PUBLICATION: Influence of nonclassical plasticity features on shock wave existence and spectral solutions

A scanned copy is available here.

ABSTRACT (from OCR so has some typos): The influence of non-classical elastic-plastic constitutive features on dynamically moving discontinuities in stress, strain, and material velocity is investigated. Non-classical behavior here includes non-normality of the plastic strain increment to the yield surface, plastic compressibility, pressure sensitivity of yield, and dependence of the elastic moduli on plastic strain. DRUGAN and SHEN’s (1987) analysis of dynamically moving discontinuities with strain as well as stress jumps in classical materials is shown to be valid for a broad class of non-associative material models until deviation from normality exceeds a critical (non-infinitesimal) level. For these non-classical materials, an inequality that bounds the magnitude of the stress jump is derived, which is information not obtainable from a standard spectral analysis of a shock. For the special case of stress discontinuities with continuous strain, or for quasi-static deformations, this inequality is shown to rule out jumps in specific projections of the stress tensor unless the non-normality is sufficiently large. These results invalidate a … claim in the literature that an infinitesimal amount of non-normality permits moving surfaces of discontinuity in stress (with no strain jump) near the tip of a dynamically advancing crack tip.

Using a very general plastic constitutive law that subsumes most non-classical (and classical) descriptions currently in use, a complete closed-form solution is obtained for the plastic wave speeds and eigenvectors. A novel feature of the analysis is the clarity and completeness of the solutions. If the elastic part of the response is isotropic, one plastic wave speed equals the elastic shear wave speed, while the other two possible wave speeds depend in general on the stress and plastic strain within the shock transition layer. Concise necessary and sufficient conditions for real eigenvalues and for vanishing eigenvalues are derived. The real eigenvalues are classified by numerical sign and ordering relative to the elastic eigenvalues. The geometric multiplicity of plastic eigenvectors associated with elastic eigenvalues is shown to depend on the stress state within the shock transition layer. These solutions, several of which hold for arbitrary elastic anisotropy, are also applicable to acceleration waves and localization problems, and to materials with dependence of the elastic moduli on plastic strain. Such elastic–plastic coupling is shown to imply a non-self-adjoint fourth order tangent stiffness tensor even if the plastic constitutive law is associative.

A scanned copy is available here.

Publication: Elements of Phenomenological Plasticity: Geometrical Insight, Computational Algorithms, and Topics in Shock Physics


This 2007 Book Chapter on the basics of plasticity theory reviews the terminology and governing equations of plasticity, with emphasis on amending misconceptions, providing physical insights, and outlining computational algorithms. Plasticity theory is part of a larger class of material models in which a pronounced change in material response occurs when the stress (or strain) reaches a critical threshold level. If the stress state is subcritical, then the material is modeled by classical elasticity. The bound- ary of the subcritical (elastic) stress states is called the yield surface. Plasticity equations apply if continuing to apply elasticity theory would predict stress states that extend beyond this the yield surface. The onset of plasticity is typically characterized by a pronounced slope change in a stress–strain dia-gram, but load reversals in experiments are necessary to verify that the slope change is not merely nonlinear elasticity or reversible phase transformation.
The threshold yield surface can appear to be significantly affected by the loading rate, which has a dominant effect in shock physics applications.

In addition to providing a much-needed tutorial survey of the governing equations and their solution (defining Lode angle and other Lode invariants and addressing the surprisingly persistent myth that closest-point return satisfies the governing equations), this book chapter includes some distinctive contributions such as a simple 2d analog of plasticity that exhibits the same basic features of plasticity (such as existence of a “yield” surface with associative flow and vertex theory), an extended discussion of apparent nonassociativity, stability and uniqueness concerns about nonassociativity, and a  summary of apparent plastic wave speeds in relation to elastic wave speeds (especially noting that non-associativity admits plastic waves that travel faster than elastic waves).

For the full manuscript with errata, click 2007 Book Chapter on the basics of plasticity theory.

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.

Available Online:

http://www.mech.utah.edu/~brannon/pubs/7-BrannonBurghardtSAND-Report2009-0351.pdf

http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=948711

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 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.

Available Online:

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

 

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.

Available Online:

http://www.mech.utah.edu/~brannon/pubs/7-2010FullerBrannonStrackBishopThermodynamicsInKayenta.pdf