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# Geometrically nonlinear microstructured materials for mechanical wave tailoring

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**Geometrically nonlinear microstructured materials for mechanical wave tailoring**

**Nicholas Boechler, Department of Mechanical Engineering, University of Washington **

Designed microstructured materials, such as phononic crystals and acoustic metamaterials, have been a topic of recent interest due to their capacity to control the propagation of acoustic waves [1,2], and exhibit locally tailorable negative [3,4], extreme [5], and anistropic [6] effective properties in dynamic regimes. In such materials, dispersion stemming from structural periodicity and local resonances are leveraged to achieve their exotic properties. In addition to these mechanisms, which can be described within the context of linear wave propagation, a number of recent studies have also explored wave tailoring through nonlinear dynamics [1,2,7]. The use of nonlinearity has been shown to open an array of additional functionality ranging from amplitude dependence [8] to non-reciprocal wave propagation [9].

One roadblock to the implementation of nonlinear wave tailoring strategies is the limited palette of nonlinearities in conventional materials to choose from. One solution to this has been to leverage microstructural geometric nonlinearities. In a system that exhibits geometric nonlinearity, while the material forming the structure remains in the linear elastic regime, large displacements, large rotations, or contact can result in a stiffness that changes with the applied loading. At the macroscale, a classic example of a geometrically nonlinear structure is a fishing rod, which stiffens with increasing load. An example of a system exhibiting microstructural geometric nonlinearity is granular media [10], where the constituent particles are often modeled as elastic spheres interacting via Hertzian contact [11]. In the Hertz contact model, because of the shape of the particles and the increasing contact area with applied load, the compressed spheres stiffen such that applied force is proportional to the relative displacement between the particle centers to the 3/2 power. This interparticulate nonlinear “spring” subsequently confers the same power law effective stress strain relationship on the granular medium, which also well describes granular media’s dynamic response for long wavelength acoustic waves.

Such microstructural geometric nonlinearities have been leveraged in a wide range of dynamically responsive designer materials. In the case of granular media, this took the form of “granular crystals,” which are ordered or reduced dimensional arrays of particles (typically metallic spheres of a few millimeters in diameter) in contact [7,12]. Studies of granular crystals have demonstrated a range of mechanical wave tailoring capabilities enabled by nonlinearity, including amplitude dependence [8], external tunability [13], frequency conversion [14], non-reciprocity [15], self-localization [16]. One of the most canonical examples in this field is the existence of solitary waves in granular crystals [17], which are waves with fixed width (approximately five particles in the case of chains of spheres) and amplitude dependent velocity, which exist as a result of the balance between the nonlinearity and dispersion of the medium. Many other types of designer materials exhibiting microstructural geometric nonlinearities have also been explored. A few examples, which are highlighted in Fig. 1, include truss-like structures such as tensegrity lattices [18], systems of O-rings and metal disks [19], origami-based folding structures [20], multistable arrays [21,22], and 3D printed elastomeric architected materials [23]. In each of the prior examples, the dynamic material response is strongly affected by the type of nonlinearity. For instance, the conversion from a compression pulse to a rarefaction wave and subsequent energy delocalization into an oscillatory tail has been demonstrated in the case of tensegrity lattices [18], and dynamic energy storage [21] and release [22] in the case of multistable arrays.

Figure 1: Several examples of quasi-one-dimensional geometrically nonlinear microstructured materials, or “lattices.” (a) Granular crystal (image from Ref. [7]). (b) Tensegrity lattice (image from Ref. [18]). (c) O-ring and disk lattice (image from Ref. [19]). (d) Origami structure (image from Ref. [20]). (e) Energy trapping bistable lattice (image from Ref. [21]). (f) Bistable lattice with controlled energy release (image from Ref. [22]). (g) Elastomeric nonlinear lattice (image from Ref. [23]).

Within the scope of designed geometrically nonlinear microstructured materials, many open challenges remain. Dynamic energy partition in nonlinear systems [24], particularly in high-dimensional settings [25], has been a long-standing challenge in the nonlinear lattices community. In addition, while the use of microstructural geometric nonlinearity has enabled significant strides, the choice of available nonlinearities remains discrete. Recent advances in nonlinear topology optimization [26] may enable access to the unexplored areas of the nonlinear dynamic spectrum. Finally, the previously described designer materials are all macroscale systems. For many practical applications, including those involving mechanical waves with short wavelengths and high frequencies, unit cells with micro- to nanoscale dimension are necessitated. In addition to being a manufacturing challenge, smaller scales are of fundamental interest as there are physical phenomena that must be taken into consideration for micro- to nanoscale systems that are unique from their macroscale counterparts. For instance, in the case of granular media, many prior studies have demonstrated the importance of adhesive forces in micro- to nanoscale contact mechanics [27]. Along these lines, recent investigations, enabled by self-assembly manufacturing, have begun to explore the dynamics of micro- to nanoscale granular crystals [28].

**References:**

[1] M. I. Hussein, M. J. Leamy, and M. Ruzzene, “Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook”, Appl. Mech. Rev. 66, 040802-1 (2014)

[2] S. A. Cummer, J. Christensen, and A. Alu, “Controlling sound with acoustic metamaterials”, Nature Rev. 1, 1 (2016)

[3] Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, “Locally resonant sonic materials”, Science 289, 1734 (2000)

[4] N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, “Ultrasonic metamaterials with negative modulus, Nature Mater. 5, 425 (2006)

[5] F. Lemoult, N. Kaina, M. Fink, and G. Lerosey, “Wave propagation control at the deep subwavelength scale in metamaterials”, Nature Phys. 9, 55 (2013)

[6] S. Zhang, C. Xia, and N. Fang, “Broadband acoustic cloak for ultrasound waves”, Phys. Rev. Lett. 106, 024301 (2011)

[7] G. Theocharis, N. Boechler, and C. Daraio, “Nonlinear periodic phononic structures and granular crystals, Ch. 7 of Acoustic Metamaterials and Phononic Crystals” (Springer-Verlag, Berlin Heidelberg, 2013)

[8] C. Daraio, V. F. Nesterenko, E. B. Herbold, and S. Jin, “Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals”, Phys. Rev. E 73, 026610 (2006)

[9] B. Liang, X. S. Guo, J. Tu, D. Zhang, and J. C. Cheng, “An acoustic rectifier”, Nature Mater. 9, 989 (2010)

[10] V. F. Nesterenko, “Dynamics of Heterogeneous Materials” (Springer-Verlag, New York, NY, 2001)

[11] H. Hertz, “On the contact of elastic solids”, J. Reine Angew. Math. 92, 156 (1882)

[12] M. A. Porter, P. G. Kevrekidis, and C. Daraio, “Granular crystals: nonlinear dynamics meets materials engineering”, Phys. Today 68, 44 (2015)

[13] N. Boechler, J. K. Yang, G. Theocharis, P. G. Kevrekidis, and C. Daraio, “Tunable vibrational band gaps in one-dimensional diatomic granular crystals with three-particle unit cells”, J. Appl. Phys. 109, 074906 (2011)

[14] V. J. Sanchez-Morcillo, I. Perez-Arjona, V. Romero-Garcia, V. Tournat, and V. E. Gusev, “Second-harmonic generation for dispersive elastic waves in a discrete granular chain”, Phys. Rev. E 88, 043203 (2013)

[15] N. Boechler, G. Theocharis, and C. Daraio, “Bifurcation-based acoustic switching and rectification,” Nature Mater. 10, 665 (2011)

[16] N. Boechler, G. Theocharis, S. Job, P. G. Kevrekidis, M. A. Porter, and C. Daraio, "Discrete breathers in one-dimensional diatomic granular crystals," Phys. Rev. Lett. 104, 244302 (2010)

[17] A. N. Lazaridi and V. F. Nesterenko, “Observation of a new type of solitary waves in a one-dimensional granular medium”, J. Appl. Mech. Tech. Phys. 26, 405 (1985)

[18] F. Fraternali, G. Carpentieri, A. Amendola, R. E. Skelton, and V. F. Nesterenko, “Multiscale tunability of solitary wave dynamics in tensegrity metamaterials”, Appl. Phys. Lett. 105, 201903 (2014)

[19] Y. Xu and V. F. Nesterenko, “Propagation of short stress pulses in discrete strongly nonlinear tunable metamaterials”, Phil. Trans. R. Soc. A 372, 2023 (2014)

[20] H. Yasuda, C. Chong, E. G. Charalampidis, P. G. Kevrekidis, and J. Yang, “Formation of rarefaction waves in origami-based metamaterials”, Phys. Rev. E 93, 043004 (2016)

[21] S. Shan, S. H. Kang, J. R. Raney, P. Wang, L. Fang, F. Candido, J. A. Lewis, and K. Bertoldi, “Multistable architected materials for trapping elastic strain energy, “Adv. Mater. 27, 4296 (2015)

[22] J. R. Raney, N. Nadkami, C. Daraio, D. M. Kochmann, J. A. Lewis, and K. Bertoldi, “Stable propagation of mechanical signals in soft media using stored elastic energy”, Proc. Natl. Acad. Sci. USA 113, 9722 (2016)

[23] B. Deng, J. R. Raney, V. Tournat, and K. Bertoldi, “Elastic vector solitons in soft architected materials”, Phys. Rev. Lett. 118, 204102 (2017)

[24] M. A. Porter, N. J. Zabusky, B. Hu, and D. K. Campbell, “Fermi, Pasta, Ulam and the birth of experimental mathematics: A numerical experiment that Enrico Fermi, John Pasta, and Stanislaw Ulam reported 54 years ago continues to inspire discovery”, American Scientist 97, 214 (2009)

[25] A. Leonard and C. Daraio, “Stress wave anisotropy in centered square highly nonlinear granular systems”, Phys. Rev. Lett. 108, 214301 (2012)

[26] F. Wang, O. Sigmund, and J. S. Jensen, “Design of materials with prescribed nonlinear properties”, J. Mech. Phys. Solid 69, 156 (2014)

[27] J. Israelachvili, “Intermolecular and Surface Forces” (Elsevier, Inc., Burlington, MA, 2011)

[28] M. Hiraiwa, M. Abi Ghanem, S. Wallen, A. Khanolkar, A. A. Maznev, and N. Boechler, “Complex contact-based dynamics of microsphere monolayers revealed by resonant attenuation of surface acoustic waves”, Phys. Rev. Lett. 116, 198001 (2016)

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## Comments

## thanks for leading the discussion on the interesting topic

Nick, many thanks for leading the discussion on the interesting topic.

As we know, the heat transport of a dielectric is mainly through phonon. According to my limited knowledge on both fields (thermal conduction of solids and elastodyanmic waves in lattice structures), there are lots of similarities (at least mathematically) between the heat tranport in a dielectric solid and elastodynamic wave propagation in lattic structures as you discussed above. Do you know if there is any interaction between the two fields? For example, using experiments on elastodynamic wave propagating in lattice structure to uncover something about heat transport in a dielectric solid. Or, designning a dieletric solid with interesting thermal transport properteis based on the new discoveries of the elastodynamic wave propagating in a lattice structure.