Stretch, rotation, and intuition about strain rate approximations

The animation below shows
RED: simple shear in physical configuration
BLUE: simple shear with the polar rotation removed (i.e., the pure stretch)
GREEN: the deformation corresponding to the approximation that D-bar (given by the unrotated symmetric part of the velocity gradient) is actually the rate of reference logarithmic strain. This is found by integrating D-bar through time to obtain the apparent logarithmic strain, and then exponentiating this apparent strain to obtain an apparent reference stretch.
GRAY: rotation of the green deformation back to the spatial configuration.

As seen, all of these coincide when strains are small. The BLUEGREEN reference deformations diverge almost immediately from their REDGRAY counterparts, suggesting graphically that accounting for material rotation is generally crucial even in cases of moderate material distortion.

Now let’s consider what is happening much later to cause the BLUE deformation to diverge from the GREEN one (or, equivalently, causing the actual shear deformation in RED to diverge from the  GRAY approximation).  This divergence is the result of two things happening simultaneously:

1. large material distortion in combination with

2. rotation of the principal axes of reference stretch (i.e., rotation of the ellipse axes in the unrotated configuration).

If your engineering problem is missing either of these (i.e., if you are working with small deformations and/or your reference stretch directions are nearly stationary), then you will not see differences in your BLUEGREEN (or RED-GRAY) deformations.  In such cases, you may use D-bar as an approximation of strain rate, which is desired because (in comparison to exactly computing logarithmic strain and its rate), D-bar is relatively inexpensive to program (i.e., easy algorithm) and relatively inexpensive in run-time CPU costs (i.e., fewer computer flops).

The BLUE and GREEN reference (unrotated) deformations do not begin to diverge from each other until the material distortion (ellipse eccentricity) becomes noticeably large.  The same distinction is applies to the spatial deformations shown in RED and GRAY, which is no surprise since they are simply the rotations of the reference deformations into the spatial configuration. In numerical methods, D-bar is often used as an approximation to the strain rate, which is loosely equivalent to assuming that the stresses required to achieve the green deformation are approximately equal to those required to achieve the actual blue deformation. This approximation considerably reduces algorithm development costs as well as run-time computational costs. For materials that can resist deformation even at very large distortions (e.g., rubber), this approximation error is probably intolerably large. For materials that would have broken by the time the material stretches that much (e.g., ceramic), the consequence of the error is probably less severe since the material model would predict essentially zero stress for both the accurate deformation and the approximate one.

To account for material rotation, we enforce the constitutive model in the unrotated configuration (BLUE or GREEN), and then apply the material rotation to transform the results to the actual spatial configuration (RED or GRAY).  Thus, when you are deciding whether or not you can “get by” approximating strain rate by D-bar, you should ask yourself: Will the stresses for the GRAY deformation be sufficiently close to those in the actual RED deformation to serve my engineering needs?  If your material is a ceramic, you will probably conclude that the error is tolerable (provided that the confining pressure is low when the large material distortions occur; otherwise, if shear-like distortion is occurring at high-pressure states the errors will be larger because ceramics become stronger in compression).

3 thoughts on “Stretch, rotation, and intuition about strain rate approximations

  1. Dear Prof. Brannon,
    I have read with great interest this post. I am currently addressing this problem:
    I am trying to simulate by ABAQUS a torsion test at large strain (up to 4) of steel at very high temperature. I am planning to use the logarithmic strain and strain rate approximated by one of the tensors introduced by Prof. Bazant (ASME J. Eng. Mater. Technol.,120(2), pp. 131–136).
    Do you think it is a correct way?
    Best regards.

    Guido Chiantoni
    Tenaris R&D Dept.

    • One thing to keep in mind is that there is no single “right” strain measure. Any reference strain measure can be converted to reference stretch (i.e., U from the polar decomposition of the deformation gradient F=RU), and therefore all reference strain tensors have the same “information content” since one may be obtained from the other. Therefore, the selection of a strain measure is motivated simply by a desire to make the functional form of the constitutive model as simple as possible. If you used Green-Lagrange strain, defined in 1-D by eGL = (stretch^2-1)/2, then a linear elastic model, sigma=E*eGL, would be unstable in compression since it would predict that a non-infinite stress can compress the material to zero volume. This doesn’t mean that eGL is bad — it only means that it requires a nonlinear model to give reasonable results. When using logarithmic strain, eL=ln(stretch), a linear elastic model, sigma= E*eL (where E is the elastic modulus) has the nice property that it takes an infinite compressive stress to reduce volume to zero (stretch=0) and an infinite tensile stress to expand volume to infinity (stretch=infinity). Bazant points out in his paper (http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/373.pdf) that logarithmic strain is not the only strain definition with this property, which is the basis for his guidance on alternative approximations to logarithmic strain. To decide if those approximations are acceptable to you at strains up to 4, I recommend that you plot logarithmic strain eL=ln(stretch) vs. stretch on the same plot as one of the approximate strains recommended by Bazant. When you visually inspect whether or not they have acceptable discrepancies between each other, I doubt that you will feel very confident about using those approximations at such large stretches. Another word of caution: for simulations of such high strain in torsion, you are likely to distort the finite elements to a non-favorable aspect ratio. Watch out for that issue and also be scrupulous to do mesh convergence studies. Finally, large shear results in significant rotations of the directions of principal stretch, which seems to give most codes (commercial and research) lots of accuracy problems. In summary, Rotza Ruck, my friend cuz you’re gonna need it!

  2. Dear Prof. Brannon,
    thank you so much for your answers and suggestion. We will apply them step by step. At the same time we will try to implement the algorithm you explained in the tutorial about the rate of hencky strain ( http://csm.mech.utah.edu/content/wp-content/uploads/2011/08/RateOfPrincipalFunctionExcerpt.pdf ).
    We have worked out all the experimental problems concering hot torsion tests, so we are not used to get scared so easily!
    Best regards.

    Guido Chiantoni

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