PUBLICATION: Hypervariate Constitutive Modeling Illustrated via Aleatory Uncertainty in a Foundation Model

The free share link (available until May 24, 2018) is…

Abstract: Even if a ceramic’s homogenized properties (such as anisotropically evolving stiffness) truly can be predicted from complete knowledge of sub-continuum morphology (e.g., locations, sizes, shapes, orientations, and roughness of trillions of crystals, dislocations, impurities, pores, inclusions, and/or cracks), the necessary calculations are untenably hypervariate. Non-productive (almost derailing) debates over shortcomings of various first-principles ceramics theories are avoided in this work by discussing numerical coarsening in the context of a pedagogically appealing buckling foundation model that requires only sophomore-level understanding of springs, buckling hinges, dashpots, etc. Bypassing pre-requisites in constitutive modeling, this work aims to help students to understand the difference between damage and plasticity while also gaining experience in Monte-Carlo numerical optimization via scale-bridging that reduces memory and processor burden by orders of magnitude while accurately preserving aleatory (finite-finite-sampling) perturbations that are crucial to accurately predict bifurcations, such as ceramic fragmentation.

This publication helps to set knowledge needed to migrate cracks from initially uniform orientations (represented as dots on the left sphere) to highly textured orientations of vertical cracking (or any other texture based on the loading history).


This publication uses this simple system to explain many complicated concepts:

This paper would serve as a good project for a smart undergrad or first-year grad student to reproduce the results. It would serve as a familiarization exercise to learn basics of scale bridging, the difference between damage and plasticity, the influence of loading rate, the influence of microscale perturbations in macroscale behavior (e.g. reducing peak strength and scale effects), and binning down an excessively large number of internal variables to obtain a tractable decimated set.  All of that without needing to know anything about constitutive modeling – just a basic knowledge of springs and rigid links would be needed.

Again: see it for free (until May 24) at


Python source code is available on request.

Cite the paper as:

Brannon, R., Jensen, K., and Nayak, D., Journal of the European Ceramic Society (2018),

Welcome new computational-mechanics professor Ashley Spear!


The University of Utah is pleased to welcome a multi-talented new computational mechanics professor: Ashley Spear, newly graduated from Tony Ingraffea’s group at Cornell.

Dr. Spear's work (in collaboration with Carnegie Mellon and Lawrence Livermore Lab) using the Advanced Photon Source to obtain synchrotron-based measurments of 3D crack evolution in polycrystalline materials.

Dr. Spear’s work (in collaboration with Carnegie Mellon and Lawrence Livermore Lab) using the Advanced Photon Source to obtain synchrotron-based measurments of 3D crack evolution in polycrystalline materials.

Dr. Spear is a published expert in high-performance multiscale computing as well as experimental materials characterization, which is a perfect counterpart to the University if Utah’s existing focus on macroscale constitutive modeling with high-performance simulations using the Material Point Method.

Check out Dr. Spear’s research and CV (with contact information) at!





Public display of affection for Prof. Gib Richards

Dear Prof. Richards:

When I was still a teenager, you were my undergraduate advisor at the University of New Mexico (UNM). While seated in your office surrounded by rubber chickens, whoopee cushions, and other fanciful toys (which you had because of your side hobby of being a clown), I asked: “How can I know if I will ultimately enjoy a career in Mechanical Engineering?” You replied: “If you are willing to graduate one or two semesters late, then you can find the answer to that question by doing a co-op student internship.” To prepare me for this opportunity, your first action was to help me get a local internship at the Air Force Research Laboratory (then named Weapons Laboratory) at Kirtland AFB. You made telephone calls and otherwise worked your magic to get me into a co-op during the next summer at Los Alamos National Laboratory, where I quickly came to realize that it was the PhDs who were doing the most interesting and self-directed work. I also learned at Los Alamos that educated people have the self-discipline to NOT SMOKE CIGARETTES and to NOT USE SWEAR WORDS. My supervisor at Los Alamos furthermore advised me to go back to UNM and take as many classes as possible from Buck Schreyer, which likewise delightfully shaped my career. Thus, Prof. Richards, you deserve more credit than anyone else for pointing me in the direction of a healthy PhD track, ultimately leading to 14 years as a researcher at Sandia National Laboratories and (most recently) as a professor of Mechanical Engineering since 2007.

I still fondly recall being the first of your students to use computer-generated graphics and laser printing in my senior research report, but that didn’t distract you from fulfilling your promise to find ten grammar/spelling errors. Did you ever fail in that quest with any other student? You were the person who graded my co-op report upon my return to UNM, where you taught me that “finite elements” is only *sometimes* hyphenated, consequently launching a campaign of my own to explain hyphenation rules to others (see, for example, my blog article

In summary, Prof. Richards, you have profoundly influenced my life! I love you for everything you have done for me and for countless other students.

Sincerely, Rebecca Brannon

The Energy Efficient Soldier

Congratulations to Martin Berzins and the other four University of Utah faculty members, as well as collaborators at Boston University, Rensselaer Polytechnic, Penn State, Harvard, Brown, UC-Davis, and Polytechnic U (Turin Italy) on the recently awarded $16.4M 5-year project to use high-performance computing to aid in the development of more efficient and lighter power supplies for soldiers!  For more information, see the news release at

Funding: CSM group receives $1.1M aimed at military vehicle safety

Stills from YouTube video of buried roadside explosive

As one of four institutions collaborating with the University of Colorado — Boulder,  the CSM group in the Department of Mechanical Engineering at the University of Utah, will be developing constitutive models for soils, as well as full-scale simulation capabilities in Uintah to predict blast and ejecta from shallow buried explosives (such as roadside improvised explosive devices).  The $1.1M slated for CSM work presumes the project will last 5 years.  For more information, see the University of Colorado’s press release.

Course offering: ME 7960 (special topics) Computational Constitutive Modeling

Third invariant yield surface with uncertainty

Constitutive modeling refers to the development of equations describing the way that materials respond to various stimuli. In classical deformable body mechanics, a simple constitutive model might predict the stress required to induce a given strain; the canonical example is Hooke’s law of isotropic linear elasticity. More broadly, a constitutive model predicts increments in some macroscale state variables of interest (such as stress, entropy, polarization, etc.) that arise from changes in other macroscale state variables (strain, temperature, electric field, etc.).

Constitutive equations are ultimately implemented into a finite element code to close the set of equations required to solve problems of practical interest. This course describes a few common constitutive equations, explaining what features you would see in experimental data or structural behavior that would prompt you to select one constitutive model over another, how to use them in a code, how to test your understanding of the model, how to check if the code is applying the model as advertised in its user’s manual, and how to quantitatively assess the mathematical and physical believability of the solution.

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