Abstract: Deformation and Fracture of Heterogeneous Media using Boundary-Conforming Convected Particle Characteristic Functions in the Material Point Method

This is an abstract for

NWU2013: Advances in Computational Mechanics with Emphasis on Fracture and Multiscale Phenomena. Workshop honoring Professor Ted Belytschko’s 70th Birthday.  April 18, 2013 – April 20, 2013, Evanston, IL, USA

The organizers allocated only 10 minutes for each person’s talk (including big wigs like Tom Hughes), so we might just present this topic in the form of a puppet show with enough information to tickle the audience to chat with us about it in the hallway!


Rebecca Brannon*, Alireza Sadeghirad,  James Guilkey

Department of Mechanical Engineering

University of Utah

Salt Lake City, UT, 84112

*Email: Rebecca.Brannon@utah.edu

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Conference Poster: Modeling, Testing, and Analysis of Impulse Response of Femoral Head Reduction in Ceramic Hip Prostheses

Kakarla, D., A. P. Sanders, S. Siskey, K. Ong, N. Ames, J. O. Ochoa, and R. M. Brannon. (2012). “Modeling, Testing, and Analysis of Impulse Response of Femoral Head Reduction in Ceramic Hip Prostheses.” Transactions of the Orthopaedic Research Society 2012 Annual Meeting, San Francisco, CA, Feb. 4-7, Poster 2076.


Hip simulator wear tests including micro-separation conditions have revealed that abnormal loading events can outweigh normal loading conditions in causing wear of hard-on-hard bearings. Yet, there is a paucity of data to describe the mechanics of abnormal events such as edge loading by femoral neck impingement or femoral head subluxation. Though the magnitude of head subluxation has been measured in-vivo for a variety of human activities, there are apparently no corresponding reports of the concurrent head-liner contact forces; accurate measurements of the same may be rendered difficult by the transient, impulsive nature of edge loading. This report provides initial laboratory results of an in-vitro and in-silico study of impulsive femoral head reduction whose ultimate aim is to quantify dynamic edge-loading contact forces and stresses. The study implements an engineering model of proximal-lateral head subluxation and edge loading as could occur in a lax hip during the swing phase of gait. Rapid reduction is caused by applying a sudden cranio-caudal motion to the acetabular liner. In the laboratory, the femur’s response to this input is measured with strain gages and a laser vibrometer.


Aldridge (AKA Blake) spherical source verification test for dynamic continuum codes

This post has the following aims:

  • Provide documentation and source code for a spherically symmetric wave propagation in a linear-elastic medium.
  • Tell a story illustrating how this simple verification problem helped to validate a complicated rate-dependent and history-dependent geomechanics model.
  • Warn against believing previously reported material parameters, since they might have been the result of constitutive parameter tweaking to compensate for unrelated errors in the host code. Continue reading

Streamline visualization of tensor fields in solid mechanics

Stress net view of maximum shear lines inferred from molecular dynamics simulation of crack growth. Image from http://doi.ieeecomputersociety.org/10.1109/VIS.2005.33

Brazilian stress net before and after material failure. Colors indicate maximum principal stress (showing tension in the center of this axially compressed disk). Lines show directions of max principal stress.

A stress net is simply a graphical depiction of principal stress directions (or other directions derived from them, such as rotating them by 45 degrees to get the maximum shear lines.)  Continue reading

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: A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations

A. Sadeghirad, R. M. Brannon, and J. Burghardt

Three snapshots of the model with 248 particles in simulation of the radial expansion of a ring problem using: (a) CPDI method and (b) cpGIMP

A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram-shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4-node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems.

Available Online:



Publication: Verification Of Frame Indifference For Complicated Numerical Constitutive Models

K. Kamojjala, R. M. Brannon (2011)

Snapshot of the deformation in time

The principle of material frame indifference require spatial stresses to rotate with the material, whereas reference stresses must be insensitive to rotation. Testing of a classical uniaxial strain problem with superimposed rotation reveals that a very common approach to strong incremental objectivity taken in finite element codes to satisfy frame indifference(namely working in an approximate un-rotated frame) fails this simplistic test. A more complicated verification example is constructed based on the method of manufactured solutions (MMS) which involves the same character of loading at all points, providing a means to test any nonlinear-elastic arbitrarily anisotropic constitutive model.

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Publication: A multi-stage return algorithm for solving the classical damage component of constitutive models for rocks, ceramics, and other rock-like media

R. M. Brannon and S. Leelavanichkul

Octahedral isosurfaces for a) the unacceptable, b) the admissible, and c) the admissible

Classical plasticity and damage models for porous quasi-brittle media usually suffer from mathematical defects such as non-convergence and nonuniqueness.Yield or damage functions for porous quasi-brittle media often have yield functions with contours so distorted that following those contours to the yield surface in a return algorithm can take the solution to a false elastic domain. A steepest-descent return algorithm must include iterative corrections; otherwise,the solution is non-unique because contours of any yield function are non-unique. A multi-stage algorithm has been developed to address both spurious convergence and non-uniqueness, as well as to improve efficiency. The region of pathological isosurfaces is masked by first returning the stress state to the Drucker–Prager surface circumscribing the actual yield surface. From there, steepest-descent is used to locate a point on the yield surface. This first-stage solution,which is extremely efficient because it is applied in a 2D subspace, is generally not the correct solution,but it is used to estimate the correct return direction.The first-stage solution is projected onto the estimated correct return direction in 6D stress space. Third invariant dependence and anisotropy are accommodated in this second-stage correction. The projection operation introduces errors associated with yield surface curvature,so the two-stage iteration is applied repeatedly to converge. Regions of extremely high curvature are detected and handled separately using an approximation to vertex theory. The multi-stage return is applied holding internal variables constant to produce a non-hardening solution. To account for hardening from pore collapse (or softening from damage), geometrical arguments are used to clearly illustrate the appropriate scaling of the non-hardening solution needed to obtain the hardening (or softening) solution.

For errata (transcription errors in two of the verification solutions), please see:

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Verification Research: The method of manufactured solutions (MMS)

MMS stands for “Method of Manufactured Solutions,” which is a rather sleazy sounding name for what is actually a respected and rigorous method of verifying that a finite element (or other) code is correctly solving the governing equations.

A simple introduction to MMS may be found on page 11 of The ASME guide for verification and validation in solid mechanics. The basic idea is to analytically determine forcing functions that would lead to a specific, presumably nontrivial, solution (of your choice) for the dependent variable of a differential equation.  Then you would verify a numerical solver for that differential equation by running it using your analytically determined forcing function.  The difference between the code’s prediction and your selected manufactured solution provides a quantitative measure of error.

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