Types of elastic nonlinearity of sedimentary soils - AGU Publications

GEOPHYSICAL RESEARCH LETTERS, VOL. 29, NO. 19, 1930, doi:10.1029/2002GL014794, 2002

Types of elastic nonlinearity of sedimentary soils Olga V. Pavlenko Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia

Kojiro Irikura Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan Received 24 January 2002; revised 14 May 2002; accepted 15 May 2002; published 11 October 2002.

[1] Vertical array records of the 1995 Kobe earthquake showed clearly nonlinearity of soil behavior. Although there have been few observational validations, earlier scientific publications on soil nonlinearity appeared since 1970’s. In theoretical studies, nonlinearity has been introduced in constitutive relations as the quadratic correction to Hooke’s law, but the question of the types of soil nonlinearity was not investigated. Here we show, on the basis of data processing and nonlinear system identification technique, that sediments represent physical systems possessing mostly odd types of nonlinearity. Even-order nonlinearities become comparable with odd-order ones only in special cases, like liquefied soils. Though the even and odd nonlinearities have much in common, the frequencies of combination harmonics, shapes of seismic solitary waves, and other nonlinear phenomena are different in these two cases. On the whole, the nonlinearity in soils leads to changes in spectra of propagating seismic signals: the spectra tending to take the I NDEX TERMS: 0902 Exploration form of E( f )f – n. Geophysics: Computational methods, seismic; 0915 Exploration Geophysics: Downhole methods; 0994 Exploration Geophysics: Instruments and techniques; 7212 Seismology: Earthquake ground motions and engineering. Citation: Pavlenko, O. V., and K. Irikura, Types of elastic nonlinearity of sedimentary soils, Geophys. Res. Lett., 29(19), 1930, doi:10.1029/2002GL014794, 2002.

1. Physical Mechanism of Soil Nonlinearity – Deviations from Hooke’s Law [2] Nonlinear behavior of soils in strong ground motion includes different phenomena: changes in amplitudes and spectra of oscillations depending on ground conditions, soil consolidation or loosening, liquefaction, residual deformations, etc., which are known since 1930’s [Suyehiro, 1932]. Representative experimental data on soil nonlinearity were obtained by the Lotung vertical array in Taiwan and by vertical arrays in Japan during the 1995 Kobe earthquake. In theoretical studies on soil nonlinearity, which appeared since 1970’s, nonlinearity has been usually introduced into constitutive relations as the quadratic correction to Hooke’s law, the question of the types of soil nonlinearity has not been investigated. [3] To investigate the types of soil nonlinearity, let us consider its physical cause, i.e., nonlinearity of stress-strain relationships of soils in strong ground motion. In strong motion, Hooke’s law does not hold for sediments, and the Copyright 2002 by the American Geophysical Union. 0094-8276/02/2002GL014794

nonlinear corrections become significant. Stress-strain relationships determine the transformations of incident seismic waves into the ground response; the nonlinearity of the stress-strain relations leads to distortions in the shapes and spectra of propagating seismic waves. In general, the relation between the input and the output of a nonlinear system can be represented as the Volterra series [Marmarelis and Marmarelis, 1978]: yðtÞ ¼ k0 þ þ þ

0 R1 R1

k1 ðtÞxðt tÞdt

k2 ðt1 ; t2 Þxðt 0 0 R1 R1 R1

t1 Þxðt t2 Þdt1 dt2

k3 ðt1 ; t2 ; t3 Þxðt 0 dt1 dt2 dt3 þ . . . 0

R1

t1 Þxðt t2 Þxðt t3 Þ

0

ð1Þ

where x(t) is the input, y(t) is the output, t is time, t, t1, t2, t3 are time delays, and k0, k1(t), k2(t1, t2), k3(t1, t2, t3) are the zero, first, second, and third-order Volterra kernels of the system, respectively. The first-order kernel k1(t) describes the linear part of the system response, while other terms represent quadratic, cubic and higher-order nonlinear corrections. Thus, by analyzing the input and output of a nonlinear system, we can judge about the types and quantitative characteristics of the system nonlinearity [Marmarelis and Marmarelis, 1978; Pavlenko, 2001]. [4] As is known, real hysteretic stress-strain relationships of soils are very diverse and dependent on the granulometric composition of a soil, its humidity, etc. In water-saturated soils, the convex-up part of the initial loading curve is followed by a deviation to the stress axis, i.e., concave-up part. In such soils, we can expect shock waves beyond the limit of elasticity [Zvolinskii, 1982]. Stress-strain relations represented by the only convex-up parts are characteristic for soft soils like loess loams and sands. Liquefied soils can be described by similar stress-strain curves, but more sloping, with the slopes being close to horizontal at large strains. The sensitivity of soils to dynamic loading obviously decreases with depth, and soils at depths behave ‘‘more linearly’’ than subsurface soils.

2. Estimation of Stress-Strain Relationships of Soils in Situ Using Vertical Array Records [5] Based on these physical representations, we investigated the records of the 1995 Kobe earthquake, obtained by vertical arrays at Port Island (PI), SGK, and TKS sites, in order to estimate the stress-strain relations in soils in situ [Pavlenko and Irikura, 2001a]. Three-component acceler-

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PAVLENKO AND IRIKURA: TYPES OF ELASTIC NONLINEARITY OF SOILS

curves were generated, and item-by-item examination was applied to identify the groups of curves showing the best-fit approximation between the observed and simulated data (Figure 1). To account for temporal changes in the soil behavior, successive 1.5-second time intervals were analyzed. The calculation program was developed by the authors. It based on the algorithm described by Joyner and Chen [1975]. In our calculations, the stress-strain relations in soils satisfied the Masing criterion. [7] The obtained stress-strain relations were used to trace changes in the shear moduli in different layers [Pavlenko and Irikura, 2002]. At PI and SGK sites, where the strongest accelerations were recorded, maximum changes in the upper layers and stability in the deeper layers were obtained. A progressive reduction of the shear modulus due to liquefaction was observed at PI, and reduction and recovery of the shear modulus were revealed at SGK and TKS sites. [8] The obtained stress-strain relations represent a fairly good approximation to the in-situ case, because (1) the choice of stress-strain curves in layers was physically justified; (2) a good agreement was obtained between the observed and simulated records (Figure 1); (3) similarity of the stress-strain curves obtained for two horizontal components testifies the validity of the solution; (4) the obtained changes in the shear moduli are physically correct and agree with the results obtained by other authors [Kawase et al., 1995; Sato et al., 1996] and with laboratory tests. For the available profiling data and records, the obtained vertical distribution of stress-strain relations shows the best agreement between simulations and observations. Our calculations show that any noticeable change in the vertical distribution of the stress-strain relations breaks this good agreement.

Figure 1. The acceleration time histories of the main shock, observed and simulated, at PI (a), SGK (b), and TKS (c) sites.

ometers at these sites were installed at GL-0 m, GL-16 m, GL-32 m, GL-83 m; GL-0 m, GL-24.9 m, GL-97 m; and GL0 m, GL-25 m, GL-100 m, respectively. All the three sites were located in the vicinity of the epicenter; the distances of PI, SGK, and TKS sites to the fault line were 2 km, 6 km, and 16 km, respectively. The map showing the locations of the sites around the Osaka bay and the soil profiles are given in [Pavlenko and Irikura, 2001a]. The soil profiles look similar at the three sites: reclaimed and alluvial soil in the upper 10 – 20 meters, clays, sands, and gravel below. [6] For calculations, the soil profiles were divided into groups of layers, for which the realistic hysteretic stressstrain relations were assumed, such as, (1) relations, similar to those obtained in laboratory experiments by Hardin and Drnevich (1972), for soils at depths (PI: 18– 26 m and below 34 m; SGK: below 11 m; TKS: below 14 m); (2) the relations similar to (1), but very sloping, with slopes being close to horizontal at large strains, for soils in liquefaction (PI: the upper 13 m); and (3) relations, in which the initial convex-up part is followed by a deviation to the stress axis, for water-saturated sands (PI: 13– 18 m and 27– 32.5 m; SGK: the upper 11 m; TKS: the upper 14 m). Sets of such

3. Changes in Spectra of Signals Propagating in Soils and Types of Soil Nonlinearity [9] Time-dependent models of nonlinear soil behavior in strong ground motion were thus constructed for the three recording sites. To determine the types and quantitative characteristics of the soil nonlinearity, the obtained models were tested by the Gaussian white noise in the frequency range of 0.1– 100 Hz and by monochromatic signals (the method was described in detail by Pavlenko [2001]). Testing signals represented approximations to the imposed motion of the Kobe earthquake. [10] The results are shown in Figures 2 – 4. Functionals of the Wiener series (similar to Volterra series (1) and consisting of terms, which are orthogonal with respect to the input signal in the form of the Gaussian white noise) were estimated. We calculated the Wiener kernels h0, h1(t), h2(t1, t2), and h3(t1, t2, t3) by the method of cross-correlation functions [Marmarelis and Marmarelis, 1978; Pavlenko, 2001]. Linear, nonlinear quadratic, and nonlinear cubic components of the ground responses were calculated for PI, SGK, and TKS sites. Figures 2 and 3 show the intensities of the whole nonlinear parts, as well as the nonlinear quadratic, and nonlinear cubic components of the responses in percents of the intensities of the responses. At PI, liquefaction occurred in the upper layers, and two time intervals were analyzed (which are marked in Figure 1a), corresponding to different stages of liquefaction (Figure 2).


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values, i.e., surface layers slowly shift to one or the other side in their oscillations. The effect can be interpreted as a result of accumulation of residual shift deformations on the surface [Zvolinskii, 1982]. This asymmetry of oscillations is related to even-order nonlinearities [Marmarelis and Marmarelis, 1978]. The intensity of these quasi-static deformations, normalized by the intensity of the response, increases while liquefaction develops (Figure 2), indicating an increase in the even-order nonlinearities. [12] Below in Figures 2 and 3 diagonal values of the Wiener kernels h1(t), h2(t), and h3(t) (t, = t2 = t3) and the weighted-mean (weight being proportional to the maximum strain) stress-strain relations for the upper layers are presented. Loading and unloading parts of the relations can be represented as sums of odd (sinusoidal) and even (cosinusoidal) functions, which are also shown. The relationships between these odd and even parts (square root of the ratio of their intensities, o/e, is shown in Figures 2 and 3) determine the relationships of the odd and even components in the ground response. As liquefaction developed at PI, the stressstrain relations in the upper layers became more and more sloping, and the part of the even-order components in the ground response increased. For comparison, the soils at SGK and TKS sites, where liquefaction did not occur, possess mostly odd-order nonlinearities, which is evidently the most typical case for soils.

Figure 2. The response (2) of the system of soil layers at PI to the Gaussian white noise (1), approximating the imposed motion in the 1995 Kobe earthquake. 3 is the response predicted by a linear model; 4 is the difference between the response of the system and the response predicted by the linear model; 5 is the nonlinear correction due to quadratic nonlinearity of the system; 6 is the difference between the response of the system and the response predicted by the linear model with the nonlinear quadratic correction; 7 is the nonlinear correction due to cubic nonlinearity of the system; 8 is the difference between the response of the system and the response predicted by the linear model with the nonlinear quadratic and cubic corrections. Two time intervals are analyzed (which are marked in Figure 1a), corresponding to the initial and final stages of liquefaction.

Figure 3 shows the results of the identification of the nonlinear soil behavior at SGK and TKS sites for time intervals, which are marked in Figures 1b and 1c. [11] For nonliquefied soils, the nonlinear parts of the responses were as high as about 30% of the whole intensities of the responses, and nonlinear cubic components were substantially higher than nonlinear quadratic ones (Figures 2 and 3). As liquefaction developed at PI, the nonlinear part of the ground response increased due to the increase of the quadratic and other even-order components, whereas the cubic component remained almost at the same level (Figure 2). The zero-order Wiener kernels h0 represent quasi-static deformations of the surface. For different time intervals of the response they take positive or negative

Figure 3. The response (2) of the systems of soil layers at SGK and TKS sites to the Gaussian white noise (1), approximating the imposed motion in the 1995 Kobe earthquake at these sites. 3 – 8 designate the same as in Figure 2.

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i.e., the spectra of ‘‘output’’ signals on the surface tend to take the form of E( f )f n (n depends on the properties of the soil) (Figure 4b). Mutual interactions of spectral components of propagating signals lead to generation of combination-frequency harmonics in the low and high frequency ranges; the amplitudes of combination harmonics being related to their frequencies. As a result, the energy is redistributed over the spectral band, so that low-frequency components increase, sharp spectral peaks disappear, and the resulting spectrum of the output tends to take the ‘‘limiting’’ form of E( f )f n. This formula was obtained theoretically in nonlinear acoustics in the approximation of an infinite number of interacting waves [Kadomtsev and Karpman, 1971]. Therefore, the ‘‘limiting’’ spectrum can be attained only in cases of strong soil nonlinearity (thick sedimentary layers and/or intense imposed motions), whereas in practice we observe ‘‘intermediate’’ cases and only the tendency of increasing low-frequency components and smoothing spectral peaks. This tendency can be clearly seen in the acceleration spectra of the 1995 Kobe earthquake at the three studied sites. The physical mechanism of this phenomenon, which is known in geotechnical engineering as overdamping of high-frequency seismic waves in soils, is elastic nonlinearity of subsurface soils. [ 16 ] Acknowledgments. This study was supported by Grants no.11792026 and no.00099 (JSPS) from the Ministry of Education, Science, Culture and Technology, Japan.

Figure 4. Velocity spectra of testing monochromatic signals (a) and the Gaussian white noise (b) propagating in soil layers at PI, SGK, and TKS sites. [13] Other indicators of the types of soil nonlinearity are higher harmonics, generated during the propagation of monochromatic signals in soils. Figure 4a shows the spectra of ‘‘input’’ monochromatic signals (at depths of locations of the deepest devices at the three sites) and spectra of simulated signals at depths of locations of other recording devices. Generation of the 3d, 5th, 7th, and other odd-order harmonics indicates the odd-order nonlinearities in soils. Even-order harmonics (the 2nd and 4th) are generated only in liquefied soils, as observed at PI. Thus, the odd-order nonlinearities are typical for nonliquefied soils. Even-order nonlinearities can be detected only in soils, in which the stress-strain relations contain noticeable even components, like in liquefied soils. [14] Physical mechanisms of nonlinearity in fractured rock are the same as in soils (while quantitative manifestations are evidently different), because the stress-strain relations in both cases are hysteretic. Thus, the odd-order nonlinearities in sediments and rock are explained by the fact that loading and unloading parts of their hysteretic stress-strain relations are principally odd functions. [15] Transformations of spectra of the Gaussian white noise propagating in soils show another important tendency,

References Joyner, W. B., and T. F. Chen, Calculation of nonlinear ground response in earthquakes, Bull. Seism. Soc. Am., 65, 1315 – 1336, 1975. Kadomtsev, B. B., V. I. Karpman, Nonlinear waves, Uspekhi fiz. nauk, 103, 2, 27 – 48, 1971. Kawase, H., T. Satoh, K. Fukutake, and K. Irikura, Borehole records observed at the Port Island in Kobe during the Hyogo-ken Nanbu earthquake of 1995 and its simulation, J. Constr. Eng., AIJ, 475, 83 – 92, 1995. Marmarelis, P. Z., and V. Z. Marmarelis, Analysis of Physiological Systems, The White-Noise Approach, Plenum Press, New York and London, 1978. Pavlenko, O. V., Nonlinear Seismic Effects in Soils: Numerical Simulation and Study, Bull. Seism. Soc. Am., 91, 381 – 396, 2001. Pavlenko, O. V., and K. Irikura, Estimation of nonlinear time-dependent soil behavior in strong ground motion based on vertical array data, Pure Appl. Geophys (submitted ), 2001a. Pavlenko, O. V., and K. Irikura, Changes in shear moduli of liquefied and nonliquefied soils during the 1995 Kobe earthquake and its aftershocks at PI, SGK, and TKS vertical array sites, Bull. Seismol. Soc. Am, 92, 1592 – 1569, 2002. Sato, K., T. Kokusho, M. Matsumoto, and E. Yamada, Nonlinear seismic response and soil property during 1995 Hyogoken Nanbu earthquake, Soils and Foundations, Special Issue on Geotechnical Aspects of the January 17, 1995 Hyogoken Nanbu earthquake, 41 – 52, 1996. Suyehiro, K., Engineering seismology. Notes on American lectures, Proceedings of the American Society of civil engineers, 58, 4, 1932. Zvolinskii, N. V., Wave processes in non-elastic media, Problems of engineering seismology, 23, 4 – 19, 1982.

O. Pavlenko, Institute of Physics of the Earth, Russian Academy of Sciences, B. Gruzinskaya 10, Moscow 123995, Russia. ([email protected]) K. Irikura, Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan. ([email protected]. ac.jp)