peridynamics (PD). The standard way to perform this calculation is a particle-based approach that discretizes the strong form of the PD governing equation. This approach has rapidly been adopted by the PD community since it offers some advantages. It is computationally cheaper than other available schemes, can conveniently handle material separation, and effectively deals with nonlinear PD models. Nevertheless, PD models are still computationally very expensive compared with those based on the classical continuum mechanics theory, particularly for large-scale problems in three dimensions. This results from the nonlocal nature of the PD theory which leads to interactions of each node of a discretized body with multiple surrounding nodes. Here, we propose a new approach to significantly boost the numerical efficiency of PD models. We propose a discretization scheme that employs a simple collocation procedure and is truly meshfree; i.e., it does not depend on any background integration cells. In contrast to the standard scheme, the proposed scheme requires a much smaller set of neighboring nodes (keeping the same physical length scale) to achieve a specific accuracy and is thus computationally more efficient. Our new scheme is applicable to the case of linear PD models and within neighborhoods where the solution can be approximated by smooth basis functions. Therefore, to fully exploit the advantages of both the standard and the proposed schemes, a hybrid discretization is presented that combines both approaches within an adaptive framework. The high performance of the developed framework is illustrated by several numerical examples, including brittle fracture and corrosion problems in two and three dimensions.
Nonlocal models, including peridynamics, often use integral operators that embed lengthscales in their definition. However, the integrands in these operators are difficult to define from the data that are typically available for a given physical system, such as laboratory mechanical property tests. In contrast, molecular dynamics (MD) does not require these integrands, but it suffers from computational limitations in the length and time scales it can address. To combine the strengths of both methods and to obtain a coarse-grained, homogenized continuum model that efficiently and accurately captures materials’ behavior, we propose a learning framework to extract, from MD data, an optimal Linear Peridynamic Solid (LPS) model as a surrogate for MD displacements. To maximize the accuracy of the learnt model we allow the peridynamic influence function to be partially negative, while preserving the well-posedness of the resulting model. To achieve this, we provide sufficient well-posedness conditions for discretized LPS models with sign-changing influence functions and develop a constrained optimization algorithm that minimizes the equation residual while enforcing such solvability conditions. This framework guarantees that the resulting model is mathematically well-posed, physically consistent, and that it generalizes well to settings that are different from the ones used during training. We illustrate the efficacy of the proposed approach with several numerical tests for single layer graphene. Our two-dimensional tests show the robustness of the proposed algorithm on validation data sets that include thermal noise, different domain shapes and external loadings, and discretizations substantially different from the ones used for training.
Powders under compression form mesostructures of particle agglomerations in response to both inter- and intra-particle forces. The ability to computationally predict the resulting mesostructures with reasonable accuracy requires models that capture the distributions associated with particle size and shape, contact forces, and mechanical response during deformation and fracture. The following report presents experimental data obtained for the purpose of validating emerging mesostructures simulated by discrete element method and peridynamic approaches. A custom compression apparatus, suitable for integration with our micro-computed tomography (micro-CT) system, was used to collect 3-D scans of a bulk powder at discrete steps of increasing compression. Details of the apparatus and the microcrystalline cellulose particles, with a nearly spherical shape and mean particle size, are presented. Comparative simulations were performed with an initial arrangement of particles and particle shapes directly extracted from the validation experiment. The experimental volumetric reconstruction was segmented to extract the relative positions and shapes of individual particles in the ensemble, including internal voids in the case of the microcrystalline cellulose particles. These computationally determined particles were then compressed within the computational domain and the evolving mesostructures compared directly to those in the validation experiment. The ability of the computational models to simulate the experimental mesostructures and particle behavior at increasing compression is discussed.
Karim, Mohammad K.; Narasimhachary, Santosh N.; Radailli, Francesco R.; Amann, Christian A.; Dayal, Kaushik D.; Silling, Stewart A.; Germann, Tim G.
In this study, we present a computational study and framework that allows us to study and understand the crack nucleation process from forging flaws. Forging flaws may be present in large steel rotor components commonly used for rotating power generation equipment including gas turbines, electrical generators, and steam turbines. The service life of these components is often limited by crack nucleation and subsequent growth from such forging flaws, which frequently exhibit themselves as non-metallic oxide inclusions. The fatigue crack growth process can be described by established engineering fracture mechanics methods. However, the initial crack nucleation process from a forging flaw is challenging for traditional engineering methods to quantify as it depends on the details of the flaw, including flaw morphology. We adopt the peridynamics method to describe and study this crack nucleation process. For a specific industrial gas turbine rotor steel, we present how we integrate and fit commonly known base material property data such as elastic properties, yield strength, and S-N curves, as well as fatigue crack growth data into a peridynamic model. The obtained model is then utilized in a series of high-performance two-dimensional peridynamic simulations to study the crack nucleation process from forging flaws for ambient and elevated temperatures in a rectangular simulation cell specimen. The simulations reveal an initial local nucleation at multiple small oxide inclusions followed by micro-crack propagation, arrest, coalescence, and eventual emergence of a dominant micro-crack that governs the crack nucleation process. The dependence on temperature and density of oxide inclusions of both the details of the microscopic processes and cycles to crack nucleation is also observed. Finally, the results are compared with fatigue experiments performed with specimens containing forging flaws of the same rotor steel.
The peridynamic theory of solid mechanics is applied to modeling the deformation and fracture of micrometer-sized particles made of organic crystalline material. A new peridynamic material model is proposed to reproduce the elastic–plastic response, creep, and fracture that are observed in experiments. The model is implemented in a three-dimensional, meshless Lagrangian simulation code. In the small deformation, elastic regime, the model agrees well with classical Hertzian contact analysis for a sphere compressed between rigid plates. Under higher load, material and geometrical nonlinearity is predicted, leading to fracture. The material parameters for the energetic material CL-20 are evaluated from nanoindentation test data on the cyclic compression and failure of micrometer-sized grains.
This Laboratory Directed Research and Development project developed and applied closely coupled experimental and computational tools to investigate powder compaction across multiple length scales. The primary motivation for this work is to provide connections between powder feedstock characteristics, processing conditions, and powder pellet properties in the context of powder-based energetic components manufacturing. We have focused our efforts on multicrystalline cellulose, a molecular crystalline surrogate material that is mechanically similar to several energetic materials of interest, but provides several advantages for fundamental investigations. We report extensive experimental characterization ranging in length scale from nanometers to macroscopic, bulk behavior. Experiments included nanoindentation of well-controlled, micron-scale pillar geometries milled into the surface of individual particles, single-particle crushing experiments, in-situ optical and computed tomography imaging of the compaction of multiple particles in different geometries, and bulk powder compaction. In order to capture the large plastic deformation and fracture of particles in computational models, we have advanced two distinct meshfree Lagrangian simulation techniques: 1.) bonded particle methods, which extend existing discrete element method capabilities in the Sandia-developed , open-source LAMMPS code to capture particle deformation and fracture and 2.) extensions of peridynamics for application to mesoscale powder compaction, including a novel material model that includes plasticity and creep. We have demonstrated both methods for simulations of single-particle crushing as well as mesoscale multi-particle compaction, with favorable comparisons to experimental data. We have used small-scale, mechanical characterization data to inform material models, and in-situ imaging of mesoscale particle structures to provide initial conditions for simulations. Both mesostructure porosity characteristics and overall stress-strain behavior were found to be in good agreement between simulations and experiments. We have thus demonstrated a novel multi-scale, closely coupled experimental and computational approach to the study of powder compaction. This enables a wide range of possible investigations into feedstock-process-structure relationships in powder-based materials, with immediate applications in energetic component manufacturing, as well as other particle-based components and processes.
An ordinary state-based peridynamic material model is proposed for single sheet graphene. The model is calibrated using coarse grained molecular dynamics simulations. The coarse graining method allows the dependence of bond force on bond length to be determined, including the horizon. The peridynamic model allows the horizon to be rescaled, providing a multiscale capability and allowing for substantial reductions in computational cost compared with molecular dynamics. The calibrated peridynamic model is compared to experimental data on the deflection and perforation of a graphene monolayer by an atomic force microscope probe.
The propagation of a wave pulse due to low-speed impact on a one-dimensional, heterogeneous bar is studied. Due to the dispersive character of the medium, the pulse attenuates as it propagates. This attenuation is studied over propagation distances that are much longer than the size of the microstructure. A homogenized peridynamic material model can be calibrated to reproduce the attenuation and spreading of the wave. The calibration consists of matching the dispersion curve for the heterogeneous material near the limit of long wavelengths. It is demonstrated that the peridynamic method reproduces the attenuation of wave pulses predicted by an exact microstructural model over large propagation distances.
Nonlocal models naturally handle a range of physics of interest to SNL, but discretization of their underlying integral operators poses mathematical challenges to realize the accuracy and robustness commonplace in discretization of local counterparts. This project focuses on the concept of asymptotic compatibility, namely preservation of the limit of the discrete nonlocal model to a corresponding well-understood local solution. We address challenges that have traditionally troubled nonlocal mechanics models primarily related to consistency guarantees and boundary conditions. For simple problems such as diffusion and linear elasticity we have developed complete error analysis theory providing consistency guarantees. We then take these foundational tools to develop new state-of-the-art capabilities for: lithiation-induced failure in batteries, ductile failure of problems driven by contact, blast-on-structure induced failure, brittle/ductile failure of thin structures. We also summarize ongoing efforts using these frameworks in data-driven modeling contexts. This report provides a high-level summary of all publications which followed from these efforts.
Nonlocal models use integral operators that embed length-scales in their definition. However, the integrands in these operators are difficult to define from the data that are typically available for a given physical system, such as laboratory mechanical property tests. In contrast, molecular dynamics (MD) does not require these integrands, but it suffers from computational limitations in the length and time scales it can address. To combine the strengths of both methods and to obtain a coarse-grained, homogenized continuum model that efficiently and accurately captures materials' behavior, we propose a learning framework to extract, from MD data, an optimal nonlocal model as a surrogate for MD displacements. Our framework guarantees that the resulting model is mathematically well-posed, physically consistent, and that it generalizes well to settings that are different from the ones used during training. The efficacy of this approach is demonstrated with several numerical tests for single layer graphene both in the case of perfect crystal and in the presence of thermal noise.
Nonlocal models, including peridynamics, often use integral operators that embed lengthscales in their definition. However, the integrands in these operators are difficult to define from the data that are typically available for a given physical system, such as laboratory mechanical property tests. In contrast, molecular dynamics (MD) does not require these integrands, but it suffers from computational limitations in the length and time scales it can address. To combine the strengths of both methods and to obtain a coarse-grained, homogenized continuum model that efficiently and accurately captures materials’ behavior, we propose a learning framework to extract, from MD data, an optimal Linear Peridynamic Solid (LPS) model as a surrogate for MD displacements. To maximize the accuracy of the learnt model we allow the peridynamic influence function to be partially negative, while preserving the well-posedness of the resulting model. To achieve this, we provide sufficient well-posedness conditions for discretized LPS models with sign-changing influence functions and develop a constrained optimization algorithm that minimizes the equation residual while enforcing such solvability conditions. This framework guarantees that the resulting model is mathematically well-posed, physically consistent, and that it generalizes well to settings that are different from the ones used during training. We illustrate the efficacy of the proposed approach with several numerical tests for single layer graphene. Our two-dimensional tests show the robustness of the proposed algorithm on validation data sets that include thermal noise, different domain shapes and external loadings, and discretizations substantially different from the ones used for training.
The peridynamic theory of solid mechanics is applied to the continuum modeling of the impact of small, high-velocity silica spheres on multilayer graphene targets. The model treats the laminate as a brittle elastic membrane. The material model includes separate failure criteria for the initial rupture of the membrane and for propagating cracks. Material variability is incorporated by assigning random variations in elastic properties within Voronoi cells. The computational model is shown to reproduce the primary aspects of the response observed in experiments, including the growth of a family of radial cracks from the point of impact.
Aeroengines ingest foreign object debris such as sand, which eventually erode components through repeated impacts. Due to the wide feature space, modeling and simulations are needed to rapidly assess the erosion behavior of materials such as composites. Peridynamic simulations were performed to analyze erosion of SiC/SiC composite due to sand impacts, which gives direct insight into the impact erosion mechanism and amounts. The erosion data was strongly correlated to impact velocity and angle, providing predictive equations.
Environmental Barrier Coatings (EBC) protect ceramic matrix composites from exposure to high temperature moisture present in turbine operation through their dense top coats. However, moisture is able to diffuse and oxidize the Si bond coat to form the Thermally Grown Oxide (TGO), a layer of SiO2 where the incorporation of O causes swelling and stress. At sufficient TGO-based swelling, the EBC will fail due to increased damage such as delamination. A multiscale simulation framework has been developed to link operating conditions of a high-performance turbine to the failure modes of the EBC. Computational fluid dynamics (CFD) simulations of the E3 turbine were performed and compared to prior literature data to demonstrate the fidelity of the Loci/CHEM software to determine the flow conditions on the turbine blade surface. Boundary condition data of pressure and heat flux were then determined with the CFD simulations, providing the temperature at the bond coat. Peridynamics was used to model the microscale TGO growth. A swelling model that links moisture concentration to strain at the TGO due to the volume increase from oxidation was demonstrated, coupling moisture transport to localized strain and directly observing TGO growth and the corresponding damage. This framework is generalized and can be adapted to a range of EBC microstructures and operating conditions.
We show that machine learning can improve the accuracy of simulations of stress waves in one-dimensional composite materials. We propose a data-driven technique to learn nonlocal constitutive laws for stress wave propagation models. The method is an optimization-based technique in which the nonlocal kernel function is approximated via Bernstein polynomials. The kernel, including both its functional form and parameters, is derived so that when used in a nonlocal solver, it generates solutions that closely match high-fidelity data. The optimal kernel therefore acts as a homogenized nonlocal continuum model that accurately reproduces wave motion in a smaller-scale, more detailed model that can include multiple materials. We apply this technique to wave propagation within a heterogeneous bar with a periodic microstructure. Several one-dimensional numerical tests illustrate the accuracy of our algorithm. The optimal kernel is demonstrated to reproduce high-fidelity data for a composite material in applications that are substantially different from the problems used as training data.
This article concerns modeling unsaturated deformable porous media as an equivalent single-phase and single-force state peridynamic material through the effective force state. The balance equations of linear momentum and mass of unsaturated porous media are presented by defining relevant peridynamic states. The energy balance of unsaturated porous media is utilized to derive the effective force state for the solid skeleton that is an energy conjugate to the nonlocal deformation state of the solid, and the suction force state. Through an energy equivalence, a multiphase constitutive correspondence principle is built between classical unsaturated poromechanics and peridynamic unsaturated poromechanics. The multiphase correspondence principle provides a means to incorporate advanced constitutive models in classical unsaturated porous theory directly into unsaturated peridynamic poromechanics. Numerical simulations of localized failure in unsaturated porous media under different matric suctions are presented to demonstrate the feasibility of modeling the mechanical behavior of such three-phase materials as an equivalent single-phase peridynamic material through the effective force state concept.
We show that machine learning can improve the accuracy of simulations of stress waves in one-dimensional composite materials. We propose a data-driven technique to learn nonlocal constitutive laws for stress wave propagation models. The method is an optimization-based technique in which the nonlocal kernel function is approximated via Bernstein polynomials. The kernel, including both its functional form and parameters, is derived so that when used in a nonlocal solver, it generates solutions that closely match high-fidelity data. The optimal kernel therefore acts as a homogenized nonlocal continuum model that accurately reproduces wave motion in a smaller-scale, more detailed model that can include multiple materials. We apply this technique to wave propagation within a heterogeneous bar with a periodic microstructure. Several one-dimensional numerical tests illustrate the accuracy of our algorithm. The optimal kernel is demonstrated to reproduce high-fidelity data for a composite material in applications that are substantially different from the problems used as training data.
A technique called the splice method for coupling local to peridynamic subregions of a body is described. The method relies on ghost nodes, whose values of displacement are interpolated from nearby physical nodes, to make each subregion visible to the other. In each time step, the nodes in each subregion treat the nodes in the other subregion as boundary conditions. Adaptively changing the subregions is possible through the creation and deletion of ghost nodes. Example problems in 2D and 3D illustrate how the method is used to perform multiscale modeling of fracture and impact events within a larger structure.
The propagation of a wave pulse due to low-speed impact on a one-dimensional, heterogeneous bar is studied. Due to the dispersive character of the medium, the pulse attenuates as it propagates. This attenuation is studied over propagation distances that are much longer than the size of the microstructure. A homogenized peridynamic material model can be calibrated to reproduce the attenuation and spreading of the wave. The calibration consists of matching the dispersion curve for the heterogeneous material in the limit of long and moderately long wavelengths. It is demonstrated that the peridynamic method reproduces the attenuation of wave pulses predicted by an exact microstructural model over large propagation distances.
The peridynamic theory of solid mechanics is applied to the continuum modeling of the impact of small, high-velocity silica spheres on multilayer graphene targets. The model treats the laminate as a brittle elastic membrane. The material model includes separate failure criteria for the initial rupture of the membrane and for propagating cracks. Material variability is incorporated by assigning random variations in elastic properties within Voronoi cells. The computational model is shown to reproduce the primary aspects of the response observed in experiments, including the growth of a family of radial cracks from the point of impact.
This report outlines the fiscal year (FY) 2019 status of an ongoing multi-year effort to develop a general, microstructurally-aware, continuum-level model for representing the dynamic response of material with complex microstructures. This work has focused on accurately representing the response of both conventionally wrought processed and additively manufactured (AM) 304L stainless steel (SS) as a test case. Additive manufacturing, or 3D printing, is an emerging technology capable of enabling shortened design and certification cycles for stockpile components through rapid prototyping. However, there is not an understanding of how the complex and unique microstructures of AM materials affect their mechanical response at high strain rates. To achieve our project goal, an upscaling technique was developed to bridge the gap between the microstructural and continuum scales to represent AM microstructures on a Finite Element (FE) mesh. This process involves the simulations of the additive process using the Sandia developed kinetic Monte Carlo (KMC) code SPPARKS. These SPPARKS microstructures are characterized using clustering algorithms from machine learning and used to populate the quadrature points of a FE mesh. Additionally, a spall kinetic model (SKM) was developed to more accurately represent the dynamic failure of AM materials. Validation experiments were performed using both pulsed power machines and projectile launchers. These experiments have provided equation of state (EOS) and flow strength measurements of both wrought and AM 304L SS to above Mbar pressures. In some experiments, multi-point interferometry was used to quantify the variation is observed material response of the AM 304L SS. Analysis of these experiments is ongoing, but preliminary comparisons of our upscaling technique and SKM to experimental data were performed as a validation exercise. Moving forward, this project will advance and further validate our computational framework, using advanced theory and additional high-fidelity experiments. ACKNOWLEDGEMENTS The authors greatly appreciate the support of Mike Saavedra in machining the experimental samples. The authors would also like to thank the Dynamic Integrated Compression facility (DICE) staff for executing the Thor experiments: Brian Stoltzfus, Randy Hickman, Keith Hodge, Joshua Usher, Lena Pacheco, and Eric Breden. The authors would also like to thank the staff at the Shock Thermodynamics Applied Research (STAR) facility for executing the plate impact experiments: Scott Alexander, Bill Reinhart, Bernardo Farfan, Rocky Palomino, John Martinez, and Rafael Sanchez. Lastly, the authors would like to acknowledge the development support of Jason Sanchez in ALEGRA to incorporate our upscaling method and Michael Powell for helping with post processing scripts for results analysis.
Under high-rate loading in tension, metals can sustain much larger tensile stresses for sub-microsecond time periods than would be possible under quasi-static conditions. This type of failure, known as spall, is not adequately reproduced by hydrocodes with commonly used failure models. The Spall Kinetics Model treats spall by incorporating a time scale into the process of failure. Under sufficiently strong tensile states of stress, damage accumulates over this time scale, which can be thought of as an incubation time. The time scale depends on the previous loading history of the material, reflecting possible damage by a shock wave. The model acts by modifying the hydrostatic pressure that is predicted by any equation of state and is therefore simple to implement. Examples illustrate the ability of the model to reproduce the spall stress and resulting release waves in plate impact experiments on stainless steel.
Most previous development of the peridynamic theory has assumed a Lagrangian formulation, in which the material model refers to an undeformed reference configuration. In the present work, an Eulerian form of material modeling is developed, in which bond forces depend only on the positions of material points in the deformed configuration. The formulation is consistent with the thermodynamic form of the peridynamic model and is derivable from a suitable expression for the free energy of a material. It is shown that the resulting formulation of peridynamic material models can be used to simulate strong shock waves and fluid response in which very large deformations make the Lagrangian form unsuitable. The Eulerian capability is demonstrated in numerical simulations of ejecta from a wavy free surface on a metal subjected to strong shock wave loading. The Eulerian and Lagrangian contributions to bond force can be combined in a single material model, allowing strength and fracture under tensile or shear loading to be modeled consistently with high compressive stresses. This capability is demonstrated in numerical simulation of bird strike against an aircraft, in which both tensile fracture and high pressure response are important.
Sintering is a component fabrication process in which powder is compacted by pressing or some other means and then held at elevated temperature for a period of hours. The powder grains bond with each other, leading to the formation of a solid component with much lower porosity, and therefore higher density and higher strength, than the original powder compact. In this project, we investigated a new way of computationally modeling sintering at the length scale of grains. The model uses a high-fidelity, three-dimensional representation with a few hundred nodes per grain. The numerical model solves the peridynamic equations, in which nonlocal forces allow representation of the attraction, adhesion, and mass diffusion between grains. The deformation of the grains is represented through a viscoelastic material model. The project successfully demonstrated the use of this method to reproduce experimentally observed features of material behavior in sintering, including densification, the evolution of microstructure, and the occurrence of random defects in the sintered solid.
Peridynamic correspondence material models provide a way to combine a material model from the local theory with the inherent capabilities of peridynamics to model long-range forces and fracture. However, correspondence models in a typical particle discretization suffer from zero-energy mode instability. These instabilities are shown here to be an aspect of material stability. A stability condition is derived for state-based materials starting from the requirement of potential energy minimization. It is shown that all correspondence materials fail this stability condition due to zero-energy deformation modes of the family. To eliminate these modes, a term is added to the correspondence strain energy density that resists deviations from a uniform deformation. The resulting material model satisfies the stability condition while effectively leaving the stress tensor unchanged. Computational examples demonstrate the effectiveness of the modified material model in avoiding zero-energy mode instability in a peridynamic particle code.
Quantitative measures are proposed for characterizing the complexity of material models used in computational mechanics. The algorithms for evaluating these metrics operate on the mathematical equations in the model rather than a code implemen- tation and are different from software complexity measures. The metrics do not rely on a physical understanding of the model, using instead only a formal statement of the equations. A new algorithm detects the dependencies, whether explicit or im- plicit, between all the variables. The resulting pattern of dependencies is expressed in a set of pathways, each of which represents a chain of dependence between the vari- ables. These pathways provide the raw data used in the metrics, which correlate with the expected ease of understanding, coding, and applying the model. Usage of the ComplexityMetrics code is described, with examples.
The peridynamic theory of solid mechanics provides a natural framework for modeling constitutive response and simulating dynamic crack propagation, pervasive damage, and fragmentation. In the case of a fragmenting body, the principal quantities of interest include the number of fragments, and the masses and velocities of the fragments. We present a method for identifying individual fragments in a peridynamic simulation. We restrict ourselves to the meshfree approach of Silling and Askari, in which nodal volumes are used to discretize the computational domain. Nodal volumes, which are connected by peridynamic bonds, may separate as a result of material damage and form groups that represent fragments. Nodes within each fragment have similar velocities and their collective motion resembles that of a rigid body. The identification of fragments is achieved through inspection of the peridynamic bonds, established at the onset of the simulation, and the evolving damage value associated with each bond. An iterative approach allows for the identification of isolated groups of nodal volumes by traversing the network of bonds present in a body. The process of identifying fragments may be carried out at specified times during the simulation, revealing the progression of damage and the creation of fragments. Incorporating the fragment identification algorithm directly within the simulation code avoids the need to write bond data to disk, which is often prohibitively expensive. Results are recorded using fragment identification numbers. The identification number for each fragment is stored at each node within the fragment and written to disk, allowing for any number of post-processing operations, for example the construction of cumulative distribution functions for quantities of interest. Care is taken with regard to very small clusters of isolated nodes, including individual nodes for which all bonds have failed. Small clusters of nodes may be treated as tiny fragments, or may be omitted from the fragment identification process. The fragment identification algorithm is demonstrated using the Sierra/SolidMechanics analysis code. It is applied to a simulation of pervasive damage resulting from a spherical projectile impacting a brittle disk, and to a simulation of fragmentation of an expanding ductile ring.
Peridynamics, a nonlocal extension of continuum mechanics, is unique in its ability to capture pervasive material failure. Its use in the majority of system-level analyses carried out at Sandia, however, is severely limited, due in large part to computational expense and the challenge posed by the imposition of nonlocal boundary conditions. Combined analyses in which peridynamics is em- ployed only in regions susceptible to material failure are therefore highly desirable, yet available coupling strategies have remained severely limited. This report is a summary of the Laboratory Directed Research and Development (LDRD) project "Strong Local-Nonlocal Coupling for Inte- grated Fracture Modeling," completed within the Computing and Information Sciences (CIS) In- vestment Area at Sandia National Laboratories. A number of challenges inherent to coupling local and nonlocal models are addressed. A primary result is the extension of peridynamics to facilitate a variable nonlocal length scale. This approach, termed the peridynamic partial stress, can greatly reduce the mathematical incompatibility between local and nonlocal equations through reduction of the peridynamic horizon in the vicinity of a model interface. A second result is the formulation of a blending-based coupling approach that may be applied either as the primary coupling strategy, or in combination with the peridynamic partial stress. This blending-based approach is distinct from general blending methods, such as the Arlequin approach, in that it is specific to the coupling of peridynamics and classical continuum mechanics. Facilitating the coupling of peridynamics and classical continuum mechanics has also required innovations aimed directly at peridynamic models. Specifically, the properties of peridynamic constitutive models near domain boundaries and shortcomings in available discretization strategies have been addressed. The results are a class of position-aware peridynamic constitutive laws for dramatically improved consistency at domain boundaries, and an enhancement to the meshfree discretization applied to peridynamic models that removes irregularities at the limit of the nonlocal length scale and dramatically improves conver- gence behavior. Finally, a novel approach for modeling ductile failure has been developed, moti- vated by the desire to apply coupled local-nonlocal models to a wide variety of materials, including ductile metals, which have received minimal attention in the peridynamic literature. Software im- plementation of the partial-stress coupling strategy, the position-aware peridynamic constitutive models, and the strategies for improving the convergence behavior of peridynamic models was completed within the Peridigm and Albany codes, developed at Sandia National Laboratories and made publicly available under the open-source 3-clause BSD license.