Swan, S. and R. Brannon (2009)
Illustration of stair-stepping typical of finite sampling from a Weibull distribution
Current simulations of material deformation are a balance between computational effort and accuracy of the simulation. To increase the accuracy of the simulated material response, the simulation becomes more computationally intensive with finer meshes and shorter timesteps, increasing the time and resource requirements needed to perform the simulation. One method for improving predictions of brittle failure while minimizing computational overhead is to implement statistical variability for the material properties being simulated. This method has low computational overhead and requires a relatively small increase in resource requirements while significantly increasing the precision of simulation results. Currently, most simulation frameworks inaccurately describe brittle and heterogeneous materials as uniform bodies of equal strength and consistency. This over-simplification underscores the need to implement statistical variability to help better predict material response and failure modes for materials that contain intermittent abnormalities such as changes in hardness, strength, and grain size throughout the specimen. Uintah, the computational framework developed by the University of Utah’s C-SAFE program, has a simplistic native Gaussian distribution function that was hard-coded into select material models. The goal of this research is to create an easily duplicable method for enabling dynamic global variability according to a Weibull distribution in constitutive models in Uintah and to implement said ability into the constitutive model Kayenta. The main application of Kayenta is to simulate geological response to penetration and perforation. For the purpose of simulating failure in brittle geological samples, the Weibull distribution produces realistic statistical scatter in constituent properties that correlates well to flaws and irregularities observed in laboratory tests.
K. Kamojjala, R. M. Brannon (2011)
Snapshot of the deformation in time
The principle of material frame indifference require spatial stresses to rotate with the material, whereas reference stresses must be insensitive to rotation. Testing of a classical uniaxial strain problem with superimposed rotation reveals that a very common approach to strong incremental objectivity taken in finite element codes to satisfy frame indifference(namely working in an approximate un-rotated frame) fails this simplistic test. A more complicated verification example is constructed based on the method of manufactured solutions (MMS) which involves the same character of loading at all points, providing a means to test any nonlinear-elastic arbitrarily anisotropic constitutive model.
T.J. Fuller, R.M. Brannon, O.E. Strack, J.E. Bishop
Displacement profile for Thermo-Kayenta at the end of the simulation. the red dots represent the experimental profiles
A persistent challenge in simulating damage of natural geological materials, as well as rock-like engineered materials, is the development of efficient and accurate constitutive models.The common feature for these brittle and quasi-brittle materials are the presence of flaws such as porosity and network of microcracks. The desired models need to be able to predict the material responses over a wide range of porosities and strain rate. Kayenta  (formerly called the Sandia GeoModel) is a unifi ed general-purpose constitutive model that strikes a balance between rst-principles micromechanics and phenomenological or semi-empirical modeling strategies. However, despite its sophistication and ability to reduce to several classical plasticity theories, Kayenta is incapable of modeling deformation of ductile materials in which deformation is dominated by dislocation generation and movement which can lead to signi cant heating. This stems from Kayenta’s roots as a geological model, where heating due to inelastic deformation is often neglected or presumed to be incorporated implicitly through the elastic moduli.The sophistication of Kayenta and its large set of extensive features, however, make Kayenta an attractive candidate model to which thermal eff ects can be added. This report outlines the initial work in doing just that, extending the capabilities of Kayenta to include deformation of ductile materials, for which thermal e ffects cannot be neglected. Thermal e ffects are included based on an assumption of adiabatic loading by computing the bulk and thermal responses of the material with the Kerley Mie-Gruneisen equation of state and adjusting the yield surface according to the updated thermal state. This new version of Kayenta, referred to as Thermo-Kayenta throughout this report, is capable of reducing to classical Johnson-Cook plasticity in special case single element simulations and has been used to obtain reasonable results in more complicated Taylor impact simulations in LS-Dyna. Despite these successes, however, Thermo-Kayenta requires additional re nement for it to be consistent in the thermodynamic sense and for it to be considered superior to other, more mature thermoplastic models. The initial thermal development, results, and required refinements are all detailed in the following report.
J. Burghardt, B. Leavy, J. Guilkey, Z. Xue, R. Brannon
The capability of the generalized interpolation material point (GIMP) method in simulation of penetration events is investigated. A series of experiments was performed wherein a shaped charge jet penetrates into a stack of aluminum plates. Electronic switches were used to measure the penetration time history. Flash x-ray techniques were used to measure the density,length, radius and velocity of the shaped charge jet. Simulations of the penetration event were performed using the Uintah MPM/GIMP code with several different models of the shaped charge jet being used. The predicted penetration time history for each jet model is compared with the experimentally observed penetration history. It was found that the characteristics of the predicted penetration were dependent on the way that the jet data are translated to a discrete description. The discrete jet descriptions were modified such that the predicted penetration histories fell very close to the range of the experimental data. In comparing the various discrete jet descriptions it was found that the cumulative kinetic energy flux curve represents an important way of characterizing the penetration characteristics of the jet. The GIMP method was found to be well suited for simulation of high rate penetration events.
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.
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.
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:
R.B. Leavy; R.M. Brannon; O.E. Strack
Weibull modulus effect on radial cracking in boron carbide simulations impacted at 400 m/s.
Sphere impact experiments are used to calibrate and validate ceramic models that include statistical variability and/or scale effects in strength and toughness parameters. These dynamic experiments supplement traditional characterization experiments such as tension, triaxial compression, Brazilian, and plate impact, which are commonly used for ceramic model calibration.The fractured ceramic specimens are analyzed using sectioning, X-ray computed tomography, microscopy, and other techniques. These experimental observations indicate that a predictive material model must incorporate a standard deviation in strength that varies with the nature of the loading. Methods of using the spherical indentation data to calibrate a statistical damage model are presented in which it is assumed that variability in strength is tied to microscale stress concentrations associated with microscale heterogeneity.
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.