1 Introduction

Poisson’s ratio, ν, can be computed by measuring the transverse, ε trans , and axial, ε axial , strains, with the help of strain gauges or LVDTs, by conducting the conventional triaxial tests (i.e., strain controlled uniaxial compression tests) (Bragg and Andersland 1982; Samsuri and Herianto 2004).

$$ \nu = - \left( {\varepsilon_{trans} /\varepsilon_{axial} } \right). $$
(1)

However, it must be noted that measurement of ε trans and ε axial for soft clays and sands is quite difficult. On one hand, penetration of LVDTs in soft clays and sands yields inaccurate strains, while on the other hand, fixing micro-strain gauges, which measure axial and lateral strains using Wheatstone bridge circuits, on the surface of these soil samples is a major challenge.

Moreover, Elastic modulus, E, and shear modulus, G, are important parameters required for safe design of various civil engineering structures. Earlier researchers (Kim and Stokoe 1992; Mancuso et al. 2002; Sawangsuriya et al. 2008) have employed resonant column test and the torsional shear test to obtain the E and G, respectively. However, these test methods are cumbersome, time consuming and require skilled manpower. ν can be computed by employing Eq. 2 (Luna and Jadi 2000; Santamarina et al. 2001; Zeng and Tammineni 2006), as well.

$$ \nu = \left( {0.5 \times E/G} \right) - 1. $$
(2)

Due to these difficulties, researchers (Jain 1988; Lees and Wu 2000; Luna and Jadi 2000; Ayres and Theilen 2001; Landon et al. 2007) have employed geophysical testing methods (viz., seismic refraction and reflection, suspension logging, steady-state vibration, down-hole, seismic cross-hole, spectral analysis of surface waves, SASW, multi-channel analysis of surface waves, MASW, seismic cone penetration tests) for estimating parameters G, and ν of the soil mass by employing Eqs 3 and 4, respectively, (Santamarina et al. 2001; Zeng and Tammineni 2006; Phani 2008). It is worth mentioning here that geophysical tests propagate seismic waves through the soil mass at a very low strain level (<0.001%).

$$ G = \rho \times V_{s}^{2} $$
(3)
$$ \nu = \left( {0. 5\cdot{\text{r}}^{ 2} - 1} \right)/\left( {{\text{r}}^{ 2} - 1} \right) $$
(4)

where, ρ is mass density of the soil mass and, r is the ratio between V s and V p , the shear and compression wave velocities, respectively.

However, geophysical methods are quite expensive and require trained and skilled manpower for interpretation of the obtained results. In such a situation, application of piezoceramic elements (i.e., bender and extender elements, which can be used for generating shear and compression waves, respectively) for determining elastic moduli and Poisson’s ratio of the soil mass has been found to be quite useful (Agarwal and Ishibashi 1991; Jovicic et al. 1996; Brocanelli and Rinaldi 1998; Lohani et al. 1999; Santamarina et al. 2001; Huang et al. 2004; Bartake et al. 2008). However, how this (computed) value of the Poisson’s ratio compares vis-à-vis that obtained from the conventional triaxial tests, which yield stress–strain relationship, needs to be checked.

With this in view, investigations were conducted on soils of different types (clays and sands) in their disturbed and undisturbed forms by resorting to piezoceramic tests and conventional triaxial tests. Details of the methodology to achieve this are presented in this paper and it has been demonstrated that by using piezoceramic elements, the Poisson’s ratio and the elastic modulus of the soils can be obtained quite easily, particularly for the soft clays and sands.

2 Experimental Investigations

2.1 Sample Details

In the present study, tests were carried out on three different grades of sands and soft clays. The sands were characterized as SP type as per the USCS (Unified Soil Classification system), the details of which are given in Table 1. The soft clays, characterized as CH type, were collected from the field in their undisturbed form, using Shelby tubes. Soil specimens were then extruded from the sampling tubes using a sample extractor and a hollow cylindrical split mold (38 mm in diameter and 76 mm in length). The water content, w, and bulk density, γ t , of these specimens vary from 36 to 71% and 14–18 kN/m3, respectively (refer Table 2). İn order to prepare the sample of the sand, the sand was poured in a plastic cylinder, made of thin polythene sheet (a transperancy sheet, which is less than 1 mm thick), with the help of a glass funnel and by maintaining 30 mm height of the fall. Later, the exact length of the sample was measured on four diametrically opposite sides of the mold and the average of these values was used for determining the initial volume and hence the dry unit weight, γ d, of the sample. Using these parameters, the initial void ratio, \( e[ = ((G_{\text{s}} \cdot\gamma_{\text{w}} )/\gamma_{\text{d}} ) - 1] \), of the sample was determined. Where, G s is the specific gravity of the soil and γ w is the unit weight of water. For achieving different void ratios of the sample, the cylinder containing the sand was subjected to shaking by mounting it on a vibration table for certain duration.

Table 1 Physical characteristics of the sand samples used
Full size table
Table 2 Experimental results obtained for the clay samples
Full size table

2.2 Load-Deformation Characteristics

The test setup employed for obtaining the load-deformation characteristics of the soil samples consists of a compression testing machine (supplied by Humboldt, USA), a load cell of 10 kN capacity, and a sample cage made of three derlin rings and the connecting rods (refer Fig. 1). The middle ring can support three LVDTs (Linearly Variable Differential Transducers) at 120° apart, as depicted in Fig. 2. These LVDTs (type KL 17, supplied by KAPTL Instrumentation, India) work on the principle that when AC current (2 Vrms, 5 kHz sine-wave) is applied to primary winding, it produces a magnetic field which, in turn, induces emf-in two differentially connected, secondary windings. The magnetic core moving linearly along the axis varies the flux linkage from primary to both the secondary. The output voltage, thus obtained is linearly proportional to linear displacement. These LVDTs have flat tips, which restricts their piercing into the sample, and can record deformations in the range of 0–10 mm, with a resolution of 1 μm. The undisturbed clay samples were housed inside the cage whereas, in case of sands, a low density polyethylene sheet of thickness 0.1 mm was rolled in the form of a cylinder (height 76 mm and diameter 38 mm), placed inside the cage and was used as a mold for preparing the sample by adopting the rain-fall technique. The LVDTs record ε trans of the sample, when it is axially loaded. For obtaining ε axial , another similar LVDT was employed, which measures the sitting deformation of the soil sample. A 5-channel readout unit, which has a computer interface, was used for recording ε trans , ε axial and the applied load, P. Using these data, the stress-deformation characteristics of the samples were established.

Fig. 1
figure 1

Details of the sample cage

Full size image
Fig. 2
figure 2

Details of the positioning of LVDTs

Full size image

2.3 Shear and Compression Wave velocity Measurement

The piezoceramic elements, used in the present study were developed using a Lead Zirconate Titanate (LZT) based material, SP-5A, corresponding to the US DOD (Department of Defense) Navy type material-II. The reason behind choosing this grade of the piezoceramic material is its high dielectric constant, with high piezoelectric sensitivity, which makes it an ideal material for low power applications. Moreover, this grade of the material exhibits excellent time stability as well and the “time lag” between the voltage application and wave generation is quite low. Different properties of this piezoceramic element are listed in Table 3.

Table 3 Properties of the piezoceramic elements used
Full size table

Two sets of piezoceramic elements (15 × 12 × 0.65 mm) as depicted in Fig. 3 were fabricated. One set was polarized in same direction while the other set is polarized in the opposite direction. The centre and the two outer electrodes (made of deposited silver) were soldered to stranded wires. This enables the piezoceramic elements to act as series- or parallel-type, respectively, as depicted in Fig. 4. The solder composition is 62% tin, 36% lead and 2% silver and the soldering time was kept as short as possible to avoid any depoling of the piezoceramic elements. It can be mentioned here that a piezoceramic element can act as a bender or extender depending upon the wiring configuration (series/parallel) and direction of polarization (same/opposite) of the bimorphs (Lings and Greening 2001). Eq. 5 (Dyvik and Madhus 1985; Leong et al. 2005) was then employed for estimating the maximum free lateral deflection, ∆h, of the piezoceramic elements for an excitation voltage V (=20V). ∆h for these piezoceramic elements was numerically found to be about 1 μm, which induces a shear strain <0.001%.

$$ \Updelta h = 3d \cdot V \cdot \left( \frac{l}{h} \right)^{2} \cdot \left( {1 + \frac{{t_{1} }}{h}} \right) $$
(5)

where, d is the piezoelectric charge constant, t 1 is the thickness of the central electrode, V is the applied voltage, h is the thickness and l is the free length of the piezoceramic material.

Fig. 3
figure 3

Details of the piezoceramic elements used in present study

Full size image
Fig. 4
figure 4

Different configurations of piezoceramic elements benders (a and b) and extenders (c and d)

Full size image

The transmitter was then excited with a single sine-wave of certain amplitude and frequency, f, which is generated from a function generator developed by Bartake et al. (2008). The receiver is connected to a filter, amplifier circuitry, which is then connected to a digital oscilloscope. The oscilloscope also receives a direct sine-wave from the function generator. Sine-wave recorded by the oscilloscope was then processed to determine the time lag, t, between input and output waves. Later, V s and V p were computed by dividing the tip-to-tip distance of the bender elements (transmitter and receiver) with t. For calibration purpose, V s and V p were measured on some standard materials. Using Eq. (4), ν was computed for rubber, stainless steel, and cork and was found to be 0.5, 0.29 and 0, respectively, which match very well with the results reported in the literature (Tarantino et al. 2005; Venkatramaiah 2006; Gercek 2007).

3 Results and Discussions

Load-deformation charactersitics for the sand samples were obtained corresponding to three trials on identical samples as depicted in Fig. 5a for sample SS1. For these tests the dry density was found to be 1.5 ± 0.1 g/cc. It can be noted from the figure that, beyond certain initial nonlinear deformation, the deformations vary linearly with the applied stress. As such, the linear relationships represented by AB and CD were considered for determining ν expt (ν, obtained from the load-deformation characteristics). Corrections OC and OA were applied to ε trans and ε axial , respectively, for precise value of the ν expt . The value of ν expt , for different coarse-grained soil samples is listed in Table 4. Figure 5b depicts typical load-deformation curves, for clays. In order to determine ν expt , tangents AB and AD were drawn to the axial and transverse responses of the load-deformation charactersitics, respectively. The value of ν expt for the clay samples are listed in Table 2.

Fig. 5
figure 5

Typical load-deformation characteristics for (a) sample SS1 and (b) clayey samples

Full size image
Table 4 Experimental results for sand samples
Full size table

If ε 1, ε 2 and ε 3 are the strains computed by using the deformations recorded by the three lateral LVDTs, then the resultant strain in transverse direction, ε trans , can be obtained as follows (Gere and Timoshenko 1987):

$$ \varepsilon_{\text{trans}} = \frac{{\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} }}{3} + \frac{\sqrt 2 }{3} \cdot \root{2} \of {{(\varepsilon_{1} - \varepsilon_{2} )^{2} + (\varepsilon_{2} - \varepsilon_{3} )^{2} + (\varepsilon_{3} - \varepsilon_{1} )^{2} }}. $$
(6)

The corresponding stresses are computed as, P/A c, where P is the axial load applied on the sample and A c is the corrected area, computed as:

$$ Ac = \, \pi \cdot\frac{{(D_{c} )^{2} }}{4} $$
(7)

where, D c is the corrected diameter, after successive deformation in the sample, computed as follows:

$$ D_{\text{c}} = D_{\text{o}} + \frac{2}{3}\left( {\delta_{1}+\,\delta_{2}+ \,\delta_{3} } \right) $$
(8)

where, D o is the initial diameter of the sample and δ1 δ2 and δ3 are the transverse deformations of the soil sample as recorded by the three LVDTs.

The elastic modulus, E expt , of the sample was obtained by determining the slope of the linear portion of the load-deformation (longitudinal strain) characterisitcs. Here, it is worth mentioning that when tested at different strain rates, the linear portions of the load-deformation characterisitcs remain almost same, as depicted in Fig. 6.

Fig. 6
figure 6

Deviatric stress versus axial strain characteristics of clays corresponding to different strain rates

Full size image

For a material in an infinite, linear, elastic, isotropic and homogeneous continuum and for compression mode (plane wave), the following relationship is valid (Santamarina et al. 2001):

$$ \frac{{\delta^{2} u_{x} }}{{\delta t^{2} }} = \frac{M}{\rho } \cdot \frac{{\delta^{2} u_{x} }}{{\delta x^{2} }} (P\hbox{-} wave\,equation) $$
(9)

where, u x is the particle motion in x-direction and M is the constraint modulus. Similarly, for the shear mode (plane wave) the following relationship is valid:

$$ \frac{{\delta^{2} u_{y} }}{{\delta t^{2} }} = \frac{G}{\rho } \cdot \frac{{\delta^{2} u_{y} }}{{\delta x^{2} }} (S\hbox{-}wave\,equation) $$
(10)

where, u y is the particle motion in y-direction.

Substituting, \( u_{x} = Ae^{j(\omega t - \kappa \chi )} \) and \( u_{y} = Ae^{j(\omega t - \kappa \chi )} \) in Eqs. (9) and (10), the following can be obtained: \( \omega /\kappa = \sqrt {(M/\rho )} {\text{and }}\omega /\kappa = \sqrt {(G/\rho )} \) for P- and S- waves, respectively.

where, t is the time, A is the maximum amplitude of the motion, \( \omega = 2\pi /T \) is the temporal angular frequency, k = 2π/λ is the spatial frequency or wave number, T is the time period and λ is the wave length.

However, \( \omega /\kappa = \lambda /T \) which represents the wave velocity. Hence,

$$ V_{p} = \sqrt {(M/\rho )} $$
(11)
$$ V_{s} = \sqrt {(G/\rho )} $$
(12)

From Eqs. (11) and (12), the following can be derived:

$$ (V_{p} /V_{s} )^{2} = M/G $$
(13)

For an isotropic linear-elastic continuum:

$$ {\text{M}} = \frac{{E(1 - {{\upnu}})}}{{\left( {1 + {{\upnu}}} \right) \cdot (1 - 2{{\upnu}})}}. $$
(14)

Equation (4) can be obtained by substituting the values of M and G from Eqs. (14) and (2), respectively, into Eq. (13). Further, experimentally obtained Poisson’s ratio for different samples, ν expt , were compared to those, obtained by using Eq. 4, ν Eq. 4 , as depicted in Fig. 7. It can be observed from the figure that the data fit within 95% prediction limits. Moreover, it can be observed that as expected, ν expt is dependent on the water content of the soil mass, w, and it increases almost linearly with w, as depicted in Fig. 8. However, when ν expt is plotted against plasticity index, PI, a cloud of data is observed, as depicted in Fig. 9. A poor correaltion between these parameters is due to the fact that, unlike w, the PI, being an index property of the soil, will not be able to represent its elastic properties (viz., ν).

Fig. 7
figure 7

The comparison of Poisson’s ratios obtained from experiments and Eq. 4

Full size image
Fig. 8
figure 8

The variation of Poisson’s ratio with water content

Full size image
Fig. 9
figure 9

The variation of Poisson’s ratio with plasticity index

Full size image

Further, as depicted in Fig. 10, the elastic modulus of the samples obtained from the triaxial tests, E expt , is compared to that obtained by substituting G and ν from Eqs. (3) and (4), respectively, in Eq. 2 (defined as E Eq. 2). It can be noted that E Eq. 2 is 7.5 times higher than the E expt . This is consistent with the fact that the piezoceramic elements yield low-strains (<0.001%) in the soil mass (Dyvik and Madhus 1985; Leong et al. 2005) and hence higher elastic modulus.

Fig. 10
figure 10

Comparison between the estimated and experimental values of elastic modulus

Full size image

4 Conclusions

Investigations has been conducted on soils of different types (clays and sands) in their disturbed and undisturbed forms by resorting to piezoceramic tests and conventional triaxial tests (i.e., strain controlled uniaxial compression tests). Details of the methodology are presented in this paper and it has been demonstrated that application of piezoceramic elements yields the Poisson’s ratio and the elastic modulus of the soils quite easily, particularly for the soft clays and sands. Poisson’s ratio obtained from the triaxial testing was found to be dependent on the water content of the soil mass, w, and it increases almost linearly with w. The elastic modulus, obtained from the wave velocities was found to be 7.5 times more than that obtained from the triaxial testing.