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}  }

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

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  To help with such discussions, I am here posting two excerpts from my unpublished tensor-analysis notes.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!

The first excerpt, 150606tensorsVoigtMandelExcerpt, discusses Voigt and Mandel notation, and introduces some helpful operations on fourth-order tensors.  Here are some highlights taken from this PDF excerpt:


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!



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

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

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


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:


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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.