Publications

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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.

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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.

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.

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.

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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.

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