Nonlinear Viscoelastic Wave Propagation: An Extension of Nearly Constant Attenuation Models
Publication: Journal of Engineering Mechanics
Volume 135, Issue 11
Abstract
Hysteretic damping is often modeled by means of linear viscoelastic approaches such as “nearly constant attenuation” models (e.g. NCQ model). These models do not take into account nonlinear effects either on the stiffness or on the damping, which are well known features of soil dynamic behavior. The aim of this paper is to propose a mechanical model involving nonlinear viscoelastic behavior for isotropic materials under dynamic excitations. This model simultaneously takes into account nonlinear elasticity and nonlinear damping. On one hand, the shear modulus is a function of the excitation level; on the other, the description of viscosity is based on a generalized Maxwell body involving nonlinearity. This formulation (X-NCQ) is implemented into a one-dimensional finite-element approach for a dry soil. The validation of the model shows its ability to retrieve low amplitude seismic ground motion. For larger excitation levels, the analysis of seismic wave propagation in a nonlinear soil layer over an elastic bedrock leads to results which are physically satisfactory (lower amplitudes, larger time delays, higher frequency content).
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Acknowledgments
The writers thank Luis F. Bonilla (IRSN) for fruitful discussions. This work was partly funded by the French National Research Agency in the framework of the “QSHA” research project (“Quantitative Seismic Hazard Assessment”).
References
Aubry, D., Hujeux, J. C., Lassoudire, F., and Meimon, Y. (1982). “A double memory model with multiple mechanisms for cyclic soil behaviour.” Proc., Int. Symp. Num. Mod. Geomech, Balkema, Rotterdam, The Netherlands, 3–13.
Bard, P. Y., and Bouchon, M. (1985). “The two dimensional resonance of sediment filled valleys.” Bull. Seismol. Soc. Am., 75, 519–541.
Bard, P. Y., and Riepl-Thomas, J. (2000). “Wave propagation in complex geological structures and their effects on strong ground motion.” Wave motion in earthquake engineering, E. Kausel and G. D. Manolis, eds., WIT, Boston, 37–95.
Biot, M. A. (1958). “Linear thermodynamics and the mechanics of solids.” Proc., 3rd U.S. National Conf. of Applied Mechanics, ASME, New York, 1–18.
Bonilla, L. F., Archuleta, R. J., and Lavallée, D. (2005). “Hysteretic and dilatant behavior of cohesionless soils and their effects on nonlinear site response: Field data observations and modeling.” Bull. Seismol. Soc. Am., 95(6), 2373–2395.
Bonilla, L. F., Liu, P. C., and Nielsen, S. (2006). “1D and 2D linear and nonlinear site response in the Grenoble area.” Proc., 3rd Int. Symp. on the Effects of Surface Geology on Seismic Motion (ESG2006), LCPC, Grenoble, France.
Bonnet, G., and Heitz, J. F. (1994). “Nonlinear seismic response of a soft layer.” Proc., 10th European Conf. on Earthquake Engineering, Balkema, Rotterdam, The Netherlands, 361–364.
Bourbié, T., Coussy, O., and Zinzsner, B. (1987). Acoustics of porous media, Technip, Paris.
Bozzano, F., Lenti, L., Martino, S., Paciello, A., and Scarascia Mugnozza, G. (2008). “Self-excitation process due to local seismic amplification responsible for the reactivation of the Salcito landslide (Italy) on 31 October 2002.” J. Geophys. Res., 113, B10312.
Carcione, J. M., Cavallini, F., Mainardi, F., and Hanyga, A. (2002). “Time-domain modeling of constant-Q seismic waves using fractional derivatives.” Pure Appl. Geophys., 159(7–8), 1719–1736.
Chaillat, S., Bonnet, M., and Semblat, J. -F. (2009). “A new fast multi-domain BEM to model seismic wave propagation and amplification in 3D geological structures.” Geophys. J. Int., 177, 509–531.
Chávez-García, F. J., Stephenson, W. R., and Rodríguez, M. (1999). “Lateral propagation effects observed at Parkway, New Zealand: A case history to compare 1D vs 2D effects.” Bull. Seismol. Soc. Am., 89, 718–732.
Day, S. M., and Minster, J. B. (1984). “Numerical simulation of wavefields using a Padé approximant method.” Geophys. J. R. Astron. Soc., 78, 105–118.
El Hosri, M. S., Biarez, J., and Hicher, P. Y. (1984). “Liquefaction characteristics of silty clay.” Proc., 8th World Conf. on Earthquake Eng., Prentice-Hall, Englewood Cliffs, N.J., 277–284.
Emmerich, H., and Korn, M. (1987). “Incorporation of attenuation into time-domain computations of seismic wave fields.” Geophysics, 52(9), 1252–1264.
Fäh, D., Suhadolc, P., Mueller, S., and Panza, G. F. (1994). “A hybrid method for the estimation of ground motion in sedimentary basins: Quantitative modeling for Mexico City.” Bull. Seismol. Soc. Am., 84(2), 383–399.
Gélis, C., Bonilla, L. F., Régnier, J., Bertrand, E., and Duval, A. M. (2008). “On the use of Saenger’s finite difference stencil to model 2D P-SV nonlinear basin response: Application to Nice, France.” SEISMIC RISK 2008: Earthquake in northwest Europe, Liege, Belgium.
Gyebi, O. K., and Dasgupta, G. (1992). “Finite element analysis of viscoplastic soils with Q-factor.” Soil. Dyn. Earthquake Eng., 11(4), 187–192.
Hardin, B. O., and Drnevich, V. P. (1972). “Shear modulus and damping in soils: Design equations and curves.” J. Soil Mech. and Found. Div., 98(7), 667–692.
Hashash, Y. M. A., and Park, D. (2001). “Nonlinear one-dimensional seismic ground motion propagation in the Mississippi embayment.” Eng. Geol. (Amsterdam), 62, 185–206.
Heitz, J. F. (1992). “Propagation d’ondes en milieu non linéaire. Applications au génie parasismique et à la reconnaissance des sols.” Ph.D. thesis, Univ. Joseph Fourier, Grenoble, France.
Hughes, T. J. R. (1987). The finite element method—Linear static and dynamic finite element analysis, Prentice-Hall, Englewood Cliffs, N.J.
Iai, S., Morita, T., Kameoka, T., Matsunaga, Y., and Abiko, K. (1995). “Response of a dense sand deposit during 1993 Kushiro-Oki earthquake.” Soils Found., 35, 115–131.
Kausel, E., and Assimaki, D. (2002). “Seismic simulation of inelastic soils via frequency-dependent moduli and damping.” J. Eng. Mech., 128(1), 34–47.
Kawase, H. (2003). “Site effects on strong ground motions.” International handbook of earthquake and engineering seismology, B. Part, W. H. K. Lee, and H. Kanamori, eds., Academic, San Diego, 1013–1030.
Kjartansson, E. (1979). “Constant Q wave propagation and attenuation.” J. Geophys. Res., 84, 4737–4748.
Kramer, S. L. (1996). Geotechnical earthquake engineering, Prentice-Hall, Englewood Cliff, N.J.
Kwok, A. O. L., Stewart, J. P., and Hashash, Y. M. A. (2008). “Nonlinear ground-response analysis of Turkey flat shallow stiff-soil site to strong ground motion.” Bull. Seismol. Soc. Am., 98(1), 331–343.
Lemaître, J., and Chaboche, J. L. (1992). Mechanics of solid materials, Cambridge University Press, Cambridge, U.K.
Lenti, L., Martino, S., Paciello, A., and Scarascia-Mugnozza, G., (2009).“Evidence of two-dimensional amplification effects in an alluvial valley (Valnerina, Italy) from velocimetric records and numerical models,” Bull. Seismol. Soc. Am., 99(3), 1612–1635.
Liu, H. P., Anderson, D. L., and Kanamori, H. (1976). “Velocity dispersion due to anelasticity: Implications for seismology and mantle composition.” Geophys. J. R. Astron. Soc., 47, 41–58.
Makra, K., Chávez-García, F. J., Raptakis, D., and Pitilakis, K. (2005). “Parametric analysis of the seismic response of a 2D sedimentary valley: Implications for code implementations of complex site effects.” Soil. Dyn. Earthquake Eng., 25(4), 303–315.
Masing, G. (1926). “Eignespannungen und Verfestigung beim Messing.” Proc., 2nd Int. Congress on Applied Mechanics, Orell Füssli Verlag, Zürich and Leipzig, 332–335.
Matasovic, N. (1993). “Seismic response of composite horizontally-layered soil deposits.” Ph.D. thesis, Dept. of Civil Engineering, Univ. of California at Los Angeles, Los Angeles.
Matasovic, N., and Vucetic, M. (1995). “Seismic response of soil deposits composed of fully-saturated clay and sand layers.” Proc., 1st Int. Conf. on Earthquake Geotechnical Eng., Balkema, Rotterdam, 611–616.
Mavroeidis, G. P., and Papageorgiou, A. S. (2003). “A mathematical representation of nearfault ground motions.” Bull. Seismol. Soc. Am., 93(3), 1099–1131.
Moczo, P., and Kristek, J. (2005). “On the rheological models used for time-domain methods of seismic wave propagation.” Geophys. Res. Lett., 32, L01306.
Moeen-Vaziri, N., and Trifunac, M. D. (1988). “Scattering and diffraction of plane SH-waves by two-dimensional inhomogeneities.” Soil. Dyn. Earthquake Eng., 7(4), 179–200.
Munjiza, A., Owen, D. R. J., and Crook, A. J. L. (1998). “An proportional damping in explicit integration of dynamic structural systems.” Int. J. Numer. Methods Eng., 41, 1277–1296.
Paolucci, R. (1999). “Shear resonance frequencies of alluvial valleys by Rayleigh’s method.” Earthquake Spectra, 15, 503–521.
Park, D., and Hashash, Y. M. A. (2004). “Soil damping formulation in nonlinear time domain site response analysis.” J. Earthquake Eng., 8(6), 249–274.
Prevost, J. H. (1985). “A simple plasticity theory for frictional cohesionless soils.” Soil. Dyn. Earthquake Eng., 4(1), 9–17.
Sánchez-Sesma, F. J., and Luzón, F. (1995). “Seismic response of three-dimensional alluvial valleys for incident P, S, and Rayleigh waves.” Bull. Seismol. Soc. Am., 85, 269–284.
Schnabel, P. B., Lysmer, J., and Seed, H. B. (1972). “SHAKE-A computer program for equation response analysis of horizontally layered sites.” Rep. No. EERC 72-12, Univ. of California, Berkeley, Berkeley, Calif.
Seed, H. B., and Idriss, I. M. (1970). “Soil moduli and damping factors for dynamic response analysis.” Rep. No. EERC 70-10, Univ. of California, Berkeley, Berkeley, Calif.
Semblat, J. F. (1997). “Rheological interpretation of Rayleigh damping.” J. Sound Vib., 206(5), 741–744.
Semblat, J. F., Kham, M., Parara, E., Bard, P. Y., Pitilakis, K., Makra, K., and Raptakis, D., (2005). “Site effects: Basin geometry vs soil layering.” Soil. Dyn. Earthquake Eng., 25(7–10), 529–538.
Semblat, J. F., Duval, A. M., and Dangla, P. (2000). “Numerical analysis of seismic wave amplification in Nice (France) and comparisons with experiments.” Soil. Dyn. Earthquake Eng., 19(5), 347–362.
Semblat, J. F., Paolucci, R., and Duval, A. M. (2003). “Simplified vibratory characterization of alluvial basins.” C. R. Geosci., 335, 365–370.
Semblat, J. F., and Pecker, A. (2009). Waves and vibrations in soils: Earthquakes, traffic, shocks, construction works, IUSS, Pavia, Italy.
Strick, E. (1967). “The determination of Q, dynamic viscosity and transient creep curves from wave propagation measurements.” Geophys. J. Int., 13, 1–218.
Sugito, M. (1995). “Frequency-dependent equivalent strain for equi-linearized technique.” Proc., 1st Int. Conf. on Earthquake Geotechnical Engineering, Balkema, Rotterdam, The Netherlands, 655–660.
Van Den Abeele, K. E.-A., Johnson, P. A., and Sutin, A. (2000). “Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage. Part I: Nonlinear wave modulation spectroscopy (NWMS).” Res. Nondestruct. Eval., 12(1), 17–30.
Vucetic, M. (1990). “Normalized behaviour of clay under irregular cyclic loading.” Can. Geotech. J., 27, 29–46.
Zienkiewicz, O. C., and Taylor, R. L. (2005). The finite element method for solid and structural mechanics, 6th Ed., Butterworth-Heinemann, Stoneham, Mass.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 135 • Issue 11 • November 2009
Pages: 1305 - 1314
Copyright
© 2009 ASCE.
History
Received: Mar 8, 2007
Accepted: Jul 1, 2009
Published online: Oct 15, 2009
Published in print: Nov 2009
Notes
Note. Associate Editor: Andrew W. Smyth
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