臺灣博碩士論文加值系統

超材料的興起使得不少學者投入地震超材料的領域，進而從中探索及研究如何衰減地震表面波，地震超材料透過自然界不存在之特殊現象達到此效果，而此現象稱為帶隙 (bandgap)，本文利用此一機制設計超材料，衰減地震表面波中最具威脅性的「雷利波」。相較於以往簡化之等向性介質之波傳假設，本文將傳遞介質延伸至橫斷面等向性材料，在自然界中沉積岩即是常見橫斷面等向性材料。此外，透過理論推導雷利波與超材料耦合後之頻散關係，並藉有限元素分析軟體分析結果探討比對雷利波於等向性及橫斷面等向性材料之異同。在橫斷面等向性材料中，調整彈性模數k及m，均會改變帶隙寬度，且兩者造成截然不同之影響。最後，本文引入洛夫波，目標達到一種地震超材料可以同時對兩種地震表面波產生衰減之效果，再利用有限元素分析軟體之結果，分析後得到此超材料對雷利波及洛夫波均有消能效果。

Seismic metamaterials exhibit that unusual material parameters, that do not exist in nature. They can be negative, such as moduli, negative bulk modulus, negatives hear modulus and negative mass density. With these mechanical resonators buried beneath the surface, we can attenuate the seismic waves, especially Rayleigh surface waves. The objective of this work is to explore the possibility to control the Rayleigh waves dispersion behavior by varying the properties of transversely isotropic substrate and the resonators mechanical parameters. We derive the dispersion relation of Rayleigh waves coupled with metamaterials. We also perform the three-dimensional finite element simulations, using the simulation results to check with the analytic solutions. Moreover, we introduce transversely isotropic material to examine the effects on the displacement fields of Rayleigh waves. The results suggest that transverse isotropy has influences on the width of band gaps. Due to this result, our aim is to explore how do the material properties of the transversely isotropic material affect Rayleigh waves. We find the moduli k and m in transversely isotropic materials lead to an opposite trend of bandgap width. In the last chapter, we have simulations of performance the seismic metamaterials on Rayleigh and Love waves for the attenuation. The result demonstrates that this designed seismic metamaterials can mitigate Rayleigh and Love waves simultaneously.

Abstract i
中文摘要 iii
Acknowledgements v
Table of contents vii
List of figures ix
List of tables xiii
Symbols 1
Chapter 1 Introduction 3
1.1 Background and literature review 3
1.2 Motivation and objective 6
1.3 Overview of thesis 7
Chapter 2 Propagation of Rayleigh waves and Love waves 9
2.1 Types of seismic waves 9
2.2 Summary of He’s thesis (2019) 13
2.3 Isotropic materials and wave propagation theorem 14
2.4 Propagations of Love waves and Rayleigh waves 17
2.4.1 Love waves 17
2.4.2 Rayleigh waves 18
2.5 The propagation of Rayleigh waves in transversely isotropic medium 23
2.5.1 Transverse isotropy 23
2.5.2 The derivation of wave propagation of Rayleigh waves in transversely isotropic medium 26
Chapter 3 Rayleigh waves coupled with seismic metamaterials 29
3.1 The dispersion relation of Rayleigh waves coupled with seismic metamaterials 29
3.1.1 Isotropic medium 29
3.1.2 Transversely isotropic medium 35
3.2 3D simulation setting 37
3.3 Simulation results 41
3.3.1 Isotropic medium 41
3.3.2 Transversely isotropic medium 47
3.4 Tunable elastic constants 54
Chapter 4 Surface waves coupled with seismic metamaterials in transversely isotropic media 59
4.1 Simulation model 59
4.1.1 Seismic metamaterials 59
4.1.2 Simulation of field size model 61
Chapter 5 Conclusions and future perspectives 73
5.1 Conclusions 73
5.2 Future perspectives 74
Reference 77
Appendix A. Tutorial of COMSOL 81

Achaoui, Y., Ungureanu, B., Enoch, S., Brûlé, S. and Guenneau, S. (2016) ‘Seismic waves damping with arrays of inertial resonators’, Extreme Mechanics Letters, 8, pp. 30–37. doi: 10.1016/j.eml.2016.02.004.
Achaoui, Y., Antonakakis, T., Brûlé, S. Craster, R. V, Enoch, S. and Guenneau, S. (2017) ‘Clamped seismic metamaterials: Ultra-low frequency stop bands’, New Journal of Physics, 19(6). doi: 10.1088/1367-2630/aa6e21.
Achenbach, J. (1973) ‘Wave Propagation in Elastic Solids’, North Holland Publishing Company, pp. 30–30. doi: 10.1016/0003-682x(75)90007-9.
Achenbach, J. D. (1998) ‘Explicit solutions for carrier waves supporting surface waves and plate waves’, Wave Motion, 28(1), pp. 89–97. doi: 10.1016/S0165-2125(97)00056-5.
Boechler, N. et al. (2013) ‘Interaction of a contact resonance of microspheres with surface acoustic waves’, Physical Review Letters, 111(3). doi: 10.1103/PhysRevLett.111.036103.
Brûlé, S. Javelaud, E. H., Enoch, S. and Guenneau, S. (2014) ‘Experiments on seismic metamaterials: Molding surface waves’, Physical Review Letters. American Physical Society, 112(13). doi: 10.1103/PhysRevLett.112.133901.
Buchwald, V. T. (1961) ‘Rayleigh waves in transversely isotropic media’, Quarterly Journal of Mechanics and Applied Mathematics, 14(3), pp. 293–318. doi: 10.1093/qjmam/14.3.293.
Cheadle, S. P., Brown, R. J. andLawton, D. C. (1991) ‘Orthorhombic anisotropy: a physical seismic modeling study’, Geophysics, 56(10), pp. 1603–1613. doi: 10.1190/1.1442971.
Chien, T. Y. et al. (2019) ‘A Simple Proposition of Two-Dimensional Configuration of Seismic Metamaterials — A Promising Tool Towards Seismic Cloaking’, Journal of the Chinese Institute of Civil and Hydraulic Engineering, 31(4), pp. 395–410. doi: 10.6652/JoCICHE.201906_31(4).0010.
Colombi, A., Roux, P., Guenneau, S., Gueguen, P. and Craster, R. V. (2016) ‘Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances’, Scientific Reports. Nature Publishing Group, 6(January), pp. 1–7. doi: 10.1038/srep19238.
Colombi, A., Ageeva, V., Smith, R. J., Clare, A., Patel, R., Clark, M., Colquitt, D., Roux, P., Guenneau, S. and Craster, R. V. (2017) ‘Enhanced sensing and conversion of ultrasonic Rayleigh waves by elastic metasurfaces’, Scientific Reports, 7(1). doi: 10.1038/s41598-017-07151-6.
Daley, P. F. and Hron, F. (1977) ‘Reflection and transmission coefficients for transversely isotropic media’, Bulletin of the Seismological Society of America, 67(3), pp. 661–675.
Du, Q., Zeng, Y., Huang, G. and Yang, H. (2017) ‘Elastic metamaterial-based seismic shield for both Lamb and surface waves’, AIP Advances, 7(7). doi: 10.1063/1.4996716.
Graff, K. f. (1975) ‘Wave Motion in Elastic Solids’, Oxford University Press, pp. 71–72. doi: 10.1088/0031-9112/27/1/032.
Guo, D.- K., Design and numerical simulation of seismic metamaterials with Love waves in transversely isotorpic media, National Cheng Kung University Civil Engineering Department Master Thesis.
He, R.- T., Design and nymerical simulation of seismic metamaterials with Rayleigh waves dispersion effect in a transversely isotropic medium, National Cheng Kung University Civil Engineering Department Master Thesis.
Hill, R. (1964) ‘Theory of mechanical properties of fibre-strengthened materials: I. Elastic behaviour’, Journal of the Mechanics and Physics of Solids, 12(4), pp. 199–212. doi: 10.1016/0022-5096(64)90019-5.
Huang, G. L. and Sun, C. T. (2010) ‘Band gaps in a multiresonator acoustic metamaterial’, Journal of Vibration and Acoustics, Transactions of the ASME, 132(3), pp. 0310031–0310036. doi: 10.1115/1.4000784.
Huang, H. H. and Sun, C. T. (2009) ‘Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density’, New Journal of Physics, 11. doi: 10.1088/1367-2630/11/1/013003.
Huang, H. H. and Sun, C. T. (2011) ‘Locally resonant acoustic metamaterials with 2D anisotropic effective mass density’, Philosophical Magazine. Taylor ＆ Francis, 91(6), pp. 981–996. doi: 10.1080/14786435.2010.536174.
Huang, H. H., Sun, C. T. and Huang, G. L. (2009) ‘On the negative effective mass density in acoustic metamaterials’, International Journal of Engineering Science, 47(4), pp. 610–617. doi: 10.1016/j.ijengsci.2008.12.007.
Liu, Z., Zhang, X., Mao, Y., Zhu, Y., Yang, Z., Chan, C. T. and Sheng, P. (2000) ‘Locally resonant sonic materials’, Science, 289(5485), pp. 1734–1736. doi: 10.1126/science.289.5485.1734.
Manger, E. G. (1963) ‘Porosity and Bulk Density of Sedimentary Rocks’, Geological Survery Bulletin 1144-E, p. 62.
Maurel, A., Marigo, J.-J., Pham, K. and Guenneau, S. (2018) ‘Conversion of Love waves in a forest of trees’, Physical Review B, 98(13). doi: 10.1103/PhysRevB.98.134311.
Maznev, A. A. and Gusev, V. E. (2015) ‘Waveguiding by a locally resonant metasurface’, Physical Review B - Condensed Matter and Materials Physics, 92(11). doi: 10.1103/PhysRevB.92.115422.
Miniaci, M., Krushynska, A., Bosia, F. and Pugno, N. M. (2016) ‘Large scale mechanical metamaterials as seismic shields’, New Journal of Physics. IOP Publishing, 18(8). doi: 10.1088/1367-2630/18/8/083041.
Palermo, A., Krödel, S., Marzani, A. and Daraio, C. (2016a) ‘Engineered metabarrier as shield from seismic surface waves’, Scientific Reports. Nature Publishing Group, 6, pp. 1–10. doi: 10.1038/srep39356.
Palermo, A., Krödel, S., Marzani, A. and Daraio, C. (2016b) ‘Engineered metabarrier as shield from seismic surface waves’, Scientific Reports, 6, pp. 1–6. doi: 10.1038/srep39356.
Palermo, A., Vitali, M. and Marzani, A. (2018) ‘Metabarriers with multi-mass locally resonating units for broad band Rayleigh waves attenuation’, Soil Dynamics and Earthquake Engineering. Elsevier Ltd, 113(March), pp. 265–277. doi: 10.1016/j.soildyn.2018.05.035.
Pendry, J. B., Holden, A., Stewart, W. and Youngs, I. (1996) ‘Extremely low frequency plasmons in metallic mesostructures’, Physical Review Letters, 76(25), pp. 4773–4776. doi: 10.1103/PhysRevLett.76.4773.
Achaoui, Y., Ungureanu, B., Enoch, S., Brûlé, S. and Guenneau, S. (2016) ‘Seismic waves damping with arrays of inertial resonators’, Extreme Mechanics Letters, 8, pp. 30–37. doi: 10.1016/j.eml.2016.02.004.
Achaoui, Y., Antonakakis, T., Brûlé, S. Craster, R. V, Enoch, S. and Guenneau, S. (2017) ‘Clamped seismic metamaterials: Ultra-low frequency stop bands’, New Journal of Physics, 19(6). doi: 10.1088/1367-2630/aa6e21.
Achenbach, J. (1973) ‘Wave Propagation in Elastic Solids’, North Holland Publishing Company, pp. 30–30. doi: 10.1016/0003-682x(75)90007-9.
Achenbach, J. D. (1998) ‘Explicit solutions for carrier waves supporting surface waves and plate waves’, Wave Motion, 28(1), pp. 89–97. doi: 10.1016/S0165-2125(97)00056-5.
Boechler, N. et al. (2013) ‘Interaction of a contact resonance of microspheres with surface acoustic waves’, Physical Review Letters, 111(3). doi: 10.1103/PhysRevLett.111.036103.
Brûlé, S. Javelaud, E. H., Enoch, S. and Guenneau, S. (2014) ‘Experiments on seismic metamaterials: Molding surface waves’, Physical Review Letters. American Physical Society, 112(13). doi: 10.1103/PhysRevLett.112.133901.
Buchwald, V. T. (1961) ‘Rayleigh waves in transversely isotropic media’, Quarterly Journal of Mechanics and Applied Mathematics, 14(3), pp. 293–318. doi: 10.1093/qjmam/14.3.293.
Cheadle, S. P., Brown, R. J. andLawton, D. C. (1991) ‘Orthorhombic anisotropy: a physical seismic modeling study’, Geophysics, 56(10), pp. 1603–1613. doi: 10.1190/1.1442971.
Chien, T. Y. et al. (2019) ‘A Simple Proposition of Two-Dimensional Configuration of Seismic Metamaterials — A Promising Tool Towards Seismic Cloaking’, Journal of the Chinese Institute of Civil and Hydraulic Engineering, 31(4), pp. 395–410. doi: 10.6652/JoCICHE.201906_31(4).0010.
Colombi, A., Roux, P., Guenneau, S., Gueguen, P. and Craster, R. V. (2016) ‘Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances’, Scientific Reports. Nature Publishing Group, 6(January), pp. 1–7. doi: 10.1038/srep19238.
Colombi, A., Ageeva, V., Smith, R. J., Clare, A., Patel, R., Clark, M., Colquitt, D., Roux, P., Guenneau, S. and Craster, R. V. (2017) ‘Enhanced sensing and conversion of ultrasonic Rayleigh waves by elastic metasurfaces’, Scientific Reports, 7(1). doi: 10.1038/s41598-017-07151-6.
Daley, P. F. and Hron, F. (1977) ‘Reflection and transmission coefficients for transversely isotropic media’, Bulletin of the Seismological Society of America, 67(3), pp. 661–675.
Du, Q., Zeng, Y., Huang, G. and Yang, H. (2017) ‘Elastic metamaterial-based seismic shield for both Lamb and surface waves’, AIP Advances, 7(7). doi: 10.1063/1.4996716.
Graff, K. f. (1975) ‘Wave Motion in Elastic Solids’, Oxford University Press, pp. 71–72. doi: 10.1088/0031-9112/27/1/032.
Guo, D.- K., Design and numerical simulation of seismic metamaterials with Love waves in transversely isotorpic media, National Cheng Kung University Civil Engineering Department Master Thesis.
He, R.- T., Design and nymerical simulation of seismic metamaterials with Rayleigh waves dispersion effect in a transversely isotropic medium, National Cheng Kung University Civil Engineering Department Master Thesis.
Hill, R. (1964) ‘Theory of mechanical properties of fibre-strengthened materials: I. Elastic behaviour’, Journal of the Mechanics and Physics of Solids, 12(4), pp. 199–212. doi: 10.1016/0022-5096(64)90019-5.
Huang, G. L. and Sun, C. T. (2010) ‘Band gaps in a multiresonator acoustic metamaterial’, Journal of Vibration and Acoustics, Transactions of the ASME, 132(3), pp. 0310031–0310036. doi: 10.1115/1.4000784.
Huang, H. H. and Sun, C. T. (2009) ‘Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density’, New Journal of Physics, 11. doi: 10.1088/1367-2630/11/1/013003.
Huang, H. H. and Sun, C. T. (2011) ‘Locally resonant acoustic metamaterials with 2D anisotropic effective mass density’, Philosophical Magazine. Taylor ＆ Francis, 91(6), pp. 981–996. doi: 10.1080/14786435.2010.536174.
Huang, H. H., Sun, C. T. and Huang, G. L. (2009) ‘On the negative effective mass density in acoustic metamaterials’, International Journal of Engineering Science, 47(4), pp. 610–617. doi: 10.1016/j.ijengsci.2008.12.007.
Liu, Z., Zhang, X., Mao, Y., Zhu, Y., Yang, Z., Chan, C. T. and Sheng, P. (2000) ‘Locally resonant sonic materials’, Science, 289(5485), pp. 1734–1736. doi: 10.1126/science.289.5485.1734.
Manger, E. G. (1963) ‘Porosity and Bulk Density of Sedimentary Rocks’, Geological Survery Bulletin 1144-E, p. 62.
Maurel, A., Marigo, J.-J., Pham, K. and Guenneau, S. (2018) ‘Conversion of Love waves in a forest of trees’, Physical Review B, 98(13). doi: 10.1103/PhysRevB.98.134311.
Maznev, A. A. and Gusev, V. E. (2015) ‘Waveguiding by a locally resonant metasurface’, Physical Review B - Condensed Matter and Materials Physics, 92(11). doi: 10.1103/PhysRevB.92.115422.
Miniaci, M., Krushynska, A., Bosia, F. and Pugno, N. M. (2016) ‘Large scale mechanical metamaterials as seismic shields’, New Journal of Physics. IOP Publishing, 18(8). doi: 10.1088/1367-2630/18/8/083041.
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Palermo, A., Krödel, S., Marzani, A. and Daraio, C. (2016b) ‘Engineered metabarrier as shield from seismic surface waves’, Scientific Reports, 6, pp. 1–6. doi: 10.1038/srep39356.
Palermo, A., Vitali, M. and Marzani, A. (2018) ‘Metabarriers with multi-mass locally resonating units for broad band Rayleigh waves attenuation’, Soil Dynamics and Earthquake Engineering. Elsevier Ltd, 113(March), pp. 265–277. doi: 10.1016/j.soildyn.2018.05.035.
Pendry, J. B., Holden, A., Stewart, W. and Youngs, I. (1996) ‘Extremely low frequency plasmons in metallic mesostructures’, Physical Review Letters, 76(25), pp. 4773–4776. doi: 10.1103/PhysRevLett.76.4773.
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