Publications

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.

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We have performed a series of ten planar impact experiments on six different materials, including certain reactive powders and two inert materials, using Sandia's 89 - mm powder gun at the STAR facility. Time - resolved particle - velocity histories were determined during each of the experiments from one or more VISAR measurements. We have analyzed the results of these measurements 1) by using jump conditions to determine shock and first reshock states and 2) by comparing measured particle velocity histories to synthetic histories predicted by one - dimensional computational analyses using the CTH shock physics code with various models for inert and reactive materials . These comparisons are consistent with the conclusion for these particular reactive powders, that for the duration of shock loading either 1) there is insignificant reaction or 2) the products of any reaction are indistinguishable from the reactants under the experimental conditions. Shock and reshock states were extracted for shock pressures between 5 and 40 GPa. Densities were at or greater than the theoretical maximum zero - pressure density of the starting mixture. This result would be expected if there were no reaction or negligible reaction for the first two shock states. Two experiments were performed on one reactive powder in a "ring - down " geometry to look for evidence of vapor production on pressure release. In both cases, the measured velocity continued to increase slowly over a period of microseconds for the du ration of the experiment. This observation suggests that vapor is produced along the release path, but information about the mechanism for vapor production cannot be extracted from these data. While it is possible that vapor is produced by a shock - induced reaction involving more than one of the original constituents, a simpler interpretation is that the vapor is made up of products of shock - induced decomposition reactions and/or simple vaporization of the constituents as would be expected to take place under the conditions of these experiments. Other sources of vapor could be water adsorbed on grain surfaces and air originally in the voids. Thus it is not necessary to invoke significant recombination reactions to explain the data. However, in the absence of ring - down control experiments, the possibility remains open. These conclusions are different from those of previous workers, but reassessment of a subset of the earlier data yields results consistent with the present work, i.e., the shock compression data do not provide evidence for strong exothermic reactions.

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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.

The propagation of large amplitude nonlinear waves in a peridynamic solid is ana- lyzed. With an elastic material model that hardens in compression, sufficiently large wave pulses propagate as solitary waves whose velocity can far exceed the linear wave speed. In spite of their large velocity and amplitude, these waves leave the material they pass through with no net change in velocity and stress. They are nondissipative and nondispersive, and they travel unchanged over large distances. An approximate solution for solitary waves is derived that reproduces the main features of these waves observed in computational simulations. We demonstrate, by numerical studies, that waves interact only weakly with each other when they collide. Finally, we found that wavetrains composed of many non-interacting solitary waves form and propagate under certain boundary and initial conditions.

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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 implementation 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 implicit, 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 variables. 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.

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In this study, a mechanical model is introduced for predicting the initiation and evolution of complex fracture patterns without the need for damage variable or law. The model, a continuum variant of Newton's second law, uses integral rather than partial differential operators where the region of integration is over finite domain. The force interaction is derived from a novel nonconvex strain energy density function, resulting in a nonmonotomic material model. The resulting equation of motion is proved to be mathematically well-posed. The model has the capacity to simulate nucleation and growth of multiple, mutually interacting dynamic fractures. In the limit of zero region of integration, the model reproduces the classic Griffith model of brittle fracture. The simplicity of the formulation avoids the need for supplemental kinetic relations that dictate crack growth or the need for an explicit damage evolution law.

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As discussed in the previous chapter, the purpose of peridynamics is to unify the mechanics of continuous media, continuous media with evolving discontinuities, and discrete particles. To accomplish this, peridynamics avoids the use of partial derivatives of the deformation with respect to spatial coordinates. Instead, it uses integral equations that remain valid on discontinuities. Discrete particles, as will be discussed later in this chapter, are treated using Dirac delta functions.

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Therefore, to design a building or a bridge that stands up and is safe, one might assume that engineers must need to know a lot about these tensor fields and stress potentials, in all their mathematical glory. If not, then surely they must depend on specialists in continuum mechanics for guidance. Right?

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.