0
Research Papers

Sagittal Plane Waves in Infinitely Periodic Multilayered Composites Composed of Alternating Viscoelastic and Elastic Solids

[+] Author and Article Information
A. B. M. Tahidul Haque, Amjad Aref

Department of Civil, Structural and
Environmental Engineering,
University at Buffalo,
Buffalo, NY 14260

Ratiba F. Ghachi

Department of Civil and Architectural
Engineering,
Qatar University,
P. O. Box 2713,
Doha, Qatar

Wael I. Alnahhal

Department of Civil and Architectural
Engineering,
Qatar University
P. O. Box 2713,
Doha, Qatar

Jongmin Shim

Department of Civil, Structural and
Environmental Engineering,
University at Buffalo,
240 Ketter Hall,
Buffalo, NY 14260
e-mail: jshim@buffalo.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 8, 2017; final manuscript received January 16, 2018; published online February 2, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(4), 041001 (Feb 02, 2018) (15 pages) Paper No: JAM-17-1496; doi: 10.1115/1.4039039 History: Received September 08, 2017; Revised January 16, 2018

In order to design phononic crystals whose band-gaps are located in low-frequency ranges, researchers commonly adopt low stiffness polymeric materials as key constituents and exploit the high impedance mismatch between metals and polymers. However, there has been very little research on wave propagation at arbitrary angles in the sagittal plane of viscoelastic-elastic multilayered composites because there exist the intricate wave attenuation characteristics at the layer interfaces. The objective of our investigation is to obtain analytical dispersion relation for oblique wave motion in the sagittal plane of infinitely periodic multilayered composite composed of alternating viscoelastic and elastic solids, where the attenuation of harmonic plane waves is found to occur only in the direction perpendicular to the layers. By using this wave propagation characteristic, we directly apply the semi-analytical approach employed in elastic multilayered composites to calculate the dispersion relation of sagittal plane waves in alternating viscoelastic-elastic multilayered composites. Specifically, we consider a bilayered composite composed of alternating aluminum and polyurethane elastomer, whose complex-valued viscoelastic moduli are experimentally determined by performing dynamic mechanical analysis (DMA). The analysis shows that the alternating viscoelastic-elastic layered composite does not possess a phononic band-gap regardless of incident angles. In addition, wave motions at oblique angles are found to travel with a wide range of frequency contents compared to wave motions perpendicular to the layers. The presented analysis demonstrates that wave dispersion relation in viscoelastic-elastic layered composites is distinctly different from the corresponding elastic counterpart, and highlights the importance of the viscoelastic modeling of polymeric materials in wave dispersion analysis.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Cao, W. W. , and Qi, W. K. , 1995, “ Plane Wave Propagation in Finite Composites,” J. Appl. Phys., 78(7), pp. 4627–4632. [CrossRef]
Zhang, V. Y. , Lefebvre, J. E. , and Gryba, T. , 2006, “ Resonant Transmission in Stop Bands of Acoustic Waves in Periodic Structures,” Ultrasonics, 44(1), pp. 899–904. [CrossRef]
Hussein, M. I. , Hulbert, G. M. , and Scott, R. A. , 2006, “ Dispersive Elastodynamics of 1D Banded Materials and Structures: Analysis,” J. Sound Vib., 289(4–5), pp. 779–806. [CrossRef]
Jensen, J. S. , and Pedersen, N. L. , 2006, “ On Maximal Eigenfrequency Separation in Two-Material Structures: The 1D and 2D Scalar Cases,” J. Sound Vib., 289(4–5), pp. 967–986. [CrossRef]
Lee, C. Y. , Leamy, M. J. , and Nadler, J. H. , 2010, “ Frequency Band Structure and Absorption Predictions for Multi-Periodic Acoustic Composites,” J. Sound Vib., 329(10), pp. 1809–1822. [CrossRef]
Aly, A. H. , Mehaney, A. , and Abdel-Rahman, E. , 2013, “ Study of Physical Parameters on the Properties of Phononic Band Gaps,” Int. J. Mod. Phys. B, 27(11), p. 1350047. [CrossRef]
Ponge, M. F. , Jacob, X. , and Gibiat, V. , 2014, “ Comparison of the Transmission Properties of Self-Similar, Periodic, and Random Multilayers at Normal Incidence,” J. Acoust. Soc. Am., 135(6), pp. 3390–3397. [CrossRef] [PubMed]
Nemat-Nasser, S. , Sadeghia, H. , Amirkhizib, A. V. , and Srivastavac, A. , 2015, “ Phononic Layered Composites for Stress-Wave Attenuation,” Mech. Res. Commun., 68, pp. 65–69. [CrossRef]
Nemat-Nasser, S. , and Srivastava, A. , 2011, “ Negative Effective Dynamic Mass-Density and Stiffness: Micro-Architecture and Phononic Transport in Periodic Composites,” AIP Adv., 1(4), p. 041502. [CrossRef]
Zhu, R. , Huang, G. L. , and Hu, G. K. , 2012, “ Effective Dynamic Properties and Multi-Resonant Design of Acoustic Metamaterials,” ASME J. Vib. Acoust., 134(3), p. 031006. [CrossRef]
Wang, Y. , Song, W. , Sun, E. , Zhang, R. , and Cao, W. , 2014, “ Tunable Passband in One-Dimensional Phononic Crystal Containing a Piezoelectric 0.62Pb(Mg1/3Nb2/3)O3-0.38 PbTiO3 Single Crystal Defect Layer,” Physica E, 60, pp. 37–41. [CrossRef]
Lee, S. M. , Cahill, D. G. , and Venkatasubramanian, R. , 1997, “ Thermal Conductivity of Si-Ge Superlattices,” Appl. Phys. Lett., 70(22), pp. 2957–2959. [CrossRef]
Pernot, G. , Stoffel, M. , Savic, I. , Pezzoli, F. , Chen, P. , Savelli, G. , Jacquot, A. , Schumann, J. , Denker, U. , Monch, I. , Deneke, C. , Schmidt, O. G. , Rampnoux, J. M. , Wang, S. , Plissonnier, M. , Rastelli, A. , Dilhaire, S. , and Mingo, N. , 2010, “ Precise Control of Thermal Conductivity at the Nanoscale Through Individual Phonon-Scattering Barriers,” Nat. Mater., 9(6), pp. 491–495. [CrossRef] [PubMed]
Liang, B. , Guo, X. S. , Tu, J. , Zhang, D. , and Cheng, J. C. , 2010, “ An Acoustic Rectifier,” Nat. Mater., 9(12), pp. 989–992. [CrossRef] [PubMed]
Ma, C. , Parker, R. G. , and Yellen, B. B. , 2013, “ Optimization of an Acoustic Rectifier for Uni-Directional Wave Propagation in Periodic Mass-Spring Lattices,” J. Sound Vib., 332(20), pp. 4876–4894. [CrossRef]
Saini, G. , Pezeril, T. , Torchinsky, D. H. , Yoon, J. , Kooi, S. E. , Thomas, E. L. , and Nelson, K. A. , 2011, “ Pulsed Laser Characterization of Multicomponent Polymer Acoustic and Mechanical Properties in the Sub-GHz Regime,” J. Mater. Res., 22(3), pp. 719–723. [CrossRef]
Lee, E. H. , and Yang, H. W. , 1973, “ On Waves in Composite Materials With Periodic Structure,” Soc. Ind. Appl. Math., 25(3), pp. 492–499. [CrossRef]
He, J. J. , Djafarirouhani, B. , and Sapriel, J. , 1988, “ Theory of Light-Scattering by Longitudinal-Acoustic Phonons in Superlattices,” Phys. Rev. B, 37(8), pp. 4086–4098. [CrossRef]
Tamura, S. , and Wolfe, J. P. , 1988, “ Acoustic Phonons in Multiconstituent Superlattices,” Phys. Rev. B, 38(8), pp. 5610–5614. [CrossRef]
Esquivel Sirvent, R. , and Cocoletzi, G. H. , 1994, “ Band Structure for the Propagation of Elastic Waves in Superlattices,” J. Acoust. Soc. Am., 95(1), pp. 86–90. [CrossRef]
Rouhani, B. D. , Dobrzynski, L. , Duparc, O. , Camley, R. , and Maradudin, A. , 1983, “ Sagittal Elastic Waves in Infinite and Semi-Infinite Superlattices,” Phys. Rev. B, 28(4), pp. 1711–1720. [CrossRef]
Nougaoui, A. , and Rouhani, B. D. , 1987, “ Elastic Waves in Periodically Layered Infinite and Semi-Infinite Anisotropic Media,” Surf. Sci., 185(1–2), pp. 125–153. [CrossRef]
Nougaoui, A. , and Rouhani, B. D. , 1988, “ Complex Band Structure of Acoustic Waves in Superlattices,” Surf. Sci., 199(3), pp. 623–637. [CrossRef]
Sapriel, J. , and Rouhani, B. D. , 1989, “ Vibrations in Superlattice,” Surf. Sci. Rep., 10(4–5), pp. 189–275. [CrossRef]
Nayfeh, A. H. , 1991, “ The General Problem of Elastic Wave Propagation in Multilayered Anisotropic Media,” J. Acoust. Soc. Am., 89(4), pp. 1521–1531. [CrossRef]
Braga, A. M. B. , and Herrmann, G. , 1992, “ Floquet Waves in Anisotropic Periodically Layered Composites,” J. Acoust. Soc. Am., 91(3), pp. 1211–1227. [CrossRef]
Haque, A. B. M. T. , and Shim, J. , 2016, “ On Spatial Aliasing in the Phononic Band-Structure of Layered Composites,” Int. J. Solids Struct., 96(1), pp. 380–392. [CrossRef]
Haque, A. B. M. T. , Ghachi, R. F. , Alnahhal, W. I. , Aref, A. , and Shim, J. , 2017, “ Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method,” ASME J. Vib. Acoust., 139(5), p. 051010. [CrossRef]
Tamura, S. , and Wolfe, J. P. , 1987, “ Coupled-Mode Stop Bands of Acoustic Phonons in Semiconductor Superlattices,” Phys. Rev. B, 35(5), pp. 2528–2531. [CrossRef]
Hurley, D. C. , Tamura, S. , Wolfe, J. P. , and Morkoc, H. , 1987, “ Imaging of Acoustic Phonon Stop Bands in Superlattices,” Phys. Rev. Lett., 58(23), pp. 2446–2449. [CrossRef] [PubMed]
Calle, F. , and Cardona, M. , 1989, “ Frequency Gaps for Folded Acoustic Phonons in Superlattices,” Solid State Commun., 72(12), pp. 1153–1158. [CrossRef]
Mizuno, S. , 2003, “ Resonance and Mode Conversion of Phonons Scattered by Superlattices With Inhomogeneity,” Phys. Rev. B, 68(19), p. 193305. [CrossRef]
Pennec, Y. , Djafari-Rouhani, B. , Larabi, H. , Vasseur, J. O. , and Hladky-Hennion, A. C. , 2008, “ Low-Frequency Gaps in a Phononic Crystal Constituted of Cylindrical Dots Deposited on a Thin Homogeneous Plate,” Phys. Rev. B, 78(10), p. 104105.
Assouar, M. B. , Senesi, M. , Oudich, M. , Ruzzene, M. , and Hou, Z. L. , 2012, “ Broadband Plate-Type Acoustic Metamaterial for Low-Frequency Sound Attenuation,” Appl. Phys. Lett., 101(17), p. 173505.
Varanasi, S. , Bolton, J. S. , Siegmund, T. H. , and Cipra, R. J. , 2013, “ The Low Frequency Performance of Metamaterial Barriers Based on Cellular Structures,” Appl. Acoust., 74(4), pp. 485–495. [CrossRef]
Hayashi, T. , Morimoto, Y. , Serikawa, M. , Tokuda, K. , and Tanaka, T. , 1983, “ Experimental Study on Cut-Off Phenomenon for Layered Composite,” Bull. JSME, 26(211), pp. 23–29. [CrossRef]
Marechal, P. , Lenoir, O. , Khaled, A. , El Kettani, M. E. C. , and Chenouni, D. , 2014, “ Viscoelasticity Effect on a Periodic Plane Medium Immersed in Water,” Acta Acust. Acust., 100(6), pp. 1036–1043. [CrossRef]
Tanaka, K. , and Kon-No, A. , 1980, “ Harmonic Viscoelastic Waves Propagating Normal to the Layers of Laminated Media,” Bull. JSME, 23(181), pp. 1092–1099. [CrossRef]
Mukherjee, S. , and Lee, E. H. , 1975, “ Dispersion Relations and Mode Shapes for Waves in Laminated Viscoelastic Composites by Finite Difference Methods,” Comput. Struct., 5(5–6), pp. 279–285. [CrossRef]
Mukherjee, S. , and Lee, E. , 1978, “ Dispersion Relations and Mode Shapes for Waves in Laminated Viscoelastic Composites by Variational Methods,” Int. J. Solids Struct., 14(1), pp. 1–13. [CrossRef]
Chevalier, Y. , 1988, “ Dispersion of Harmonic Waves in Elastic and Viscoelastic Periodic Composite Materials,” Recent Developments in Surface Acoustic Waves: Proceedings of European Mechanics Colloquium, Vol. 7, D. F. Parker , and G. A. Maugin , eds., Springer, Berlin, pp. 260–268. [CrossRef]
Zhao, Y. P. , and Wei, P. J. , 2009, “ The Band Gap of 1D Viscoelastic Phononic Crystal,” Comput. Mater. Sci., 46(3), pp. 603–606. [CrossRef]
Cooper, H. F. , and Reiss, E. L. , 1966, “ Reflection of Plane Viscoelastic Waves From Plane Boundaries,” J. Acoust. Soc. Am., 39(6), pp. 1133–1138. [CrossRef]
Shaw, R. P. , and Bugl, P. , 1969, “ Transmission of Plane Waves Through Layered Linear Viscoelastic Media,” J. Acoust. Soc. Am., 46(3), pp. 649–654. [CrossRef]
Borcherdt, R. D. , 2009, Viscoelastic Waves in Layered Media, Cambridge University Press, Cambridge, UK. [CrossRef]
Schoenberg, M. , 1971, “ Transmission and Reflection of Plane Waves at an Elastic-Viscoelastic Interface,” Geophys. J. R. Astron. Soc., 25(1–3), pp. 35–47. [CrossRef]
Silva, W. , 1976, “ Body Waves in a Layered Anelastic Solid,” Bull. Seismol. Soc. Am., 66(5), pp. 1539–1554. https://pubs.geoscienceworld.org/bssa/article-lookup/66/5/1539
Borcherdt, R. D. , 1977, “ Reflection and Refraction of Type-II S Waves in Elastic and Anelastic Media,” Bull. Seismol. Soc. Am., 67(1), pp. 43–67. https://pubs.geoscienceworld.org/ssa/bssa/article/67/1/43/117599/
Borcherdt, R. D. , 1982, “ Reflection-Refraction of General P- and Type-I S-Waves in Elastic and Anelastic Solids,” Geophys. J. R. Astron. Soc., 70(3), pp. 621–638. [CrossRef]
Krebes, E. S. , 1983, “ The Viscoelastic Reflection/Transmission Problem: Two Special Cases,” Bull. Seismol. Soc. Am., 73(6), pp. 1673–1683. https://pubs.geoscienceworld.org/ssa/bssa/article/73/6A/1673/118527/
Borcherdt, R. D. , and Wennerberg, L. , 1985, “ General P, Type-I S and Type-II S Waves in Anelastic Solids; Inhomogeneous Wave Fields in Low-Loss Solids,” Bull. Seismol. Soc. Am., 75(6), pp. 1729–1763. https://pubs.er.usgs.gov/publication/70012378
Sharma, M. D. , 2015, “ Snell's Law at the Boundaries of Real Elastic Media,” Math. Stud., 84(3–4), pp. 75–94. http://www.indianmathsociety.org.in/mathstudent-part-2-2015.pdf
Sharma, M. D. , 2011, “ Wave Propagation in a Dissipative Poroelastic Medium,” IMA J. Appl. Math., 78(1), pp. 59–69. https://academic.oup.com/imamat/article/78/1/59/756444
Buchen, P. W. , 1971, “ Plane Waves in Linear Viscoelastic Media,” Geophys. J. R. Astron. Soc., 23(5), pp. 531–542. [CrossRef]
Borcherdt, R. D. , 1973, “ Energy and Plane Waves in Linear Viscoelastic Media,” J. Geophys. Res., 78(14), pp. 2442–2453. [CrossRef]
Brinson, H. F. , and Brinson, L. C. , 2008, Polymer Engineering Science and Viscoelasticity: An Introduction, Springer, New York. [CrossRef]
Banergee, B. , 2011, An Introduction to Metamaterials and Waves in Composites, CRC Press, Boca Raton, FL.
Coquin, G. , 1964, “ Attenuation of Guided Waves in Isotropic Viscoelastic Materials,” J. Acoust. Soc. Am., 36(6), pp. 1074–1080. [CrossRef]
Garcia-Barruetabena, J. , Cortes, F. , Manuel Abete, J. , Fernandez, P. , Jesus Lamela, M. , and Fernandez-Canteli, A. , 2013, “ Relaxation Modulus-Complex Modulus Interconversion for Linear Viscoelastic Materials,” Mech. Time-Depend. Mater., 17(3), pp. 465–479. [CrossRef]
Pike, R. , and Sabatier, P. , 2002, Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, London.
Lockett, F. J. , 1962, “ The Reflection and Refraction of Waves at an Interface Between Viscoelastic Materials,” J. Mech. Phys. Solids, 10(1), pp. 53–64. [CrossRef]
Cooper, H. F. , 1967, “ Reflection and Transmission of Oblique Plane Waves at a Plane Interface Between Viscoelastic Media,” J. Acoust. Soc. Am., 42(5), pp. 1064–1069. [CrossRef]
Sharma, M. D. , and Vashishth, A. K. , 2011, “ Intrinsic Attenuation From Inhomogeneous Waves in a Dissipative Anisotropic Poroelastic Medium,” Earth, Planets Space, 63(2), pp. 89–101. [CrossRef]
Graff, K. F. , 1991, Wave Motion in Elastic Solids, Dover Publications, Mineola, NY.
Kittel, C. , 2004, Introduction to Solid State Physics, 8th ed., Wiley, New York.
Ashcroft, N. W. , and Mermin, N. D. , 1976, Solid State Physics, Saunders College, Philadelphia, PA.
Birkhoff, G. , and MacLane, S. , 1977, A Survey of Modern Algebra, 4th ed., Macmillan Publishing, New York.
Jensen, F. B. , Schmidt, H. , Porter, M. B. , and Kuperman, W. A. , 2011, Computational Ocean Acoustics, Springer, New York. [CrossRef]
Brito-Santana, H. , Wang, Y. S. , Rodriguez-Ramos, R. , Bravo-Castillero, J. , Guinovart-Diaz, R. , and Tita, V. , 2015, “ A Dispersive Nonlocal Model for In-Plane Wave Propagation in Laminated Composites With Periodic Structures,” ASME J. Appl. Mech., 82(3), p. 031006. [CrossRef]
Brito-Santana, H. , Wang, Y. S. , Rodriguez-Ramos, R. , Bravo-Castillero, J. , Guinovart-Diaz, R. , and Tita, V. , 2015, “ A Dispersive Nonlocal Model for Shear Wave Propagation in Laminated Composites With Periodic Structures,” Eur. J. Mech. A, 49(1), pp. 35–48. [CrossRef]
Theocharis, G. , Richoux, O. , Garcia, V. R. , Merkel, A. , and Tournat, V. , 2014, “ Limits of Slow Sound Propagation and Transparency in Lossy, Locally Resonant Periodic Structures,” New J. Phys., 16(1), p. 093017. [CrossRef]
Yu, G. K. , and Wang, X. L. , 2014, “ Acoustical ‘Transparency’ Induced by Local Resonance in Bragg Bandgaps,” J. Appl. Phys., 115(4), p. 044913. [CrossRef]
Groby, J. P. , Huang, W. , Lardeau, A. , and Auregan, Y. , 2015, “ The Use of Slow Waves to Design Simple Sound Absorbing Materials,” J. Appl. Phys., 117(12), p. 124903. [CrossRef]
Groby, J. P. , Pommier, R. , and Auregan, Y. , 2016, “ Use of Slow Sound to Design Perfect and Broadband Passive Sound Absorbing Materials,” J. Acoust. Soc. Am., 139(4), pp. 1660–1671. [CrossRef] [PubMed]
Potel, C. , and de Belleval, J. F. , 1993, “ Propagation in an Anisotropic Periodically Multilayered Medium,” J. Acoust. Soc. Am., 93(5), pp. 2669–2677. [CrossRef]
Sigalas, M. M. , and Soukoulis, C. M. , 1995, “ Elastic-Wave Propagation Through Disordered and/or Absorptive Layered Systems,” Phys. Rev. B, 51(5), pp. 2780–2789. [CrossRef]
Bousfia, A. , El Boudouti, E. H. , Djafari-Rouhani, B. , Bria, D. , Nougaoui, A. , and Velasco, V. R. , 2001, “ Omnidirectional Phononic Reflection and Selective Transmission in One-Dimensional Acoustic Layered Structures,” Surf. Sci., 482–485(2), pp. 1175–1180. [CrossRef]
Bria, D. , Djafari-Rouhani, B. , Bousfia, A. , El Boudouti, E. H. , and Nougaoui, A. , 2001, “ Absolute Acoustic Band Gap in Coupled Multilayer Structures,” Europhys. Lett., 55(6), pp. 841–846. [CrossRef]
Shen, M. R. , and Cao, W. W. , 1999, “ Acoustic Band-Gap Engineering Using Finite-Size Layered Structures of Multiple Periodicity,” Appl. Phys. Lett., 75(23), pp. 3713–3715. [CrossRef]
Shen, M. R. , and Cao, W. W. , 2000, “ Acoustic Bandgap Formation in a Periodic Structure With Multilayer Unit Cells,” J. Phys. D: Appl. Phys., 33(10), pp. 1150–1154. [CrossRef]
Santos, P. V. , Mebert, J. , Koblinger, O. , and Ley, L. , 1987, “ Experimental Evidence for Coupled-Mode Phonon Gaps in Superlattice Structures,” Phys. Rev. B, 36(2), pp. 1306–1309. [CrossRef]
Naciri, T. , Navi, P. , and Granacher, O. , 1990, “ On Harmonic Wave Propagation in Multilayered Viscoelastic Media,” Int. J. Mech. Sci., 32(3), pp. 225–231. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

(a) Infinitely periodic M-layered composite that is periodic along the x2-axis. The primitive lattice vector a2 defines the unit cell of the composite with a periodic length a2=||a2||. (b) Unit cell of the M-layered composite in the two-dimensional (2D) sagittal plane where wave can propagate in any direction by making angle θ with the x2-axis. The periodic length of the inclined wave is given by (a2 cos θ). (c) Reciprocal space of the M-layered composite showing the first Brillouin zone on the κ1−κ2 plane. The rectangle on the right represents the IBZ which is bounded by κ2∈[0,π/a2] in the κ2-axis. The propagating wavevector κ describes wave motion at an angle θ which has periodic length π/(a2 cos θ) in the reciprocal space.

Grahic Jump Location
Fig. 2

(a) P-wave with a wavevector κp,1 (i.e., κp,1=κp,1R+iκp,1I) incidents at the interface between viscoelastic layers 1 and 2 with a propagation angle θp,1 and an attenuation angle ζp,1. (b) Reflected and transmitted P-waves in layers 1 and 2, respectively. The propagation and attenuation wavevectors are represented by κp,jR and κp,jI (for j = 1, 2), respectively. (c) Reflected and transmitted SV-waves in layers 1 and 2, respectively. The propagation and attenuation wavevectors are represented by κs,jR and κs,jI (for j = 1, 2), respectively. Note that the angles of propagation and attenuation waves are denoted by θr,j and ζr,j (for j = 1, 2 and r=p,s), respectively.

Grahic Jump Location
Fig. 3

(a-1) Incident P-wave having a wavevector κp,1R at an angle θp,1 from the elastic medium at the interface between elastic and viscoelastic layers 1 and 2. Reflected and transmitted (a-2) P-waves and (a-3) SV-waves having characterized by the propagation wavevectors κr,jR (for j = 1, 2 and r=p,s) for both layers and the attenuation wavevectors κr,2I for the viscoelastic layer. (b-1) Incident P-wave having a wavevector κp,1R at an angle θp,1 from the viscoelastic medium at the interface between elastic and viscoelastic layers 1 and 2. Reflected and transmitted (b-2) P-waves and (b-3) SV-waves having characterized by the propagation wavevectors κr,jR (for j = 1, 2 and r=p,s) for both layers and the attenuation wavevectors κr,2I for the viscoelastic layer. Note that the propagation angles in the layers are denoted by θr,j and the attenuation angles in the viscoelastic layer are shown by ζr,2.

Grahic Jump Location
Fig. 4

Viscoelastic properties of polyurethane elastomer obtained by DMA. (a) Frequency-dependent modulus λ̂(ω). (b) Frequency-dependent modulus λ̂(ω). The storage and the loss moduli are represented by circle- and square-marked solid lines, respectively. Note that the shaded areas denote one standard deviation from two DMA tests. The constant modulus of pseudo-elastic approximation is shown by dotted lines.

Grahic Jump Location
Fig. 5

A schematic view of representing the complex-valued dispersion relation (κ=[κ1 κ2R]T+i [0 κ2I]T and f) in the two-dimensional plots (i.e., κθR−f and κ2I−f). The phase dispersion relation is shown by a solid line in the wave propagation plane inclined at an angle θ on the κ1−κ2R plane. The inclined length of the phase dispersion plane is π/a2 cos θ, 0 which is the projection length of π/a2 on the κ2R-axis. As a demonstration, three exemplary points of κθR−f are projected on the κ2I−f plane, which represents the corresponding attenuation relation.

Grahic Jump Location
Fig. 6

Complete dispersion analysis results of the pseudo-elastic composite showing attenuation relations κ2I−f for wave motions at (a-1) θ=0 deg, (b-1) θ=15 deg, (c-1) θ=30 deg, and (d-1) θ=60 deg. The phase dispersion relations κθR−f are illustrated for wave motions at (a-2) θ=0 deg, (b-2) θ=15 deg, (c-2) θ=30 deg, and (d-2) θ=60 deg. Note that the range of the wavevector κθR∈[0,π/(a2  cos θ)] within the IBZ varies with the propagation angle.

Grahic Jump Location
Fig. 7

Complete dispersion analysis results of the viscoelastic-elastic composite showing attenuation relations κ2I−f for wave motions at (a-1) θ=0 deg, (b-1) θ=15 deg, (c-1) θ=30 deg, and (d-1) θ=60 deg. The phase dispersion relations κθR−f are illustrated for wave motions at (a-2) θ=0 deg, (b-2) θ=15 deg, (c-2) θ=30 deg, and (d-2) θ=60 deg. Note that the range of the wavevector κθR∈[0,π/(a2  cos θ)] within the IBZ varies with propagation angles.

Grahic Jump Location
Fig. 8

Transmission coefficient Ct of the pseudo-elastic composite calculated from Eq. (33) for wave motion at (a-1) θ=0 deg, (b-1) θ=15 deg, (c-1) θ=30 deg, and (d-1) θ=60 deg. Group slowness Sg obtained using Eq. (31) for wave motion at (a-2) θ=0 deg, (b-2) θ=15 deg, (c-2) θ=30 deg, and (d-2) θ=60 deg.

Grahic Jump Location
Fig. 9

Transmission coefficient Ct of periodic the viscoelastic-elastic composite calculated from Eq. (33) for wave motion at (a-1) θ=0 deg, (b-1) θ=15 deg, (c-1) θ=30 deg, and (d-1) θ=60 deg. Group slowness Sg obtained using Eq. (31) for wave motion at (a-2) θ=0 deg, (b-2) θ=15 deg, (c-2) θ=30 deg, and (d-2) θ=60 deg.

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In