1 Introduction

Damping in soil is of vital importance for the dynamic response analyses of soil layers. Hardin and Drnevich (1972a) defined the soil damping ratio via stress–strain hysteresis loop and discussed the parameter effects in soil damping. Stokoe et al. (1999) future analyzed the effects of different ground shaking parameters, material properties and soil state on soil damping ratio, especially on small-strain damping ratio. Also, Stokoe initiated the comparison of damping measurement methods and pointed out a critical value of shear strain amplitude (0.002%) for choosing the resonant column test method or free vibration test method. Nonetheless, in order to consider the region difference of soil and the specific critical value about SHANTOU remold saturated soft clay, this study applied GZZ-10 resonant column apparatus for small-strain damping measurement (10−4% < γ < 10−1%) by the methods of both resonant column test and free vibration test. Furthermore, differences of these two methods were discussed around experimental data and measurement principles.

Based on numerous laboratory tests and analysis of elliptic stress–strain hysteresis loops, Hardin and Drnevich (1972b) first modeled the damping curve with shear strain. After that, some researchers continued to developed the Hardin–Drnevich model for damping data fitting (Chen et al. 2007; Huang et al. 2000). However, there were two underlying assumptions of Hardin–Drnevich model that need to be addressed herein: the viscoelastic soil assumption and the elliptic loop assumption. With consideration of the potential abrupt structural failure of soil caused by repeated energy consumption of the small-strain damping, damping at small strains should be taken seriously.

Although according to Hardin and Drnevich (1972a) damping ratio should be equal to zero when the shear strain amplitude is zero, the fitting curve of experimental data herein demonstrated that the non-zero minimum damping ratio needs to exist for zero shear amplitude so as to better fit the small-strain damping (10−4% < γ < 10−1%). Therefore, a modified small-strain damping model was given in a later section, combining Davidenkov model proposed by Martin and Seed (1982) with non-zero minimum damping ratio effectively.

2 Experimental Sample Preparation and Overview of Test Methods

The original clay used as specimens is SHANTOU soft clay, of which the primary physical indexes are listed below in Table 1. In accordance with the rules of the unified soil classification system soil classification method, the SHANTOU soft clay in the tests is classified as CH clay soil (i.e., clay of high plasticity).

Table 1 Primary physical indexes of tested clay
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Then cylinder SHANTOU soft clay samples (D × H = 3.91 cm × 8 cm) were prepared for resonant column test and free vibration test under different confining pressures of 100, 200, 300 and 400 kPa in GZZ-10 resonant column apparatus (0–120 Hz). 17 groups of soil samples were tested, obtaining 317 data points of resonant column test and 55 data points of free vibration. Note. the effect of different confining pressures was eliminated by the consideration of relative shear strains (γ0), which is not study object in this paper.

Generally, resonant column test method and free vibration test method are both for small-strain damping measurement in laboratory. Nevertheless, the principles of the two methods are different. Specially, the resonant column test method measures damping from the dynamic response using half-power bandwidth method (Fig. 1), while the free vibration test method evaluates damping from the dynamic response using free-vibration attenuation curve (Fig. 2).

Fig. 1
figure 1

Damping measurement in the resonant column test using half-power bandwidth method (Hardin and Drnevich 1972a)

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Fig. 2
figure 2

Damping measurement in the free vibration test using free-vibration attenuation curve (Hardin and Drnevich 1972a; Xie 2011)

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On the basis of the width measurement of dynamic response curve around the resonance peak (Fig. 1), the half-power bandwidth method provided one approach to damping ratio as:

$$D=\frac{{f_{2} - f_{1} }}{{2f_{r} }}$$
(1)

where D = damping ratio; f1 and f2 = two frequencies where the amplitude is 0.707 times the amplitude of resonance frequency point as illustrated in Fig. 1; fr = resonant frequency. By instantly shutting off the torsion applied to the static sample, the free-vibration attenuation curve was recorded as Fig. 2 and the damping ratio herein could be presented as:

$${\text{D}} = \left[ {\frac{{\ln^{2} (z_{1} /z_{2} )}}{{4\pi^{2} + \ln^{2} (z_{1} /z_{2} )}}} \right]^{{\frac{1}{2}}}$$
(2)

where z1 and z2 = strain amplitudes of two successive cycles in free-vibration attenuation curve.

3 Comparison Between Resonant Column Test Method and Free Vibration Test Method in Terms of Small-Strain Damping Measurement

The small-strain damping measurements of SHANTOU soft clay were illustrated in Figs. 3 and 4.

Fig. 3
figure 3

Relationship between damping ratio and shear strain by methods of resonant column test

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Fig. 4
figure 4

Relationship between damping ratio and shear strain by methods of free vibration test

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Figure 3 shows the damping measurement at small strains (10−4% < γ < 10−1%) by resonant column test method. Valid range of damping ratio should be consistent with the implied assumption in half-power bandwidth method that the damping ratio should be below the value of 1. However, when the strain amplitude is greater than 0.01% (Fig. 3), the measurement values of certain damping ratios are—out of range—greater than 1, which leads to an exponential attenuation curve (over-damping state). This would obviously result in serious error of damping measurement when applying half-power bandwidth method.

In comparison, the measurement values of damping ratios by the free vibration test method (Fig. 4) not only meet the valid range but also have a relatively concentrated distribution. In this sense, this paper recommends free vibration test as the damping measurement method at small strains (10−4% < γ < 10−1%) for SHANTOU remolded saturated soft clay. Moreover, shear strain amplitude at 0.01% could be regarded as one critical value of SHANTOU remolded saturated soft clay: when strain amplitude is between 10−4 and 10−2%, both methods could be applied to small-strain damping measurement; when the amplitude is greater than 10−2%, only free vibration test is recommended herein for damping measurement at small strains(10−2% < γ < 10−1%).

4 Modified Model for Fitting The Small-Strain Damping of SHANTOU Remolded Saturated Soft Clay

Hardin and Drnevich (1972b) assumed an empirical constant α as Eq. 5 and defined the damping ratio (Eq. 6) according to elliptic stress–strain hysteresis loop. While Davidenkov model improves fitting effects to some extend by providing three fitting parameters, damping evaluation still needs to rely on the constant assumption (Eq. 5), which postulates the damping ratio D is equal to zero for strain amplitude at zero (Eq. 6). But this assumption cannot meet the need of fitting experimental data. Some researchers have challenged the zero damping at zero strain amplitude by introducing the non-zero minimum damping ratio Dmin in order to obtain acceptable fit (Nie 2008; Zhang 2008).

$$\alpha = \frac{{\pi G_{\hbox{max} } D}}{{2(G_{\hbox{max} } - G)}}$$
(3)
$$D = D_{\hbox{max} } \left( {1 - \frac{G}{{G_{\hbox{max} } }}} \right)$$
(4)

where Dmax = the maximum damping ration (Dmax = 2α/π); Gmax = initial (maximum) dynamic shear modulus.

In this paper, based on the laboratory tests of small-strain damping of SHANTOU remolded saturated soft clay through free vibration test method, a modified Davidenkov damping model with the consideration of non-zero minimum damping ratio is described as:

$$D = D_{\hbox{min} }+\left( {D_{\hbox{max} } - D_{\hbox{min} } } \right)\left[ {\frac{{\left( {\frac{\gamma }{{\gamma_{0} }}} \right)^{2B} }}{{1 + \left( {\frac{\gamma }{{\gamma_{0} }}} \right)^{2B} }}} \right]^{A}$$
(5)

where Dmax = 0.26623, Dmin = 0.04057, A = 0.45196, B = 1.0268, γ0 = 0.0123791% for small-strain damping of SHANTOU remolded saturated soft clay.

5 Conclusions

  1. 1.

    The free vibration test method is more favorable than resonant test method for damping measurement at small strains (10−4% < γ < 10−1%), especially for the small- strain damping at the strain greater than 10−2% (the critical value of strain amplitude for the damping analysis of SHANTOU remolded saturated soft clay).

  2. 2.

    The modified Davidenkov damping model was given by introducing the non-zero minimum damping ratio and the three parameters were obtained for SHANTOU remolded saturated soft clay as:

$$D = 0.04057{ + }0.22566\left[ {\frac{{\left( {\frac{\gamma }{0.0123791\% }} \right)^{2.0536} }}{{1 + \left( {\frac{\gamma }{0.0123791\% }} \right)^{2.0536} }}} \right]^{0.45196}$$
(6)