Maximum shear modulus and modulus degradation curves of an unsaturated tropical soil

Abstract The maximum shear modulus (G0) and the modulus degradation curve (G/G0 versus γ) are important information in the evaluation of the soil mechanical behavior, both for dynamic and static loads. Dynamic tests (resonant column and cyclic triaxial tests) are not routinely performed in geotechnical practice in Brazil, and the geotechnical literature on the dynamic behavior of unsaturated tropical soils is limited. This paper presents and discusses seismic dilatometer (SDMT), resonant column, and triaxial test with bender elements and internal instrumentation to determine G0 and the modulus degradation curve in an unsaturated tropical sandy soil profile. It was observed that G0 tends to increase non-linearly with soil suction and net stress (σ - ua). It was also observed that the in situ G0 values determined with the SDMT were higher than those from laboratory tests (bender elements and resonant column). The modulus degradation curves determined with resonant column were used to define the reference curve via SDMT for the studied site. Soil suction influence in shear modulus degradation curves determined with unsaturated triaxial compression tests with local instrumentation is also presented and discussed.


Introduction
The maximum shear modulus (G 0 ) and the modulus degradation curve (G/G 0 versus γ) are important information to analyze the mechanical behavior of soils.It is necessary to determine these parameters and this curve due to the increase demand for nuclear facilities, offshore structures, and machine foundation design.The ground motion of the site is significantly affected by the local site condition during an earthquake, and the average shear wave velocity (Vs) up to 30 m is the key variable for site characterization in geotechnical earthquake engineering (Bang & Kim, 2007;ICC, 1997).Moreover, the G 0 values can be used for a static deformation analysis such as slope stability, settlement estimative, an evaluation for ground improvements, as well as assessment of collapsible soils (Burland et al., 1977;Kim & Park, 1999;Rocha et al., 2022).Tests to determine soil dynamic parameters are not currently performed in Brazil and the geotechnical literature on the dynamic behavior of tropical soils is limited.
The crosshole test is the most effective technique for determining Vs, and to calculate the maximum shear modulus (G 0 ) via Elasticity theory.Recently, the seismic dilatometer (SDMT) has being widely used since it allows the site characterization together with the determination of Vs profiles, consequently G 0 (Marchetti et al., 2008).Resonant column tests and the bender elements incorporated to triaxial tests can be used to determine Vs under controlled conditions in laboratory, such as confining stress, strain amplitude and soil suction influence.
The soil behavior is highly non-linear and has an important influence on the selection of design parameters for simple routine geotechnical projects (Atkinson, 2000).So, the direct application of G 0 to evaluated deformations problems is not applicable, and the shear modulus decay curve is necessary.The non-linear soil stress-strain behavior can be estimated with in situ and laboratory tests.In situ tests, like the crosshole and downhole can be used to determine shear modulus at small strains; dilatometer, pressuremeter, and plate load tests for medium strains; cone penetration and standard penetration tests for largely deformed soils (Amoroso, 2011;Atkinson, 2000;Ishihara, 2001).Laboratory tests, such as the bender elements or the resonant column, the cyclic triaxial or torsional shear tests, or even monotonic triaxial tests, or double specimen direct simple shear can be used to estimate the non-linear soil behavior (Amoroso, 2011).
A large portion of Brazil is covered by unsaturated tropical soils and the geotechnical literature about dynamic parameters of these soils is limited since dynamic tests are not currently carried out.The term tropical soil includes both lateritic and saprolitic soils.Saprolitic soils are residual and retain the macro fabric of the parent rock.Lateritic soils can be either residual or transported and are distinguished by the occurrence of the laterization process, which is an enriching of a soil with iron and aluminum and their associated oxides (cementation), caused by weathering in regions which are hot, acidic, and at least seasonally humid (Nogami & Villibor, 1981).Cementation and soil suction affects the soil behavior of unsaturated tropical soils, both in situ and in laboratory (Fernandes et al., 2022;Giacheti et al., 2019;Rocha et al., 2021).The contribution of microstructure (cementation) and soil suction to the soil stiffness depends on the strain level the soil will experience (Atkinson, 2000).These characteristics increase the overconsolidation stress and cohesion intercept (Vaughan et al., 1988) and the most existing empirical correlation should be employed with caution (Robertson, 2016).
In this paper, SDMT, triaxial tests with bender elements and internal instrumentation, as well as resonant column carried out in an unsaturated tropical soil are presented and discussed.G 0 values determined by these different techniques were compared.The modulus degradation curves (G/G 0 versus γ) determined via resonant column tests were used to define the reference curve for the SDMT based on the approach proposed by Amoroso et al. (2014).In addition, the effect of the unsaturated soil condition on the modulus degradation curves obtained from triaxial tests with suction control and internal instrumentation are presented and discussed.

Study site
SDMT, resonant column and triaxial tests with bender elements were conducted at the Experimental Research Site at the São Paulo State University (Unesp), located in the city of Bauru, State of São Paulo, Brazil.The study site includes a colluvial Neo-Cenozoic deposit up to about 13 m depth, followed by a residual soil formed during the Quaternary (De Mio, 2005).The soil profile consists in an unsaturated clayey fine sand with lateritic behavior up to about 13 m depth.The MCT Classification System (Mini, Compacted, and Tropical) proposed by (Nogami & Villibor, 1981) for tropical soils was used to define and classify the soils with regards to the lateritic behavior.These soils have undergone pedogenic and morphogenetic processes (Giacheti et al., 2019).Consequently, this soil has high porosity, high saturated hydraulic conductivity (10 -5 to 10 -6 m/s), and a cohesive-frictional behavior.A major geotechnical problem for this soil is collapsibility caused by soil wetting.
Several site characterization programs including Standard Penetration Tests (SPT), Standard Penetration Tests with Torque (SPT-T), Seismic Cone Penetration (SCPT), Flat Dilatometer (DMT), Pressuremeter (PMT), and Seismic tests (crosshole -CH and downhole -DH) were carried at the site.Sample pits were also excavated to retrieve undisturbed and disturbed soil blocks.Soil samples from these blocks were tested in laboratory for soil characterization and determination of mechanical properties and parameters.Figure 1 summarizes laboratory and in-situ tests carried out at the study site: grain size distribution (with and without dispersant), some index properties, SCPT, SPT, PMT and Seismic tests data along the soil profile.

In situ and laboratory tests
Four SDMTs were carried out at the study site up to 15 m depth.The SDMT testing procedures were conducted in accordance with Marchetti et al. (2006).A multi-function penetrometer with a 150 kN thrust capacity (Model Pagani TG 63 -150 DP), which was anchored to the ground by helical augers, was used to carry this in situ test.The SDMT blade was pushed into the ground at a constant rate of 20 mm/s.The readings A-pressure and B-pressure were taken at intervals of 200 mm, and then these pressures were corrected for membrane stiffness and converted into p 0 and p 1 .The three intermediate DMT parameters (I D : material index; K D : horizontal stress index; E D : dilatometer modulus) were calculated from the p 0 and p 1 values.Field measurements of the shear wave velocity (Vs) were taken at 0.5 or 1 m depth interval.
The resonant column tests were presented by Giacheti (1991).The triaxial tests were performed with internal instrumentation and bender elements by Fernandes (2022).The modulus degradation curves were determined from the resonant column, triaxial and SDMT test data.
Cylindrical specimens of about 36 mm in diameter and 80 mm in height were used in the resonant column tests.They were rigidly fixed to the base of the triaxial chamber by means of a blade embedded in the porous stone (Giacheti, 1991).Table 1 shows some geotechnical indexes, the confining stresses, and the moisture content conditions for each of them.The multi-stage technique was employed in the resonant column tests, as described by Anderson & Stokoe (1978).A very low amplitude torsional excitation was applied to the top of the specimen and the shear wave velocities were determined over the logarithmic time interval up to 1,000 minutes or up to 10,000 minutes in some cases for each confining stress stage.Subsequently, the excitation force was gradually increased and the variation of the ratio of shear modulus to strain amplitude was determined.
The saturated and unsaturated triaxial tests with internal instrumentation and bender elements were performed using 50 mm diameter specimens with the height ranging from 100 to 120 mm.The determination of Vs (and hence G 0 ) via bender elements was performed for the samples collected at 1.5, 5, 7, 11, and 13 m depth.The phase angle between the waves and the frequency domain method (Ferreira, 2002) were used to determine the wave propagation time.Suction values of 0 (saturated), 50, 200 and 400 kPa were imposed for samples collected at 1.5 and 5 m depth and suction values of 0 (saturated), 50, 100 and 200 kPa for samples collected at 7, 11 and 13 m depth.The applied confining stresses were 25, 50, 100, and 200 kPa for all samples tested.The axial (ε a ) and radial (ε r ) strains were measured by internal instrumentation (LVDTs with axial and radial displacement measurement) for the sample collected at 2 m depth, with a confining stress of 50 kPa and suction values equal to 0, 50, 200, and 400 kPa.The shear strain for individual soils elements (ε s ) can be calculated from Equation 1 based on ε a and ε r , and it was transformed in shear strain (γ) with Equation 2. The modulus of elasticity (E) was obtained from the triaxial test data and a Poisson ratio (μ) equal to 0.2 was assumed to determine the modulus degradation curve (Equation 3).

Modulus degradation curve via SDMT
The modulus degradation curve can be estimated from a reference degradation curve determined in the laboratory by cyclic testing (Marchetti et al., 2008).This curve can be defined from two points obtained by means of SDMT: (1) maximum soil shear modulus (G 0 ), and ( 2) shear modulus at the working condition (G DMT ).Amoroso et al. ( 2014) presents a procedure to estimate the modulus degradation curve via SDMT based on the findings of Marchetti et al. (2008).This procedure is schematically represented by Figure 2, and it consists of the following steps: Table 1.Some information of the previously performed resonant column tests [adapted from Giacheti (1991)].

Depth (m)
Confinant stresses (kPa)  • Determine G 0 based on Vs from SDMT, at the same depth of the available reference modulus degradation curve; • Calculate G DMT based on the constrained modulus obtained from SDMT data (M DMT ) (Equation 4) and normalized by its maximum shear modulus (G 0 ).
( ) Where µ is the Poisson ratio.
Therefore, the ratio G DMT /G 0 obtained from SDMT and the estimated shear strain γ DMT were used to plot the corresponding hyperbolic curve at each investigated test site.
According to Amoroso et al. ( 2014), the "typical range" of shear strain (γ DMT ) associated to the working strain moduli G DMT can be approximately assumed as 0.01 to 0.45% for sand, 0.1 to 1.9% for silt and clay, and higher than 2% for soft clay.The authors considers that this approach can provide a first estimate of the modulus degradation curve (G/G 0 versus γ) of the soil.

SDMT
Figure 3 shows the intermediate DMT parameters (I D , K D, and E D ) and the shear wave velocity, and consequently maximum shear modulus profiles for the study site.I D , K D and E D were calculated by Marchetti's equations (Marchetti, 1980).Shear wave velocity determined with SDMT, and total mass density (ρ) determined using undisturbed soil samples collected in a sample pit excavated at the study site were used to calculate G 0 values based on Elastic Theory (Equation 6).
Where Vs is shear wave velocity, and ρ is the total mass density.
Figure 3 shows a good agreement between the Vs and G 0 profiles determined by all the performed tests.The values of Vs and G 0 increase with depth up to 10 m and this trend becomes almost constant after that depth.

Triaxial tests with bender elements (BE) and internal instrumentation
The shear wave velocity (Vs) and consequently G 0 were determined using bender elements.G 0 shows a tendency to increase non-linearly with the suction and with the net stress (σ -u a ) for sandy soils, and tends asymptotically to a limit (Nyunt et al., 2011).A hyperbolic function was used to evaluate the influence of the suction and the net stress variables on the maximum shear modulus (Equation 7). Figure 4 shows the variation of G 0 with suction and with net stress, as well as the fitting for the samples collected at 1.5, 5, 7, 11, and 13 m depth.As the soil suction increases from 0 to 400 kPa, the shear moduli at small strain (assumed equal to 0.001%) also increase.Table 2 shows the fitting parameters for the Equation 7 at the 100 kPa net stress for all the investigated depths.It can be seen from this figure that the experimental data are well represented by a non-linear relationship (Equation 7) between G 0 and the variables suction and net stress, except for the net stresses of 25 and 50 kPa for the depth of 7 m, which showed a quasi-linear behavior.
Figure 5 shows the absolute values of the shear modulus at a small shear strain of 0.001% and at a finite strain of 1% for the sample collected at 2 m depth.It also can be seen from this figure that the soil shows a relatively high and fast variation of the shear modulus as the shear strain increases from 0.001 to 1%.Giacheti (1991) presented the maximum shear modulus (G 0 ) as a function of time of confinement by performing resonant column tests and observed that the samples tested (Table 1) showed an almost linear increase of G 0 with logarithmic time, practically from the beginning of drainage, which is a typical behavior for sands.Figure 6 shows the value of G 0 at 1,000 minutes of confinement at different confinement stresses for the samples collected at 0.95, 4.8, and 8.85 m depth (Giacheti, 1991).Figure 7 shows the modulus degradation curves (G/G 0 versus γ) determined for different confining stresses for the samples collected at 0.95, 4.8, and 8.85 m depth (Giacheti, 1991).

Resonant column (RC) tests
Figure 7 shows the modulus degradation curves for the samples collected at 0.95, 4.8, and 8.85 m depth at different confinement stress (σ 3 ).It can be seen from Figure 7 that the modulus G presents a small reduction for shear strains higher than 10 -4 %, which is accentuated from γ greater than 10 -3 %.Furthermore, a lower influence of confining stresses and depth on G values is observed for the range of strains investigated, with a tendency for a lower degradation of the modulus with increasing confining stress (σ 3 ).

G 0 from in situ and laboratory tests
In order to compare G 0 values determined by SDMT, resonant column (RC) and bender elements (BE), in situ confining stresses were defined based on at-rest earth pressure coefficient (K 0 ) estimated from the Jaky (1948) equation.
All values were considered for the bender element tests with suction equal to 50, 100, and 200 kPa for the investigated depths (1.5, 5, 7, 11, and 13 m depth) since the SDMT and resonant column tests were performed in the natural soil condition.These suction values were defined from the suction monitoring by tensiometers, and watermark sensors presented by Giacheti et al. (2019).In addition, an average G 0 profile from four SDMTs was adopted.
Figure 8 shows the differences between the G 0 values determined by the average SDMT, average SDMT plus and   minus one standard deviation (SD), RC and BE tests.The average SDMT values were higher than those determined by RC and BE.These values were 8% and 35% higher than those determined via BE, and 6% and 28% higher than those determined via RC.It is important to mention that the G 0 values determined by RC and BE are positioned at the lower limit or slightly outside the range.These differences may be related to possible disturbances during the sampling process and specimens preparation, errors in the estimation of the in situ confining stresses (Ferreira et al., 2011) as well as the influence of soil suction in G 0 (Nyunt et al., 2011).

Modulus degradation curve
4.2.1 SDMT and resonant column Amoroso et al. (2014) suggest a method to estimate the modulus degradation curve (G/G 0 versus γ) by using SDMT, based on the parameters G 0 , G DMT and γ DMT , as previously discussed in item 3.1.This approach allows a preliminary definition of the modulus degradation curve, which needs to be interpreted in conjunction with reference G/G 0 versus γ determined in the laboratory via cyclic triaxial or resonant column tests.So, the degradation curves presented in Figure 7 were considered representative and an average degradation curve was assumed.Table 3 presents the average values of G 0 , M DMT , G DMT /G 0 determined by means of the four SDMTs performed, as well as the shear strain imposed with the expansion of the DMT blade (γ DMT ) determined from the average degradation curve assumed by the resonant column tests.
Figure 9 shows the value of the G DMT /G 0 ratio determined via SDMT, the shear strain fitted from the resonant column data (γ DMT -gray symbol), and the average stiffness degradation curve obtained by Equation 3 (in gray dashed line).The G DMT / G 0 value is equal to 0.051 for the studied site, which is slightly lower than the typical values reported in the literature (shaded areas), which can be associated to the natural cementation of the particles and the unsaturated condition, typical of tropical soil sites.At the working condition (G DMT ), the stiffness due to cementation and soil suction is lost.On the other hand, the value of the shear strain imposed by SDMT blade pushing into the soil (γ DMT ) for the studied soil is in the range of values commonly reported in the literature for sands and silty sands to sandy silts (0.1 to 0.5%) (Amoroso et al., 2014).It is important to mention that the multistage technique (Anderson & Stokoe, 1978) used to perform the resonant column tests can generate accumulated deformations and disturbances in the structure of the specimen, resulting in cementation bond breakage, realignment of grains, and changes in void ratios (Barros, 1997); therefore, the modulus degradation for the soil of the study site can be lower than those presented in Figures 7 and 9.

G/G 0 versus γ determined via unsaturated triaxial tests with internal instrumentation
The modulus degradation curves were also determined via unsaturated triaxial tests with internal instrumentation (LVDTs) for the sample collected at 2 m depth for soil suctions equal to 0, 50, 200, and 400 kPa (Figure 10).The objective is to evaluate the soil suction influence in the pattern and shape of the shear modulus degradation curve.Figure 10a shows that the shear modulus reduction curves are not monotonically related to the change in soil suction and reaches a maximum for a suction value equal to 200 kPa.The modulus degradation curve for the suction equal to 400 kPa is equivalent or slightly lower than those of suction equal to 50 kPa.It shows that the G/G 0 versus γ curves firstly rises and then falls in a certain range with the increase in soil suction.This trend is different from that found by Ng et al. (2021) for compacted unsaturated lateritic sandy clays and for eight different soil types as reported by Dong et al. (2018).However, Ng & Xu (2012) observed that the G/G 0 curves shift towards higher shear strain values with increasing soil suction for yellowish-brown completely decomposed tuff (CDT).
In order to describe the non-linear soil behavior, several researchers have proposed a mathematical model to capture the features of modulus reduction curve (Darendeli, 2001;Iwasaki et al., 1978;Kokusho, 1980;   Seed et al., 1986;Vucetic & Dobry, 1991).According to Amoroso et al. (2014), the G/G 0 versus γ curves proposed by Darendeli (2001) include all other reference curves.Darendeli (2001) equation was used to represent the modulus reduction curves, as follows: Where d is the constant that represents the curvature of the modulus reduction curve, and the γ ref is the reference strain controls the location where G decreases to half of its maximum value as the shear strain increases.
Figure 10b shows the modulus reduction curves for the tested samples.The dashed line at G/G 0 = 0.5 reflects the positions of the reference shear strain for each modulus reduction curve.Figure 10b also allows to examine the dependencies of the reference strain (γ ref ), and coefficient of curvature (d) on soil suction.The relationships between reference strain and suction are shown in Figure 11a, whereas the relationships between coefficient de curvature and suction are shown in Figure 11b for the soil from the study site.The reference strain increases with soil suction up to 200 kPa and decreases  for soil suction equal to 400 kPa.The coefficient of curvature presents a slight reduction between saturated and soil suction equal to 50 kPa and remains approximately constant as the soil suction increases from 50 kPa to 400 kPa, indicating no suction dependence on this parameter.

Conclusion
The main conclusions drawn from the study are as follows: • The maximum shear modulus of the studied soil increases nonlinearly with suction and net confining stress based on the bender elements test data; • The maximum shear modulus values from the SDMT were higher than those determined in the laboratory via resonant column and bender elements tests.This behavior can be related to possible soil disturbances during the sampling and preparation of the specimens, errors in estimating the in situ confining stresses, as well as the influence of soil suction on G 0 ; • The average modulus degradation curve defined from resonant column was used to obtain the modulus degradation curve from SDMT.The approach proposed by Amoroso et al. (2014) for SDMT is interesting and can be used as a first tentative to represent the modulus degradation curve for the soil from the study site; • The modulus degradation curves from the suctioncontrolled, internally instrumented triaxial tests are not monotonically related to the change in soil suction and it was maximum for a suction value equal to 200 kPa, and this behavior is different from which was observed by other researchers.Additional tests on samples collected at other depths as well as for other suction values should be done to explore and confirm the relation between soil suction and modulus degradation curves for the studied soil.

Figure 1 .
Figure 1.Summary of in situ and laboratory tests carried out at the study site [adapted from Rocha & Giacheti (2018)].
sat is the maximum saturated shear modulus, s is the soil suction, and a, b, and c are empirical parameters of the fit.G 0 and G 0,sat are expressed in MPa and the net stress (σ -u a ) and suction in kPa.

Figure 3 .
Figure3.SDMT data at the study site.

Figure 4 .
Figure 4. Variation of G 0 with net stress (σ -u a ) and with suction for the undeformed soil samples collected at (a) 1.5 m, (b) 5 m, (c) 7 m, (d) 11 m, and (e) 13 m depth.

Figure 5 .
Figure 5. Modulus degradation curves for the tested samples from 2 m depth with suction values equal to 0, 50, 200, and 400 kPa.

Figure 9 .
Figure 9. Modulus degradation curve via SDMT from Amoroso et al. (2014) method and the resonant column test data for the study site.

Figure 8 .
Figure 8. G 0 values determined by SDMT, bender elements, and resonant column for the study site.

Figure 11 .
Figure 11.Soil suction influence on: (a) reference strain (γ ref ); (b) curvature coefficient a for the soil from the study site.

Table 2 .
Fitting parameter for the Equation 7 at 100 kPa net stress.

Table 3 .
Best-fit parameters for Amoroso et al. (2014) method for the study site.