Continuum homogenization and RVEs

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For now (to help with a conversation that I’m having with a few collaborators) this post provides only the following “infographic” to illustrate the concept of approximating a periodic discrete system with an effective continuum over a sufficiently large scale. (More information will be added about this topic as needed and/or as requested).

Below is shown a five-link chain (in red-blue-green-orange-black). Immediately this colorful chain is a dark-gray plot of the exact (mesocale) lineal density, which is defined at a location “x” to be the mass within an infinitesimal segment dx at that location divided by the segment’s length dx. This local density is shown as the dark-gray shaded plot in the upper-left corner, and it is the slope of the black line in the graph of the lower-left corner.

The continuum concept. Homogenization and identification of an RVE is based on the value of x for which perturbations fall below numerical round-off error for a given application.

The continuum concept. Homogenization and identification of an RVE is based on the value of x for which perturbations fall below numerical round-off error for a given application.

The exact homogenized (macroscale) lineal density at a location “x” is defined as the exact total mass falling inside the span from zero to x,  divided by the chain’s length (x itself).  While the mesoscale density is the local slope at location “x” of the black line in the graph, the macroscale density is the secant slope at location “x” of the same black line.   The continuum (red-dashed) approximation of the local mass distribution ignores local fluctuations from the fact that the chain is actually heterogeneous. For short chain lengths, the exact macroscale density is significantly different from the continuum density, but this discrepancy asymptotes toward zero as the chain length is increased.

 

The theoretical representative volume element (RVE) size corresponds to the size for which the discrepancy (like the plot in the lower-right corner of the infographic) falls below some tolerable threshold, which is determined by considering the tolerable error in an engineering simulation.

These concepts apply to other properties besides density. For example, the macroscale elastic stiffness would be defined as the force applied to the chain divided by the corresponding induced displacement. Like density, this macroscale property varies with the number of links in the chain, but it asymptotes to a steady value as the chain length increases.

Density has a nice asymptotic continuum limit that isn’t sensitive to dilutely distributed statistical perturbations in the local (microscopic) density.  If, for example, 1 in 10000 links is made of light aluminum while the others are made of heavy steel, then the continuum density will be nevertheless close to that of a chain that is made entirely of steel links. The continuum elastic stiffness is likewise not highly sensitive to slight variations in local constituent (link) stiffness.  A chain’s failure strength, on the other hand, is profoundly affected by existence of even a miniscule fraction of weaker links. A mostly steel chain that contains relatively few aluminum links would have a continuum strength equal to the strength of the weaker (aluminum) link. That’s because (in the limit) an infinitely long chain would contain at least one aluminum link. For short chains that are made of, say, 10 links (each of which has a 1 in 10000 chance of being made of aluminum), the average macroscale strength would be higher on average than the strength of longer chains. The strength data for short chains would also be more variable.

These observations give insight into what a modeler must pay attention to when using continuum macroscale properties in simulations of engineering structures.  To design for the structure’s daily (i.e., normal and therefore usually elastic) usage conditions, homogenized continuum properties would be fine. However, continuum strength properties would need to be appropriately perturbed based on the size of the finite elements. This explicit incorporation of statistical variability in continuum properties is required when those perturbations strongly influence the engineering objective of the analysis (such as computing failure risk). In fact, it can be argued that such revisions are crucial to predict fracture and fragmentation whenever the finite-element size is smaller than a few kilometers. For more details on scale-dependent and statistically variable macroscale properties, see Publication: Aleatory quantile surfaces in damage mechanics and the more recent 2015 IJNME article, “Aleatory uncertainty and scale effects in computational damage models for failure and fragmentation” by Strack, Leavy and Brannon.

Material Property Terminology Tutorial

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If you do a web search on the difference between various terms in materials engineering, you will encounter a mind-boggling array of misinformation. The following infographic summarizes basic differences between the following terms: stiffness, compliance, yield strength, rupture strength, ultimate strength, hardening, softening, ductility, rupture strain, resilience, and toughness.  Of course no real stress-strain diagram looks like any of these, but the sketches are exaggerated to help illustrate the terminology.

MaterialPropertyTerminologyShadow

Copyright statement: This infographic may be used freely as long as it isn’t altered in any way.

Keep reading for important clarifications!

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I love my MoM

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Some mechanical-engineering programs, like ours here at the University of Utah, use the phrase “Strength of Materials” to refer to what the majority of other mechanical engineers more accurately refer to as “Mechanics of Materials” (often locally shortened to “MoM”).   The latter designation is more appropriate because this class typically is focused on an introduction to elementary elasticity with only very lightweight coverage of failure criteria (and almost never any post-failure theories, which are typically covered in upper-division and grad courses). Our University of Utah class, ME EN 3300, is locally called “Strengths” even though strength is barely covered (and only at an idealized level such as Tresca and von Mises criteria).

The following infographic furthermore shows that engineering textbooks appropriately and overwhelmingly favor MoM over Strengths (to see details, click to open in separate page and then zoom to fit the page):

LoveMom

Thanks go to Dr. Ashley Spear for stimulating this commentary/flame.

Simulation of sand/soil/clay thrown explosively into obstacles

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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.
    FragmentsHittingPlateAreniscaDrained
  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!
    CentrifugeRigidParticlesNoWall

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

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StressSpaceVisualization

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.

http://link.springer.com/article/10.1007%2Fs00707-015-1407-2

Bibdata:

@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={http://dx.doi.org/10.1007/s00707-015-1407-2},  publisher={Springer Vienna},  author={Homel, Michael A. and Guilkey, James E. and Brannon, Rebecca M.},  pages={1-32},  language={English}  }

Fourth-order tensor tutorial excerpts: Voigt and Mandel representations as well as isotropy topics

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I often get asked: when are you going to finish your tensor-analysis book?

Well, I have so many pressures on my time, that this hobby gets pushed to the bottom of the stack.  This said, I find myself often discussing fourth-order tensor operations, the distinction between Voigt and Mandel components, and scalar measures of anisotropy.  To help with such discussions, I am here posting two excerpts from my unpublished tensor-analysis notes.   Enjoy this enthralling topic!

EDIT: you can now cite this topic from a real publication. It is in chapter 26 of my 2018 book: Rotation, Reflection, and Frame change.

The first excerpt from the still unpublished notes, 150606tensorsVoigtMandelExcerpt (for which the cited references may be found in 150606tensorsVoigtMandelExcerptReferences), discusses Voigt and Mandel notation, and introduces some helpful operations on fourth-order tensors.  Here are some highlights taken from this PDF excerpt (all of which are included in the rotation book if you need to cite something):

VoigtNotation

What we have labeled as the 9×1 “contravariant Voigt array (without factors of 2)” is typically called a “stress-like” array in the composites community, while the 9×1 “covariant Voigt array (with factors of 2)” is called a “strain-like” array.  When these arrays actually represent stress and strain, their last three entries are zero because of symmetry. Likewise, the 9×9 array is, in this context, the elastic stiffness so its last three columns and last three rows are zero because of minor symmetry. Accordingly, in constitutive modeling, you typically see this matrix relationship truncated down to only 6 dimensions. When using computer code to work out components of such tensors, we recommend keeping all 9 dimensions just to serve as a visual cue that you have indeed enforced symmetries properly in your equations.

It is mystifying that the composites community doesn’t seem to even realize that the Voigt representations would be more properly referred to as covariant and contravariant, so please add a comment to this post if you have ever seen any composites articles use this mathematically proper terminology. These are not just matrix equations. There is a tensor basis that goes with Voigt representations, and that basis is a set of mutually orthogonal tensors. The basis is not normalized, so that leads to co/contravariant representations, in which the factors of 2 are metrics.  Whenever you have an orthogonal but not normalized basis, the obvious thing to do is to normalize it! That is what gives the following Mandel form, which you should note has no more of that ugly distinction between contravariant (stress-like) arrays and covariant (strain-like) arrays. Both arrays are treated the same!

MandelNotation

VoigtAndMandelBases

In this list, the very last set of basis tensors are the ones that pair with ordinary components of a tensor. For example, the 11 component of an ordinary second-order tensor is paired with a basis tensor whose component matrix is all zeros everywhere except a 1 in the 11 spot.  The 12 component goes with a basis tensor that has all zeros everywhere except for a 1 in the 12 spot, and so on.  The Voigt and Mandel representations merely represent a change of basis so that the first six basis tensors span the manifold of all possible symmetric tensors, while the last three basis tensors span the space of all possible skew tensors.

If you are not convinced that the Mandel representation is the better choice, try comparing it with Voigt for the components of the fourth-order identity tensor. The result is the identity matrix in Mandel form (not so for Voigt). Also, you really need the Mandel form to find eigenvalues and eigentensors.  Ordinary spectral analysis of the Voigt representation is completely meaningless — you need to use Mandel form to get meaningful eigenvalues and eigenmodes of a stiffness tensor.

Page 658 (PDF page 30) of my second excerpt, 150606tensorsFourthOrderOperationsAndMeasureOfIsotropyExcerpt, shows how to find the isotropic (IFOET) part of a fourth-order tensor (which is NOT generally some multiple of the identity), and how to define a scalar measure of anisotropy in the range from zero to one, as determined from the “angle” that the tensor makes with the linear manifold of isotropic tensors.  Doesn’t this sound exciting?  Here is an infographic summary of this process of finding the isotropic part of a fourth-order tensor, as well as setting the scalar measure of anisotropy (equal to 1 minus the scalar measure of isotropy):

The isotropic part of a fourth-order tensor, as well as a scalar measure of anisotropy ranging from 0 to 1.

The isotropic part of a fourth-order tensor, as well as a scalar measure of anisotropy ranging from 0 to 1.

In this infographic, the acronym IFOET stands for “isotropic fourth-order engineering tensor”  (labeled “ISO” elsewhere in the image).  In the scalar measure of isotropy, the denominator is the L2 norm of the original fourth-order tensor, equal to the square root of the sum of the squares of the tensor’s Mandel components (which is another benefit of Mandel over Voigt because getting the magnitude of a Voigt tensor would require insertions of factors of 2 and 4 — Yuck!). As you can see, the isotropy is 100% if the original tensor is isotropic, and it ranges down to 0% isotropy if the isotropic part of the original tensor is zero.

Linear algebra applied to sundials

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CSM alumnus, Scot Swan, offers Sundials_and_Linear_Algebra,  which is a short (informal) writeup on the equations that are used for making standard horizontal dials.   Challenge: see if Scot’s write up is consistent with the calculator at http://www.anycalculator.com/horizontalsundial.htm.

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Undergraduate researcher applies binning to study aleatory uncertainty in nonlinear buckling foundation models

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20150721_KatharinJensenAvatar

Sophomore undergraduate, Katharin Jensen, has developed an easily understood illustration of the effect of aleatory uncertainty, which means natural point-to-point variability in systems. She has put statistical variability on the lengths of buckling elements in the following system:

bistableSystem

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