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Nonlinear elastic wave propagation in a phononic material with periodic solid-solid contact interface. (English) Zbl 1524.74283

Summary: Phononic materials enable enhanced dynamic properties, and offer the ability to engineer the material response. In this work we study the wave propagation in such a structure when introduced with nonlinearity. Our system is comprised of pre-compressed material with periodic solid-solid contacts, which exhibit a quadratic nonlinearity for small displacements. We suggest a new approach to modeling this system, where we discretize the unit cell in order to derive an approximate analytical solution using a perturbation method, which we are then able to easily validate numerically. With these methods, we study the band structure in the system and the second harmonic generation originating from the nonlinearity. We qualitatively analyze the second harmonic response of the system in terms of the single-crack response with linear band structure considerations. Significant band structure manipulation by changing system parameters is demonstrated, including possible in-situ tuning. The system also exhibits effective frequency doubling, i.e. the transmitted wave is primarily comprised of the second harmonic wave, for a certain range of frequencies. We demonstrate very high robustness to disorder in the system, in terms of band structure and second harmonic generation. These results have possible applications as frequency-converting devices, tunable engineered materials, and in non-destructive evaluation.

MSC:

74J30 Nonlinear waves in solid mechanics
74M15 Contact in solid mechanics

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[1] Nemat-Nasser, S.; Srivastava, A., Negative effective dynamic mass-density and stiffness: Micro-architecture and phononic transport in periodic composites, AIP Adv., 1, 4, Article 041502 pp. (2011)
[2] Tan, K.; Huang, H.; Sun, C., Blast-wave impact mitigation using negative effective mass density concept of elastic metamaterials, Int. J. Impact Eng., 64, 20-29 (2014)
[3] Lu, Y.; Yang, Y.; Guest, J. K.; Srivastava, A., 3-D Phononic crystals with ultra-wide band gaps, Sci. Rep., 7, Article 43407 pp. (2017)
[4] Lemoult, F.; Kaina, N.; Fink, M.; Lerosey, G., Wave propagation control at the deep subwavelength scale in metamaterials, Nat. Phys., 9, 1, 55 (2013)
[5] Bilal, O. R.; Foehr, A.; Daraio, C., Reprogrammable phononic metasurfaces, Adv. Mater., 29, 39, Article 1700628 pp. (2017)
[6] Overvelde, J. T.; De Jong, T. A.; Shevchenko, Y.; Becerra, S. A.; Whitesides, G. M.; Weaver, J. C.; Hoberman, C.; Bertoldi, K., A three-dimensional actuated origami-inspired transformable metamaterial with multiple degrees of freedom, Nature Commun., 7, Article 10929 pp. (2016)
[7] Wang, P.; Lu, L.; Bertoldi, K., Topological phononic crystals with one-way elastic edge waves, Phys. Rev. Lett., 115, Article 104302 pp. (2015)
[8] Mousavi, S.; Khanikaev, A. B.; Wang, Z., Topologically protected elastic waves in phononic metamaterials, Nature Commun., 6, 8682 (2015)
[9] Matlack, K. H.; Serra-Garcia, M.; Palermo, A.; Huber, S. D.; Daraio, C., Designing perturbative metamaterials from discrete models, Nat. Mater., 17, 4, 323 (2018)
[10] Pal, R. K.; Ruzzene, M., Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect, New J. Phys., 19, 2, Article 025001 pp. (2017) · Zbl 1512.81093
[11] Nash, L. M.; Kleckner, D.; Read, A.; Vitelli, V.; Turner, A. M.; Irvine, W. T.M., Topological mechanics of gyroscopic metamaterials, Proc. Natl. Acad. Sci., 112, 47, 14495-14500 (2015)
[12] Fleury, R.; Sounas, D. L.; Sieck, C. F.; Haberman, M. R.; Alù, A., Sound isolation and giant linear nonreciprocity in a compact acoustic circulator, Science, 343, 6170, 516-519 (2014)
[13] Nesterenko, V. F., Propagation of nonlinear compression pulses in granular media, J. Appl. Mech. Tech. Phys., 24, 5, 733-743 (1983)
[14] Lazaridi, A. N.; Nesterenko, V. F., Observation of a new type of solitary waves in a one-dimensional granular medium, J. Appl. Mech. Tech. Phys., 26, 3, 405-408 (1985)
[15] Porter, M. A.; Daraio, C.; Herbold, E. B.; Szelengowicz, I.; Kevrekidis, P. G., Highly nonlinear solitary waves in periodic dimer granular chains, Phys. Rev. E, 77, Article 015601 pp. (2008)
[16] Porter, M. A.; Daraio, C.; Szelengowicz, I.; Herbold, E. B.; Kevrekidis, P., Highly nonlinear solitary waves in heterogeneous periodic granular media, Physica D, 238, 6, 666-676 (2009) · Zbl 1160.37410
[17] Kim, E.; Chaunsali, R.; Xu, H.; Jaworski, J.; Yang, J.; Kevrekidis, P. G.; Vakakis, A. F., Nonlinear low-to-high-frequency energy cascades in diatomic granular crystals, Phys. Rev. E (3), 92, 6, 1-7 (2015), arXiv:1505.05556
[18] Ganesh, R.; Gonella, S., From modal mixing to tunable functional switches in nonlinear phononic crystals, Phys. Rev. Lett., 114, 5, 1-5 (2015)
[19] Ganesh, R.; Gonella, S., Nonlinear waves in lattice materials: Adaptively augmented directivity and functionality enhancement by modal mixing, J. Mech. Phys. Solids, 99, 272-288 (2017)
[20] Ganesh, R.; Gonella, S., Experimental evidence of directivity-enhancing mechanisms in nonlinear lattices, Appl. Phys. Lett., 110, 084101, 1-6 (2017)
[21] Zhang, J.; Romero-García, V.; Theocharis, G.; Richoux, O.; Achilleos, V.; Frantzeskakis, D. J., Second-harmonic generation in membrane-type nonlinear acoustic metamaterials, Crystals, 6, 8 (2016)
[22] Nassar, H.; Xu, X.; Norris, A.; Huang, G., Modulated phononic crystals: Non-reciprocal wave propagation and Willis materials, J. Mech. Phys. Solids, 101, 10-29 (2017)
[23] Grinberg, I.; Vakakis, A. F.; Gendelman, O. V., Acoustic diode: Wave non-reciprocity in nonlinearly coupled waveguides, Wave Motion, 83, 49-66 (2018) · Zbl 1469.76110
[24] Cottone, F.; Vocca, H.; Gammaitoni, L., Nonlinear energy harvesting, Phys. Rev. Lett., 102, 8, Article 080601 pp. (2009)
[25] Gendelman, O.; Manevitch, L.; Vakakis, A. F.; M’closkey, R., Energy pumping in nonlinear mechanical oscillators: Part I-dynamics of the underlying Hamiltonian systems, J. Appl. Mech., 68, 1, 34-41 (2001) · Zbl 1110.74452
[26] Vakakis, A., Inducing passive nonlinear energy sinks in vibrating systems, J. Vib. Acoust., 123, 3, 324-332 (2001)
[27] Ablowitz, M. J.; Ladik, J. F., Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys., 17, 6, 1011-1018 (1976) · Zbl 0322.42014
[28] Gendelman, O. V.; Manevitch, L. I., Discrete breathers in vibroimpact chains: Analytic solutions, Phys. Rev. E, 78, Article 026609 pp. (2008)
[29] Gendelman, O. V., Exact solutions for discrete breathers in a forced-damped chain, Phys. Rev. E, 87, Article 062911 pp. (2013)
[30] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Method for solving the Korteweg-devries equation, Phys. Rev. Lett., 19, 1095-1097 (1967) · Zbl 1061.35520
[31] Jiang, D.; Pierre, C.; Shaw, S., Nonlinear normal modes for vibratory systems under harmonic excitation, J. Sound Vib., 288, 4, 791-812 (2005)
[32] Liao, S., Homotopy Analysis Method in Nonlinear Differential Equations (2012), Springer · Zbl 1253.35001
[33] Nayfeh, A. H., Perturbation Methods (1973), John Wiley & Sons · Zbl 0265.35002
[34] Nayfeh, A. H.; Hassan, S. D., The method of multiple scales and non-linear dispersive waves, J. Fluid Mech., 48, 3, 463-475 (1971) · Zbl 0239.76021
[35] Porter, M. A.; Kevrekidis, P. G.; Daraio, C., Granular crystals: Nonlinear dynamics meets materials engineering, Phys. Today, 68, 11, 44-50 (2015)
[36] Daraio, C.; Nesterenko, V., Strongly nonlinear wave dynamics in a chain of polymer coated beads, Phys. Rev. E, 73, 2, Article 026612 pp. (2006)
[37] Daraio, C.; Nesterenko, V. F.; Herbold, E. B.; Jin, S., Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals, Phys. Rev. E (3), 73, 2, 1-10 (2006), arXiv:0506513
[38] Manktelow, K.; Narisetti, R. K.; Leamy, M. J.; Ruzzene, M., Finite-element based perturbation analysis of wave propagation in nonlinear periodic structures, Mech. Syst. Signal Process., 39, 32-46 (2013)
[39] Narisetti, R. K.; Leamy, M. J.; Ruzzene, M., A perturbation approach for predicting wave propagation in one-dimensional nonlinear periodic structures, J. Vib. Acoust., 132, 031001, 1-11 (2010)
[40] Theocharis, G.; Boechler, N.; Daraio, C., Nonlinear phononic structures and metamaterials, (Deymier, P. A., Acoustic Metamaterials and Phononic Crystals, vol. 298 (2013), Springer)
[41] Ganesh, R.; Gonella, S., Invariants of nonlinearity in the phononic characteristics of granular chains, Phys. Rev. E, 90, 2, 1-9 (2014)
[42] Cabaret, J.; Tournat, V.; Béquin, P., Amplitude-dependent phononic processes in a diatomic granular chain in the weakly nonlinear regime, Phys. Rev. E, 86, Article 041305 pp. (2012)
[43] Sánchez-Morcillo, V. J.; Pérez-Arjona, I.; Romero-García, V.; Tournat, V.; Gusev, V. E., Second-harmonic generation for dispersive elastic waves in a discrete granular chain, Phys. Rev. E, 88, Article 043203 pp. (2013)
[44] Wallen, S.; Boechler, N., Shear to longitudinal mode conversion via second harmonic generation in a two-dimensional microscale granular crystal, Wave Motion, 68, 22-30 (2017) · Zbl 1524.74093
[45] Molinari, A.; Daraio, C., Stationary shocks in periodic highly nonlinear granular chains, Phys. Rev. E, 80, Article 056602 pp. (2009)
[46] Goddard, J. D.; Enderby, J. E., Nonlinear elasticity and pressure-dependent wave speeds in granular media, Proc. R. Soc. Lond. Ser. A, 430, 1878, 105-131 (1990) · Zbl 0712.73008
[47] James, G.; Kevrekidis, P. G.; Cuevas, J., Breathers in oscillator chains with Hertzian interactions, Physica D, 251, 39-59 (2013) · Zbl 1278.37053
[48] Suksangpanya, N.; Yaraghi, N. A.; Pipes, R. B.; Kisailus, D.; Zavattieri, P., Crack twisting and toughening strategies in Bouligand architectures, Int. J. Solids Struct., 150, 83-106 (2018)
[49] Mirkhalaf, M.; Zhou, T.; Barthelat, F., Simultaneous improvements of strength and toughness in topologically interlocked ceramics, Proc. Natl. Acad. Sci., 115, 37, Article 201807272 pp. (2018)
[50] Ostrovsky, L.; Johnson, P., Dynamic nonlinear elasticity in geomaterials, Riv. Nuovo Cimento, 24, 7, 1-46 (2001)
[51] Bera, B.; Mitra, S. K.; Vick, D., Understanding the micro structure of Berea sandstone by the simultaneous use of micro-computed tomography (micro-CT) and focused ion beam-scanning electron microscopy (FIB-SEM), Micron, 42, 5, 412-418 (2011)
[52] TenCate, J. A., Slow dynamics of earth materials: An experimental overview, Pure Appl. Geophys., 168, 12, 2211-2219 (2011)
[53] Remillieux, M. C.; Ulrich, T. J.; Goodman, H. E.; Ten Cate, J. A., Propagation of a finite-amplitude elastic pulse in a bar of Berea sandstone: A detailed look at the mechanisms of classical nonlinearity, hysteresis, and nonequilibrium dynamics, J. Geophys. Res., 122, 11, 8892-8909 (2017)
[54] Shokouhi, P.; Rivière, J.; Guyer, R. A.; Johnson, P. A., Slow dynamics of consolidated granular systems: Multi-scale relaxation, Appl. Phys. Lett., 111, 25 (2017)
[55] Remillieux, M. C.; Guyer, R. A.; Payan, C.; Ulrich, T. J., Decoupling nonclassical nonlinear behavior of elastic wave types, Phys. Rev. Lett., 116, 115501, 1-5 (2016)
[56] Solodov, I. Y., Ultrasonics of non-linear contacts: propagation, reflection and NDE-applications, Ultrasonics, 36, 383-390 (1998)
[57] Biwa, S.; Nakajima, S.; Ohno, N., On the acoustic nonlinearity of solid-solid contact with pressure-dependent interface stiffness, J. Appl. Mech., 71, 4, 508 (2004) · Zbl 1111.74329
[58] Biwa, S.; Hiraiwa, S.; Matsumoto, E., Experimental and theoretical study of harmonic generation at contacting interface, Ultrasonics, 44, e1319-e1322 (2006)
[59] Kim, J.-Y.; Lee, J.-S., A micromechanical model for nonlinear acoustic properties of interfaces between solids, J. Appl. Phys., 101, Article 043501 pp. (2007)
[60] Aleshin, V.; Delrue, S.; Trifonov, A.; Matar, O. B.; Abeele, K. V.D., Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction, Part I: Theoretical background, Ultrasonics, 82, 11-18 (2018)
[61] Delrue, S.; Aleshin, V.; Truyaert, K.; Matar, O. B.; Abeele, K. V.D., Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction, Part II: Numerical implementation, Ultrasonics, 82, 19-30 (2018)
[62] Biwa, S.; Ishii, Y., Second-harmonic generation in an infinite layered structure with nonlinear spring-type interfaces, Wave Motion, 63, 55-67 (2016) · Zbl 1469.74007
[63] Ishii, Y.; Biwa, S.; Adachi, T., Second-harmonic generation in a multilayered structure with nonlinear spring-type interfaces embedded between two semi-infinite media, Wave Motion, 76, 28-41 (2018) · Zbl 1524.35619
[64] Ahmed, S.; Saka, M., A sensitive ultrasonic approach to NDE of tightly closed small cracks, ASME. J. Pressure Vessel Technol., 120, 4, 384-392 (1998)
[65] Aleshin, V.; Van Den Abeele, K., Micro-potential model for stress-strain hysteresis of micro-cracked materials, J. Mech. Phys. Solids, 53, 4, 795-824 (2005) · Zbl 1120.74309
[66] Tattersall, H., The ultrasonic pulse-echo technique as applied to adhesion testing, J. Phys. D: Appl. Phys., 6, 7, 819-832 (1973)
[67] Achenbach, J. D., Flaw characterization by ultrasonic scattering methods, (Achenbach, J. D.; Rajapakse, Y., Solid Mechanics Research for Quantitative Non-Destructive Evaluation (1987), Springer Netherlands: Springer Netherlands Dordrecht), 67-81
[68] Richardson, J. M., Harmonic generation at an unbonded interface-I. planar interface between semi-infinite elastic media, Internat. J. Engrg. Sci., 17, 1, 73-85 (1979) · Zbl 0392.73016
[69] Rudenko, O., Nonlinear acoustic properties of a rough surface contact and acoustodiagnostics of a roughness height distribution, Acoust. Phys., 40, 593-596 (1994)
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