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: 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: Survey of Four Damage Models for Concrete

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.

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Research: Instability of *ANY* nonassociative plasticity model

The CSM group has independently confirmed  a case study demonstrating the truth of a claim in the literature that any non-associative rate-independent model admits a non-physical dynamic achronistity instability. By stimulating a non-associative material in the “Sandler-Rubin wedge” (above yield but below the flow surface), plastic waves are generated that travel faster than elastic waves, thus introducing a negative net work in a closed strain cycle that essentially feeds energy into a propagating wave to produce unbounded increases in displacement with time.

Sandler-Rubin instability: an infinitesimal pulse grows as it propagates

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Publication: Conjugate stress and strain caveats /w distortion and deformation distinction

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.”

Tutorial: the thermoelastic square

A very kewl mnemonic device for recalling thermodynamic identities (the Gibbsian relations, the Maxwell relations, the contact or Legendre transformations, etc.) I am working on a new version of this document that will clarify why property definitions for solids do NOT, in general, reduce to those for fluids when the tensors are isotropic. Stay tuned…

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Tutorial: Define Your Strain!

This single-page document emphasizes the need for experimentalists and theorists alike to ALWAYS define their strain measure. For every percent increase in strain, the most popular measures of strain will disagree by as much as 1.5%. This might not sound like much, but try running a simple shear Von Mises strain cycle using log strain and engineering strain. You will find that the engineering strain calculation produces anomalous PRESSURES because volumetric strain does NOT equal the trace of strain EXCEPT for logarithmic strain.

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Tutorial: Rotation

A REALLY BIG (long download time) tutorial on how to describe rotation. Topics include coordinate transformations, tensor transformations, converting an axis and angle of rotation into a rotation tensor, Euler angles, quaternions, and generating a uniformly random rotation tensor. This document also discusses the common numerical problem of “mixing” rotations in such a way that the mixed rotation is physically reasonable. The pages in the document that deal with random rotations contain some complicated figures, so don’t worry if your pdf reader pauses for a while on those pages. As a matter of fact, watching the pdf viewer render the figures is like an informative movie because it draws the random dots in the same order as I computed them. By watching the rendering, you can see the nonuniform clustering quite clearly.] (Last posted here 020509, but a formal publication is anticipated)

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