Author Archives: Rebecca Brannon

Publication: Validating Theories for Brittle Damage

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R.M. Brannon, J.M. Wells, and O.E. Strack

Realistic-looking, uneven damage zones in Brazilian simulations compare favorably with laboratory data for observable damage

Validating simulated predictions of internal damage within armor ceramics is preferable to simply assessing a models ability to predict penetration depth, especially if one hopes to perform subsequent ‘‘second strike’’ analyses. We present the results of a study in which crack networks are seeded by using a statistically perturbed strength, the median of which is inherited from a deterministic ‘‘smeared damage’’ model, with adjustments to reflect experimentally established size effects. This minor alteration of an otherwise conventional damage model noticeably mitigates mesh dependencies and, at virtually no computational cost, produces far more realistic cracking patterns that are well suited for validation against X-ray computed tomography (XCT) images of internal damage patterns. For Brazilian, spall, and indentation tests, simulations share qualitative features with externally visible damage. However, the need for more stringent quantitative validation, software quality testing, and subsurface XCT validation, is emphasized.

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Publication: On a viscoplastic model for rocks with mechanism-dependent characteristic times

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A. F. Fossum and R. M. Brannon

Rate Dependance

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.

Available Online:

http://www.mech.utah.edu/~brannon/pubs/7-2006FossumBrannonMechanismDependentViscoplasticity.pdf

 

Request for Letter of Reference from Dr. Brannon

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Seeking a letter of reference from Dr. Brannon? Then please request one using RequestForLetterOfReferenceFromProfBrannon, which lets you know your own role (supplying information and having a chance to say what YOU think is worthy of recommendation) and to release liability.

For a nice philosophical discussion of the fallibility of reference letters, see http://chronicle.com/article/Why-You-Can-t-Trust-Letters/2132.

Publication: KAYENTA: Theory and User’s Guide

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R.M. Brannon, A.F. Fossum, and O.E. Strack

Kayenta continuous yield surface. (a) three-dimensional view in principal stress space, (b) the meridional “side” view (thick line), and (c) the octahedral view

The physical foundations and domain of applicability of the Kayenta constitutive model are presented along with descriptions of the source code and user instructions. Kayenta, which is an outgrowth of the Sandia GeoModel, includes features and fitting functions appropriate to a broad class of materials including rocks, rock-like engineered materials (such as concretes and ceramics),and metals. Fundamentally, Kayenta is a computational framework for generalized plasticity models. 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. Kayenta supports optional anisotropic elasticity associated with ubiquitous joint sets. Kayenta support optional deformation-induced anisotropy through kinematic hardening (inwhich the initially isotropic yield surface is permitted to translate in deviatoric stress space to model Bauschinger effects). The governing equations are otherwise isotropic. Because Kayenta is a unification and generalization of simple models, it can be run using as few as 2 parameters (for linear elasticity) to as many as 40 material and control parameters in the exceptionally rare case when all features are used. For high-strain-rate applications, Kayenta support rate dependence through an overstress model. Isotropic damage is model through loss of stiffness and strength.

Available Online:
http://www.mech.utah.edu/~brannon/pubs/7-2009Kayenta_Users_Guide.pdf
http://dx.doi.org/10.1111/j.1744-7402.2010.02487.x

Publication: Survey of Four Damage Models for Concrete

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R.M. Brannon and S. Leelavanichkul

RHT Model: Contour plots of damage: side, front, and back view of the target (top to bottom).

Four conventional damage plasticity models for concrete, the Karagozian and Case model (K&C),the Riedel-Hiermaier-Thoma model (RHT), the Brannon-Fossum model (BF1), and the Continuous Surface Cap Model (CSCM) are compared. The K&C and RHT models have been used in commercial finite element programs many years, whereas the BF1 and CSCM models are relatively new. All four models are essentially isotropic plasticity models for which plasticity is regarded as any form of inelasticity. All of the models support nonlinear elasticity, but with different formulations.All four models employ three shear strength surfaces. The yield surface bounds an evolving set of elastically obtainable stress states. The limit surface bounds stress states that can be reached by any means (elastic or plastic). To model softening, it is recognized that some stress states might be reached once, but, because of irreversible damage, might not be achievable again. In other words, softening is the process of collapse of the limit surface, ultimately down to a final residual surface for fully failed material. The four models being compared differ in their softening evolution equations, as well as in their equations used to degrade the elastic stiffness. For all four models, the strength surfaces are cast in stress space. For all four models, it is recognized that scale effects are important for softening, but the models differ significantly in their approaches. The K&C documentation, for example, mentions that a particular material parameter affecting the damage evolution rate must be set by the user according to the mesh size to preserve energy to failure. Similarly, the BF1 model presumes that all material parameters are set to values appropriate to the scale of the element, and automated assignment of scale-appropriate values is available only through an enhanced implementation of BF1 (called BFS) that regards scale effects to be coupled to statistical variability of material properties. The RHT model appears to similarly support optional uncertainty and automated settings for scale-dependent material parameters. The K&C, RHT, and CSCM models support rate dependence by allowing the strength to be a function of strain rate, whereas the BF1 model uses Duvaut-Lion viscoplasticity theory to give a smoother prediction of transient effects. During softening, all four models require a certain amount of strain to develop before allowing significant damage accumulation. For the K&C, RHT, and CSCM models, the strain-to-failure is tied to fracture energy release, whereas a similar effect is achieved indirectly in the BF1 model by a time-based criterion that is tied to crack propagation speed.

Available Online:

http://www.mech.utah.edu/~brannon/pubs/7-2009BrannonLeelavanichkulSurveyConcrete.pdf

Publication: A multi-stage return algorithm for solving the classical damage component of constitutive models for rocks, ceramics, and other rock-like media

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R. M. Brannon and S. Leelavanichkul

Octahedral isosurfaces for a) the unacceptable, b) the admissible, and c) the admissible

Classical plasticity and damage models for porous quasi-brittle media usually suffer from mathematical defects such as non-convergence and nonuniqueness.Yield or damage functions for porous quasi-brittle media often have yield functions with contours so distorted that following those contours to the yield surface in a return algorithm can take the solution to a false elastic domain. A steepest-descent return algorithm must include iterative corrections; otherwise,the solution is non-unique because contours of any yield function are non-unique. A multi-stage algorithm has been developed to address both spurious convergence and non-uniqueness, as well as to improve efficiency. The region of pathological isosurfaces is masked by first returning the stress state to the Drucker–Prager surface circumscribing the actual yield surface. From there, steepest-descent is used to locate a point on the yield surface. This first-stage solution,which is extremely efficient because it is applied in a 2D subspace, is generally not the correct solution,but it is used to estimate the correct return direction.The first-stage solution is projected onto the estimated correct return direction in 6D stress space. Third invariant dependence and anisotropy are accommodated in this second-stage correction. The projection operation introduces errors associated with yield surface curvature,so the two-stage iteration is applied repeatedly to converge. Regions of extremely high curvature are detected and handled separately using an approximation to vertex theory. The multi-stage return is applied holding internal variables constant to produce a non-hardening solution. To account for hardening from pore collapse (or softening from damage), geometrical arguments are used to clearly illustrate the appropriate scaling of the non-hardening solution needed to obtain the hardening (or softening) solution.

For errata (transcription errors in two of the verification solutions), please see:

Errata for two verification publications

Available Online:
http://www.mech.utah.edu/~brannon/pubs/7-2009BrannonLeelavanichkul-IJF.pdf
http://dx.doi.org/10.1007/s10704-009-9398-4

Publication: Decomposition and Visualization of Fourth-Order Elastic-Plastic Tensors

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A.G. Neeman; R.M. Brannon; B. Jeremic; A. Van Gelderand;  A. Pang

Top view (Z from above) of eigentensors for Drucker-Prager material, time step 124, colored by minimum stretch eigenvalue.

Visualization of fourth-order tensors from solid mechanics has not been explored in depth previously. Challenges include the large number of components (3x3x3x3 for 3D), loss of major symmetry and loss of positive definiteness(with possibly zero or negative eigenvalues). This paper presents a decomposition of fourth-order tensors that facilitates their visualization and understanding. Fourth-order tensors are used to represent a solid’s stiffness.The stiffness tensor represents the relationship between increments of stress and increments of strain. Visualizing stiffness is important to understand the changing state of solids during plastification and failure. In this work,we present a method to reduce the number of stiffness components to second-order 3×3 tensors for visualization.The reduction is based on polar decomposition, followed by eigen-decomposition on the polar “stretch”. If any resulting eigenvalue is significantly lower than the others, the material has softened in that eigen-direction. The associated second-order eigentensor represents the mode of stress (such as compression, tension, shear, or some combination of these) to which the material becomes vulnerable. Thus we can visualize the physical meaning of plastification with techniques for visualizing second-order symmetric tensors.

Available Online:

Publication: http://www.mech.utah.edu/~brannon/pubs/7-2008NeemanBrannonJeremicVanGelderPang.pdf

Poster: http://www.mech.utah.edu/~brannon/pubs/7-09NeemanBrannonEtAlNEESposter.pdf

Publication: Advances in X-ray Computed Tomography Diagnostics of Ballistic Impact Damage

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J.M. Wells and R.M. Brannon

Dynamic indentation of SiC-N ceramic by a tungsten carbide sphere. Left: experimentally observed impact crater and radial cracking (both highlighted for clarity). Middle: BFS model prediction of externally visible damage. Right: prediction of internal damage (suitable for validation against XCT data).

With the relatively recent introduction of quantitative and volumetric X-ray computedtomography (XCT) applied to ballistic impact damage diagnostics, significant inroads have beenmade in expanding our knowledge base of the morphological variants of physical impactdamage. Yet, the current state of the art in computational and simulation modeling of terminalballistic performance remains predominantly focused on the penetration phenomenon, withoutdetailed consideration of the physical characteristics of actual impact damage. Similarly, armorceramic material improvements appear more focused on penetration resistance than on improved intrinsic damage tolerance and damage resistance. Basically, these approaches minimizeour understanding of the potential influence that impact damage may play in the mitigation orprevention of ballistic penetration. Examples of current capabilities of XCT characterization,quantification, and visualization of complex impact damage variants are demonstrated anddiscussed for impacted ceramic and metallic terminal ballistic target materials. Potential benefitsof incorporating such impact damage diagnostics in future ballistic computational modeling arealso briefly discussed.

Available Online:

http://dx.doi.org/10.1007/s11661-007-9304-5
http://www.mech.utah.edu/~brannon/pubs/7-2007WellsBrannonAdvancesInXrayComputedTomographyDiagnosticsOfBallisticDamage.pdf

CPDI shape functions for the Material Point Method

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In a conventional MPM formulation, the shape functions on the grid are the same as in a traditional FEM solution. In the CPDI, the shape functions on the grid are replaced by alternative (and still linearly complete*) shape functions, given by piecewise linear interpolations of the traditional FEM shape functions to the boundaries of the particles.  This change provides FEM-level accuracy in moderately deforming regions while retaining the attractive feature of MPM that particles can move arbitrarily relative to one another in massively deforming regions (provided, of course, that the deformation is updated in a manner compatible with the constitutive model).

In the images below, the shaded regions are the traditional FEM “tent” linear shape functions in 1-D, and the solid lines are the CPDI interpolated shape functions, which clearly change based on particle position relative to the grid.  Both the traditional FEM tent functions and these new CPDI functions are linearly complete (i.e., they can exactly fit any affine function). The tremendous advantage of CPDI is that the basis functions are extraordinarily simple over a particle domain, thus facilitating exact and efficient evaluation of integrals over particle domains.

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