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# Journal Club Theme of November 2012: Harnessing Instabilities in Response to Stimuli

After a few JClubs focused on elastic instabilities of materials and structures, I would like to continue the exciting discussions on instabilities in a broader context - harnessing instabilities in response to stimuli. Besides mechanical forces and stresses, many stimuli such as temperature, PH, light, electric field, magnetic field and chemical potential can induce complex modes of instabilities in diverse materials and structures. Examples range broadly from Turing patterns for morphogenesis in biology to ferroelectric ceramics used in modern technology. Discovering, understanding, and exploiting these instabilities are of both fundamental and practical importance to science and technology. Here, instead of giving an extensive review of the field, I will introduce our recent works on instabilities of dielectrics in response to electric fields. Comments with examples of instabilities in response to other stimuli would be more than welcome and greatly appreciated.

**Figure 1.** Evolution (left to right) of a water drop in a dielectric polymer under a ramping electric field [Wang et al, Nature Communications, 3, 1157 (2012)].

It is well known that surfaces and defects of solids can become unstable under mechanical loads, i.e. wrinkling or creasing on surfaces and evolution or cavitation of defects in solids. If we replaced mechanical loads with another stimulus, say electric field, would the surface or defect undergo similar modes of instabilities? How to understand instabilities under coupled physical fields and large deformation? Are these instabilities detrimental or beneficial? I would like to use the following papers to initiate our discussion, and invite Imechanicians to share your thoughts about this field and post relevant works here.

• Qiming Wang, Lin Zhang, Xuanhe Zhao, Creasing to Cratering instability in polymers under ultrahigh electric fields Physical Review Letters, 106, 118301 (2011). Supporting Information Video 1.

• Qiming Wang, Zhigang Suo, Xuanhe Zhao, Bursting Drops in Solids Caused by High Voltages Nature Communications, 3, 1157 (2012). Supporting Information Video 1 Video 2 Video 3.

- Xuanhe Zhao's blog
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## Comments

## creasing instability with influence of surface energy

Thanks Xuanhe for introducing this very insteresting example of illustrating harnessing instability. I would like to bring one more example to the discussion, which is about creasing instability.

When a soft material is compressed beyond a critical strain,the free surface suddenly forms creases, singular regions of self contact. In one of our recent studies, we found that the nucleation and growth of creases can be understoond in close analogy to classical nulceation theory for a thermodynamic phase transition. When the compression is high, forming a crease reduces elastic energy but increases the surface area. thus surface energy provides a nucleation barrier. We did a simple scaling analysis on crease nucleation and growth with considering surface energy effect, which agrees with experimental measurements.

The paper has been published in PRL and attached below

http://www.seas.harvard.edu/suo/papers/289.pdf

## Crease or wrinkle under dominant surface tension effect?

Shengqiang, thanks for sharing this inspiring paper!

In many cases, the effect of surface tension on instabilities of solids is relatively small (e.g. small capillary number), where surface defects or roughness provide critical sites for nucleation and the delay of the critical strain is also small. However, in many other cases, especially by modifying modulus, surface tension, and dimension of the samples (like in your work), one can significantly enhance the effect of surface tension. Now, quantitatively, surface tension can greatly delay the critical strain for propagation of the instability or "channeling" in your paper. A question for you. Would surface tension, if become dominant, qualitatively change the phenomenon, say transit from creasing to wrinkling in your work?

A more general question. Is there a robust and general computational method to predict these modes of instabilities that account for the effect of surface tension and/or coupling of physical fields?

## Good point. There is a need

Good point. There is a need a need for numerical tools capable of predicting the large-deformation, three-dimensional, coupled response. For the case of soft dielectric elastomers several approaches have been undertaken. First, simplified finite-element computational procedures have been proposed that make geometrical assumptions, reducing the electrical problem to one dimension (Wissler and Mazza, 2005, 2007; Zhao and Suo, 2008; Zhou et al., 2008). These techniques are useful for basic configurations; however, fully three-dimensional procedures are needed to guide the design of more complex designs. Recently, finite-element implementations have been reported using in-house codes both in quasi-static (Vu and Steinmann, 2007; Vu et al., 2007) and dynamic settings (Park et al., 2012); however, these codes are not available to the community. In my opinion, implementation of the theory within a widely-available finite-element software is a crucial step toward facilitating interactions between industry and researchers and guiding the design of complex three-dimensional systems. Unfortunately, this task is not straightforward within commercial finite-element packages, since additional nodal degrees of freedom are required. Few efforts in this direction have been reported, namely using Comsol (Rudykh and deBotton, 2012). Although Comsol is amenable to the implementation of the coupled electromechanical theory, its difficulty in dealing with large deformations is well-known, and as such, it is not well-suited for problems involving dielectric elastomers. To overcome these issues, we recently implemented the fully-coupled theory governing the behavior of dielectric elastomers in the commercial finite-element code Abaqus/Standard, taking full advantage of the capability to actively interact with the software through user-defined subroutines. Abaqus is an attractive platform because it is a well-known code, widely-available, stable, portable, and particularly suitable for analyses involving large deformations.

We hope this will opens the door to further simulation-based study of complex dielectric elastomeric structures.

## ABAQUS standard or explicit?

Katia, thanks for this nice summary of existing computational tools on dielectric polymers! Implementing field-coupled theories into commercially available software is definitely an attractive approach in the field, yet it is still a challenging task. Look forward to seeing your codes soon! One specific question for you. Why do you choose Abaqus standard, which may incur convergence problem especially when handling instabilities, instead of Abaqus explicit? For instance, in implementing strain gradient theory into Abaqus, it seems many people preferred Abaqus explicit.

I would also invite Imechanicians to share your thoughts on computational techniques for instabilities in response to other stimuli, such as those in magnetostrictive materials, swelling gels, shape memory alloys/polymers, light sensitive polymers et al.

## numerical simulations of instabilities

Thanks Xuanhe and Katia for bringing up this important and interesting issue. It's clear that computation does have an important role to play in these complex, coupled field problems with instabilities. I have a couple of thoughts related to the approaches people are using to study instabilities. First, while there isn't an extensive literature of FEM approaches for elastomers, all of the approaches I have seen so far are quasistatic. If we simplify the problem to consider only mechanical instabilities, there are still issues with quasistatic approaches, namely that the critical (stress/strain, etc) for instability can be predicted, but it becomes challenging to resolve the evolution of the instability as arc-length and other standard approaches may not be completely robust - this could be problematic, for example, if crack-like propagation (as seen in Xuanhe's excellent recent Nature Communication papers shows) is to be studied. In contrast, dynamic simulations naturally capture the time evolution of the instability, and is the reason why we adopted this line of thinking in our recent publication:

(http://www.sciencedirect.com/science/article/pii/S0020768312001783)

Another thought is that material incompressibility has not been addressed in the existing FEM formulations of elastomers, and thus the constitutive behavior that is being modeled may not be realistic. Incompressibility is a challenging issue (again looking at it from a purely mechanical point of view) which will very much complicate the ability of any numerical calculation to capture instabilities like wrinkling or creasing. Finally, rate-dependent material response (i.e. viscoelasticity) has also rarely been accounted for, usually in a linearized form. These issues thus go beyond the need for implementing coupled physics field equations in commercial FEM codes, and point out that new computational formulations, because the issues arise in an electromechanical, and not purely mechanical, setting, are also needed.

I have recently done some work that addresses all of the above issues (though, as Katia pointed out above, not implemented in a commercial FEM code). I would be happy to post that paper once it is accepted.

## Terrific insights and

Terrific insights and paper! Thanks, Harold. Large deformation, field-coupling, geometrical singularity, incompressibility and viscoelasticity indeed represent some of the great challenges in the field. Look forward to your new work out soon!

## Computational Modeling of Electromechanical Instabilities

Hi Xuanhe - the paper I referred to previously has been published online. I detailed this in a separate post:

(http://imechanica.org/node/13755)

## Standard vs Explicit

Yes, it is sometimes painful to use the standard methods to simulate instabilites, especially when there is contact. Also, it might be hard to identify the mode with the lowest energy with standard method, if there are several modes with similar energy levels. However, for the explicit methods, when there is a very small length scale in the problem, like the situation of thin films, the small size of the elements may make the simulation very time consuming. Incompressibility will also make the limit time step smaller. Another disadvantage of the explicit method is that it can only get the stable solutions, but not the unstable ones. While for the standard method, with some special algorithm, like arc length method, it is possible to get the unstable solutions.

## why explicit dynamic method cannot get the unstable solution?

It's interesting, but I do not quite understand why explicit dynamic method cannot get the unstable solution. To my knowledge, explicit method can be as accurate as standard method if we can properly calibrate some data to minimize the dynamic effect.

## unstable solutions

By 'unstable solutions', I mean the solutions which maximize the energy, although they are also solutions to the equilibrium equations. In dynamic simulations, due to the kinetic energy, the state cannot stay on the energy maximum, but always goes to the minimum. Actually, the normal standard method can't capture the unstable solutions either, and some special algorithms, like arc length method, are needed to get the unstable solutions. Indeed, for the stable solutions, the explicit method can be as accurate as the standard method.

## Surface tension in soft materials

Dear Xuanhe,

Thanks for the interesting question. Actually, in our case, we did not find the transition from creasing instability to wrinkling instability. The critical strain for wrinkling instability is always much higher than creasing instability (see the inset in Fig.2). However, I am not sure if such transition can happen in other circumstances.

For numerical simulations, Katia and I have really tried to embed the effect of surface tension into abaqus, but have not succeeded so far.

## a possible transition

Interesting discussions. Surface tension alone may not lead to transition from creasing to wrinkling. But other surface effects may. For example, a stiff superficial layer would most likely wrinkle first. Such surface is not uncommon in soft materials. In a recent paper, we showed that swelling of a hydrogel with a stiff surface layer wrinkles first and then evolves to form creases. A preprint of the paper is available at http://www.ae.utexas.edu/~ruihuang/papers/Bilayer2012.pdf.

RH

## Post-wrinkling instability

Rui, thanks for sharing your thought and this interesting paper!

Post-wrinkling of bilayer structure is definitely a fascinating but complicated problem in solid mechanics. Depending on properties, dimensions and pre-deformation of the layers, one may get a wide variety of post-wrinkling modes such as double pairing, folds, hierachical wrinkles... ... In collaboration with Prof Hutchinson and Prof Cao, we recently discovered yet another new mode, localized ridge.

Jianfeng Zang, Xuanhe Zhao, Yanping Cao, John W. Hutchinson,

Localized Ridge Wrinkling of

Stiff Films on Compliant Substrates, Journal of the Mechanics and Physics of

Solids, 60, 1265-1279 (2012)

## Soft robotics controled by instability

In the following paper, a soft pneumatic composite tentacle is fabricated. The pneumatic tentacle can bend by asymmetrically pressurizing the pneumatic channels. The interesting thing is that instead of bending homogeneously, the tentacle always starts bending at one end. It turns out that this is related to the instability of the structure, and due to the boundary condition, the instability condition is always first satisfied at the end. The instability triggers a big deformation and big bending curvature of the soft robot. Thus, by careful design, we can use instability to control the behavior of soft robots.

http://www.seas.harvard.edu/suo/papers/279.pdf

## harness instability to achieve extraordinary functions

Lihua, Christoph, thanks for sharing your thoughts and papers. Harnessing instabilities in response to external stimuli to achieve extraordinary functions is definitely an exciting new area. By harnessing the electro-creasing-cratering instability, we recently invented a new technology called "Dynamic Electrostatic Lithography"

Qiming Wang, Mukarram Tahir, Jianfeng Zang, and Xuanhe Zhao, Dynamic Electrostatic Lithography: Multiscale On-demand Patterning on Large-Area Curved Surfaces, Advanced Materials , 24, 1947–1951 (2012)

## EMI in dielectric elastomers: To like or not to like?

Thanks Xuanhe for bringing up the wonderful topic of 'harnessing instabilities in response to stimuli'.

I have read with great interest the above discussions!

Let me briefly share my own experiences with the topic of instabilities in soft matter:

The first period of research on dielectric elastomer actuators was characterized by a common theme: how to avoid instabilities that usually lead to failure such as dielectric breakdown.

In our paper "Roentgen's electrode-free elastomer actuators without electromechancial pull-in instability" (http://www.pnas.org/content/107/10/4505.full.pdf), we showed that spraying electrical charge onto dielectric elastomer membranes allows for giant electrically induced deformation, as the electromechanical instability is not an issue for this form of operation.

But on a second thought, is it always necessary to avoid or suppress the electromechanical instability? Without doubt it is easier to control electromechanical transducers with a linear response. But can we harness the instability and the highly nonlinear behavior to achieve performance that would not be possible with linear systems?

In the recent two papers

http://www.seas.harvard.edu/suo/papers/251.pdf

and

http://www.seas.harvard.edu/suo/papers/271.pdf

we show that the electromechanical instability in dielectric elastomer actuators can be harnessed to achieve a voltage induced area expansion of far above 1000%. The operation of systems near the verge of instability opens up new options for the design of actuators that can be triggered by small voltages to achieve large deformations.

For me personally the most important lesson I have learned from these research experiences is the following:

"Instabilities make things more complex, which is often not desirable for applications. On the other hand they allow for specific forms of extreme performance that would be very hard or impossible to achieve with linear systems".

## Microstructures of plants and internal rupture of seals

Thanks Xuanhe for this timely post!

Yet another example of an advantageous use of instabilities --- which has partly motivated our recent work on cavitation in elastomers (Lopez-Pamies et al., 2011) --- is the process by which plants can create their tubular microstructures; the beautiful paper by Takano et al. (1995) contains some impressive figures in this regard. The recent paper by Yamabe et al. (2011) reminds us, however, that the same type of instabilities can be detrimental --- and not advantageous --- depending on the specifics of the material system of interest.

References:

Lopez-Pamies, O., Idiart, M.I., Nakamura, T. 2011. Cavitation in elastomeric solids: I --- A defect-growth theory. Journal of the Mechanics and Physics of Solids 59, 1464-1487.

Takano, M., Takahashi, H., Suge, H. 1995. Mechanical Stress and Gibberellin: Regulation of Hollowing Induction in the Stem of a Bean Plant, Phaseolus vulgaris L. Plant Cell Physiol. 36, 101-108.

Yamabe, J., Matsumoto, T., Nishimura, S. 2011. Application of acoustic emission method to detection of internal fracture of sealing rubber material by high-pressure hydrogen decompression. Polymer Testing 30, 76-85.