# Publication: Aleatory quantile surfaces in damage mechanics

ABSTRACT: In statistical damage mechanics, a deterministic failure limit surface is replaced with a scale-dependent family of quantile surfaces. An idealized homogeneous isotropic matrix material containing cracks of random size and orientation is used to elucidate expected mathematical character
of aleatory uncertainty and scale effects for initiation of damage in a brittle material. Scope is limited to statistics and scale dependence for the ONSET (not subsequent progression) of shear-driven failure. Exact analytical solutions for probability of such failure (with an interesting pole-point visualization) are derived for axisymmetric extension or compression of a single-crack sample. A semi-analytical bound on the failure CDF is found for a multi-crack specimen by integrating the single-crack probability over an exponential crack size distribution for which the majority of ﬂaws are small enough to be safe from failure at any orientation. Resulting tails of the predicted failure distribution differ from Weibull theory,
especially in the third invariant.

Selected cool pictures (see the article for more images):

# 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

# Publication: A model for statistical variation of fracture properties in a continuum mechanics code

NEWS FLASH: The print version of the Meyer-Brannon paper on statistical variation of fracture patterns in a continuum code (CTH) is now available at http://dx.doi.org/10.1016/j.ijimpeng.2010.09.007.

Perforation with Aleatory Uncertainty of high-pressure strength in an Eulerian Simulation.

# Some topics in rock and brittle media modeling

The following slides are taken primarily from a standard collection that has been used over the last several years to introduce mechanics researchers to concepts such as third-invariant dependence of material failure, softening, mesh dependency, the need for regularization through introduction of a length scale, Weibull statistics in strength data, etc.

# Delft Short Course: excerpts of discussion of basis and frame indifference

This posting links to a pdf,  DelftExcerpts, which contains slides taken from a 2004 short course given in TU Delft (Netherlands) on the mathematics of tensor analysis.  Following a review of the mathematics of line integrals, inexact differentials, and integrability, this set of slides provides some insight into the distinction between a global basis change (equivalent to the “space rotation” in the slides) and superimposed rotation.  It also provides an introduction to the principle of material frame indifference (PMFI) as it applies to restricting allowable forms and input/output variables of computational constitutive models.

# Publication: Elements of Phenomenological Plasticity: Geometrical Insight, Computational Algorithms, and Topics in Shock Physics

This 2007 Book Chapter on the basics of plasticity theory reviews the terminology and governing equations of plasticity, with emphasis on amending misconceptions, providing physical insights, and outlining computational algorithms. Plasticity theory is part of a larger class of material models in which a pronounced change in material response occurs when the stress (or strain) reaches a critical threshold level. If the stress state is subcritical, then the material is modeled by classical elasticity. The bound- ary of the subcritical (elastic) stress states is called the yield surface. Plasticity equations apply if continuing to apply elasticity theory would predict stress states that extend beyond this the yield surface. The onset of plasticity is typically characterized by a pronounced slope change in a stress–strain dia-gram, but load reversals in experiments are necessary to verify that the slope change is not merely nonlinear elasticity or reversible phase transformation.
The threshold yield surface can appear to be significantly affected by the loading rate, which has a dominant effect in shock physics applications.

In addition to providing a much-needed tutorial survey of the governing equations and their solution (defining Lode angle and other Lode invariants and addressing the surprisingly persistent myth that closest-point return satisfies the governing equations), this book chapter includes some distinctive contributions such as a simple 2d analog of plasticity that exhibits the same basic features of plasticity (such as existence of a “yield” surface with associative flow and vertex theory), an extended discussion of apparent nonassociativity, stability and uniqueness concerns about nonassociativity, and a  summary of apparent plastic wave speeds in relation to elastic wave speeds (especially noting that non-associativity admits plastic waves that travel faster than elastic waves).

For the full manuscript with errata, click 2007 Book Chapter on the basics of plasticity theory.

# 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

# Annulus Twist as a verification test

Illustrated below is the solution to an idealized problem of a linear elastic annulus (blue) subjected to twisting motion caused by rotating the T-bar an angle $\alpha$.  The motion is presumed to be applied slowly enough that equilibrium is satisfied.

This simple problem is taken to be governed by the equations of equilibrium $\vec{\nabla}\cdot\sigma=0$, along with the plane strain version of Hooke’s law in which Cauchy stress is taken to be linear with respect to the small strain tensor (symmetric part of the displacement gradient).  If this system of governing equations is implemented in a code, the code will give you an answer, but it is up to you to decide if that answer is a reasonable approximation to reality. This observation helps to illustrate the distinction between verification (i.e., evidence that the equations are solved correctly) and validation (evidence that physically applicable and physically appropriate equations are being solved).  The governing equations always have a correct answer (verification), but that answer might not be very predictive of reality (validation).

# Rate of Hencky (logarithmic) strain and similar tasks

The logarithmic (Hencky) strain is evaluated by taking the log of the symmetric stretch tensor in continuum mechanics. Doing so requires transforming to the principal stretch basis, taking logs of the principal stretch eigenvalues, and transforming the result back to the lab basis.  While this procedure is a bit tedious, it certainly is straightforward.

The harder — almost freakishly daunting — question is: how do you get the rate of the logarithmic strain?  This rate must include contributions from both the rate of the stretch eigenvalues and the rate of the stretch eigenvectors, which is difficult to handle when there are repeated eigenvalues causing extra ambiguity of eigenvectors.  Continue reading