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),

Simulation of sand/soil/clay thrown explosively into obstacles

Here are a couple of cool movies created by CSM researcher, Biswajit Banerjee, in preparation for our project review this week:

  1. Clods of soil impact a plate:  A major advantage of the Material Point Method (developed as part of this research effort) is that it automatically allows material interaction without needing a contact algorithm.
  2. Extrapolated buried explosive ejecta. The sample is in a centrifuge to get higher artificial gravity, so the particles move to the side because of the Coriolis effect!

PUBLICATION: Continuum effective-stress approach for high-rate plastic deformation of fluid-saturated geomaterials with application to shaped-charge jet penetration


AUTHORS: Michael A. Homel · James E. Guilkey · Rebecca M. Brannon

ABSTRACT: A practical engineering approach for modeling the constitutive response of fluid-saturated porous geomaterials is developed and applied to shaped-charge jet penetration in wellbore completion. An analytical model of a saturated thick spherical shell provides valuable insight into the qualitative character of the elastic– plastic response with an evolving pore fluid pressure. However, intrinsic limitations of such a simplistic theory are discussed to motivate the more realistic semi-empirical model used in this work. The constitutive model is implemented into a material point method code that can accommodate extremely large deformations.Consistent with experimental observations, the simulations of wellbore perforation exhibit appropriate dependencies of depth of penetration on pore pressure and confining stress.


@article{  year={2015},  issn={0001-5970},  journal={Acta Mechanica},  doi={10.1007/s00707-015-1407-2},  title={Continuum effective-stress approach for high-rate plastic deformation of fluid-saturated geomaterials with application to shaped-charge jet penetration},  url={},  publisher={Springer Vienna},  author={Homel, Michael A. and Guilkey, James E. and Brannon, Rebecca M.},  pages={1-32},  language={English}  }

PUBLICATION: An efficient binning scheme with application to statistical crack mechanics

This paper has an algorithm that alleviates the computational burden of evaluating summations involving thousands or millions of terms, each of which is statistically variable.  It is a simple binning strategy that replaces the large (thousand or million-member) population of terms with a much smaller representative (~10 member) weighted population. This binning method typically gives ~500x computational efficiency boost.


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Errata for two verification publications


This posting provides errata for an analytical solution that appeared in the following two publications:

Brannon, R. M. and S. Leelavanichkul (2010) A multi-stage return algorithm for solving the classical damage component of constitutive models for rocks, ceramics, and other rock-like media. Int. J. Fracture v. 163(1), pp. 133-149.

K.C. Kamojjala, R. Brannon, A. Sadeghirad, and J. Guilkey (2013) Verification tests in solid mechanics, Engineering with Computers, 1-21.

As pointed out by Dr. Andy Tonge, they both contain a the same transcription error that was not in the original unpublished working document where the details of the solution are archived.  The following excerpt from the original unpublished working document contains correct formulas:  PlasticityVerification2excerpt.   See the red comment boxes in this file for details.

Publication (Abstract and Erratum): Second-order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces


Convected particle domain interpolation (CPDI) is a recently developed extension of the material point method, in which the shape functions on the overlay grid are replaced with alternative shape functions, which (by coupling with the underlying particle topology) facilitate efficient and algorithmically straightforward evaluation of grid node integrals in the weak formulation of the governing equations. In the original CPDI algorithm, herein called CPDI1, particle domains are tracked as parallelograms in 2-D (or parallelepipeds in 3-D). In this paper, the CPDI method is enhanced to more accurately track particle domains as quadrilaterals in 2-D (hexahedra in 3-D). This enhancement will be referred to as CPDI2. Not only does this minor revision remove overlaps or gaps between particle domains, it also provides flexibility in choosing particle domain shape in the initial configuration and sets a convenient conceptual framework for enrichment of the fields to accurately solve weak discontinuities in the displacement field across a material interface that passes through the interior of a grid cell. The new CPDI2 method is demonstrated, with and without enrichment, using one-dimensional and two-dimensional examples.

Bib data:

Sadeghirad, A., R. M. Brannon, J.E. Guilkey (2013) Second-order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces, Int. J. Num. Meth. Engr., vol. 95, 928-952


Bibtex entry:

author = {A. Sadeghirad and R.M. Brannon and J.E. Guilkey},
title = {Second-order convected particle domain interpolation ({CPDI2}) with
enrichment for weak discontinuities at material interfaces},
journal = {Intl. J. Num. Meth. Engng.},
year = {2013},
volume = {95},
pages = {928–952}

Erratum:  Eq. 33 should be

Corrected Eq. 33

Publication: Verification tests in solid mechanics

ABSTRACT: Code verification against analytical solutions is a prerequisite to code validation against experimental data. Though solid-mechanics codes have established basic verification standards such as patch tests and convergence tests, few (if any) similar standards exist for testing solid-mechanics constitutive models under nontrivial massive deformations. Increasingly complicated verification tests for solid mechanics are presented, starting with simple patch tests of frame-indifference and traction boundary conditions under affine deformations, followed by two large-deformation problems that might serve as standardized verification tests suitable to quantify accuracy, robustness, and convergence of momentum solvers used in solid-mechanics codes. These problems use an accepted standard of verification testing, the method of manufactured solutions (MMS), which is rarely applied in solid mechanics. Body forces inducing a specified deformation are found analytically by treating the constitutive model abstractly, with a specific model introduced only at the last step in examples. One nonaffine MMS problem subjects the momentum solver and constitutive model to large shears comparable to those in penetration, while ensuring natural boundary conditions to accommodate codes lacking support for applied tractions. Two additional MMS problems, one affine and one nonaffine, include nontrivial traction boundary conditions.

Some eye-catching graphics (see the paper itself for details):

2013verificationPic1 Continue reading