Publication: Experimental Assessment of Unvalidated Assumptions in Classical Plasticity Theory

R. Brannon, J.A. Burghardt, D. Bronowski, and S. Bauer

Common isotropic yield surfaces. Von Mises and Drucker-Prager models are often used for metals. Gurson’s function, and others like it, are used for porous media. Tresca and Mohr-Coulomb models approximate the yield threshold for brittle media. Fossum’s model, and others like it, combine these features to model realistic geological media.

This report investigates the validity of several key assumptions in classical plasticity theory regarding material response to changes in the loading direction. Three metals, two rock types, and one ceramic were subjected to non-standard loading directions, and the resulting strain response increments were displayed in Gudehus diagrams to illustrate the approximation error of classical plasticity theories. A rigorous mathematical framework for fitting classical theories to the data,thus quantifying the error, is provided. Further data analysis techniques are presented that allow testing for the effect of changes in loading direction without having to use a new sample and for inferring the yield normal and flow directions without having to measure the yield surface. Though the data are inconclusive, there is indication that classical, incrementally linear, plasticity theory may be inadequate over a certain range of loading directions. This range of loading directions also coincides with loading directions that are known to produce a physically inadmissible instability for any nonassociative plasticity model.

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Publications: Nonuniqueness and instability of classical formulations of nonassociative plasticity

A plot of the frequency-dependent wave propagation velocity for the case study problem with an overlocal plasticity model, with the elastic and local hardening wave speeds shown for reference (left). Stress histories using an overlocal plasticity model with a nonlocal length scale of 1m and a mesh resolution of 0.125m (right)

The following series of three articles (with common authors J. Burghardt and R. Brannon of the University of Utah) describes a state of insufficient experimental validation of conventional formulations of nonassociative plasticity (AKA nonassociated and non-normality).  This work provides a confirmation that such models theoretically admit negative net work in closed strain cycles, but this simple prediction has never been validated or disproved in the laboratory!

  1. An early (mostly failed) attempt at experimental investigation of unvalidated plasticity assumptions (click to view),
  2. A simple case study confirming that nonassociativity can cause non-unique and unstable solutions to wave motion problems (click to view),
  3. An extensive study showing that features like rate dependence, hardening, etc. do not eliminate the instability and also showing that it is NOT related to conventional localization (click to view).

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Publication: On the thermodynamic requirement of elastic stiffness anisotropy in isotropic materials

Measure of anisotropy for Zircon, Quartz, Uranium, Titanium, Hornblende, and Copper.

T. Fuller and R.M. Brannon

In general, thermodynamic admissibility requires isotropic materials develop reversible deformation induced anisotropy (RDIA) in their elastic stiffnesses. Taking the elastic potential for an isotropic material to be a function of the strain invariants, isotropy of the elastic stiffness is possible under distortional loading if and only if the bulk modulus is independent of the strain deviator and the shear modulus is constant. Previous investigations of RDIA have been limited to applications in geomechanics where material non-linearityand large deformations are commonly observed. In the current paper, the degree of RDIA in other materials is investigated. It is found that the resultant anisotropy in materials whose strength does not vary appreciably with pressure, such as metals, is negligible, but in materials whose strength does vary with pressure, the degree of RDIA can be significant. Algorithms for incorporating RDIA in a classical elastic–plastic model are provided.

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

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.

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Research: Weibull fragmentation in the Uintah MPM code

Tough disk impacting brittle disk

Below are links to two simulations of disks colliding. The first is elastic and the second uses a fracture model with spatially variable strength based on a scale-dependent Weibull realization. Both take advantage of the automatic contact property of the MPM.

WeibConstMovie:  disks colliding without fracture

WeibPerturbedGood: disks colliding with heterogeneous fracture

This basic capability to support statistically variable strength in a damage model has been extended to the Kayenta plasticity model in Uintah.

Publication: Initial inclusion of thermodynamic considerations in Kayenta

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 [1] (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.

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Publication: KAYENTA: Theory and User’s Guide

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.

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