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Soft metamaterials with dynamic viscoelastic functionality tuned by pre-deformation. (English) Zbl 1425.74110
Summary: The small amplitude dynamic response of materials can be tuned by employing inhomogeneous materials capable of large deformation. However, soft materials generally exhibit viscoelastic behaviour, i.e. loss and frequency-dependent effective properties. This is the case for inhomogeneous materials even in the homogenization limit when propagating wavelengths are much longer than phase lengthscales, since soft materials can possess long relaxation times. These media, possessing rich frequency-dependent behaviour over a wide range of low frequencies, can be termed metamaterials in modern terminology. The sub-class that are periodic are frequently termed soft phononic crystals although their strong dynamic behaviour usually depends on wavelengths being of the same order as the microstructure. In this paper we describe how the effective loss and storage moduli associated with longitudinal waves in thin inhomogeneous rods are tuned by pre-stress. Phases are assumed to be quasi-linearly viscoelastic, thus exhibiting time-deformation separability in their constitutive response. We illustrate however that the effective incremental response of the inhomogeneous medium does not exhibit time-deformation separability. For a range of nonlinear materials it is shown that there is strong coupling between the frequency of the small amplitude longitudinal wave and initial large deformation.
74D05 Linear constitutive equations for materials with memory
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[1] Ying Z, Ni Y. (2017) Advances in structural vibration control application of magneto-rheological visco-elastomer. Theor. Appl. Mech. Lett. 7, 61-66. (doi:10.1016/j.taml.2017.01.003)
[2] Rus D, Tolley MT. (2015) Design, fabrication and control of soft robots. Nature 521, 467-475. (doi:10.1038/nature14543)
[3] Li CH et al. (2016) A highly stretchable autonomous self-healing elastomer. Nat. Chem. 8, 618-624. (doi:10.1038/nchem.2492)
[4] Richards MS, Barbone PE, Oberai AA. (2009) Quantitative three-dimensional elasticity imaging from quasi-static deformation: a phantom study. Phys. Med. Biol. 54, 757. (doi:10.1088/0031-9155/54/3/019)
[5] Destrade M, Ogden RW, Saccomandi G. (2009) Small amplitude waves and stability for a pre-stressed viscoelastic solid. Z. Angew. Math. Phys. 60, 511-528. (doi:10.1007/s00033-008-7147-6) · Zbl 1169.74024
[6] Bigoni D. (2012) Nonlinear solid mechanics: bifurcation theory and material instability. Cambridge, UK: Cambridge University Press. · Zbl 1269.74003
[7] Destrade M, Saccomandi G. (2007) Waves in nonlinear pre-stressed materials, vol. 495. Berlin, Germany: Springer Science & Business Media. · Zbl 1135.74011
[8] Li GY, He Q, Mangan R, Xu G, Mo C, Luo J, Michel D, Cao Y. (2017) Guided waves in pre-stressed hyperelastic plates and tubes: application to the ultrasound elastography of thin-walled soft materials. J. Mech. Phys. Solids 102, 67-79. (doi:10.1016/j.jmps.2017.02.008)
[9] Huang X, Zhou S, Sun G, Li G, Xie YM. (2015) Topology optimization for microstructures of viscoelastic composite materials. Comput. Methods Appl. Mech. Eng. 283, 503-516. (doi:10.1016/j.cma.2014.10.007) · Zbl 1423.74748
[10] Ansari R, Aghdam MH. (2016) Micromechanics-based viscoelastic analysis of carbon nanotube-reinforced composites subjected to uniaxial and biaxial loading. Compos. B Eng. 90, 512-522. (doi:10.1016/j.compositesb.2015.10.048)
[11] Bakhvalov N, Panasenko GP. (1989) Homogenisation: averaging processes in periodic media: mathematical problems in the mechanics of composite materials. Dordrecht, The Netherlands: Kluwer Academic Publishers. · Zbl 0692.73012
[12] Jianmin Q, Cherkaoui M. (2006) Fundamentals of micromechanics of solids. New York, NY: John Wiley and Sons.
[13] Milton GW. (2002) The theory of composites. Cambridge, UK: Cambridge University Press.
[14] Yi YM, Park SH, Youn SK. (1998) Asymptotic homogenization of viscoelastic composites with periodic microstructures. Int. J. Solids Struct. 35, 2039-2055. (doi:10.1016/S0020-7683(97)00166-2) · Zbl 0933.74054
[15] Parnell WJ, Abrahams ID. (2006) Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH waves. Wave Motion 43, 474-498. (doi:10.1016/j.wavemoti.2006.03.003) · Zbl 1231.74373
[16] Parnell WJ, Abrahams ID. (2008) A new integral equation approach to elastodynamic homogenization. Proc. R. Soc. A 464, 1461-1482. (doi:10.1098/rspa.2007.0254) · Zbl 1136.74034
[17] Joyce D, Parnell WJ, Assier RC, Abrahams ID. (2017) An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems. Proc. R. Soc. A 473, 20170080. (doi:10.1098/rspa.2017.0080) · Zbl 1404.74142
[18] Cummer SA, Christensen J, Alù A. (2016) Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 16001. (doi:10.1038/natrevmats.2016.1)
[19] Ma G, Sheng P. (2016) Acoustic metamaterials: from local resonances to broad horizons. Sci. Adv. 2, e1501595. (doi:10.1126/sciadv.1501595)
[20] Torrent D, Sánchez-Dehesa J. (2008) Acoustic cloaking in two dimensions: a feasible approach. New J. Phys. 10, 063015. (doi:10.1088/1367-2630/10/6/063015)
[21] Groby JP, Huang W, Lardeau A, Aurégan Y. (2015) The use of slow waves to design simple sound absorbing materials. J. Appl. Phys. 117, 124903. (doi:10.1063/1.4915115)
[22] Zigoneanu L, Popa BI, Cummer SA. (2014) Three-dimensional broadband omnidirectional acoustic ground cloak. Nat. Mater. 13, 352-355. (doi:10.1038/nmat3901)
[23] Rowley WD, Parnell WJ, Abrahams ID, Voisey SR, Lamb J, Etaix N. (2018) Deepening subwavelength acoustic resonance via metamaterials with universal broadband elliptical microstructure. Appl. Phys. Lett. 112, 251902. (doi:10.1063/1.5022197)
[24] Parnell WJ. (2012) Nonlinear pre-stress for cloaking from antiplane elastic waves. Proc. R. Soc. A 468, 563-580. (doi:10.1098/rspa.2011.0477) · Zbl 1364.74049
[25] Norris AN, Parnell WJ. (2012) Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids. Proc. R. Soc. A 468, 2881-2903. (doi:10.1098/rspa.2012.0123) · Zbl 1371.74039
[26] Chang Z, Guo HY, Li B, Feng XQ. (2015) Disentangling longitudinal and shear elastic waves by neo-Hookean soft devices. Appl. Phys. Lett. 106, 161903. (doi:10.1063/1.4918787)
[27] Zhang P, Parnell WJ. (2018) Hyperelastic antiplane ground cloaking. J. Acoust. Soc. Am. 143, 2878-2885. (doi:10.1121/1.5036629)
[28] Parnell WJ. (2007) Effective wave propagation in a prestressed nonlinear elastic composite bar. IMA J. Appl. Math. 72, 223-244. (doi:10.1093/imamat/hxl033) · Zbl 1118.74025
[29] Bertoldi K, Boyce M. (2008) Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations. Phys. Rev. B 78, 184107. (doi:10.1103/PhysRevB.78.184107)
[30] Gei M, Movchan AB, Bigoni D. (2009) Band-gap shift and defect-induced annihilation in prestressed elastic structures. J. Appl. Phys. 105, 063507. (doi:10.1063/1.3093694)
[31] Barnwell EG, Parnell WJ, Abrahams ID. (2016) Antiplane elastic wave propagation in pre-stressed periodic structures; tuning, band gap switching and invariance. Wave Motion 63, 98-110. (doi:10.1016/j.wavemoti.2016.02.001)
[32] Barnwell EG, Parnell WJ, Abrahams ID. (2017) Tunable elastodynamic band gaps. Extreme Mech. Lett. 12, 23-29. (doi:10.1016/j.eml.2016.10.009)
[33] Zhang P, Parnell WJ. (2017) Soft phononic crystals with deformation-independent band gaps. Proc. R. Soc. A 473, 20160865. (doi:10.1098/rspa.2016.0865) · Zbl 1404.74137
[34] Galich PI, Fang NX, Boyce MC, Rudykh S. (2017) Elastic wave propagation in finitely deformed layered materials. J. Mech. Phys. Solids 98, 390-410. (doi:10.1016/j.jmps.2016.10.002)
[35] Getz R, Kochmann DM, Shmuel G. (2017) Voltage-controlled complete stopbands in two-dimensional soft dielectrics. Int. J. Solids Struct. 113, 24-36. (doi:10.1016/j.ijsolstr.2016.10.002)
[36] Wu B, Zhou W, Bao R, Chen W. (2018) Tuning elastic waves in soft phononic crystal cylinders via large deformation and electromechanical coupling. J. Appl. Mech. 85, 031004. (doi:10.1115/1.4038770)
[37] Chen Y, Wu B, Su Y, Chen W. (2019) Tunable two-way unidirectional acoustic diodes: design and simulation. J. Appl. Mech. 86, 031010. (doi:10.1115/1.4042321)
[38] Torrent D, Parnell WJ, Norris AN. (2018) Loss compensation in time-dependent elastic metamaterials. Phys. Rev. B 97, 014105.
[39] Wineman A. (2009) Nonlinear viscoelastic solids - a review. Math. Mech. Solids 14, 300-366. (doi:10.1177/1081286509103660) · Zbl 1197.74021
[40] De Pascalis R, Abrahams ID, Parnell WJ. (2014) On nonlinear viscoelastic deformations: a reappraisal of Fung’s quasi-linear viscoelastic model. Proc. R. Soc. A 470, 20140058. (doi:10.1098/rspa.2014.0058)
[41] Jridi N, Arfaoui M, Hamdi A, Salvia M, Bareille O, Ichchou M, Ben Abdallah J. (2018) Separable finite viscoelasticity: integral-based models vs. experiments. Mechanics of time-dependent materials, 1-31. (doi:10.1007/s11043-018-9383-2)
[42] De Pascalis R, Parnell WJ, Abrahams ID, Shearer T, Daly DM, Grundy D. (2018) The inflation of viscoelastic balloons and hollow viscera. Proc. R. Soc. A 474, 20180102. (doi:10.1098/rspa.2018.0102) · Zbl 1407.74024
[43] De Pascalis R, Napoli G, Saccomandi G. (2019) Kink-type solitary waves within the quasi-linear viscoelastic model. Wave Motion 86, 195-202. (doi:10.1016/j.wavemoti.2018.12.004)
[44] De Pascalis R, Abrahams ID, Parnell WJ. (2015) Simple shear of a compressible quasilinear viscoelastic material. Int. J. Eng. Sci. 88, 64-72. (doi:10.1016/j.ijengsci.2014.11.011) · Zbl 1423.74184
[45] Balbi V, Shearer T, Parnell WJ. (2018) A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation. Proc. R. Soc. A 474, 20180231. (doi:10.1098/rspa.2018.0231) · Zbl 1407.74021
[46] Simo JC. (1987) On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60, 153-173. (doi:10.1016/0045-7825(87)90107-1) · Zbl 0588.73082
[47] Pipkin AC, Rivlin RS. (1961) Small deformations superposed on large deformations in materials with fading memory. Arch. Ration. Mech. Anal. 8, 297-308. (doi:10.1007/BF00277445) · Zbl 0111.36105
[48] Hayes M, Rivlin R. (1969) Propagation of sinusoidal small-amplitude waves in a deformed viscoelastic solid. I. J. Acoust. Soc. Am. 46, 610-616. (doi:10.1121/1.1911738) · Zbl 0169.28001
[49] Hayes M, Rivlin R. (1972) Propagation of sinusoidal small-amplitude waves in a deformed viscoelastic solid. II. J. Acoust. Soc. Am. 51, 1652-1663. (doi:10.1121/1.1913011) · Zbl 0239.73029
[50] Hayes M, Rivlin R. (1974) Plane waves in linear viscoelastic materials. Q. Appl. Math. 32, 113-121. (doi:10.1090/qam/1974-32-02) · Zbl 0282.73012
[51] Zapas L, Wineman A. (1985) Superposition of small shear deformations on large uniaxial extensions for viscoelastic materials. Polymer 26, 1105-1109. (doi:10.1016/0032-3861(85)90237-X)
[52] Rivlin RS. (1949) Large elastic deformations of isotropic materials VI. Further results in the theory of torsion, shear and flexure. Phil. Trans. R. Soc. Lond. A 242, 173-195. (doi:10.1098/rsta.1949.0009) · Zbl 0035.41503
[53] Kim BK, Youn SK. (2001) A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation. Arch. Appl. Mech. 71, 748-763. (doi:10.1007/s004190100186) · Zbl 1050.74014
[54] Kim BK, Youn SK, Lee WS. (2004) A constitutive model and FEA of rubber under small oscillatory load superimposed on large static deformation. Arch. Appl. Mech. 73, 781-798. (doi:10.1007/s00419-004-0325-x) · Zbl 1145.74326
[55] Morman K, Nagtegaal J. (1983) Finite element analysis of sinusoidal small-amplitude vibrations in deformed viscoelastic solids. Part I: theoretical development. Int. J. Numer. Methods Eng. 19, 1079-1103. (doi:10.1002/(ISSN)1097-0207) · Zbl 0509.73088
[56] Gurtin ME, Sternberg E. (1962) On the linear theory of viscoelasticity. Arch. Ration. Mech. Anal. 11, 291-356. (doi:10.1007/BF00253942) · Zbl 0107.41007
[57] Hunter S. (1983) Mechanics of continuous media, 2nd edn. New York: John Wiley and Sons. · Zbl 0509.73004
[58] Chen T. (2000) Determining a Prony series for a viscoelastic materials from time varying strain data, 21 p. NASA.
[59] Shatalov M, Marais J, Fedotov I, Tenkam MD. (2011) Longitudinal vibration of isotropic solid rods: from classical to modern theories. In Advances in computer science and engineering (ed. M Schmidt), pp. 187-214. IntechOpen. (doi:10.5772/15662)
[60] Graff KF. (1991) Wave motion in elastic solids. New York, NY: Dover Publications.
[61] Christensen R. (1975) Wave propagation in layered elastic media. J. Appl. Mech. 42, 153-158. (doi:10.1115/1.3423507) · Zbl 0311.73012
[62] Ting T, Mukunoki I. (1979) A theory of viscoelastic analogy for wave propagation normal to the layering of a layered medium. J. Appl. Mech. 46, 329-336. (doi:10.1115/1.3424550) · Zbl 0406.73030
[63] Zhao Y, Wei P. (2009) The band gap of 1D viscoelastic phononic crystal. Comput. Mater. Sci. 46, 603-606. (doi:10.1016/j.commatsci.2009.03.040)
[64] Moiseyenko RP, Laude V. (2011) Material loss influence on the complex band structure and group velocity in phononic crystals. Phys. Rev. B 83, 064301. (doi:10.1103/PhysRevB.83.064301)
[65] Wang YF, Wang YS, Laude V. (2015) Wave propagation in two-dimensional viscoelastic metamaterials. Phys. Rev. B 92, 104110. (doi:10.1103/PhysRevB.92.104110)
[66] Zhu X, Zhong S, Zhao H. (2016) Band gap structures for viscoelastic phononic crystals based on numerical and experimental investigation. Appl. Acousts. 106, 93-104. (doi:10.1016/j.apacoust.2016.01.007)
[67] Hussein MI, Frazier MJ. (2010) Band structure of phononic crystals with general damping. J. Appl. Phys. 108.9, 093506.
[68] Hussein MI, Frazier MJ. (2013) Metadamping: an emergent phenomenon in dissipative metamaterials. Journal of Sound and Vibration 332.20, 4767-4774.
[69] Garrido M, Correia JR, Keller T. (2016) Effect of service temperature on the shear creep response of rigid polyurethane foam used in composite sandwich floor panels. Constr. Build. Mater. 118, 235-244. (doi:10.1016/j.conbuildmat.2016.05.074)
[70] Zajac M, Kahl H, Schade B, Rödel T, Dionisio M, Beiner M. (2017) Relaxation behavior of polyurethane networks with different composition and crosslinking density. Polymer 111, 83-90. (doi:10.1016/j.polymer.2017.01.032)
[71] Fung YC. (1981) Biomechanics: mechanical properties of living tissues. New York, NY: Springer.
[72] Mooney M. (1940) A theory of large elastic deformation. J. Appl. Phys. 11, 582-592. (doi:10.1063/1.1712836) · JFM 66.1021.04
[73] De Pascalis R. (2010) The Semi-Inverse Method in solid mechanics: theoretical underpinnings and novel applications. PhD thesis.
[74] Rogerson GA, Sandiford KJ. (2000) The effect of finite primary deformations on harmonic waves in layered elastic media. Int. J. Solids Struct. 37, 2059-2087. (doi:10.1016/S0020-7683(98)00347-3) · Zbl 0987.74037
[75] Kayestha P, Wijeyewickrema AC, Kishimoto K. (2010) Time-harmonic wave propagation in a pre-stressed compressible elastic bi-material laminate. Eur. J. Mech. A/Solids 29, 143-151. (doi:10.1016/j.euromechsol.2009.08.005)
[76] Vaughan H. (1979) Effect of stretch on wave speed in rubberlike materials. Q. J. Mech. Appl. Math. 32, 215-231. (doi:10.1093/qjmam/32.3.215) · Zbl 0421.73016
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