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.
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.
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.
The Uintah computational framework (UCF) has been adopted for simulation of shaped charge jet penetration and subsequent damage to geological formations. The Kayenta geomechanics model, as well as a simplified model for shakedown simulations has been incorporated within the UCF and is undergoing extensive development to enhance it to account for fluid in pore space.
A generic penetration simulation using Uintah
The host code (Uintah) itself has been enhanced to accommodate material variability and scale effects. Simulations have been performed that import flash X-ray data for the velocity and geometry of a particulated metallic jet so that uncertainty about the jet can be reduced to develop predictive models for target response. Uintah’s analytical polar decomposition has been replaced with an iterative algorithm to dramatically improve accuracy under large deformations. Continue reading
A.F. Fossum and R.M. Brannon (2006)
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.
MMS stands for “Method of Manufactured Solutions,” which is a rather sleazy sounding name for what is actually a respected and rigorous method of verifying that a finite element (or other) code is correctly solving the governing equations.
A simple introduction to MMS may be found on page 11 of The ASME guide for verification and validation in solid mechanics. The basic idea is to analytically determine forcing functions that would lead to a specific, presumably nontrivial, solution (of your choice) for the dependent variable of a differential equation. Then you would verify a numerical solver for that differential equation by running it using your analytically determined forcing function. The difference between the code’s prediction and your selected manufactured solution provides a quantitative measure of error.
The publication, “Caveats concerning conjugate stress and strain measures (click to download)” contains an analytical solution for the stress in a fiber reinforced composite in the limit as the matrix material goes to zero stiffness. Because the solution is exact for arbitrarily large deformations, it is a great test case for verification of anisotropic elasticity codes, and it nicely illustrates several subtle concepts in large-deformation continuum mechanics.
Also see related viewgraphs entitled “The distinction between large distortion and large deformation.”