iMechanica - Comments for "Journal Club Theme of February 2012: Elastic Instabilities for Form and Function"
https://imechanica.org/node/11812
Comments for "Journal Club Theme of February 2012: Elastic Instabilities for Form and Function"enHarnessing postbuckling behavior of cylindrical shell
https://imechanica.org/comment/26011#comment-26011
<a id="comment-26011"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11812">Journal Club Theme of February 2012: Elastic Instabilities for Form and Function</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>A recent study by our group on tailoring the postbuckling behavior of cylindrical shell. We provide a preliminary study on how postbuckling behavior can be modified and potentially tailored by three methods. Enjoy our efforts!</span></p>
<p><span>Full text can be found in: </span><a href="http://authors.elsevier.com/sd/article/S0263823114001670" target="_blank">http://authors.elsevier.com/sd/article/S0263823114001670</a></p>
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</ul>Fri, 11 Jul 2014 12:46:07 +0000Nan Hucomment 26011 at https://imechanica.orgbuckling analysis of 1D beam column
https://imechanica.org/comment/20115#comment-20115
<a id="comment-20115"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11812">Journal Club Theme of February 2012: Elastic Instabilities for Form and Function</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi,
</p>
<p>
I have modelled 1D beam-column problem with axial and lateral loads by variational methods. There are 4 elements with modnodes and two degrees of freedom at each node. The governing equation is as given below.
</p>
<p>
([Ke]-P[Kg])(Δ)=(F)
</p>
<p>
Where [Ke] = elastic stiffness matrix
</p>
<p>
P = buckling load
</p>
<p>
[Kg] = Geometric stiffness matrix
</p>
<p>
(Δ) = Global DOF matrix, and
</p>
<p>
(F) = lateral loads at each node.
</p>
<p>
I want to calculate buckling load and corresponding eigen vector. I am unable to solve this system of equations (I lack knowledge in solving this sytem of equations). Can any body help me in modelling it as an incremental solution.
</p>
<p>
Best regards,
</p>
<p>
Brahmendra S Dasaka.
</p>
<p>
</p>
<p>
</p>
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</ul>Fri, 05 Oct 2012 05:12:27 +0000dasakabcomment 20115 at https://imechanica.orgHere is the link of paper
https://imechanica.org/comment/19122#comment-19122
<a id="comment-19122"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18529#comment-18529">The relevance of equilibria in the problem of evolution</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Here is the link of paper Prof.Amit Acharya refers to
</p>
<p>
<a href="http://www.imechanica.org/node/12588">http://www.imechanica.org/node/12588</a>
</p>
<p>
</p>
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</ul>Sun, 10 Jun 2012 01:52:07 +0000Likun Tancomment 19122 at https://imechanica.orgLooks Great
https://imechanica.org/comment/18591#comment-18591
<a id="comment-18591"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18535#comment-18535">Instabilities & APS</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
These sessions look great. I hope my post-doc attended. (And I'm now even more sorry I didn't go to the MM this year.)
</p>
<p>
Eric Mockensturm
</p>
<p><a href="mailto:emm10@psu.edu">emm10@psu.edu</a></p>
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</ul>Thu, 01 Mar 2012 13:18:03 +0000ericmockcomment 18591 at https://imechanica.orgA Few Last Ones
https://imechanica.org/comment/18584#comment-18584
<a id="comment-18584"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11812">Journal Club Theme of February 2012: Elastic Instabilities for Form and Function</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
A few last references you might like...
</p>
<p>
</p>
<p>
J. Jiang and E. Mockensturm, “A motion amplifier using an axially driven buckling beam: I. Design and experiments,” Nonlinear Dynam, vol. 43, no. 4, pp. 391–409, 2006.
</p>
<p>
J. Jiang and E. Mockensturm, “A motion amplifier using an axially driven buckling beam: II. Modeling and analysis,” Nonlinear Dynam, vol. 45, pp. 1–14, 2006.
</p>
<p>
N. Goulbourne, E. Mockensturm, and M. Frecker, “Dynamic Actuation of Electro-Elastic Spherical Membranes Using Dielectric Elastomers,” in ASME 2005 International Mechanical Engineering Congress and Exposition, 2005, vol. 2005, pp. 227–237.
</p>
<p>
E. Mockensturm and N. Goulbourne, “Dynamic response of dielectric elastomers,” Int J Nonlinear Mech, vol. 41, no. 3, pp. 388–395, 2006.
</p>
<p>
</p>
<p>
Eric Mockensturm
</p>
<p> </p>
<p><a href="mailto:emm10@psu.edu">emm10@psu.edu</a></p>
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</ul>Wed, 29 Feb 2012 15:43:38 +0000ericmockcomment 18584 at https://imechanica.orgrandom perturbation via a kinetics approach
https://imechanica.org/comment/18574#comment-18574
<a id="comment-18574"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18495#comment-18495">Random perturbation approach</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear velandkar,
</p>
<p>
Thank you for pointing out this approach, which has been my favorite. In addition to the two early works I did with Zhigang on viscous layer, my group has extended the approach to viscoelastic substrates, which is now summarized in a book chapter (to be published; a preprint is available at <a href="http://imechanica.org/node/12022">http://imechanica.org/node/12022</a>). As noted by Zhigang, this approach has its own limitations, but I would love to see it being used more frequently in complementary to the other approaches (equilibrium and bifurcation analysis).
</p>
<p>
RH
</p>
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</ul>Wed, 29 Feb 2012 02:32:00 +0000Rui Huangcomment 18574 at https://imechanica.orgSymmetry and bifurcation
https://imechanica.org/comment/18538#comment-18538
<a id="comment-18538"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18536#comment-18536">Hi Zhigang,
This is a</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Thank you, Ryan, for your help. My students and I will try to read what you have suggested and come back to you. Best, Zhigang</p>
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</ul>Mon, 27 Feb 2012 06:48:57 +0000Zhigang Suocomment 18538 at https://imechanica.orgHi Zhigang,
This is a
https://imechanica.org/comment/18536#comment-18536
<a id="comment-18536"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18317#comment-18317">libraries of bifurcation-tracking codes</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Hi Zhigang,</p>
<p> </p>
<p>This is a subject close to my heart, and one I think is important.</p>
<p> </p>
<p>Something that is missing (as far as I could see) from this thread is the fact that GENERICALLY bifurcation (where two paths of equilibrum cross) only occurs when SYMMETRY is present in the problem. In the generic case only turning points (or limit load, sometimes called "saddle-node" bifurcations) should be expected to occur. When symmetry exists, then it is likely that bifurcations occur where multiple paths cross at a single bifurcation point. The book "Imperfect bifurcation in structures and materials" by Ikeda and Murota, 2nd Ed. Springer 2010 is one recent reference for these ideas.</p>
<p> </p>
<p>So, if your problem has symmetry (common in nature, at least approximately, and very common in engeneered systems) you need to take advantage of the results of symmetry group theroy applied to the bifurcation problem.</p>
<p> </p>
<p>I only know of two code (similar to AUTO and LOCA) that do this. One is called SYMCON and was developed by Karin Gatermann and Andreas Hohmann. Unfortunately this code is, to my knowledge, no longer under active use or development (tragically, Gatermann died at a young age). [The reference is Gatermann and Hohmann, Impact of computing in science and engineering, 3 (1991)].</p>
<p> </p>
<p>The other code is my own. I have developed a reasonably general purpose code that takes advantage of many of the group theory results in bifurcation problems and used it to study Martensitic Phase transformations in Shape Memory Alloys. (e.g., Elliott, Triantafyllidis, and Shaw. JMPS 2011, 216-236). The code is reasonably general, but has many features specific to the study of materials from the atomistic modeling perspective. The code has gone under a few names, starting with simply LatticeStatices, then BFBSymPac, and now (hopefully the final name): SyBFB -- "Symmetry aware Branch-Following and Bifurcation" Package. [Pronounced "Sib-fib", with both i's being short.]</p>
<p> </p>
<p>The code is not widely distributed, yet. I have plans for an effort to make it more general and user-friendly and then release it under an open source license. However, at the moment it is available from me directly by specific request.</p>
<p> </p>
<p>If you are interested in these ideas and have more questions, I'm happy to discuss this great topic further.</p>
<p> </p>
<p>Cheers,</p>
<p> </p>
<p>Ryan S. Elliott</p>
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</ul>Mon, 27 Feb 2012 03:55:05 +0000ellio167comment 18536 at https://imechanica.orgInstabilities & APS
https://imechanica.org/comment/18535#comment-18535
<a id="comment-18535"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11812">Journal Club Theme of February 2012: Elastic Instabilities for Form and Function</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
As this Journal Club month draws to a close, I would like to draw your attention to the <strong>Extreme Mechanics</strong> symposium that begins on Tuesday, February 28th at this year's APS March Meeting in Boston, MA.
</p>
<p>
</p>
<ol><li>Tuesday (8a-11a): <a href="http://meetings.aps.org/Meeting/MAR12/sessionindex2/?SessionEventID=168742">Rods</a></li>
<li>Tuesday (11:15a-2:15p): <a href="http://meetings.aps.org/Meeting/MAR12/sessionindex2/?SessionEventID=168769">Plates</a></li>
<li>Tuesday (2:30p-5:30p): <a href="http://meetings.aps.org/Meeting/MAR12/sessionindex2/?SessionEventID=168773">Origami, Creasing, & Folding</a></li>
<li>Wednesday (8a-11a): <a href="http://meetings.aps.org/Meeting/MAR12/sessionindex2/?SessionEventID=168776">Structures for Form & Function</a> </li>
<li>Wednesday (11:15a-2:15p): <a href="http://meetings.aps.org/Meeting/MAR12/sessionindex2/?SessionEventID=164510">Shells & Snapping</a> </li>
<li>Thursday (8a-11a): <a href="http://meetings.aps.org/Meeting/MAR12/sessionindex2/?SessionEventID=168786">Biological Systems & Structures</a></li>
<li>Thursday (11:15a-2:15p): <a href="http://meetings.aps.org/Meeting/MAR12/sessionindex2/?SessionEventID=168789">Fluid Structure Interactions & Swelling</a></li>
<li>Thursday (2:30p-5:30p): <a href="http://meetings.aps.org/Meeting/MAR12/sessionindex2/?SessionEventID=169350">Fracture, Friction, & Frequencies</a> </li>
</ol><p>Hope to see you there!</p>
<p>-Doug </p>
<p>
</p>
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</ul>Mon, 27 Feb 2012 01:11:59 +0000Douglas P Holmescomment 18535 at https://imechanica.orgRe: The relevance of equilibria in the problem of evolution
https://imechanica.org/comment/18530#comment-18530
<a id="comment-18530"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18529#comment-18529">The relevance of equilibria in the problem of evolution</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Amit: Thank you for your comment. Indeed, long-time dynamics can involve much more than equilibria. The mathematics is compelling and inviting. In 1990s, inspired by evrything dynamic and chaotic, I looked at many examples of evolving systems in solid mechanics materials science [1,2], but always went back to the behavior of equilibria. Perhaps it is a good time to look again at other types of dynamic behavior in solid mechanics and materials science. Your examples will be helpful.
</p>
<ol><li>
Z. Suo, "<a href="http://www.seas.harvard.edu/suo/papers/066.pdf">Motions of microscopic surfaces in materials</a>,"<em>Advances in Applied Mechanics</em> 33 193-294 (1997).</li>
<li>Z. Suo, <a href="http://imechanica.org/node/4868">Evolving small structures</a>. Class notes.</li>
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</ul>Sun, 26 Feb 2012 18:40:00 +0000Zhigang Suocomment 18530 at https://imechanica.orgThe relevance of equilibria in the problem of evolution
https://imechanica.org/comment/18529#comment-18529
<a id="comment-18529"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18526#comment-18526">Re: Random perturbation approach</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Zhigang,
</p>
<p>
Bifurcation analysis of the type you outline can be a very powerful tool for following limiting dynamics when the equilibria being tracked are stable (in a sense that can be made precise). However, in the context of getting a 'general view' of what a dynamical system can do, even in the limit of very large times, concentrating only on equilibria can be misleading. We all know of stable limit cycles to which dynamics may converge, or even attractors, both of which require different ideas than simply looking at equilibria.
</p>
<p>
A good simple example is the Ginzburg-Landau equation (I believe you have worked on these too) or a related system. In the following paper
</p>
<p>
<a href="http://www.imechanica.org/node/7312">http://www.imechanica.org/node/7312</a>
</p>
<p>
we characterize analytically the whole (infinite-dimensional) class of equilibria of the dynamics involving a parameter - the applied strain (g) as loading.
</p>
<p>
As pretty as this result is, in the context of the actual GL dynamics, you start from generic initial conditions and these equilibria are not picked up. Even more interestingly, what is picked up would be indistinguishable from equilibria on a computer but such profiles cannot be predicted by the equilibrium equations (with the parameter)!! All this is demonstrated in
</p>
<p>
<a href="http://www.imechanica.org/node/11819">http://www.imechanica.org/node/11819</a>
</p>
<p>
Finally, I point readers (especially young ones) to the following paper:
</p>
<p>
<a href="http://dml.cz/bitstream/handle/10338.dmlcz/134168/MathBohem_127-2002-2_3.pdf">http://dml.cz/bitstream/handle/10338.dmlcz/134168/MathBohem_127-2002-2_3...</a>
</p>
<p>
Look at example 2.2 - this is the standard relaxation oscillation example where for most times, the equilibria are relevant limits (think of x in the example as Zhigang's p above). A little variation allows adapting this dynamics to the usual up-down-up stress-strain or electromechancial responses that we are so familiar with from mechanics. However, then look at Example 5.1 and you see how the equilibria become completely irrelevant (they are unstable). I suspect a similar thing happens in the GL case too.
</p>
<p>
Also, these are not academic examples - if one does MD or complicated nonlinear systems as arise in continuum mechanics (e.g. models capable of predicting microstructure with length scales, e.g. <a href="http://www.imechanica.org/node/9906">http://www.imechanica.org/node/9906</a>), one should expect such things to be the rule rather than the exception.
</p>
<p>
Thus, there is more to life and evolution than equilibria, as important as the stable latter ones are. And, it is very worth learning about dynamics too - for practical and realistic reasons, because in many situations that, and not statics), is the only way to undestand limit *slow* evolution. A first cut at some progress for practical applications is in
</p>
<p>
<a href="http://www.imechanica.org/node/10998">http://www.imechanica.org/node/10998</a>
</p>
<p>
and a student, Likun Tan, will present her work in this forum related to such matters soon.
</p>
<p>
In connection to bifurcation analysis of equilibria in the context of solid mechanics, perhaps readers here should be aware of the very nice works of Ryan Elliott (U. of Minnesota) and Tim Healey (Cornell).
</p>
<p>
- Amit
</p>
<p>
</p>
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</ul>Sun, 26 Feb 2012 17:29:28 +0000Amit Acharyacomment 18529 at https://imechanica.orgWhat's Next? Instability: diversity and unification
https://imechanica.org/comment/18527#comment-18527
<a id="comment-18527"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18465#comment-18465">What's Next?</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Although instability has long been part of the education of mechanicians, several recent trends demand that we rethink what we teach.
</p>
<p>
<strong>Computation</strong>. The development of comutation has greatly extended our ability to analyze nonlinear systems. In particular, bifurcation analysis has undergone considerable development in general nonlinear analysis. However, our course on nonlinear continuum mechanics has mostly confined within the formulation of nonlinear equations, rather than describing diverse nonlinear phenomena.
</p>
<p>
<strong>Microfabrication</strong>. The development of microfabication has allowed us to create complex structures. Instability is a property of structure, but we don't have a systemtic way to design a structure to produce desirable instability. Insatbility is no longer a mode of failure; it is a feature.
</p>
<p>
<strong>A catalog of many kinds of instability</strong>. Instability may be classified one way or another, but perhaps it is useful to know a large number of them, before we attempt to classify them or unify them. Here is a partial list:
</p>
<ul><li>Buckling of thin structures</li>
<li>Imperfection sensitivity</li>
<li>Snap </li>
<li>Wrinkles</li>
<li>Creases</li>
<li>Cavitation</li>
<li>Shear bands</li>
<li>Phase transition (nucleation and growth)
</li>
<li>Symmetry breaking</li>
<li>Fracture
</li>
</ul><p>
We may witness a change of emphasis in mechanics education in coming years.
</p>
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</ul>Sun, 26 Feb 2012 12:27:00 +0000Zhigang Suocomment 18527 at https://imechanica.orgRe: Random perturbation approach
https://imechanica.org/comment/18526#comment-18526
<a id="comment-18526"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18495#comment-18495">Random perturbation approach</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Indeed, one may simulate the full dynamics of an initial-value problem. However, the amount of work might be extremely large, and the numerical results might be too much to survey. An alternative approach is bifurcation analysis. The two approaches can be compared by looking at an ODE:
</p>
<p>
dx/dt = f(x,p),
</p>
<p>
where x is a vector describing the state of a system, t is time, and p is a parameter. For a fixed value of the parameter p, the function x(t) describes the evolution of the state of the system.
</p>
<p>
<strong>Dynamic simmulation. </strong>For a fixed value of parameter p, given an initial condition x(0), we can numerically determine x(t). To survey the behavior of the system, we will need vary p and vary the initial condition.
</p>
<p>
<strong>Bifurcation analysis</strong>. A state of equilibrium is determined by
</p>
<p>
f(x,p) = 0.
</p>
<p>
For a fixed value of parameter p, this is an algebraic equation for x. Each solution to this algebraic equation gives a state of equilibrium of the system. When f(x,p) is nonlinear function of x, multiple solutions are possible for a given p. Once you plot all solutions as p varies, you get a general view of the system. The amount of work can be considerablly less than full dynamic simulation.
</p>
<p>
These two approaches are described well in many textbooks. Here is a readable one: <a href="http://www.amazon.com/dp/144191739X/ref=rdr_ext_tmb">Practical Bifurcation and Stability Analysis</a> by Rudiger Seydel.
</p>
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</ul>Sun, 26 Feb 2012 11:48:00 +0000Zhigang Suocomment 18526 at https://imechanica.orgRandom perturbation approach
https://imechanica.org/comment/18495#comment-18495
<a id="comment-18495"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11812">Journal Club Theme of February 2012: Elastic Instabilities for Form and Function</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p> </p>
<p>
I am a newcomer to this area, and as an experimentalist, not particularly knowledgeable about numerical simulations. Yet, I wonder why a "sledge hammer" approach could not work for simulating instabilities. For example randomly perturb the structure over a large range of frequencies and then deform the structure gradually to let any microdeformations build up nonlinearly. That is exactly what happens in real life: small defects cause microbending and then eventually buckling. Could this be a fruitful approach for simulations?
</p>
<p>
One major advantage could be simulating instabilities that are involve not just elasticity, but other coupled physical phenomena. One example is research by Profs. Suo and Rui Huang on films on viscous substrates [1,2] in which the fluid-structure interaction is crucial. One could use the solid mechanical equations for the film, and Stokes equations for the viscous layer, give the film a broad spectrum perturbation, and let the simulation proceed. This approach might have true predictive capability, i.e. predicting the instability without prior knowledge of the mode.
</p>
<p>
I can see one limitation: snap buckling instabilities would be difficult to capture. But might this approach be fruitful for instabilities in which amplitude remains continuous at the instability?<span> </span>Indeed, we have done some work (not yet published, but to be presented in the upcoming APS meeting) and are able to quantitatively simulate the "radial wrinkles" problem[3,4] using a random perturbation.
</p>
<p>
There are some examples of using random perturbations with excellent success[5,6]. Could someone comment on why this approach is not more popular, especially in problems where the mode cannot be predicted by standard eigenvalue analysis?
</p>
<p>
</p>
<ol><li><span>Huang, R.; Suo, Z. "Wrinkling of a compressed elastic film on a viscous layer", J. Appl. Phys. 2002, 91, 1135.</span></li>
<li><span>Liang, J.; Huang, R.; Yin, H.; Sturm, J. C.et al. "Relaxation of compressed elastic islands on a viscous layer", Acta Materialia 2002, 50, 2933.</span></li>
<li><span>Geminard, J. C.; Bernal, R.; Melo, F. "Wrinkle formations in axi-symmetrically stretched membranes", Eur. Phys. J. E 2004, 15, 117.</span></li>
<li><span>Cerda, E. "Mechanics of scars", J. Biomech. 2005, 38, 1598.</span></li>
<li><span>Yin, J.; Cao, Z. X.; Li, C. R.; Sheinman, I.et al. "Stress-driven buckling patterns in spheroidal core/shell structures", PNAS 2008, 105, 19132.</span></li>
<li><span>Li, B.; Jia, F.; Cao, Y.-P.; Feng, X.-Q.et al. "Surface Wrinkling Patterns on a Core-Shell Soft Sphere", Phys. Rev. Lett. 2011, 106.</span></li>
</ol></div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/11812%23comment-form">Log in</a> or <a href="/user/register?destination=node/11812%23comment-form">register</a> to post comments</span></li>
</ul>Fri, 24 Feb 2012 13:53:30 +0000velankarcomment 18495 at https://imechanica.orgI think people here might
https://imechanica.org/comment/18476#comment-18476
<a id="comment-18476"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18302#comment-18302">approach to deal with instability numerically</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
I think people here might be interested in our robust and efficient FEM algorithm for buckling simulation. Please see the following link.</p>
<p><a href="http://imechanica.org/node/4124">http://imechanica.org/node/4124</a>
</p>
<p>
</p>
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</ul>Thu, 23 Feb 2012 06:45:34 +0000Bin Liucomment 18476 at https://imechanica.orgHarnessing instabilities: what's next?
https://imechanica.org/comment/18474#comment-18474
<a id="comment-18474"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18465#comment-18465">What's Next?</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
I have the good fortune of interacting with people from a number of industries.
</p>
<p>
One particular problem comes to mind at this stage - the problem of air transport of frozen fish.
</p>
<p>
Frozen fish are usually transported in styrofoam containers that are leak proof, thermally insulating, and structurally stable enough that they hold their shape during transport. The flip side of having these desirable properties is that they occupy a lot of space and, once delivered to a destination country, are essentially wasted because it is not cost effective to return them to their countries of origin. Go to any major fish market to see the millions of styrofoam containers that have been discarded.
</p>
<p>
One solution is to make biodegradable containers. Another is to make containers that are foldable (but still leakproof and insulating and antifouling/easily cleaned) and reusable. Instabilities are one way of achieving such a foldable structure (think, for example, of windshield light reflectors or even umbrellas).
</p>
<p>
But more than solid mechanics has to go into the design of these structures.
</p>
<p>
-- Biswajit
</p>
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</ul>Thu, 23 Feb 2012 01:13:45 +0000Biswajit Banerjeecomment 18474 at https://imechanica.orgWhat's Next?
https://imechanica.org/comment/18465#comment-18465
<a id="comment-18465"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11812">Journal Club Theme of February 2012: Elastic Instabilities for Form and Function</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
With a week remaining in February, allow me to pose the following question: <em>Where do you see this field going in the future?</em>
</p>
<p>
We have spent a lot of time talking about specific instabilities that perform specific functions, but it would be beneficial to generate a long-range vision. I'd be interested to hear everyone's thoughts.
</p>
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</ul>Wed, 22 Feb 2012 13:01:11 +0000Douglas P Holmescomment 18465 at https://imechanica.orgRiks method in ABAQUS
https://imechanica.org/comment/18349#comment-18349
<a id="comment-18349"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18301#comment-18301">How is arclength defined in ABAQUS?</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
In ABAQUS, arc length is defined by the combination of the increments of displacement and load. You can get energy for both stable and unstable solutions (fixed points). I got pretty good results of snap-through instability by using Riks method, and I am able to deal with turning points. You may want to take a look at the following ABAQUS documentation on the implementation of Riks method.
</p>
<p>
<a href="http://mse-license1.mse.drexel.edu:2080/v6.8/books/stm/default.htm?startat=ch02s03ath18.html#stm-anl-modifiedriks">http://mse-license1.mse.drexel.edu:2080/v6.8/books/stm/default.htm?startat=ch02s03ath18.html#stm-anl-modifiedriks</a>
</p>
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</ul>Sat, 11 Feb 2012 04:50:00 +0000Lihua Jincomment 18349 at https://imechanica.orgSoft machines: increased functionality and extreme performance
https://imechanica.org/comment/18348#comment-18348
<a id="comment-18348"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18318#comment-18318">Operate a device on the verge of instability</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Thanks everybody for this intriguing discussion! So far the comments on this jClub theme have been focused on theoretical aspects to a large degree. Let me throw in some thoughts from an experimental perspective, mainly in the context of dielectric elastomers.
</p>
<p>
<br />
Without question the utilization of mechanical instabilities comprises a great opportunity to increase the functionality of mechanical systems in general and especially of soft systems, which are far less developed in research and practice. Particularly soft machines with a highly nonlinear response or even with a response structure including instabilities have not found many practical applications yet. They are comparably difficult to control and simulation techniques are less developed. To antagonize the hesitancy of industrial engineers to utilize soft machines with complex behavior we have to collectively highlight the advantages of such systems and design prototypes with extreme performance.
</p>
<p>
<br />
So far in this jClub we have discussed the rapid timescale of instabilities, the possibility to maintain a desired state in a bistable system without expending energy and some other aspects. We should also consider the possibility to trigger events that require relatively large amounts of energy with small signals. If we store mechanical energy in a system and operate it near the verge of instability a small amount of "control" energy may be sufficient to trigger an event that would require a lot of external energy without the utilization of an instability. That aspect may be very useful to build sensors, mechanical triggers or amplifiers. Especially for systems that require two distinctly different mechanical deformation states (such as Braille displays or haptic systems in general that require "on" and "off" states), easily switchable bistable systems will be very useful.
</p>
<p>
<br />
Maybe in our effort to increase the functionality of soft machines by harnessing instabilities we should also look into different fields and draw analogies. In electronics an ohmic resistor effects a linear dependence of electrical current on applied voltage. In contrast, a diode exhibits a highly nonlinear response and particularly a tunnel diode (<a href="http://en.wikipedia.org/wiki/Tunnel_diode">http://en.wikipedia.org/wiki/Tunnel_diode</a>) even shows a N-shaped correlation between electrical current and voltage. Each of these devices can be used for specific purposes and each modification of the current-voltage characteristics results in an altered functionality of the device. Similarly, we can try to utilize modified versions of the stress-strain, pressure volume, ... characteristics of mechanical systems to span their functionality over as diverse application fields as we have for ohmic resistors and diodes.
</p>
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</ul>Sat, 11 Feb 2012 01:14:02 +0000Christoph Keplingercomment 18348 at https://imechanica.orgInstability: Friend or Foe?
https://imechanica.org/comment/18336#comment-18336
<a id="comment-18336"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11812">Journal Club Theme of February 2012: Elastic Instabilities for Form and Function</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear all,
</p>
<p>
This is indeed a fascinating topic. As Doug rightly pointed out, researchers are now making a friend out of a foe in instability. Instability changes the state of a structure. If the final state still serves a function, then instability is harnessed. Otherwise, instability destroys.
</p>
<p>
A rubber-like polymer exhibits softening behavior at small strains, and stiffens infinitely near the limiting strain (whereby most of the long-chain polymers are fully stretched). The stiffening behavior may create a safe haven for the final state of any instability to fall into: It limits the maximum strain of the polymer.</p>
<p>For a dielectric elastomer, instability occurs in the form of electromechanical instability. Subject to a voltage, a thinning dielectric elastomer induces an even larger electric field. At some point, positive feedback ensues. For the same voltage, the elastomer snaps from a thick state with small actuation strain (typically < 30%), to a thin state with extremely large actuation strain (near the limiting strain, typical values for elastomers are in the order of 1000%). This extremely large strain almost always renders the elastomer to fail by dielectric breakdown, thereby crippling its function as an actuator. To turn this foe into a friend, one may either: Find a dielectric with an extremely large dielectric strength (> 1000 MV/m), or find a way to bring the limiting strain closer to the point of instability. The latter seems to be an easier way out.
</p>
<p>
A direct way is to apply pre-stretch on the elastomer. This brings the start point closer to the limit. Pelrine et. al., in his now classic 2000 Science Paper [1], has inadvertently exploited this fact to achieve an actuation strain of > 100%. A second way is to choose a polymer that has "shorter" chains, which limits its mechanical strain to a level where it allows the snap to survive dielectric breakdown. Ha et. al. has exploited this by designing polymer networks with short chains, interspersed within long chain polymers [2]. Both works have essentially made a friend out of instability, by allowing the polymer to safely snap into a large strain region, without dielectric breakdown taking place.
</p>
<p>
We crystallized these observations by using a model for dielectric elastomers [3]. Our model allows one to construct phase diagrams indicating regions of safe and unsafe snapping. By modifying the mechanical and electrical properties of the dielectric elastomer, one may achieve large actuations of above 500% by simply pre-stretching it. This provides a guide to materials selection and design to achieve large actuation strains.
</p>
<p>
But there is a caveat: Large actuation strains create large leakage currents. Experimental observations have shown that the current that leaks through a polymer dielectric increases exponentially with the applied field [4]. It is possible for a dielectric elastomer with an extremely good actuation performance, to be an energy-wasting device [5]. Excessive leakage currents may also put the dielectric at the verge of dielectric breakdown.
</p>
<p>
There exist layers of considerations in harnessing instabilities. An apparent foe may be turned into a friend, but it may in fact still be a foe. One certainly has to balance practical considerations with the desired function, and therein lies one exhilarating challenge for problems in instability.
</p>
<p>
</p>
<p>
[1] Pelrine, R.; Kornbluh, R.; Pei, Q.; Joseph, J. Science <strong>287</strong>, 836–839 (2000).
</p>
<p>
[2] Ha, S. M.; Yuan, W.; Pei, Q. B.; Pelrine, R. Adv. Mater. <strong>18</strong>, 887–891 (2006).
</p>
<p>
[3] Soo Jin Adrian Koh, Tiefeng Li, Jinxiong Zhou, Xuanhe Zhao, Wei Hong, Jian Zhu, Zhigang Suo.<a href="http://www.seas.harvard.edu/suo/papers/247.pdf"> Mechanisms of large actuation strain in dielectric elastomers</a>. Journal of Polymer Science Part B: Polymer Physics <strong>49</strong>, 504-515 (2011).
</p>
<p>
[4] T. A. Gisby, S. Q. Xie, E. P. Calius, and I. A. Anderson, Proc. SPIE <strong>7642</strong>, 764213 (2010).
</p>
<p>
[5] Choon Chiang Foo, Shengqiang Cai, Soo Jin Adrian Koh, Siegfried Bauer, Zhigang Suo.<a href="http://www.seas.harvard.edu/suo/papers/265.pdf"> Model of dissipative dielectric elastomers</a>. Journal of Applied Physics <strong>111</strong>, 034102 (2012).
</p>
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</ul>Fri, 10 Feb 2012 07:48:00 +0000Adrian S. J. Kohcomment 18336 at https://imechanica.orgDesign principals for functionality
https://imechanica.org/comment/18324#comment-18324
<a id="comment-18324"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18318#comment-18318">Operate a device on the verge of instability</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Thanks for your additional notes on this topic Zhigang.
</p>
<p>
When designing materials or devices to utilize elastic instabilities for functionality, I think the first step is finding a connection between what specific mechanical instabilities can provide and what features are desirable for a particular application.
</p>
<p>
The snap-through instability of a structure, for example, can describe a rapid jump between two stable configurations. The timescale of this "switch" is on the order of milliseconds and the lengthscale may be on the order of the structure's initial deflection. Therefore, one obvious benefit of utilizing this instability is its rapid timescale. Another benefit is instead of expending energy to force a structure with one stable configuration to continuously maintain an alternative configuration, one can design systems that expend a smaller amount of energy to predictably switch from one stable configuration to another.
</p>
<p>
These types of design parameters may be useful in creating "switchable adhesive" devices. For instance, it has been shown that a surface with posts [1] has different adhesive properties than a surface of holes [2]. Therefore, an interesting device may be one that switches between two types of interfaces to control adhesion. One example of a such a surface used switchable microlenses to change surface topography [3].
</p>
<p>
I think the key is identifying the benefits of specific mechanical instabilities and combining those ideas with desirable functionality.
</p>
<p>
[1] A.J. Crosby, M. Hageman, A. Duncan, <a href="http://pubs.acs.org/doi/abs/10.1021/la051721k">Controlling Polymer Adhesion with "Pancakes"</a>, Langmuir, <strong>21</strong>, 25, (2005).
</p>
<p>
[2] T. Thomas and A.J. Crosby, <a href="http://www.tandfonline.com/doi/abs/10.1080/00218460600646610#preview">Controlling Adhesion with Surface Hole Patterns</a>, J. Adhesion, <strong>82</strong>, 3, (2006).
</p>
<p>
[3] D.P. Holmes and A.J. Crosby, <a href="http://onlinelibrary.wiley.com/doi/10.1002/adma.200700584/abstract">Snapping Surfaces</a>, Advanced Materials, <strong>19</strong>, 21, (2007).
</p>
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</ul>Thu, 09 Feb 2012 14:57:31 +0000Douglas P Holmescomment 18324 at https://imechanica.orgOperate a device on the verge of instability
https://imechanica.org/comment/18318#comment-18318
<a id="comment-18318"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11812">Journal Club Theme of February 2012: Elastic Instabilities for Form and Function</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Thank you, Doug, for an interesting jClub theme. Thank you also for selecting our paper on dielectric elastomers for discussion. I’d like to follow up on your discussion, as well as the comments by Xuanhe and Katia. I’ll limit this comment to dielectric elastomers.
</p>
<p>
While elastomers can readily be stretched several times their initial length by a mechanical force, achieving large deformations by applying a voltage has been difficult. This difficulty is understood as follows. As the thickness decreases in response to an applied voltage, the electric field increases. This leads to a positive feedback between the reduction in thickness and the increase in electric field, leading to electromechanical instability.
</p>
<p>
It was therefore especially intriguing when an elastomer was demonstrated to attain voltage-induced strain over 100%. The large actuation was achieved by pre-stretching a sheet. We now understand that the pre-stretch in this case eliminates the electromechanical instability. But the elastomer is still near the verge of the instability, so that the voltage can induce large deformation.
</p>
<p>
In nearly all reported cases of observing large voltage-induced deformation, the devices operate near the verge of instability. That is, the operation of large-actuation dielectric elastomers is closely linked to electromechanical instability. Here instability is not failure; it is a feature.
</p>
<p>
An analysis of a particularly simple setup and an experimental demonstration are given in two recent papers:
</p>
<ul><li>
Soo Jin Adrian Koh, Tiefeng Li, Jinxiong Zhou, Xuanhe Zhao, Wei Hong, Jian Zhu, Zhigang Suo.<a href="http://www.seas.harvard.edu/suo/papers/247.pdf"> Mechanisms of large actuation strain in dielectric elastomers</a>. Journal of Polymer Science Part B: Polymer Physics 49, 504-515 (2011).
</li>
<li>
Jiangshui Huang, Tiefeng Li, Choon Chiang Foo, Jian Zhu, David R. Clarke, Zhigang Suo. <a href="http://www.seas.harvard.edu/suo/papers/267.pdf"> Giant, voltage-actuated deformation of a dielectric elastomer under dead load</a>. Applied Physics Letters 100, 041911 (2012).
</li>
</ul><p>
In your initial post, you have discussed examples of using instability to design devices. Xuanhe and Katia gave several more examples. Perhaps we can talk more about such experience, so that some basic principles will emerge.
</p>
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</ul>Thu, 09 Feb 2012 08:14:00 +0000Zhigang Suocomment 18318 at https://imechanica.orglibraries of bifurcation-tracking codes
https://imechanica.org/comment/18317#comment-18317
<a id="comment-18317"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18315#comment-18315">Re: Numerics for nonlinear equations with multiple solutions</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Ilinca: Thank you very much for sharing your experience. What a coincidence! On the bus to work this morning, I was reading a <a href="http://scholar.google.com/scholar?cluster=13324702584735473&hl=en&as_sdt=0,5">paper</a> by the developers of LOCA. (I'm on sabbatical leave in Karlsruhe.)
</p>
<ul><li>The paper has a nice review of general-purpose codes for bifurcation tracking: a large number of codes have been developed.
</li>
<li>The paper also emphasizes that LOCA is a bifurcation library, which is separated from application codes.
</li>
<li>The paper has a concise summary of algorithms.
</li>
<li>The paper also shows how to combine LOCA with a PDE code to analyze Rayleigh-Benard convection.</li>
</ul><p>
This iMechanica thread has helped me greatly. You and others have confirmed that various aspects of bifurcation tracking have been integrated with FEM.
</p>
<p>
Perhaps more specific question for people using ABAQUS is how these features can be implemented in ABAQUS. Let's hope people with that kind of experience will jump in and share their experience.
</p>
<p>
One more question for you. You said things can be interesting during a snap. In our problems, the system snaps to a stable state of equilibrium, so we don't study transitent closely. Can please you point to examples where transients after snap is interesting to study?
</p>
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</ul>Thu, 09 Feb 2012 07:51:00 +0000Zhigang Suocomment 18317 at https://imechanica.orgRe: Numerics for nonlinear equations with multiple solutions
https://imechanica.org/comment/18315#comment-18315
<a id="comment-18315"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18285#comment-18285">Numerics for a nonlinear equation with multiple solutions</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal">
Zhigang,
</p>
<p class="MsoNormal">
As many already pointed out, arc length strategies are<br />
available in most FE codes. What these strategies allow for is a “continuation”<br />
on a solution path from a point that you already determined on that path. Most<br />
typically one starts from 0 and identifies the “primary” equilibrium path but<br />
if you happen to know a solution on a different branch, you can certainly start<br />
from there. Stability of the configurations on this path can also be monitored<br />
easily. Many codes also offer capabilities for branch switching. All these<br />
options are fairly common and reliable, at least for a scalar p as you describe<br />
your nonlinear function. Most codes will also provide the energy.
</p>
<p class="MsoNormal">
Note however that if solution branches exist that are NOT<br />
connected to the primary branch, if you start from 0 (or whatever your unloaded<br />
or reference state is) you cannot reach those via a continuation method. Physically,<br />
a system has to jump dynamically onto that unconnected branch. There is also no<br />
method (at least none that I am aware of) that can tell HOW many solutions<br />
branches the system has. And given that a typical FE discretization can lead to<br />
very high dimensional systems, this can become a significant challenge.
</p>
<p class="MsoNormal">
Also note that in problems like the snap-through (e.g.,<br />
curved beams or panels) the actual system response after it snaps is transient.<br />
In these problems, if the primary branch has a segment with unstable<br />
configurations, from a physical point of view the system has to “jump” over<br />
this segment and retrieve another (remote) stable configuration. <span> </span>So it is no longer a problem of identifying<br />
an equilibrium point.
</p>
<p class="MsoNormal">
If p is not a scalar, there also exist multi parameter<br />
continuation algorithms. Sandia has a code, LOCA that can do that <a href="http://www.cs.sandia.gov/LOCA/">http://www.cs.sandia.gov/LOCA/</a> To the best of my knowledge, commercial FE codes like ABAQUS<br />
or ANSYS do not have an equivalent option.<span> </span>I have no personal experience with LOCA so I cannot comment<br />
on how reliable it is for very complex problems. In my research I am interested<br />
in finding the critical states (snap-through events) of systems with multiple<br />
parameters. The solution manifold in these cases is extremely rich. Finding the<br />
primary solution path and identifying the stability of those configurations is<br />
definitely doable. The biggest challenge is to find robust time integrators to<br />
obtain the transient post snap response. In many cases snap-through is also<br />
closely related to chaos, at least in the type of systems I am looking at.
</p>
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</ul>Thu, 09 Feb 2012 03:25:12 +0000ilincacomment 18315 at https://imechanica.orgMaterial instabilities
https://imechanica.org/comment/18312#comment-18312
<a id="comment-18312"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18302#comment-18302">approach to deal with instability numerically</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Zhigang,
</p>
<p>
A few remarks, drawing from the recent work of a post-doc. For nonlinear FE structural analysis with material and/or geometric nonlinearities, the Generalized Displacement ControlMethod (GDCM) appears to be the preferred method-of-choice: tracks the load-displacement path well in the presence of snap-back, limit points and/or softening behavior. The algorithm is due to <a href="http://www1.aiaa.org/content.cfm?pageid=406&gTable=JAPaperImportPre97&gid=10529">Yang and Shieh (1990)</a> and is widely used in structural analysis codes. In our work, we used GDCM for modeling reinforced concrete structures with max-ent and it performed robustly in capturing the softening branch: constitutive behavior of concrete included material degradation through a rotated smeared crack band model. Many good reviews on the GDCM can be found via a search on google. If there is interest, can ask the post-doc (he is back in Italy now) for further details.
</p>
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</ul>Wed, 08 Feb 2012 20:39:00 +0000N. Sukumarcomment 18312 at https://imechanica.orgArclength in ABAQUS
https://imechanica.org/comment/18310#comment-18310
<a id="comment-18310"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18308#comment-18308">Re: Arclength in ABAQUS</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Prof. Suo,
</p>
<p>
Both ANSYS and ABAQUS have the capability to define user material. The user can develop a user material subroutine in FORTRAN and make the software use this subroutine in its stress calculations at Gauss points. Here is a PDF file for ANSYS:
</p>
<p>
<a href="http://www.ansys.spb.ru/pdf/present/usermat.pdf">http://www.ansys.spb.ru/pdf/present/usermat.pdf</a>
</p>
<p>
A similar capability should be available in ABAQUS as well. There are certain limitations for this capability. For example certain elements should be used, the type of integration might be limited (hypoelastic material versus hyperelastic ones), the user should work with macros instead of GUI and so on. The following keywords can be helpful to find information on the net:
</p>
<p>
User Material Subroutine ANSYS/ABAQUS
</p>
<p>
Regards
</p>
<p>
Mohsen
</p>
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</ul>Wed, 08 Feb 2012 16:24:24 +0000M. Jahanshahicomment 18310 at https://imechanica.orgRe: Arclength in ABAQUS
https://imechanica.org/comment/18308#comment-18308
<a id="comment-18308"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18304#comment-18304">Arclength in ABAQUS</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Mohsen: Thank you! Can you point to a reference or give more detail?</p>
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</ul>Wed, 08 Feb 2012 14:54:58 +0000Zhigang Suocomment 18308 at https://imechanica.orgArclength in ABAQUS
https://imechanica.org/comment/18304#comment-18304
<a id="comment-18304"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18301#comment-18301">How is arclength defined in ABAQUS?</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Prof. Suo,
</p>
<p>
It should be possible to define a user material routine. In such a routine,<br />
one can implement the tangent operator, stress integration procedures at Gauss<br />
points and the overall convergence loop into which a customized arclenght method<br />
can be integrated.
</p>
<p>
Mohsen
</p>
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</ul>Wed, 08 Feb 2012 11:39:36 +0000M. Jahanshahicomment 18304 at https://imechanica.orgapproach to deal with instability numerically
https://imechanica.org/comment/18302#comment-18302
<a id="comment-18302"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18287#comment-18287">My experience of simulating mechanical instabilities</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Shengqiang: Thank you for sharing your experience and thoughts. In the book <a href="http://www.amazon.com/dp/144191739X/ref=rdr_ext_tmb">Practical Bifurcation and Stability Analysis</a> by Rudiger Seydel, the numerical analysis of a nonlinear equation is broken down into the following tasks:
</p>
<ol><li>Find one solution on a branch of solutions.</li>
<li>Trace the branch of solutions.</li>
<li>Watch out for a point of bifurcation.</li>
<li>Upon reaching the point of bifurcation, switch from one branch of solutions to another. </li>
</ol><p>
"A nonlinear equation" means a set of nonlinear algebraic equations with a parameter:
</p>
<p>
f(u, p) = 0
</p>
<p>
where u is a vector of n components, p is a scalar, and f is a set of n functions. For a given value of p, the vector u that satisfies f(u, p) = 0 <br />
is a solution. As p changes, the solutions form a branch.
</p>
<p>
In the context of mechanics, we may think of u as a set of generalized coordinates that describe the configuration of a system, p as an applied load, and f (u, p) = 0 as conditions of equilibrium. The set of nonlinear algebraic equations can result from a descretization of a nonlinear differential equation.
</p>
<p>
It seems that the numerical methods described in the book should apply to finite element analysis. For example, the methods should readily handle diffused modes of instability, such as buckling and snap-through instability.
</p>
<p>
Localized instability, however, will pose specific issues. You have mentioned crease. One may also mention cavitation, shear bands, and fracture. In the localized instability, one may need to add additional ingredients to the original PDE. For example, in dealing with cavitation, one may as well add a small cavity to begin with. I also really like your approach to simulate the formation of crease:
</p>
<p>
Shengqiang Cai, Katia Bertoldi, Huiming Wang, and Zhigang Suo. <a href="http://www.seas.harvard.edu/suo/papers/233.pdf"> Osmotic collapse of a void in an elastomer: breathing, buckling and creasing.<br /></a> Soft Matter 6, 5770-5777(2010).
</p>
<p>
You simply add a crease-like defect into the mesh. To some extent, your appoach addresses task 1. One can still use generic methods for the remaining tasks.
</p>
<p>
As you pointed out, "adding new ingredient" to the original PDE require experimental observations and physical insight.
</p>
<p>
Perhaps we already have a sensible approach to deal with instability numerically. We just stick to the generic tasks as much as we can. For any "additional ingredients", we ask why we need them, and whether we can abstract them to solve other problems.
</p>
<p>
I'd like to hear more from you and others concerning the general approach to simulate and discover instability.
</p>
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</ul>Wed, 08 Feb 2012 10:46:00 +0000Zhigang Suocomment 18302 at https://imechanica.orgHow is arclength defined in ABAQUS?
https://imechanica.org/comment/18301#comment-18301
<a id="comment-18301"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/18291#comment-18291">Riks method for the simulation of snap-through instability</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Lihua: Thank you for describing your experience. Let's hope others will also jump in and share their experience.
</p>
<p>
The Riks method is discussed in Section 4.5 in the book <a href="http://www.amazon.com/dp/144191739X/ref=rdr_ext_tmb">Practical Bifurcation and Stability Analysis</a> by Rudiger Seydel. It is one approach to choose a parameter that allows the computer to trace a branch of solutions that contains a turning point. In one formulation, the parameter is chosen as the arclength defined in the space of all displacements and the loading parameter (i.e., the n+1 space).
</p>
<p>
How does ABAQUS implement the Riks method? In particular, how is the arclength defined in ABAQUS? Once you use the Riks method to trace a branch of states, does ABAQUS allow you to calculate energy for an unstable state? Using ABAQUS, do you still have any issue to deal with turning points?
</p>
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</ul>Wed, 08 Feb 2012 10:10:00 +0000Zhigang Suocomment 18301 at https://imechanica.orgError | iMechanica