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# Journal Club Theme of February 2012: Elastic Instabilities for Form and Function

Welcome to February 2012's Journal club, which will include a discussion on elastic instabilities for form and function. Not long ago, the loss of structural stability through buckling generally referred to failure and disaster. It was a phenomenon to be designed around, and rarely did it provide functionality*. The increasing focus on soft materials, from rubbers and gels to biological tissues, encouraged scientists to revisit the role of elastic instabilities in the world around us and inspired their utilization in advanced materials. Now the field of elastic instabilities, or extreme mechanics, brings together the disciplines of physics, mechanics, mathematics, biology, and materials science to extend our understanding of structural instabilities for both form and function. In this journal club, we're going to look at research on the wrinkling, crumpling, and snapping of soft or slender structures.

Swelling-Induced Buckling & Wrinkling

T. Tanaka, S-T. Sun, Y. Hirokawa, S. Katayama, J. Kucera, Y. Hirose, and T. Amiya, Mechanical instability of gels at the phase transition, *Nature*, **325**, 796, (1987).

J. Dervaux, Y. Couder, M-A Guedueau-Boudeville, M. Ben Amar, Shape Transition in Artificial Tumors: From Smooth Buckles to Singular Creases, *Physical Review Letters*, **107**, 018103, (2011).

Pattern formation, especially unexpected and aesthetically pleasing patterns, has fascinated scientists for centuries. While studying the kinetics of swelling gels (physically or chemically crosslinked polymer networks immersed in a fluid, typically water), Tanaka et al. noticed a pattern emerge as a confined gel underwent extensive swelling. The pattern appeared to be hexagonal in nature and the length scale of its features changed as swelling progressed. Their research suggested a relationship between the stress generated from osmotic pressure, the kinetics of the swelling process, and the dominant length scale of the observed mechanical instability. The results sparked a surge of research on the bending, creasing, and wrinkling of swollen gels. Recent work on swelling and buckling has highlighted pattern control, thin film mechanics, non-Euclidean geometry, and morphogenesis.

The field of morphogenesis has long discussed the connection between structural stability and the shapes of growing objects. Coming out of a mathematical framework known as catastrophe theory which seeks to provide a qualitative understanding of complex systems, Rene Thom developed a methodological approach to connecting morphogenesis with structural stability [1,2]. Recently, an interesting analogy between swelling hydrogels and artificial tumor growth was discussed by Dervaux et al. The complex biochemical processes behind the development of solid tumors can lead to multilayered structures which grow at different rates. On a qualitative level, the mechanics of this phenomenon can be experimentally modeled by swelling a multilayer gel whose geometry and properties lead to different swelling rates. It is known that applying stress to a symmetric plate leads to buckling and symmetry breaking [3,4], and by combining this selective swelling with confined circular discs, Dervaux et al. were able to observe a symmetry breaking that included smooth undulations or sharp creases. Their experimental results follow a nonlinear poroelastic model, and highlight a transition between wrinkling and crease formation at a free boundary. Although the migration of solvent within an extensible matrix is not equivalent to the solid material growth in soft tissues, these experiments provide a simple analogy to the stress-induced growth of soft structures.

Crumpling

A.D. Cambou and N. Menon, Three-dimensional structure of a sheet crumpled into a ball, *Proceedings of the National Academy of Sciences*, **108** (36), 14741, (2011).

Crumpling is a commonly observed phenomenon that can be seen by simply compressing a piece of paper. Consider the thought experiment in which a thin sheet is confined within a sphere and the size of the sphere is being continuously decreased. The sheet of paper will bend and ultimately exhibit sharp creases**. The thinner the sheet, the higher the curvature of the folded regions [5]. The three-dimensional structure that emerges with these crumpled sheets is quite complex, but as Dominique Cambou and Menon point out in a recent article, there exist many consistencies within the crumpled structure, regardless of how the crumpling is achieved. The authors studied the three-dimensional structure of thin sheets of aluminum that were arbitrarily crumpled into nearly-spherical balls. The complete structure of the thin sheet was resolved using an X-ray tomography scanner. One of the most striking findings is that much of the geometry of the crumpled ball is homogenous and isotropic. The folds or regions of high-curvature were uniformly distributed, and a large fraction of the sheet is mostly flat. The local orientation of the sheet within the crumpled structure was found to be homogenous, except for the outermost layers. Interestingly, the flat regions within the structure order in a nematic manner into parallel stacks. The results presented suggest that much of the structure of the far-from-equilibrium shape of an crumpled thin sheet may be well suited for a complete statistical mechanics description.

Snapping

C. Keplinger, T. Li, R. Baumgartner, Z. Suo, and S. Bauer, Harnessing snap-through instability in soft dielectrics to achieve giant voltage-triggered deformation, *Soft Matter*, **8**, 285, 2012.

The snapping instability gained some notoriety in 2005 when Forterre et al. proposed that a geometric snap-buckling instability was the governing mechanics behind the rapid closure of the Venus flytrap's leaves [6]. This response is a fascinating example of an elastic instability utilized for biological function. The highlight of the flytrap's snapping leaves is its rapid timescale (100ms, one of the fastest movements in the plant kingdom), but in more general terms, the snap-through instability describes a transition between two non-local stability states. Euler buckling represents a globally stable catastrophe (i.e. there is always some stable state) and is attributed to a loss of stability where the post-critical state is infinitesimally close to the pre-critical state. Snap-through, on the other hand, refers to a globally unstable catastrophe, known as a dual cusp catastrophe, where the pre- and post-critical states are separated by some finite distance. Once the critical criteria for stability loss is met; there are no stable, intermediate states, and a jump between states occurs.

Since research on snap-buckling began on arches (used in structural engineering) and shells (used in designing airplanes) the term catastrophe was quite appropriate. But, recent research has harnessed this instability to create rapid, responsive structures [7,8]. In an article in Soft Matter, Keplinger et al. combine the mechanical snap-through instability commonly observed while inflating a balloon with dielectric elastomers to achieve massive voltage-triggered deformations. Building upon theory developed by Zhao and Suo [9], Keplinger et al. begin by inflating a soft dielectric membrane near the verge of instability, and then trigger the instability with an applied voltage. A dramatic thinning of the membrane thickness (which would lead to electric breakdown) accompanies the snap-through instability of a balloon. Their system was designed to avert this expected electrical breakdown by including a subsequent pressure drop within the chamber beneath the membrane. The result is a dielectric elastomer that can achieve an area expansion of 1692% while averting electrical breakdown using a snap-through instability.

Future Directions

The history of buckling instabilities goes back centuries, and while many of its fundamental mechanics are well understood, there remains an assortment of open questions in a wide range of fields. Soft and slender structures and complex geometries are more easy to fabricate and analyze than ever before. It is likely that this community has only scratched the surface of using elastic instabilities in the design of advanced materials and in the explanation of complex, natural phenomena.

* One example in which Euler buckling is utilized to provide functionality is the steering column in a car which is designed to buckle under significant loads to prevent it from further injuring the driver in a crash.

** An interesting aspect of this phenomenon is that these focused, sharp structures do not form in the one-dimensional case of confining a thin wire.

**Further Reading**

[1] R. Thom, Structural Stability and Morphogenesis, Advanced Book Program, (1972).

[2] R. Gillmore, Catastrophe Theory for Scientists and Engineers, Dover Publications, Inc., p254-294, (1981).

[3] L. Freund, Substrate curvature due to thin film mismatch strain in the nonlinear deformation range, *J. Mech. Phys. Solids*, **48**, 1159-1174, (2000).

[4] D.P. Holmes, M. Roche, T. Sinha, and H.A. Stone, Bending and twisting of soft materials by non-homogenous swelling, *Soft Matter*, **7**, 5188, (2011).

[5] T.A. Witten, Stress focusing in elastic sheets, *Reviews of Modern Physics*, **79**, (2007).

[6] Y. Forterre, J.M. Skotheim, J. Dumais, L. Mahadevan, How the Venus flytrap snaps, **433**, 421, (2005).

[7] D.P. Holmes and A.J. Crosby, Snapping Surfaces, *Advanced Materials*, **19**(21), 3589, (2007).

[8] Lienhard, J. et al., Flectofin: a hingeless flapping mechanism inspired by nature, *Bioinspiration and Biomimetics*, **6**(4), (2011).

[9] X. Zhao and Z. Suo, Theory of Dielectric Elastomers Capable of Giant Deformation of Actuation, *Physical Review Letters*, **104**, 178302, (2010).

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## Comments

## Harnessing or suppressing instabilities

Doug,

Thanks for posting such an interesting topic. Investigating mechanical instabilities of thermodynamic systems has been one of the research thrusts of my group since the very beginning. The goals of our study are twofold:

to understand the physical mechanism behind new instabilitiesandto achieve extraordinary functions by harnessing or suppressing instabilities. Some examples of our work are listed as follow:Discovery of electro-creasing to cratering instabilities in deformable dielectrics,

PRL, 106, 118301 (2011).

Quantitative explanation of necking instability in double-network hydrogels

JMPS, 60, 319 (2012).

Enhancement of electric energy density of dielectrics over 10 times by suppressing mechanical instability

APL, 99, 171906 (2011)

Soft Matter,7, 6583 (2011)

## Doug, thanks for posting

Doug,

thanks for posting on this exciting topic. Over the past few years soft materials have really driven the scientific community into new directions. In this scenario mechanical instabilities assumed a new role since they give us the opportunity to design materials and devices with switchable functionalities. A couple of more instability-related papers:

J.H. Jang, C.Y. Koh, K. Bertoldi, M.C. Boyce and E.L. Thomas.

Combining Pattern Instability and Shape-Memory Hysteresis for Phononic Switching. Nano Letters, 9, 2113-2119, 2009.K. Bertoldi and M.C. Boyce.

Mechanically-Triggered Transformations of Phononic Band Gaps in Periodic Elastomeric Structures. Physical Review B, 77, 052105, 2008.Cullen, D. K., Browne, K. D., Xu, Y., Adeeb, S., Wolf, J. A., McCarron R. M., Yang, S., and Smith, D. H.,

Blast-Induced Color Change in Photonic Crystals Corresponds with Brain Pathology, accepted to J. Neurotrauma, 2012From my point of view, this new scenario is even more exciting, since physicists, mechanicians, material scientists and biologists are finally working synergistically!

## Numerics for a nonlinear equation with multiple solutions

I’m reading the book entitled Practical Bifurcation and Stability Analysis by Rudiger Seydel. It seems that systematic numerical methods exist to solve a nonlinear equation of multiple solutions. Once you have these solutions, you can test the stability of each solution, switch from one branch of solutions to another, or snap from one state to another. You can decide to call one solution localized and another diffused.

The book looks practical, focusing on issues and solutions, and is mainly written for engineers and scientists who will use the methods. The methods are generic, independent of field of application.

Here is my question: to what extent have mechanicians implemented these methods into our general-purpose finite-element codes? I’d love to hear about your experience.

## Re: Numerics for nonlinear equations with multiple solutions

Zhigang,

As many already pointed out, arc length strategies are

available in most FE codes. What these strategies allow for is a “continuation”

on a solution path from a point that you already determined on that path. Most

typically one starts from 0 and identifies the “primary” equilibrium path but

if you happen to know a solution on a different branch, you can certainly start

from there. Stability of the configurations on this path can also be monitored

easily. Many codes also offer capabilities for branch switching. All these

options are fairly common and reliable, at least for a scalar p as you describe

your nonlinear function. Most codes will also provide the energy.

Note however that if solution branches exist that are NOT

connected to the primary branch, if you start from 0 (or whatever your unloaded

or reference state is) you cannot reach those via a continuation method. Physically,

a system has to jump dynamically onto that unconnected branch. There is also no

method (at least none that I am aware of) that can tell HOW many solutions

branches the system has. And given that a typical FE discretization can lead to

very high dimensional systems, this can become a significant challenge.

Also note that in problems like the snap-through (e.g.,

curved beams or panels) the actual system response after it snaps is transient.

In these problems, if the primary branch has a segment with unstable

configurations, from a physical point of view the system has to “jump” over

this segment and retrieve another (remote) stable configuration. So it is no longer a problem of identifying

an equilibrium point.

If p is not a scalar, there also exist multi parameter

continuation algorithms. Sandia has a code, LOCA that can do that http://www.cs.sandia.gov/LOCA/ To the best of my knowledge, commercial FE codes like ABAQUS

or ANSYS do not have an equivalent option. I have no personal experience with LOCA so I cannot comment

on how reliable it is for very complex problems. In my research I am interested

in finding the critical states (snap-through events) of systems with multiple

parameters. The solution manifold in these cases is extremely rich. Finding the

primary solution path and identifying the stability of those configurations is

definitely doable. The biggest challenge is to find robust time integrators to

obtain the transient post snap response. In many cases snap-through is also

closely related to chaos, at least in the type of systems I am looking at.

## libraries of bifurcation-tracking codes

Dear Ilinca: Thank you very much for sharing your experience. What a coincidence! On the bus to work this morning, I was reading a paper by the developers of LOCA. (I'm on sabbatical leave in Karlsruhe.)

This iMechanica thread has helped me greatly. You and others have confirmed that various aspects of bifurcation tracking have been integrated with FEM.

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.

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?

## Hi Zhigang, This is a

Hi Zhigang,

This is a subject close to my heart, and one I think is important.

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.

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.

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)].

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.]

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.

If you are interested in these ideas and have more questions, I'm happy to discuss this great topic further.

Cheers,

Ryan S. Elliott

## Symmetry and bifurcation

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

## My experience of simulating mechanical instabilities

Following Zhigang's question, I'd like to share some of my experience of simulating mechanical instabilities by using FEM softwares, such as ABAQUS.

After prescribing certain boundary conditions on a geometry, ABAQUS usually gives an equilibrium solution without testing its stability.

For example, in ABAQUS, if you simulate a bar under compression, without any treatment, you can easily get a significantly compressed geometry far beyond the Euler buckling point.

However, in many cases, we know the possible instability mode in the geometry before doing simulations, through experimental obervations or some intuition.

Therefore, I ususally use linear perturbation function embedded in ABAQUS to perturb a predeformed geometry. (I learned the skills from Katia.)

Through solving eigenvalue problems, the software can report the critical load for different mode of instabilities.

So far as I know, this is also the typical procedure adopted by most people.

However, the linear perturbation is just a subset of perturbations. It is principly impossible to use linear perturbaion method to predict all the instability mode, and creasing instability serves as a good example.

Particular perturbations are needed to simulate different instabilities and some experimental inspirations are very important.

Generally speaking, I have no idea of how to use a FEM software to find all the equilibrium states of a gemoetry and test the stability of each of them.

However, in some cases, the FEM calcuation indeed stops when the configuration is in the vicinity of a branching point. I sometimes can get some clues from the abortion of the calculations.

## approach to deal with instability numerically

Dear Shengqiang: Thank you for sharing your experience and thoughts. In the book Practical Bifurcation and Stability Analysis by Rudiger Seydel, the numerical analysis of a nonlinear equation is broken down into the following tasks:

"A nonlinear equation" means a set of nonlinear algebraic equations with a parameter:

f(u, p) = 0

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

is a solution. As p changes, the solutions form a branch.

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.

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.

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:

Shengqiang Cai, Katia Bertoldi, Huiming Wang, and Zhigang Suo. Osmotic collapse of a void in an elastomer: breathing, buckling and creasing.

Soft Matter 6, 5770-5777(2010).

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.

As you pointed out, "adding new ingredient" to the original PDE require experimental observations and physical insight.

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.

I'd like to hear more from you and others concerning the general approach to simulate and discover instability.

## Material instabilities

Zhigang,

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 Yang and Shieh (1990) 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.

## I think people here might

I think people here might be interested in our robust and efficient FEM algorithm for buckling simulation. Please see the following link.

http://imechanica.org/node/4124

## Riks method for the simulation of snap-through instability

Thank Doug for the interesting topic, and thank Zhigang for the question. I think simulation is a difficulty for instability problems.

I have been using finite element software ABAQUS to simulate instability. Besides what Shengqiang mentioned, I also used Riks method to simulate snap-through instability.

For example, we want to use finite element to simulate the snap-through of a spherical balloon under inner pressure. We know the pressure is supposed to increase, then decrease, and increase again with the volume of the balloon, which is called a snap-through instability. If we use pressrue as the loading parameter, and normal Newton's method, the calculation stops at the first peak, since the slope becomes zero. If we change to use volume as the controlling parameter, the problem can be solved and we can get the whole non-monotonic pressure-volume relation.

As a more general strategy, in the Riks method both displacement and load are considered as unknowns, and the arc length is chosen to describe the calculation process, even if pressure is set as the controlling parameter. Still Newton's method is used to solve the nonlinear equilibrium equation for the incrementation. Since arc length is used to measure the solution progress, the situation of vanishing slope can be conquered, and then displacement and load can be solved simultaneously, no matter whether the solution is stable or unstable. Thus both loading and unloading (increase or decrease of pressure) are possible.

In ABAQUS, to use the Riks method, you can simply choose static-Riks in the step module. Riks method in ABAQUS is only limited to proportional loading.

I can think of some other solutions to instability simulations: (1) dynamic procudure, (2) using damping to stablize the structure. Unfortunately I don't have too much experience on these. Is there anyboy who can say something about these?

## How is arclength defined in ABAQUS?

Dear Lihua: Thank you for describing your experience. Let's hope others will also jump in and share their experience.

The Riks method is discussed in Section 4.5 in the book Practical Bifurcation and Stability Analysis 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).

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?

## Arclength in ABAQUS

Dear Prof. Suo,

It should be possible to define a user material routine. In such a routine,

one can implement the tangent operator, stress integration procedures at Gauss

points and the overall convergence loop into which a customized arclenght method

can be integrated.

Mohsen

## Re: Arclength in ABAQUS

Dear Mohsen: Thank you! Can you point to a reference or give more detail?

## Arclength in ABAQUS

Dear Prof. Suo,

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:

http://www.ansys.spb.ru/pdf/present/usermat.pdf

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:

User Material Subroutine ANSYS/ABAQUS

Regards

Mohsen

## Riks method in ABAQUS

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.

http://mse-license1.mse.drexel.edu:2080/v6.8/books/stm/default.htm?startat=ch02s03ath18.html#stm-anl-modifiedriks

## Operate a device on the verge of instability

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.

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.

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.

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.

An analysis of a particularly simple setup and an experimental demonstration are given in two recent papers:

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.

## Design principals for functionality

Thanks for your additional notes on this topic Zhigang.

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.

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.

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].

I think the key is identifying the benefits of specific mechanical instabilities and combining those ideas with desirable functionality.

[1] A.J. Crosby, M. Hageman, A. Duncan, Controlling Polymer Adhesion with "Pancakes", Langmuir,

21, 25, (2005).[2] T. Thomas and A.J. Crosby, Controlling Adhesion with Surface Hole Patterns, J. Adhesion,

82, 3, (2006).[3] D.P. Holmes and A.J. Crosby, Snapping Surfaces, Advanced Materials,

19, 21, (2007).## Soft machines: increased functionality and extreme performance

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.

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.

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.

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 (http://en.wikipedia.org/wiki/Tunnel_diode) 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.

## Instability: Friend or Foe?

Dear all,

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.

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.

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.

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.

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.

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.

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.

[1] Pelrine, R.; Kornbluh, R.; Pei, Q.; Joseph, J. Science

287, 836–839 (2000).[2] Ha, S. M.; Yuan, W.; Pei, Q. B.; Pelrine, R. Adv. Mater.

18, 887–891 (2006).[3] Soo Jin Adrian Koh, Tiefeng Li, Jinxiong Zhou, Xuanhe Zhao, Wei Hong, Jian Zhu, Zhigang Suo. Mechanisms of large actuation strain in dielectric elastomers. Journal of Polymer Science Part B: Polymer Physics

49, 504-515 (2011).[4] T. A. Gisby, S. Q. Xie, E. P. Calius, and I. A. Anderson, Proc. SPIE

7642, 764213 (2010).[5] Choon Chiang Foo, Shengqiang Cai, Soo Jin Adrian Koh, Siegfried Bauer, Zhigang Suo. Model of dissipative dielectric elastomers. Journal of Applied Physics

111, 034102 (2012).## What's Next?

With a week remaining in February, allow me to pose the following question:

Where do you see this field going in the future?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.

## Harnessing instabilities: what's next?

I have the good fortune of interacting with people from a number of industries.

One particular problem comes to mind at this stage - the problem of air transport of frozen fish.

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.

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).

But more than solid mechanics has to go into the design of these structures.

-- Biswajit

## What's Next? Instability: diversity and unification

Although instability has long been part of the education of mechanicians, several recent trends demand that we rethink what we teach.

Computation. 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.Microfabrication. 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.A catalog of many kinds of instability. 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:We may witness a change of emphasis in mechanics education in coming years.

## Random perturbation approach

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?

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.

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? 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.

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?

## Re: Random perturbation approach

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:

dx/dt = f(x,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.

Dynamic simmulation.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.Bifurcation analysis. A state of equilibrium is determined byf(x,p) = 0.

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.

These two approaches are described well in many textbooks. Here is a readable one: Practical Bifurcation and Stability Analysis by Rudiger Seydel.

## The relevance of equilibria in the problem of evolution

Zhigang,

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.

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

http://www.imechanica.org/node/7312

we characterize analytically the whole (infinite-dimensional) class of equilibria of the dynamics involving a parameter - the applied strain (g) as loading.

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

http://www.imechanica.org/node/11819

Finally, I point readers (especially young ones) to the following paper:

http://dml.cz/bitstream/handle/10338.dmlcz/134168/MathBohem_127-2002-2_3...

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.

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. http://www.imechanica.org/node/9906), one should expect such things to be the rule rather than the exception.

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

http://www.imechanica.org/node/10998

and a student, Likun Tan, will present her work in this forum related to such matters soon.

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).

- Amit

## Re: The relevance of equilibria in the problem of evolution

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.

Advances in Applied Mechanics33 193-294 (1997).## Here is the link of paper

Here is the link of paper Prof.Amit Acharya refers to

http://www.imechanica.org/node/12588

## random perturbation via a kinetics approach

Dear velandkar,

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 http://imechanica.org/node/12022). 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).

RH

## Instabilities & APS

As this Journal Club month draws to a close, I would like to draw your attention to the

Extreme Mechanicssymposium that begins on Tuesday, February 28th at this year's APS March Meeting in Boston, MA.Hope to see you there!

-Doug

## Looks Great

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.)

Eric Mockensturm

emm10@psu.edu

## A Few Last Ones

A few last references you might like...

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.

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.

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.

E. Mockensturm and N. Goulbourne, “Dynamic response of dielectric elastomers,” Int J Nonlinear Mech, vol. 41, no. 3, pp. 388–395, 2006.

Eric Mockensturm

emm10@psu.edu

## buckling analysis of 1D beam column

Hi,

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.

([Ke]-P[Kg])(Δ)=(F)

Where [Ke] = elastic stiffness matrix

P = buckling load

[Kg] = Geometric stiffness matrix

(Δ) = Global DOF matrix, and

(F) = lateral loads at each node.

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.

Best regards,

Brahmendra S Dasaka.

## Harnessing postbuckling behavior of cylindrical shell

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!

Full text can be found in: http://authors.elsevier.com/sd/article/S0263823114001670