# Journal Club Theme of January 2014: Electro- and magneto-active deformable composites

Dielectric Elastomers (DE) are promising materials for developing soft machines (e.g., a human-made octopus). The principle of actuation has gained to DE the name "artificial muscles" since they can undergo large deformation when excited by an electric field. Another class of active materials that can be actuated by external field is Magnetoactive Elastomers (MAE). Although MAE and DE share mathematical similarities, the physics is different. Electric field can induce polarization in elastomers, and hence generate electrostatic stresses within the material. As a response to the electrically induced stresses, the material deforms. For MAE the situation is different: elastomers are magnetically inactive, and a similar to DE effect is achieved by mixing the elastomer with magnetically active particles (e.g., carbonyl iron, nickel or Terfenol-D). Thus, due to the magnetic interaction of the particles embedded in a soft matrix, the composite can deform and modify overall stiffness as a response to a magnetic field. The performance of these composites can be further enhanced by optimizing microstructures. Indeed, a similar idea applies for designing DE composites with enhanced properties. These composites, once manufactured can solve the bottle-neck problem of DE technology - the need in extremely high electric fields for meaningful actuations, and potentially lead to a breakthrough in the technology.

**Composites for Better Performance**

DE composites with random distribution of fillers showed a promising enhancement in electromechanical coupling [2]. According to theoretical and numerical studies, the enhancement is only a beginning, and DE composites with periodic microstructures will exceed the response of homogeneous materials (e.g., acrylic elastomer VHB 4910) by orders of magnitude [1]. In particular, Rudykh et al. [1] showed that neatly assembled layered materials (with sub-layered microstructures) benefits from existence of soft mode of deformation and local amplification of electrostatic stresses. This combination gives rise to the dramatic enhancement in the overall performance (see Fig. 2).

Thanks to the mathematical analogies between MAE and DE, these theoretical and numerical findings apply indeed for MAE. Recently, studies have begun exploring homogenized properties of the composites with random distribution of particles [3, 4]. Although, this type of composites is not competitive with the enhanced periodic composites, the random composites are easier to manufacture, and the models can provide useful insights for designing DE and MAE composites. An important feature of the active composites is the ability to modify their stiffness by external fields. Remarkably, the stiffening effect in MAEs with random distributions can be enhanced if the composite is prepared in the presence of a magnetic field. As a result, the particles form chain like structures resulting in an anisotropic behavior of the composite MAE. Rudykh and Bertoldi [5] have developed a micromechanical model for these anisotropic MAE and obtained a closed form expressions for modeling and optimizing these composites. Danas et al. derived a more complicated model and fitted the material parameters by using experimental data [6]. Cao and Zhao [7] analyzed layered structures and demonstrated a strong effect of the mesostructures on the stiffening effect in DE and MAEs.

**Instabilities for New Functionalities**

Elastic instabilities open the path for even more intriguing opportunities for controlling large variety of composite properties by triggering and controlling instability-induced microstructures (such as wrinkled interfaces in layered composites [8] , or periodically buckled porous materials [9]). These microstructure transformations can be used, for example, to manipulate wave propagation [10, 11]. This application was highlighted in previous discussions at iMechanica (e.g., http://imechanica.org/node/14431).

The presence of electric/magnetic fields significantly influences the stability of the composites. Depending on a particular microstructure, the role of the external stimuli may be either stabilizing or destabilizing. To analyze this phenomenon, Rudykh and deBotton [12] derived a general criterion for onset of macroscopic instabilities for a plane-strain settings. Rudykh and Bertoldi [5] extended the critical condition for fully 3D loading conditions. Bertoldi and Gei [14] explored microscopic and macroscopic instabilities in layered structures. These studies [5, 12, 14] formulated the analyses in terms of electrical displacement D as a primary field (magnetic induction B, in the magnetomechanical case). Although the use of electric displacement leads to a simpler mathematical problem, it is not natural from an experimental viewpoint. In experimental practice, it is significantly simpler to prescribe the electrostatic potential on a surface than to prescribe the exact charge distribution. Therefore, Rudykh et al. [13] formulated the problem in terms of electrostatic potential as the primary field variable. The study of electromechanical instability is direct since bifurcation criteria are prescribed in terms of the electric field. The study shows how different length-scale patterns can be induced by combinations of mechanical and electrical loadings.

These coupled multiphysics analyses are limited to a particular class of periodic composites, namely layered composites. Indeed, there is the whole world of various microstructures to explore, to discover new supereffective composites and fascinating microstructure transformations.

**References**

1. S. Rudykh, A. Lewinstein, G. Uner, and G. deBotton. Analysis of microstructural induced enhancement of electromechanical coupling in soft dielectrics. Appl. Phys. Lett., 102:151905, 2013. http://dx.doi.org/10.1063/1.4801775.

2. C. Huang and Q. M. Zhang. Enhanced dielectric and electromechanical responses in high dielectric constant allpolymer percolative composites. Adv. Funct. Mater., 14:501-506, 2004. http://dx.doi.org/10.1002/adfm.200305021.

3. P. Ponte Castañeda and E. Galipeau. Homogenization-based constitutive models for magnetorheological elastomers at finite strain. J. Mech. Phys. Solids, 59:194-215, 2011. http://dx.doi.org/10.1016/j.jmps.2010.11.004.

4. E. Galipeau, S. Rudykh, G. deBotton, and P. Ponte Castañeda. Magnetoactive elastomers with periodic and random microstructures. Int. J. Solids Struct., 2014. http://dx.doi.org/.

5. S. Rudykh and K. Bertoldi. Stability of anisotropic magnetorheological elastomers in finite deformations: a micromechanical approach. J. Mech. Phys. Solids, 61:949-967, 2013. http://dx.doi.org/10.1016/j.jmps.2012.12.008.

6. K. Danas, S. V. Kankanala, and N. Triantafyllidis. Experiments and modeling of iron-particle-filled magnetorheological elastomers. J. Mech. Phys. Solids, 60:120-138, 2012. http://dx.doi.org/10.1016/j.jmps.2011.09.006.

7. C. Cao and X. Zhao. Tunable stiffness of electrorheological elastomers by designing mesostructures. Appl. Phys. Lett., 103(4), 2013. http://dx.doi.org/10.1063/1.4816287.

8. Y. Li, N. Kaynia, S. Rudykh, and M. C. Boyce. Wrinkling of interfacial layers in stratified composites. Adv. Eng. Mater., 15(10):921-926, 2013. http://dx.doi.org/10.1002/adem.201200387.

9. K. Bertoldi, M. C. Boyce, S. Deschanel, S.M. Prange, and T. Mullin. Mechanics of deformation-triggered pattern transformations and superelastic behavior in periodic elastomeric structures. J. Mech. Phys. Solids, 56 (8):2642-2668, 2008. http://dx.doi.org/10.1016/j.jmps.2008.03.006.

10. S. Rudykh and M.C. Boyce. Transforming wave propagation in layered media via instability-induced interfacial wrinkling. Phys. Rev. Lett. (in press), 2013. http://dx.doi.org/.

11. K. Bertoldi and M. C. Boyce. Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations. Phys. Rev. B, 78:184107, 2008. http://dx.doi.org/10.1103/PhysRevB.78.184107.

12. S. Rudykh and G. deBotton. Stability of anisotropic electroactive polymers with application to layered media. Z. Angew. Math. Phys., 62:1131-1142, 2011. http://dx.doi.org/10.1007/s00033-011-0136-1.

13. S. Rudykh, K. Bhattacharya, and G. deBotton. Multiscale instabilities in soft heterogeneous dielectrics. Proc. R. Soc. A, 470:20130618, 2014. http://dx.doi.org/10.1098/rspa.2013.0618.

14. K. Bertoldi and M. Gei. Instabilities in multilayered soft dielectrics. J. Mech. Phys. Solids, 59:18-42, 2011. http://dx.doi.org/10.1016/j.jmps.2010.10.001.

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## Hello Stephan Thank you

Hello Stephan

Thank you very much for the interesting brief and references on Dielectric and Magnetoactive elastomers. I had a couple of questions about these:

(a) I see that some of the applications of these materials are in biomechanics (like artificial muscles etc), aerospace (like flapping wing MAVs) etc. I think, these are applications where one might also need to look into aspects like fatigue failure etc. It is very likely that these materials will be subjected to cyclic loads. Is there any notable work that has addressed these issues pertaining to these materials

(b) If I understand right, MAEs are made by randomly embedding particles into a soft matrix. Apart from the large deformation of the overall material, are there any detailed studies on how the particles influence each around. Such understanding could help in better tailoring of the microstructure, right?

Thanks

Ajay

## fatigue; particle interactions in MAEs

Hello Ajay,

thanks a lot for your comments and questions.

(a) Regarding possible fatigue failure, I have to say that I am not an expert in fatigue, but we have many really good experts in this field at iMechanica. I hope they will join the discussion.

As far as I understand, the main issue with DE is (once again) high electric fields that we need. So people mainly focus on trying to get large actuations with smaller actuation field. I did a quick google check, and there are some works that mention that DE a reliable and have long fatigue life; but I did not find a detailed study of this important subject. If you know a good paper on this subject, please share.

(b) I belive that periodic microstructures can do better than "random" composite, but it is importnat to understand how the particles interact and give rise to the magneto-elastic effects in the "random composites"

I would recommend excellent works of

[1] Galipeau and Ponte Castañeda (2012). The effect of particle shape and distribution on the macroscopic behavior of magnetoelastic composites. Int. J. Solids Struct, 49(1), 1-17

[2] Javili et al 2013 Computational homogenization in magnetomechanics. Int. J. Solids Struct. 50, 4197–4216.

Happy new year!

Stephan

## Thanks Stephan, for the

Thanks Stephan, for the interesting discussions. We worked on these types of problems about 10 years ago, and what we learned are

1. Composite can do order of magnitude better than homogeneous phase, from very simple physical argument: Appl. Phys. Lett. 81, 1860 (2002); this was built on layered structure.

2. It is field fluctuation that is most important, not the average field: Appl. Phys. Lett. 84, 3124 (2004); this point of view I think is very important, though may not be widely appreciated.

3. Interfacial interaction can be critical, resulting in further enhancement: Phys. Rev. Lett. 90, 217601 (2003).

We have some other papers, but these three summarize key insight we have learned.

I am glad that many works appeared later, and formulated the problem in large deformation seting, and on mathematically more rigours footing. Instatbility is another mechanism that we have not thought about then.

One more point on perioid versus random composite. It is important to recognize that the periodic composite gains order of magnitude enhancement because of separation of length scale, built on sequential lamination. Conceptually, similar structure can be contructuerd for random composite as well, and if so, it is hard to predict which one will be optimal. That will be interesting case to study.

## Hello Jiangyu, Many thanks

Hello Jiangyu,

Many thanks for your comment and references. I agree that there is a lot that can be learned from linear models (actually, the idea of the Rank-2 layered material with the scale separation came from the exact solution in the liner regime; we extended it to finitely deformed materials after that). But we definitely should be very careful applying these ideas to non-linear materials. I think that simple layered material may not be very useful for enhanced actuation. The only soft mode of deformation in these materials is the shear mode, where the deformation is mostly accommodated in soft matrix – but it is not clear how to achieve this mode by applying external electric field.

Thanks for pointing out the importance of the fluctuation, in that aspect we are not far from what you found tens years ago – it is extremely important for the enhancement of the actuation. We also find that in finitely deformed DE composites: the field fluctuation is a key.

I think it would be very interesting to combine the large deformations with instabilities and even with the interface effect…

Finally, the idea for the periodic composites is that the local effects add up working cooperatively, while random microstructure don’t have the cooperative feature and may exhibit weaker performance one the macroscopic level, when the local effects are averaged.

Thanks again for your comments and excellent work!