Publications

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This paper summarizes the results of a theoretical and experimental program at Sandia National Laboratories aimed at identifying and modeling key physical features of rocks and rock-like materials at the laboratory scale over a broad range of strain rates. The mathematical development of a constitutive model is discussed and model predictions versus experimental data are given for a suite of laboratory tests. Concurrent pore collapse and cracking at the microscale are seen as competitive micromechanisms that give rise to the well-known macroscale phenomenon of a transition from volumetric compaction to dilatation under quasistatic triaxial compression. For high-rate loading, this competition between pore collapse and microcracking also seems to account for recently identified differences in strain-rate sensitivity between uniaxial-strain 'plate slap' data compared to uniaxial-stress Kolsky bar data. A description is given of how this work supports ongoing efforts to develop a predictive capability in simulating deformation and failure of natural geological materials, including those that contain structural features such as joints and other spatial heterogeneities.

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In this article we describe stress nets, a technique for exploring 2D tensor fields. Our method allows a user to examine simultaneously the tensors eigenvectors (both major and minor) as well as scalar-valued tensor invariants. By avoiding noise-advection techniques, we are able to display both principal directions of the tensor field as well as the derived scalars without cluttering the display. We present a CPU-only implementation of stress nets as well as a hybrid CPU/GPU approach and discuss the relative strengths and weaknesses of each. Stress nets have been used as part of an investigation into crack propagation. They were used to display the directions of maximum shear in a slab of material under tension as well as the magnitude of the shear forces acting on each point. Our methods allowed users to find new features in the data that were not visible on standard plots of tensor invariants. These features disagree with commonly accepted analytical crack propagation solutions and have sparked renewed investigation. Though developed for a materials mechanics problem, our method applies equally well to any 2D tensor field having unique characteristic directions.

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To establish mechanical properties and failure criteria of silicon carbide (SiC-N) ceramics, a series of quasi-static compression tests has been completed using a high-pressure vessel and a unique sample alignment jig. This report summarizes the test methods, set-up, relevant observations, and results from the constitutive experimental efforts. Results from the uniaxial and triaxial compression tests established the failure threshold for the SiC-N ceramics in terms of stress invariants (I{sub 1} and J{sub 2}) over the range 1246 < I{sub 1} < 2405. In this range, results are fitted to the following limit function (Fossum and Brannon, 2004) {radical}J{sub 2}(MPa) = a{sub 1} - a{sub 3}e -a{sub 2}(I{sub 1}/3) + a{sub 4} I{sub 1}/3, where a{sub 1} = 10181 MPa, a{sub 2} = 4.2 x 10{sup -4}, a{sub 3} = 11372 MPa, and a{sub 4} = 1.046. Combining these quasistatic triaxial compression strength measurements with existing data at higher pressures naturally results in different values for the least-squares fit to this function, appropriate over a broader pressure range. These triaxial compression tests are significant because they constitute the first successful measurements of SiC-N compressive strength under quasistatic conditions. Having an unconfined compressive strength of {approx}3800 MPa, SiC-N has been heretofore tested only under dynamic conditions to achieve a sufficiently large load to induce failure. Obtaining reliable quasi-static strength measurements has required design of a special alignment jig and load-spreader assembly, as well as redundant gages to ensure alignment. When considered in combination with existing dynamic strength measurements, these data significantly advance the characterization of pressure-dependence of strength, which is important for penetration simulations where failed regions are often at lower pressures than intact regions.

An important challenge encountered during post-processing of finite element analyses is the visualizing of three-dimensional fields of real-valued second-order tensors. Namely, as finite element meshes become more complex and detailed, evaluation and presentation of the principal stresses becomes correspondingly problematic. In this paper, we describe techniques used to visualize simulations of perturbed in-situ stress fields associated with hypothetical salt bodies in the Gulf of Mexico. We present an adaptation of the Mohr diagram, a graphical paper and pencil method used by the material mechanics community for estimating coordinate transformations for stress tensors, as a new tensor glyph for dynamically exploring tensor variables within three-dimensional finite element models. This interactive glyph can be used as either a probe or a filter through brushing and linking.

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The mathematical and physical foundations and domain of applicability of Sandia's GeoModel are presented along with descriptions of the source code and user instructions. The model is designed to be used in conventional finite element architectures, and (to date) it has been installed in five host codes without requiring customizing the model subroutines for any of these different installations. Although developed for application to geological materials, the GeoModel actually applies to a much broader class of materials, including rock-like engineered materials (such as concretes and ceramics) and even to metals when simplified parameters are used. Nonlinear elasticity is supported through an empirically fitted function that has been found to be well-suited to a wide variety of materials. Fundamentally, the GeoModel is a generalized plasticity model. As such, it includes a yield surface, but the term 'yield' is generalized to include any form of inelastic material response including microcrack growth and pore collapse. The geomodel supports deformation-induced anisotropy in a limited capacity through kinematic hardening (in which the initially isotropic yield surface is permitted to translate in deviatoric stress space to model Bauschinger effects). Aside from kinematic hardening, however, the governing equations are otherwise isotropic. The GeoModel is a genuine unification and generalization of simpler models. The GeoModel can employ up to 40 material input and control parameters in the rare case when all features are used. Simpler idealizations (such as linear elasticity, or Von Mises yield, or Mohr-Coulomb failure) can be replicated by simply using fewer parameters. For high-strain-rate applications, the GeoModel supports rate dependence through an overstress model.

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The theory, numerical algorithm, and user documentation are provided for a new ''Centroidal Voronoi Tessellation (CVT)'' method of filling a region of space (2D or 3D) with particles at any desired particle density. ''Clumping'' is entirely avoided and the boundary is optimally resolved. This particle placement capability is needed for any so-called ''mesh-free'' method in which physical fields are discretized via arbitrary-connectivity discrete points. CVT exploits efficient statistical methods to avoid expensive generation of Voronoi diagrams. Nevertheless, if a CVT particle's Voronoi cell were to be explicitly computed, then it would have a centroid that coincides with the particle itself and a minimized rotational moment. The CVT code provides each particle's volume and centroid, and also the rotational moment matrix needed to approximate a particle by an ellipsoid (instead of a simple sphere). DIATOM region specification is supported.

The theory and algorithm for the Material Point Method (MPM) are documented, with a detailed discussion on the treatments of boundary conditions and shock wave problems. A step-by-step solution scheme is written based on direct inspection of the two-dimensional MPM code currently used at the University of Missouri-Columbia (which is, in turn, a legacy of the University of New Mexico code). To test the completeness of the solution scheme and to demonstrate certain features of the MPM, a one-dimensional MPM code is programmed to solve one-dimensional wave and impact problems, with both linear elasticity and elastoplasticity models. The advantages and disadvantages of the MPM are investigated as compared with competing mesh-free methods. Based on the current work, future research directions are discussed to better simulate complex physical problems such as impact/contact, localization, crack propagation, penetration, perforation, fragmentation, and interactions among different material phases. In particular, the potential use of a boundary layer to enforce the traction boundary conditions is discussed within the framework of the MPM.

A theory is developed for the response of moderately porous solids (no more than {approximately}20% void space) to high-strain-rate deformations. The model is consistent because each feature is incorporated in a manner that is mathematically compatible with the other features. Unlike simple p-{alpha} models, the onset of pore collapse depends on the amount of shear present. The user-specifiable yield function depends on pressure, effective shear stress, and porosity. The elastic part of the strain rate is linearly related to the stress rate, with nonlinear corrections from changes in the elastic moduli due to pore collapse. Plastically incompressible flow of the matrix material allows pore collapse and an associated macroscopic plastic volume change. The plastic strain rate due to pore collapse/growth is taken normal to the yield surface. If phase transformation and/or pore nucleation are simultaneously occurring, the inelastic strain rate will be non-normal to the yield surface. To permit hardening, the yield stress of matrix material is treated as an internal state variable. Changes in porosity and matrix yield stress naturally cause the yield surface to evolve. The stress, porosity, and all other state variables vary in a consistent manner so that the stress remains on the yield surface throughout any quasistatic interval of plastic deformation. Dynamic loading allows the stress to exceed the yield surface via an overstress ordinary differential equation that is solved in closed form for better numerical accuracy. The part of the stress rate that causes no plastic work (i.e-, the part that has a zero inner product with the stress deviator and the identity tensor) is given by the projection of the elastic stressrate orthogonal to the span of the stress deviator and the identity tensor.The model, which has been numerically implemented in MIG format, has been exercised under a wide array of extremal loading and unloading paths. As will be discussed in a companion sequel report, the CKP model is capable of closely matching plate impact measurements for porous materials.

Several concepts (and assumptions) from the literature for porous metals and ceramics have been synthesized into a consistent model that predicts an admissibility limit on a material's porous yield surface. To ensure positive plastic work, the rate at which a yield surface can collapse as pores grow in tension must be constrained.

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A mathematical framework is developed for the study of materials containing axisymmetric inclusions or flaws such as ellipsoidal voids, penny-shaped cracks, or fibers of circular cross-section. The general case of nonuniform statistical distributions of such heterogeneities is attacked by first considering a spatially uniform distribution of flaws that are all oriented in the same direction. Assuming an isotropic substrate, the macroscopic material properties of this simpler microstructure naturally should be transversely isotropic. An orthogonal basis for the linear subspace consisting of all double-symmetric transversely-isotropic fourth-order tensors associated with a given material vector is applied to deduce the explicit functional dependence of the material properties of these aligned materials on the shared symmetry axis. The aligned and uniform microstructure seems geometrically simple enough that the macroscopic transversely isotropic properties could be derived in closed form. Since the resulting properties are transversely isotropic, the analyst must therefore be able to identify the appropriate coefficients of the transverse basis. Once these functions are identified, a principle of superposition of strain rates ay be applied to define an expectation integral for the composite properties of a material containing arbitrary anisotropic distributions of axisymmetric inhomogeneities. A proposal for coupling plastic anisotropy to the elastic anisotropy is presented in which the composite yield surface is interpreted as a distortion of the isotropic substrate yield surface; the distortion directions are coupled to the elastic anisotropy directions. Finally, some commonly assumed properties (such as major symmetry) of the Cauchy tangent stiffness tensor are shown to be inappropriate for large distortions of anisotropic materials.

A set of model interface guidelines, called MIG, is presented as a means by which any compliant numerical material model can be rapidly installed into any parent code without having to modify the model subroutines. Here, {open_quotes}model{close_quotes} usually means a material model such as one that computes stress as a function of strain, though the term may be extended to any numerical operation. {open_quotes}Parent code{close_quotes} means a hydrocode, finite element code, etc. which uses the model and enforces, say, the fundamental laws of motion and thermodynamics. MIG requires the model developer (who creates the model package) to specify model needs in a standardized but flexible way. MIG includes a dictionary of technical terms that allows developers and parent code architects to share a common vocabulary when specifying field variables. For portability, database management is the responsibility of the parent code. Input/output occurs via structured calling arguments. As much model information as possible (such as the lists of required inputs, as well as lists of precharacterized material data and special needs) is supplied by the model developer in an ASCII text file. Every MIG-compliant model also has three required subroutines to check data, to request extra field variables, and to perform model physics. To date, the MIG scheme has proven flexible in beta installations of a simple yield model, plus a more complicated viscodamage yield model, three electromechanical models, and a complicated anisotropic microcrack constitutive model. The MIG yield model has been successfully installed using identical subroutines in three vectorized parent codes and one parallel C++ code, all predicting comparable results. By maintaining one model for many codes, MIG facilitates code-to-code comparisons and reduces duplication of effort, thereby reducing the cost of installing and sharing models in diverse new codes.

A unifying framework is developed for the analysis of brittle materials. Heretofore diverse classes of models result from different choices for unspecified coefficient and distribution functions in the unified theory. Material response is described in terms of expectation integrals of transverse symmetry tensors. First, a canonical body containing cracks of all the same orientation is argued to possess macroscopic transverse isotropy. An orthogonal basis for the linear subspace consisting of all double-symmetric transversely-isotropic fourth-order tensors associated with a given material vector is introduced and applied to deduce the explicit functional dependence of the compliance of such contrived materials on the shared crack orientation. A principle of superposition of strain rates is used to write the compliance for a more realistic material consisting of cracks of random size and orientation as an expectation integral of the transverse compliance for each orientation times the joint distribution function for the size and orientation. Utilizing an evolving (initially exponential) size- dependence in the joint distribution, the general theory gives unprecedented agreement with measurements of the dynamic response of alumina to impact loading, especially upon release where the calculations predict the development of considerable deformation- induced anisotropy, challenging the conventional notion of shocks as isotropic phenomena.