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Types of elastic nonlinearity of sedimentary soils
Olga V. Pavlenko1
Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia

and Kojiro Irikura
Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan

Abstract. Vertical array records of the 1995 Kobe earthquake
showed clearly nonlinearity of soil behavior. There have beenfew
observational
validations,although,scientificpublications on soil nonlinearity appeared since 1970's. In
theoretical studies, nonlinearity has been introduced inconstitutive relations as the quadratic correction to Hooke's
law, however, the question of the types of soil nonlinearity
was not investigated. Here we show, on the basis of dataprocessing and nonlinear system identification technique, that
sediments represent physical systems possessing mostly odd
types of nonlinearity. Even-order nonlinearities becomecomparable with odd-order ones only in special cases, likeliquefied soils. Though even and odd nonlinearities havemuch in common, frequencies of combination harmonics,
shapes of seismic solitary waves, and other nonlinear
phenomena are different in these two cases. On the whole,nonlinearity of soils leads to changes in spectra of
propagating seismic signals: the spectra tend to take the formof E(f)~f –n.

1. Physical mechanism of soil nonlinearity –deviations from Hooke's law
Nonlinear behavior of soils in strong ground motions
includes different phenomena: changes in amplitudes andspectra of oscillations depending on ground conditions, soil
consolidation
orloosening,liquefaction,
residualdeformations, 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 byvertical arrays in Japan during the 1995 Kobe earthquake. In
theoretical studies on soil nonlinearity, which appeared since1970's, nonlinearity has been usually introduced into
constitutive relations as the quadratic correction to Hooke's
law [e.g., Aleshin et al., 1981; Gushchin and Shalashov, 1981;
Lund, 1983; Parker, 1988; Dimitriu, 1990; Groshkov et al.,
1990; Nikolaev et al., 1995]; the question of the types of soil
nonlinearity has not been investigated.
To investigate the types of soil nonlinearity, let us consider
the physical cause of soil nonlinearity, i.e., nonlinearity of thestress-strain relationships of soils in strong ground motions. In
strong motion, Hooke’s law does not hold for sediments, andnonlinear corrections of higher-orders become significad

2∞∞+ ∫ ∫ k 2 (τ 1 ,τ 2 ) x(t − τ 1 ) x (t − τ 2 )dτ 1dτ 2 +0 0∞∞∞

+ ∫ ∫∫ k 3 (τ 1 ,τ 2 ,τ 3 ) x (t − τ 1 ) x (t − τ 2 ) x(t − τ 3 )dτ 1dτ 2 dτ 3 +...
0 0 0(1)where x(t) is an input, y(t) is the output, t is time, τ,τ1,τ2,τ3 aretime delays, k0, k1(τ), k2(τ1,τ2), k3(τ1,τ2,τ3) are the zero-order,
first-order, second-order, and third-order Volterra kernels ofthe system. The first-order kernel k1(τ) describes the linear
part of the system response, while other terms represent
quadratic, cubic and other nonlinear corrections. Thus,
analysing input and output of a nonlinear system, we canjudge about the types and quantitative characteristics of thesystem nonlinearity [Marmarelis and Marmarelis, 1978].
As is known, real stress-strain relationships of soils arevery diverse and depend on the granulometric composition ofa soil, its humidity, etc. In water-saturated soils, the initialconvex-up part of the stress-strain curve is followed by adeviation 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 loessloams and sands. Liquefied soils can be described by similar"soft"-type stress-strain curves, but more sloping, with slopes
close to horizontal at large strains. Sensitivity of soils todynamic loading obviously decreases with depth, and soils at
depths behaive "more linearly" than subsurface soils.

2. Estimation of stress-strain relationships ofsoils in situ using vertical array records
Based on these physical representations, we investigatedrecords of the 1995 Kobe earthquake and its aftershocks
provided by vertical arrays at Port Island (PI), SGK, and TKS
sites, in order to estimate the stress-strain relations in situ
[Pavlenko and Irikura, 2001,a]. For calculations, the soil
profiles were divided into groups of layers, for which
physically justified types of stress-strain relations wereassumed (Figure 1). Sets of such curves were generated, anditem-by-item examination was applied to find groups ofcurves showing the best-fit approximation between the theobserved and simulated data. To account for temporalchanges in soil behavior, successive 1.5-second time intervals
were analysed. The calculation algorithm based on themodified lumped mass method [Joyner and Chen, 1975] wasused. The obtained stress-strain relations (Figures 2,3,4) wereused to trace changes in shear moduli in different layers[Pavlenko and Irikura, 2001,b]. At Port Island and SGK sites,
where the strongest accelerations were recorded, maximum
changes in upper layers and stability in deeper parts wereobtained. A progressive reduction of the shear modulus due toliquefaction was observed at Port Island, and reduction andrecovery of the shear modulus were noticed at SGK and TKS
sites.
The obtained stress-strain relations (Figures 2,3,4)
represent a fairly good approximation to the reality, because(1) the choice of stress-strain curves in layers was physicallyjustified; (2) a good agreement was obtained between theobserved and simulated records; (3) similarity of the stressstrain curves obtained for two horizontal components testifies
to the validity of the solution; (4) the obtained changes inshear moduli are physically correct and agree with results
obtained by other authors [Kawase et al., 1995; Sato et al.,

31996] and with laboratory tests. Since the choice of curves inlayers was physically justified, the obtained vertical
distribution of the types of nonlinear behavior of soils isunique. Our calculations show that there is no other vertical
distribution of stress-strain relations, which could give asimilar good agreement between simulations and observations[Pavlenko and Irikura, 2001,ab].

3. Changes in spectra of signals propagating insoils, types of soil nonlinearity
Thus, time-dependent models of nonlinear soil behavior instrong ground motions are constructed for three recordingsites. To determine the types and quantitative characteristicsof the soil nonlinearity, the obtained models were tested by
the Gaussian white noise (at 0.1 - 100 Hz) and by
monochromatic signals (testing signals represented
approximations to the imposed motion of the Kobe
earthquake).
The results are shown in Figures 5 and 6. Functionals of
the Wiener series (similar to Volterra series and consisting ofterms, which are orthogonal with respect to the input signal in
the form of the Gaussian white noise) were calculated[Marmarelis and Marmarelis, 1978], and linear, nonlinear
quadratic, and nonlinear cubic components of the groundresponse were estimated for PI, SGK, and TKS sites (Figure5). At PI, liquefaction occurred in the upper layers, and fourtime intervals are analysed, corresponding to different stagesof liquefaction (Figure 5a). At SGK and TKS sites, similar
results were obtained for successive time intervals, andexamples are presented in Figure 5b.
As liquefaction developed at Port Island, the nonlinear partof the ground response increased, due to the increase of thequadratic and other even-order components, whereas thecubic component remained at the same level (Figure 5, a).
Zero-order kernels h0 represent quasi-static deformations of
the surface. For different time intervals of the response, theytake positive or negative values, i.e., surface layers slowly
shift to one or the other side in their oscillations. The effectcan be interpreted as a result of accumulation of residual shift
deformations on the surface [Zvolinskii, 1982]. Thisasymmetry of oscillations is connected with even-ordernonlinearities [Marmarelis and Marmarelis, 1978]. The
intensity of these quasi-static deformations, normalized by theintensity of the response, increases, while liquefaction
developes (Figure 5a), indicating an increase in the evenorder nonlinearities.
Below in the figure, diagonal values of the Wiener kernels
h1(τ), h2(τ1,τ2), and h3(τ1,τ2,τ3) and the weighted-mean (weightbeing proportional to the maximum strain) stress-strain
relations for the upper layers are presented. Loading andunloading parts of these relations can be represented as a
Fourier-series, consisting of even (cosinusoidal) and odd(sinusoidal) parts, which are also shown in Figure 5a, b. The
relations between these even and odd parts (square root of theratio of their intensities is shown in Figure 5a, b) determine
the relations of the even and odd components in the groundresponse. As liquefaction developes at PI, the stress-tsrain
relations in the upper layers become more and more sloping,
and the part of the even-order components in the groundresponse increases. For comparison, soils at SGK and TKS
sites, where liquefaction did not occurr, possess mostly oddorder nonlinearities, which is evidently the most typical casefor soils.

4Another indicators of the types of soil nonlinearity arehigher harmonics, generated during the propagation of
monochromatic signals in soils. Figure 6 shows spectra of"input" monochromatic signals and spectra of simulated
signals at depths of locations of recording devices at PI, SGK,and TKS. Generation of the 3d, 5th, 7th, and other odd-orderharmonics testifyes to the odd-order nonlinearities in soils.
Even-order harmonics (the 2d and 4th) are generated only inliquefied soils at PI. Thus, odd-order nonlinearities are typicalfor nonliquefied soils. Even-order nonlinearities can bedetected only in soils, in which stress-strain relations containnoticeable even components, like in liquefied soils.
"Hysteretic effects" in rock were investigated analyticallyand in laboratory experiments [Kadish et al., 1996; Koen etal., 1996]. Hysteresis loops were approximated byparallelograms, and, if the parallelogram was narrow, themodel predicted zero spectral density at even multiplies of thesource frequency (which is in agreement with ourconclusions) and approximate "pairing" of amplitudes for odd
multiplies (which seems to be an artefact of the model).
Transformations of spectra of the Gaussian white noisepropagating in soils show another important tendency: spectraof "output" signals on the surface tend to take the form ofE(f)~f -n (n depends on the properties of soils) [Kadomtsev andKarpman, 1971]. Interactions of spectral components ofpropagating signals lead to generation of combinationfrequency harmonics in the low and high frequency ranges;
amplitudes of combination harmonics being related to theirfrequencies [Kadomtsev and Karpman, 1971]. As a result,energy is redistributed over the spectral band, so that lowfrequency components increase, shapr spectral peaks
dissappear, and the resulting spectrum of the output tends totake the "limiting" form of E(f)~f -n (Figure 6a). This formula
was obtained theoretically in the approximation of an infinite
number (or chaotic phases) of interacting waves [Kadomtsevand Karpman, 1971]. Therefore, the "limiting" spectrum canbe attained only in cases of a very strong soil nonlinearity(i.e., thick sedimentary layers and intense imposed motion ina wide frequency band), whereas in practice we observe"intermediate" cases and only the tendency of increasing lowfrequency components and smoothing spectral peaks.
Acceleration spectra of the 1995 Kobe earthquake are shownin Figure 6b, and the tendency is clearly seen. Physicalmechanizms of these phenomena, which are known ingeotechnical engineering as overdamping of high-frequency
seismic waves in soils, is elastic nonlinearity of subsurface
soils.
Acknowledgments. We thank Yoshio Soeda, Hideki Tamai, andNaoyuki Nakatsu (Kansai Electric Power Co.) for acceleration
records of SGK and TKS sites and Kobe City for acceleration recordsof Port Island. This study was partially supported by Grant-in-Aid for
Scientific Research, No.11792026 and No.00099 (JSPS), from theMinistry of Education, Science, Sports, and Culture, Japan.

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O. Pavlenko, Institute of Physics of the Earth, Russian Academyof Sciences, B. Gruzinskaya 10, Moscow 123995, Russia(olga@synapse.ru; olga@egmdpri01.dpri.kyoto-u.ac.jp).
K. Irikura, Disaster Prevention Research Institute, Kyoto
University,Gokasho,Uji,Kyoto
611-0011,Japan.
(irikura@egmdpri01.dpri.kyoto-u.ac.jp).
(Received ; revised ;accepted )1Now at Disaster Prevention
University, Kyoto, Japan.

ResearchInstitute,

Copyright 2001 by the American Geophysical Union
Paper number 2001GL000000.
0148-0227/01/2001GL000000$09.00

Kyoto

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Running Heads:
PAVLENKO AND IRIKURA: TYPES OF ELASTIC NONLINEARITY OF SOILS

Figure Captions:Figure 1. Nonlinear system identification with the Gaussianwhite noise in a wide frequency band at Port Island, fourstages of liquefaction.
Figure 2. Nonlinear system identification with the Gaussian
white noise in a wide frequency band at SGK and TKS sites.
Figure 3. Velocity spectra of testing monochromatic andGaussian-white-noise signals propagating in soils at Port
Island, SGK, and TKS sites.
Figure 4. Acceleration spectra of the 1995 Kobe earthquake
at Port Island, SGK, and TKS sites.
Figure 1. Figure or plate captions consist of a paragraph containing several sentences. Captions must be set at both 8.4 cm (3.3inches, 20 picas) and 17.3 cm (6.8 inches, 41 picas) wide.
Figure 1. Nonlinear system identification with the Gaussian white noise in a wide frequency band at Port Island,four stages of liquefaction.
Figure 2. Nonlinear system identification with the Gaussian white noise in a wide frequency band at SGK and TKS
sites.
Figure 3. Velocity spectra of testing monochromatic and Gaussian-white-noise signals propagating in soils at Port
Island, SGK, and TKS sites.
Figure 4. Acceleration spectra of the 1995 Kobe earthquake at Port Island, SGK, and TKS sites.