Dynamic shear modulus and damping ratio characteristics of undisturbed marine soils in the Bohai Sea, China

Zhang Yan, Zhao Kai , Peng Yanjv and Chen Guoxing

1. Institute of Geotechnical Engineering, Nanjing Tech University, Nanjing 210009, China

2. Civil Engineering and Earthquake Disaster Prevention Center of Jiangsu Province, Nanjing 210009, China

3. National Institute of Natural Hazards, Ministry of Emergency Management, Beijing 100085, China

Abstract: This paper presents results from a series of stress-controlled undrained cyclic triaxial tests on the undisturbed marine silty clay, silt, and fine sand soils obtained from the Bohai Sea, China. Emphasis is placed on the major factors for predominating the dynamic shear modulus (G) and damping ratio (λ) in the shear strain amplitude (γa) from 10-5 to 10-2, involving depth, sedimentary facies types, and water content of marine soils. The empirical equations of the small-strain shear modulus (Gmax) and damping ratio (λmin) using a single-variable of depth H are established for the three marine soils. A remarkable finding is that the curves of shear modulus reduction (G/Gmax) and the damping ratio (λ) with increasing γa of the three marine soils can be simply determined through a set of explicit expressions with the two variables of depth H and water content W. This finding is validated by independent experimental data from the literature. At the similar depths, the G value of the marine soils of terrestrial facies is the largest, followed by the neritic facies, and the G value of the marine soils of abyssal facies is the smallest. The sedimentary facies types of the marine soils have slight effect on the λ value. Another significant finding is that the shear modulus reduction curves plotted against the γa of the three marine soils at the similar depths are significantly below those of the corresponding terrigenous soils, while the damping curves plotted against γa are just the opposite. The results presented in this paper serve as a worthful reference for the evaluation of seabed seismic site effects in the Bohai Sea due to lack of experimental data.

Keywords: Bohai Sea; marine sediments; dynamic shear modulus; damping ratio; sedimentary facies

1 Introduction

The Bohai Sea is an inner sea located in the northernmost part of the eastern mainland of China, existing strong historical earthquakes due to the existence of the Yingkou-Weifang section of the Tanlu fault zone in the NNE direction and the Zhangjiakou-Bohai fault zone in the NW direction. The high seismic intensity in the Bohai Sea makes the seismic safety of the ports, oil and gas offshore platforms, and other infrastructure systems in the region a challenging issue. Therefore, the cyclic behavior of undisturbed marine soils subjected to cyclic loadings associated with earthquake events is of great concern. The dynamic shear modulusGand damping ratioλare the required fundamental parameters for site seismic response analysis (Miaoet al., 2018a, 2018b; Ruanet al., 2019; Chenet al., 2021; Zhaoet al., 2021).

Extensive experimental investigations on theGandλbehavior of soils have been performed by using, for example, resonant-column testes (Yuanet al., 2000; Sunet al., 2013; Brosseet al., 2017), cyclic simple-shear testes (Mccartneyet al., 2017; Ahmadinezhadet al., 2019), cyclic triaxial testes (Chenet al., 2017, 2019b; Wanget al., 2020), torsional-shear testes (Chenet al., 2016; Bedret al., 2019), shake table testes (Sadrekarimi, 2013), and dynamic centrifuge tests (Rayhani and El Naggar, 2008; Wang and Brennan, 2019). Among these, the major factors affecting theGandλvalues are identified, e.g., cyclic shear strain amplitudeγa, initial effective confining pressurecσ′ (or soil depthH), and plasticity indexPI; while frequency of cyclic loading, number of loading cycles, over-consolidation ratio, void ratio, degree of saturation, and grain characteristics are of less important (Darendeli, 2001). However, these studies are usually limited to the terrigenous soils. Because of the significantly difference of marine and terrigenous soils in the sedimentary environment, their undrained cyclic behaviors exist large difference. Thus, it is not appropriate to use the experientialGandλdata of terrigenous soils for seabed site seismic response analysis. Due to the high cost and difficulty of offshore operation, the undisturbed marine soil samples are quite limited, and the variation features of the measuredGandλvalues with increasingγafor marine soils have been less studied. Therefore, it is a challenging issue to study quantitatively the variation ofGandλwith both depth and other major factors for marine soils concerned the seismic safety of critical infrastructure systems in the Sea regions.

Based on the 97 undisturbed marine soil samples taken from the Bohai Sea, including silty clay, silt, and fine sand soils up to a depth of 120 m below the seabed surface, the stress-controlled undrained cyclic triaxial tests on the marine soils are performed in this paper. The influences of the depthH, water contentW, and sedimentary facies types on theGandλof the three marine soils are then studied. Finally, the empirical equations of the small-strain shear modulus (Gmax) and damping ratio (λmin) using the single-variableHas well as theG/Gmax–γaandλ–γacurves using the two-variableHandWfor the three marine soils are proposed in this paper.

2 Test procedure

2.1 Geological background and undisturbed soil samples

The undisturbed marine soil samples, including silty clay, silt, and fine sand soils, were taken from the nine boreholes in the Bohai Sea (Fig. 1). The depth contour in Fig. 1 refers to the curve formed by points with the same water depth. The offshore drilling was completed on the offshore platform to reduce the influence of wind and waves. The soil samples were extracted by using thin-wall sampler combined with static pressure method to maintain the undisturbed characteristics of marine soils. In the geological history of the Bohai Sea, there have been several cycles of marine transgressions and regressions. The Quaternary glacial and interglacial periods alternated, sea levels rose and fell sharply, and the marine strata overlapped with the terrestrial strata, which reflects the periodicity and reincarnation of sea water advancing and retreating in the Bohai Sea. The sedimentary facies types of marine soils in the Bohai Sea include abyssal facies, neritic facies, and terrestrial facies. Table 1 shows the variation range of water contents and unit weights of marine soils with different sedimentary facies types in different depth intervals. In general, the water contents and unit weights of the three marine soils decrease and increase sequentially with the increasing depth, following the law of gradual compaction under the action of self-weight. The borehole positions in the Bohai Sea and the sedimentary facies type of each sample tested are given in Table 2. The depthHin Tables 1 and 2 refers to the depth of soil sample below the seabed surface.

2.2 Test apparatus and method

In this study, the Young′s modulusEand damping ratioλof various marine soil samples over the axial strainεrange of 10-5to 10-2were measured by using the HX-100 cyclic triaxial apparatus manufactured by SBEL company, USA (Fig. 2). The shear wave velocityVsmeasurement of each specimen was independently performed using a pair of piezoceramic bender elements installed in the top and bottom platens of the cell of the HX-100 cyclic triaxial apparatus.

Fig. 1 The borehole locations in the Bohai Sea, China

Fig. 2 Cyclic triaxial apparatus with the shear velocity measuring system

According to the ASTM test standard (ASTM D3999-11, 2013), the undisturbed marine specimen was prepared into a solid cylindrical specimen with diameter of 3.91 cm and length of 8.0 cm by using the rotary cutting method. The prepared specimen was covered with a rubber membrane and installed in the triaxial apparatus. The mean effective confining stress is calculated byis the vertical effective stress related toHandk0is the coefficient of lateral earth pressure at rest which can be considered as constant for the specific soil (Baxteret al., 2008; Chenet al., 2020). Therefore, isotropic consolidation for each specimen was conducted with the initial effective confining pressure corresponding to the depthHof undisturbed soil sample (Table 2).

Table 1 Unit weights and water contents distribution of the marine soil samples taken from the Bohai Sea, China

Table 2 Sedimentary facies types and depths of marine soil specimens tested in this paper

After consolidation, theVsmeasurement of each specimen was conducted. A set of sinusoid signals from 1 to 40 kHz, rather than a single signal, was used as the excitation, and the received signals corresponding to these excitation frequencies were examined in whole to better identify the travel time of the shear wave. The time domain first arrival method was used in this paper to determine the velocity of propagation of shear wave through the sample (Chenet al., 2020; Ruanet al., 2021). Figure 3 shows the typical received signal diagram of shear wave velocity test. PointPrepresents the first arrival point, which is obvious and distinguishable. Thus, theGmax(at theγa(10-6)) can be calculated as following:

whereρis the soil density.

After theVsmeasurement, the multi-stage stresscontrolled undrained cyclic triaxial test was carried on each specimen tested. A uniform sinusoidal load at a frequency within the range of 0.1 to 2 Hz can be used to replace random seismic wave for cyclic triaxial tests (ASTM D3999-11, 2013). It is widely accepted that the cyclic loading with frequency of 1.0 Hz is applied for the cyclic triaxial testing on the dynamic shear modulusGand damping ratioλ(Burbanket al., 2013; Wanget al., 2020; Xionget al., 2018; Chenet al., 2019b; Duet al., 2021).

The 10-cycle sinusoidal cyclic stress with the frequency of 1 Hz was conducted at each loading stage. The applied stress amplitudes increased stage by stage, and the corresponding cyclic axial strain amplitudes started from –0.001% and increased gradually stage by stage until 1.0%. The excess pore water pressure in the saturated specimen was released after each loading stage, and a 30-minute intervals reconsolidation process was conducted to regain the initial consolidation state before the next loading stage.

3 Test results and analysis

3.1 G-γa and λ-γa curve features

The stress-strain hysteresis loop is characterized by three parameters, i.e.Gmax, strain-dependent secant shear modulusGand damping ratioλ. In Fig. 4, the slope of the line between the extreme points of the shear stressstrain hysteresis loop caused by one loading cycle is the secant shear modulusG. For the cyclic triaxial tests, the following relationship between the secant Young′s modulusEandGcan be used:

Fig. 3 Typical time history of output voltage from the receiver of testing system

Fig. 4 Diagram of hysteretic shear stress-strain relationship for estimating shear modulus and damping ratio

whereσais the uniform amplitude of the applied sinusoidal cyclic axial stressσdat each loading stage,εis the cyclic axial strain with varying amplitudeεa;γis the cyclic shear strain with varying amplitudeγa;vis the dynamic Poisson′s ratio assumed to be 0.5 for fully saturated specimens (Chenet al., 2019a). The material damping ratioλcan be expressed as:

where Δwandware the hysteresis loop area and the shadow triangle areas in Fig. 4, respectively.

Fig. 5 Continued

Fig. 5 Correlation among: (a) shear modulus G, shear strain amplitude γa, and depth H; and (b) damping ratio λ, shear strain amplitude γa, and depth H for the three marine soils

Fig. 6 Variation of the small-strain shear modulus Gmax with depth H for the three marine soils

Figure 5 shows the measuredG–γaandλ–γacurves of silty clay, silt, and fine sand soils in the different regions of the Bohai Sea. The results indicate that the marine soils exhibit almost linear at lower strain levels (γa<5×10-5), and strong nonlinear at higher strain levels (γa>1×10-4), i.e. theGdecreases rapidly and theλincreases with the increasingγa.Due to soil particles slide under cyclic loading, the strain energy released in unloading stage is less than that accumulated in loading stage. Therefore, at a higher strain level, soil particles slip and rearrange, and thus theλvalues become larger. With the increasing depthH, theGandλvalues of soil increase and decrease at the sameγa, respectively. For theγaabove 10-2levels, theGvalues approach 0 for the marine soils ofH< 20 m; while theGvalues will not decay to 0 for the marine soils ofH> 20 m. The contact between the adjacent soil particles becomes closer with the increase of effective confining pressure, which leads to less energy dissipation. Therefore, the marine soils have largerGand lowerλvalues with increasingH. The above results show that for a specific marine soil in the same sea region, theGandλvalues are strongly dependent on theγaandH.

3.2 Effect of soil depth on G and λ

Figure 6 shows the relationship between theGmaxand soil depthHof the three marine soils in the different regions of the Bohai Sea. TheGmaxincreases exponentially with the increasingH. For the same type of marine soils taken from the different regions of the Bohai Sea with the same depth, theGmaxvalues in different regions up to a depth of 20 m below the seabed surface has little difference, while the difference ofGmaxvalues below 20 m depth from the seabed surface becomes obvious. In general, in the depth range of 0–120 m, theGmaxdifference of silty clay soils is the largest, followed by silt soils, and that of fine sand soils is the smallest. Note that theGmaxvalues of silty clay and silt soils increase with the increasing plasticity indexPI(Hardin and Black, 1968), and thePIvalues of silty clay soils are higher than those of silt soils. This is why theGmaxdifference of silty clay soils is greater than that of silt soils in the different regions of the Bohai Sea. In addition, thePI= 0 for sand soils, theGmaxvalues of sand soils are mainly affected by the relative densityDrand depth. Generally, theDrvalues of sand soils increase with the depth, thus theGmaxvalues of sand soils are more significantly affected by the depth. Therefore, the difference between theGmaxof fine sand soils in the different regions of the Bohai Sea is less than that of silty clay and silt soils. It is a difficult task to determine the in-situ stress state of marine soils, however, it is closely dependent on the depthHfor a certain marine soil. Therefore, the robust mechanical model ofGmaxrelated to depthHis preferred to replace the stress-related model:

whereKandnare material-dependent best-fitting parameters (Table 3), and theKis theGmaxvalue at the reference depthH10= 10 m;GmaxandKare in kPa. In addition, the exponentnreflects the contact behavior and fabric change related to the effective stress change (Chaet al., 2014; Lyuet al., 2021). TheKandnvalues are closely related to the compression coefficientCcmeasured in the oedometer cell due to the relative displacement and fabric change between the soil particles. With the increasingCc, theKdecreases and thenincreases. Clay soils with lower plasticity and denser sand soils composed of rounder particles are associated with largerKand smallernvalues (Chaet al., 2014). For the three marine soils in the different regions of the Bohai Sea, thePIvalues of silty clay soils are larger than those of silt soils, while thePIvalues are zero for fine sand soils. Therefore, as shown in Table 3, theKvalues of silty clay soils are the smallest, followed by those of silt soils, and theKvalues of fine sand soils are the largest. However, the magnitude ofnvalues is the opposite to the magnitude ofKvalues for the three marine soils. Lyuet al. (2021) compiled the test results of the shear wave velocity of sand and silt soils in the literature, and gave thenvalues of various sand soils are between 0.17 and 0.19. Thenvalues of fine sand soils in the different regions of the Bohai Sea are between 0.51 and 0.57, which are far greater than those of sand soils given by Lyuet al. (2021). This is because the sand soils given by Lyuet al. (2021) are mainly composed of crushed particles with larger grain sizes, and their compressibility are lower. The fine sand soils in the Bohai Sea are composed of fine particles with higher compressibility. Thenvalues of silt soils given by Lyuet al. (2021) range from 0.23 to 0.32, whereas in the different regions of the Bohai Sea, thenvalues of the silt soils vary from 0.60 to 0.62, which may be due to the higher compressibility and plasticity of the silt soils in the Bohai Sea compared with those given by Lyuet al. (2021). TheR2values of a nonlinear regression for Eq. (6) for the three marine soils in the different regions of the Bohai Sea are greater than 0.95 in Table 3. This implies that a strong correlation exists between theGmaxandH.

Table 3 Parameters of the Gmax and γr prediction equations for the three marine soils

Figure 7 plots theG/Gmaxversusγacurves of the three marine soils in the different regions of the Bohai Sea. TheG/Gmax–γadata are all distributed in a narrow strip. At the low shear strain level (γa<5×10-5), theG/Gmaxdecreases slowly with the increase ofγa; while theγa>5×10-5, theG/Gmaxdecreases rapidly with the increasingγa. At the same level ofγa, the tangent slope ofG/Gmax–γacurves significantly reduces with the increasingH. The increase ofHcan induce the more contact between the soil particles and therefore more moving paths between the soil particles. This can lead to less energy dissipation and largerG/Gmaxvalues.

TheG/Gmaxversusγacurve can be described by the Davidenkov backbone curve (Martin and Seed, 1982) as follows:

where the coefficientsAandBare best-fitting values for the soil in question, and theγris reference shear strain for the soil in question, here, theγris theγavalue at which theG/Gmax= 0.5 (Martin and Seed, 1982; Darendeli, 2001; Chenet al., 2019b).

As shown in Fig. 8, there is no universal variation tendency of theA,Bvalues with theHfor the marine soils in the Bohai Sea. Note that the water contentWis the basic physical property index reflecting the physical state of soil, e.g., the magnitude ofWhas a significant effect on theGmaxandG/Gmaxof soil (Zhanget al., 2018; Xenaki and Athanasopoulos, 2008). Figure 9 shows the effect of water content on theG/Gmax–γacurve of silty clay soils with similar depths and same sedimentary facies type. Although the depth of sample S28 is greater than that of S17, the water content of S28 is much higher than that of S17, resulting in little difference in G/Gmax between samples S28 and S17. The depths of samples S11 and S22 are similar, but the water content of S11 is significantly higher than that of S22, as a result, the G/Gmax value of S11 is obviously lower than that of S22. The shape of the G/Gmax–γa curve is closely related to the coefficients A and B. Therefore, there may be certain relationship between the water content W and the coefficients A and B.clay soils with similar depths and same sedimentary facies type. Although the depth of sample S28 is greater than that of S17, the water content of S28 is much higher than that of S17, resulting in little difference inG/Gmaxbetween samples S28 and S17. The depths of samples S11 and S22 are similar, but the water content of S11 is significantly higher than that of S22, as a result, theG/Gmaxvalue of S11 is obviously lower than that of S22. The shape of theG/Gmax–γacurve is closely related to the coefficientsAandB. Therefore, there may be certain relationship between the water contentWand the coefficientsAandB.

Fig. 7 Variation of the shear modulus reduction G/Gmax with shear strain amplitude γa of the three marine soils

Fig. 8 Variation of the parameters A and B with depth H for the three marine soils

Figure 10 shows that the values ofAdecrease with increasingWof the marine soils in the Bohai Sea behaves obvious nonlinear feature. A finding of the laboratory investigation is that the relationship between theAandWof the marine soils follows the exponential function:

where the parameterssandtare best-fitting coefficients for the soil in question, andWis in percent. Another finding in Fig. 10 is that the values ofBincrease approximately linearly with increasingWof the marine soils in the Bohai Sea:

Fig. 9 Effect of water content on G/Gmax–γa curves for the silty clay with similar depths and same sedimentary facies

Fig. 10 Variation of the parameters A and B with water content W for the three marine soils

Fig. 11 Variation of the reference shear strain γr with depth curves for the three marine soils

where the parametersaandbare best-fitting coefficients for the soil in question, andWis in percent. The four best-fitting coefficientss,t,a, andbwithR2≥ 0.90 are listed in Table 4. This implies that theAandBare significantly correlated withWfor the three marine soils in the Bohai Sea.

Figure 11 shows that theγrincreases exponentially with increasingHfor the three marine soils in the Bohai Sea. The relationship betweenγrandHof the marine soils follows the exponential function:

whereγr1is theγrvalue atH10= 10 m,mis the soil specific best-fitting constant. The values ofγr1andmwithR2≥ 0.95 are given in Table 3. This implies that a strong correlation exists between theγrandHof the three marine soils in the Bohai Sea.

Table 4 Parameters of the A and B best-fitting equations for the three marine soils

Given that the measurement ofG/Gmaxis more accurate than that ofλ, Chen and Liu (2005) proposed the empirical equation ofλas a function ofG/Gmax: w

hereλminis small-strain damping ratio for the soil in question;λ0andβare soil specific shape coefficients ofλversus (1–G/Gmax) curve. As shown in Fig. 12, the decrease ofλminandλ0is approximatively linear with increasingH, while the increase ofβis almost linearly with increasingH. Accordingly, the empirical relationships of theλmin,λ0, andβwith a single-variableHcan be described by Eqs. (12)–(14):

The six best-fitting parametersA1,B1,A2,B2,A3, andB3withR2≥ 0.86 are listed in Table 5. This implies that the values ofλmin,λ0, andβare significantly correlated withHfor the three marine soils in the Bohai Sea.

In Fig. 13, the proposed correlations are validated by the independent experimental data of Lvet al. (2003) for the silty clay, silt, and fine sand soils taken from the borehole ZKA1 in the Bozhong Basin of the Bohai Sea in Fig. 1. The predictedGvalues for the three marine soils in the borehole ZKA1 can be obtained by combining Eq. (6) with Eq. (7) and Eqs. (8)–(10). The predictedλvalues of the three marine soils in the borehole ZKA1 can be obtained using both Eq. (7) (combined with Eq. (6) and Eqs. (8)–(10)) and Eq. (11) (combined with Eqs. (12)–(14)). The parameters using in the predictedGandλvalues are the same as those for the three marine soils in the Bozhong basin given in Tables 3–5. This independent validation shows that the deviation between the predicted and measuredGandλvalues is generally within ±15% for the three marine soils. It is encouraging that theGandλof various marine soils theoretically predicted by the proposed correlations (Eqs. (7) and (11)) coincides well with theGandλvalues observed experimentally. The significant implication is that the predictedG/Gmax–γaandλ–γacurves of the three marine soils in this paper serve as a worthful reference for the seabed site seismic response analysis in the Bohai Sea due to lack of available laboratory data.

Fig. 14 Comparison of the measured G values of the three marine soils for the different sedimentary facies

Table 5 Parameters of the λmin, λ0, and β best-fitting equations for the three marine soils

Fig. 12 Variation of the parameters λmin, λ0, and β with depth H for the three marine soils

Fig. 13 Validation of the G and λ prediction equations for independent experimental data of Lv et al. (2003)

3.3 Effect of sedimentary facies types on G and λ

Sedimentary facies of soils can comprehensively reflect the characteristics of sedimentary environment, which essentially affects the dynamic properties of marine soils. The Yellow River, Haihe River, Liaohe River, and Luanhe River flow into the Bohai Sea, bringing the material source of the sediments in the Bohai sea. Since the late Quaternary, there have been three transgressions in the Bohai Sea. The sedimentary environment of the abyssal facies has weak hydrodynamic effect and low sedimentary energy. After the substances suspended in seawater sink to the seabed, they are gradually compacted with the increase of the overlying sediments, without drying and denudation, the connections between the soil particles are not tight, and the pores between the soil particles are filled with water, resulting in poor resistance to the deformation caused by earthquake and wave loads. In the sedimentary environment of the terrestrial facies, the sediments have been exposed to the earth′s surface, affected by drying and denudation, the soil particles are closely connected, and the resistance to the deformation is greatly improved. The nature of marine soils of the neritic facies are between those of the abyssal facies and terrestrial facies (He, 2006).

The three marine soils with similar depths were selected from the 97 undisturbed samples to investigate the correlation between the sedimentary facies types and theGandλvalues. Figure 14 shows theG–γacurves of the three marine soils with similar depths in the Bohai Sea. In general, for silty clay, silt, and fine sand soils at the similar depths, theGvalue of the marine soils of the terrestrial facies is the largest, followed by the neritic facies, and that of the marine soils of the abyssal facies is the smallest. In addition, the effect of the sedimentary facies types on theGvalue is also related to the shear strain amplitudeγa. When theγaranges from 10-5to 10-3levels, the effect of the sedimentary facies types on theGvalues of marine soils with similar depths is strong; while theγaranges from 10-3to 10-2levels, the difference of theGvalues of marine soils with similar depths for the different sedimentary facies types is slight. In addition, the effect of sedimentary facies types on theGvalues of marine silty clay and silt soils is obvious, but the effect on theGvalue of marine fine sand soils is relatively small. The physical and mechanical properties of marine soils are closely related to the sedimentary environment. The water contents of the marine soils of the abyssal facies and neritic facies are generally higher than that of the terrestrial facies (Table 1). The saturated soil with lower water content has fewer pores, and the soil particles are closely connected with each other, forming the dense skeleton structure, which is conducive to the propagation of shear wave. With the increase of water content, the water film will be formed on the surface of soil particles, which leads to the decrease of cohesive force between soil particles. Meanwhile, the barrier effect of pore water on the transmission of shear wave gradually increases. Therefore, theGvalues of the marine soils of the abyssal facies and neritic facies are lower than that of the terrestrial facies with similar depths.

As shown in Fig. 15, there is no unique tendency for the effect of the sedimentary facies types on theλ–γacurves of the three marine soils with similar depths. In general, in the three depth specific intervals, theλof the marine silty clay soils of neritic facies remains at relatively large values, while the relative magnitude of theλvalues of the marine soils of terrestrial facies and abyssal facies are not fixed. Theλvalues of the marine silt soils of neritic facies are lower than those of the marine silt soils of terrestrial facies and abyssal facies at the similar depths. For the fine sand soils with similar soil depths in the Bohai Sea, theλvalues of the marine fine sand soils of terrestrial facies are always less than those of the marine fine sand soils of abyssal facies and neritic facies regardless of theγalevels. Chenet al. (2017) found that sedimentary facies types have obvious effect on both theGandλvalues of the terrigenous soils in Suzhou urban area, China. The results of this paper show that the sedimentary facies types have obvious effect on theGvalues of the three marine soils in the Bohai Sea, but have no significant effect on theλvalues. This may be the difference between the marine soils and the terrigenous soils, which needs further study.

Fig. 15 Comparison of the measured λ values of the three marine soils for the different sedimentary facies

Fig. 16 Comparison of the G/Gmax values of the three marine soils and the corresponding terrigenous soils (data from Yuan et al. (2000))

3.4 Comparison of G/Gmax and λ curves for marine and terrigenous soils

The sedimentary environment of marine soils is quite different from those of the terrigenous soils. The dynamic characteristics of the shallow marine soils are different from those of the terrigenous soils due to the long-term effects of offshore storms and waves. The plots in Fig. 16 depict the measuredG/Gmax–γacurves of marine silty clay, silt, and fine sand soils at the 10-20 m depths of any borehole (see Fig. 1) in the Bohai Sea. TheG/Gmax–γacurves of the corresponding terrigenous soils at the similar depths recommended by Yuanet al. (2000) are also plotted in Fig. 16 for comparison. A marked finding is that theG/Gmaxcurves of the three marine soils in the Bohai Sea are below those of the corresponding terrigenous soils, which means that the nonlinearity of the three marine soils is stronger than those of the terrigenous soils. Similarly, the plots in Fig. 17 compare the correspondingλcurves of marine silty clay, silt, and fine sand soils at the 10–20 m depths of any borehole (see Fig. 1) in the Bohai Sea and those of the corresponding terrigenous soils at the same depths recommended by Yuanet al. (2000). As shown in Fig. 17, except few experimental data ofλfor marine silty clay soils at the shear strain levels above 0.5%, theλcurves of marine silty clay, silt, and fine sand soils in the Bohai Sea are above those of the corresponding terrigenous soils, especially, theλcurves of the marine fine sand soils are much higher than that of the terrigenous fine sand soils. Compared with terrigenous soils, marine soils have the characteristics of higher water content, higher porosity ratio, and lower unit weight. The connections between marine soil particles are relatively sparse, and there are less pathways for shear wave propagation, resulting in more energy dissipation. Therefore, theG/Gmaxandλvalues of marine soils are lower and higher than that of terrigenous soils with similar depths, respectively.

Fig. 17 Comparison of the λ values of the three marine soils and the corresponding terrigenous soils (data from Yuan et al. (2000))

4 Conclusions

This paper presents interesting results from stress-controlled undrained cyclic triaxial tests on the undisturbed silty clay, silt, and fine sand soils taken from the Bohai Sea, China. Focus has been placed on the effects of soil depthH, water contentW, and sedimentary facies types on the secant shear modulusGand damping ratioλof the three marine soils. The main conclusions are as follows:

(1) TheGandλvalues of marine soils are highly dependent on the soil depthH. The small-strain shear modulus (Gmax) is an exponential function of the singlevariableH. The empirical equation of shear modulus reduction (G/Gmax) is proposed using the Davidenkov backbone curve with the parametersA,B, andγr, in which the parametersAandBexponentially decreases and linearly increases with the increasingW, respectively, while the parameterγrexponentially increases with the increasingH. Theλis a polynomial function ofλminand (1–G/Gmax) with the parametersλ0andβ, in which theλmin,λ0, andβare all linear functions ofH. The applicability of the proposedGandλprediction equations is validated by the independent experimental data in the literature.

(2) Among the three sedimentary facies types of the three marine soils at the similar depths, the marine soils with the largestGvalue are of the terrestrial facies, while the one with the smallestGvalue is of the abyssal facies. Slight effect of the sedimentary facies types can be found on theλvalue of the marine soils.

(3) The strain-dependentG/Gmaxandλvalues of the three marine soils at the same strain levels are significantly less and larger than that of the corresponding terrigenous soils, respectively. This implies that the nonlinearity of the three marine soils tested is stronger than that of the corresponding terrigenous soils.