Strongly Nonlinear Discrete Metamaterials: Origin of new Wave ... - Core

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ScienceDirect Physics Procedia 70 (2015) 815 – 818

2015 International Congress on Ultrasonics, 2015 ICU Metz

Strongly nonlinear discrete metamaterials: origin of new wave dynamics Vitali F. Nesterenko Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla CA 92093-0411 USA b Materials Science and Engineering Program, University of California at San Diego, La Jolla CA 92093 0418 USA

Abstract Discrete metamaterials with strongly nonlinear interactions between elements demonstrate unique behaviour qualitatively different than predicted based on classical linear and weakly nonlinear wave dynamics. The paper provides analysis of theoretical and experimental observations of phenomena related to a new wave dynamics of strongly nonlinear metamaterials. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2015 V.F. Nesterenko. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). responsibility of Scientific the Scientific Committee 2015 ICU Metz. Peer-reviewunder under Peer-review responsibility of the Committee of ICUof2015 Keywords: strongly nonlinear; discrete; metamaterials; Hertzian chain; solitary wave

1. Introduction One-dimensional chains of masses with strongly nonlinear interaction between them (e.g., nonlinearizable, strongly nonlinear Hertz law) have zero sound speed in the absence of precompression (“sonic vacuum”) and support strongly nonlinear solitary waves. They are qualitatively different than Korteweg-de Vries (KdV) solitary waves (Nesterenko (1883)). The chain of particles with strongly nonlinear interaction is mathematically similar to Fermi-Pasta-Ulam (FPU) problem (Fermi, Pasta, and Ulam (1955)) only in the sense that in both cases the interaction force depends on relative displacements of neighboring particles. No periodic linear solutions (phonons), which are direct consequence of linear elastic interaction between particles, exist in “sonic vacuum”, unlike in weakly nonlinear FPU chain. Basic excitations in “sonic vacuum” are strongly nonlinear solitary waves and not phonons. Static precompression reduces “sonic vacuum” to FPU chain, if pulse amplitude is small in comparison with initial strain. Theoretically sound speed in “sonic vacuum” can be indefinitely small helping to reduce size of metamaterials to process incoming stress pulses. If the amplitude of incoming pulse is significantly larger than initial prestress then strongly nonlinear solitary wave, shock wave or periodic wave will be excited instead of sound wave depending on the shape and duration of incoming pulse.

1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015 doi:10.1016/j.phpro.2015.08.166

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Vitali F. Nesterenko / Physics Procedia 70 (2015) 815 – 818

2. Experimental realizations of “sonic vacuum” The first strongly nonlinear metamaterial with nonlinerizable interaction force between elements was chain of elastic grains interacting according to Hertz law (Nesterenko (1983)). This system attracted extensive attention from experimentalists (Lazaridi and Nesterenko (1985), Nesterenko (2001), Shukla et al., (1993), Coste, et al., (1997), Job et al., (2005), Potekin et al., (2013), Leonard et al., (2013)). Chains with cylindrical particles allowing tunability by rotation of particles (Li et al., 2012) and woodpile phononic crystals (Kim et al. (2014)) were recently proposed. A metamaterial made of the rigid plates and soft o-rings was introduced by Herbold and Nesterenko (2007). The compressive force acting on o-ring exhibits a strongly nonlinear two power law being more nonlinear than Hertzian interaction at global strains above 0.3. This metamaterial have the following advantages in comparison with granular chains made from metal or ceramic particles: (1) stronger nonlinearity allows higher degree of tunability by static force (Xu and Nesterenko (2014)); (2) rubber o-rings are able to absorb higher impact energy with fast complete recovery after impact (Lee and Nesterenko (2014)), (3) o-rings have unique capability to tailor interaction force between elements by changing the shape of o-rings, including softening regime and by placing sequence of o-rings (one inside the other) or layers of materials inside them to create practically any force dependence on displacement between elements; (4) flexibility in selecting practically any masses of the elements by reducing the thickness of the plates at the same diameter of the system; (5) o-rings are designed for use in heavy machinery and are commercially available in great variety of sizes and shapes. 3. Solitary wave in “sonic vacuum”, discrete versus continuum approach 3.1. Solitary waves in uniform chains, role of dissipation Solitary wave in a strongly nonlinear discrete system with equal masses of elements, interacting according to Hertz law, is qualitatively different from KdV solitary wave or from solitary wave in FPU chain in the following ways: (1) speed of solitary wave depends nonlinearly on the particle velocity, (2) speed of solitary wave in “sonic vacuum” can be infinitely larger than sound speed and strongly depends on energy of the wave, (3) width of solitary wave in “sonic vacuum” is independent on amplitude. This solitary wave can be considered as quasiparticle with effective mass equal about 1.4m, where m is the mass of grains. Solitary wave as stationary solution of wave equation based on long wave approximation agrees with results of numerical calculations for the discrete Hertzian chain (Nesterenko (1983), Hinch and Saint-Jean (1999), Chatterjee (1999), Sen et bal., 2008), satisfactory agreement is observed even for large exponents in the interaction law (Ahnert and Pikovsky (2009)). Dissipation in contact interaction of particles is intrinsically nonlinear (Brilliantov (2015), Lee and Nesterenko (2014)). Nevertheless the viscous linear dissipation agrees with experimental results at least in investigated range of dynamic stresses (Potekin et al. (2013) and Wang and Nesterenko (2015)). The introduction of viscous dissipation not only results in expected reduction of wave amplitude, but also in an unexpected phenomena, such as a formation of two wave structure under delta force excitation demonstrated in numerical calculations by Rosas et al. (2007). 3.2. Solitary waves in dimer chains, role of dissipation Pulse mitigation by dimer chains depends on their mesostructure at the same global properties (Nesterenko (2001)). Strongly nonlinear dimer chains demonstrate very unusual behaviour - a true solitary waves were observed only at certain discrete mass ratios of light to heavy spherical particles. In numerical calculations the maximum nondissipative decay of compression stress pulse was observed at mass ratio 0.59 (Jayaprakash et al. 2011). Better mitigation of stress pulses due to this mechanism of decay in chain with mass ratio 0.55, than in uniform chains, was confirmed in experiments by Potekin et al. (2013) and by Wang and Nesterenko (2015). Interesting that at larger damping coefficients the uniform chain mitigated the same impact better than the system with mass ratio 0.55 (Wang and Nesterenko (2015). Thus one mass system is preferable system for the higher level of damping (e.g., granular chains in liquid) because it “forcefully” supports the high gradients in the narrow pulse by strongly nonlinear dispersion thus enhancing viscous dissipation.

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4. Excitation of train of solitary waves in uniform chains The impact by a striker with significantly larger mass than mass of particle in chain results in the excitation of a train of solitary waves (Nesterenko (2001)). The possible approach to predict amplitudes of solitary waves in the well developed train can be based on imaginary steps in the transformation of linear momentum and energy of striker to the solitary waves considered as quasiparticles being in rest with effective mass equal 1.4 m. In this scenario each solitary wave is independently created by impact between striker with mass Mimp and quasiparticle in rest. Based on this approach the corresponding equation for the momenta in n-th solitary wave Pn is

Pn =

2 P0 ( B − 1) n −1

(B + 1)n

,

(1)

where P0 is the initial linear momentum of impactor, and n=1,2,3.., are numbers of solitary waves, number 1 corresponds to the leading solitary wave, and B=Mimp/meff. This imaginary path conserves linear momentum and energy of the system, but it is not unique. The similar approach was used by Tichler et al. (2013) to predict relative amplitudes in the train of solitary waves generated at the interface of two sonic vacua. The predicted results for the solitary waves based on Eq. 1 and found in direct numerical calculations of discrete chain assembled from steel spheres and cylinders of equal mass 2.086 g impacted by striker with mass 39.136 g and velocity 0.457 m/s are in a satisfactory agreement (Wang and Nesterenko (2015)). For example the values of momenta obtained from Eq. 1 and in numerical calculations for corresponding three solitary waves are: for n=1 (2.48 10-3 kgms-1 and 2.61 10-3 kgms-1); for n=2 (2.14 10-3 kgms-1 and 2.09 10-3 kgms-1); and for n=3 (1.84 10-3 kgms-1 and 1.71 10-3 kgms-1). Because effective mass of solitary wave, considered as quasiparticle depends on the interaction law (e.g., exponent in power law, Nesterenko (2001)), amplitudes of solitary waves in the train depend on interaction law also. 5. Anomalous systems The dynamic behavior of strongly nonlinear discrete metamaterials with anomalous strain softening behaviour represents unique opportunity to mitigate impact without dissipation mechanisms. It was demonstrated in numerical calculations that compression pulse quickly disintegrates into a leading rarefaction solitary wave followed by an oscillatory train (Herbold and Nesterenko (2013)). A new class of strongly nonlinear metamaterials was proposed based on tensegrity concepts (Fraternali et al., 2014). These metamaterials can be tuned into elastic hardening (“sonic vacuum” type) or elastic softening (anomalous) regimes by adjusting local and global prestress. 6. Interfaces between strongly nonlinear metamaterials “Sonic vacuum” systems have zero acoustic impedances. Thus classical approach to consider interaction of pulses with their interfaces based on acoustic impedances, can’t be applied. The transmission of the pulse through interface of two sonic vacua may be dramatically different depending on the direction of the incident pulse (Nesterenko (2001), Tichler et al., (2013)). Moreover a strong sensitivity of the reflected and transmitted energy from the interface of the two granular media to the initial precompression was observed in numerical calculations and in experiments, this phenomenon was called the acoustic diode effect (Nesterenko, et al., (2005)). It can be used for energy trapping and shock disintegration (Daraio et al., 2006). Nonclassical behavior was observed in numerical calculations of reflection of incident compression solitary pulse from the interface between elastically hardening and softening materials having correspondingly low-high acoustic impedances (Fraternali et al., (2014)). Conclusions Discrete strongly nonlinear systems (e.g., granular materials, systems with strongly nonlinear elements (O-rings, tensegrity structures)) represent a new class of metamaterials with qualitatively new wave dynamics. In case of elastically stiffening behaviour they support new type of solitary wave, compact, space scale independent on amplitude, dictated by mesoscale and interaction law. Strong nonlinearity allows highly tunable behavior, sensitive

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to low amplitude external mechanical field. Multiscale systems based on tensegrity concept allows tuning from strongly nonlinear elastically stiffening behaviour to elastically softening regime. The latter behaviour may allow design of ultimate impact mitigation devices without relying on energy dissipation. Absence of acoustic impedance in case of “sonic vacuum”, or its negligible influence at low precompression results in new interfacial phenomena, e.g. continuum-discrete-continuum pulse transformation at interface, which can be used for targeted energy transfer. References Ahnert and Pikovsky, 2009. Compactons and Chaos in Strongly Nonlinear Lattices, Phys. Rev., 79, 026209. Brilliantov, N.V., Pimeniva, A.V., and Goldobin, D.S.. 2015. A Dissipative Force Between Colliding Viscoelastic Bodies: Rigorous Approach. EPL 109, 14005. Chatterjee, A., 1999. Asymptotic Solution for Solitary Waves in a Chain of Elastic Spheres. Phys. Rev. E., 59, 5912-5919. 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