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: 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.

Available Online:

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:

Available Online:

Engineered microstructures for optimal energy absorbtion: design, validation, and verification

Breaking from conventional monolithic, layered, or woven designs for protective structures (bumpers, armor, etc.), micromanufacturing technology is now maturing to the point where precisely engineered microstructures may soon be possible.  In anticipation of such advances, novel microstructures are being here designed to optimize the ability of protective structures to thwart impact loadings. Preliminary work shows that a variety of specially designed microstructures can distribute structural damage away from an impact site rather than allowing damage to be concentrated at the impact zone. The merits of these design concept are investigated numerically and experimentally in the scope of safety net design.

UofU contributors/collaborators:
S. Leelavanichkul (Research fellow, Mechanical Engineering, UofU)
A. Cherkaev (Prof. of Mathematics, UofU)

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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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.”