Experimental study on dynamic deformation properties of muck soil under low frequency cyclic loading

2.1. Tested soils

The studied soils were taken from a construction site in Shenzhen which was located in the offshore area of the alluvial plain of the Pearl River estuary. The sampling depth was 14 meters below sea level. The undisturbed muck soils were obtained by the professional thin-walled earth borrowers. The physical and mechanical properties are as follows: the natural density $\rho$ is 1.63 g/cm³, the water content $w$ is 65.2 % (The partial samples water content exceed 70 %), the relative density of soil particles $d_s$ is 2.69, the void ratio $e$ is 1.72, the plasticity index $I_P$ is 20.9, the liquidity index $I_L$ is 1.06, compression coefficient $a$ is 1.43 MPa^-1, the cohesion $c$ is 7.0 kPa, internal friction angle $\theta$ is 2.1°.

2.2. Experimental instrument and scheme

2.2.1. Experimental instrument

The dynamic triaxial experiment was performed with the SPAX-2000 testing system produced by GCTS of USA, shown in Fig. 1. The system consists of seven parts, including pressure control panel, pressure chamber, confining pressure/volume computer servo controller, back pressure/pore pressure volume computer servo controller, digital servo controller and acquisition system, variant constant pressure oil pump, control and software.

Fig. 1SPAX-2000 true triaxial testing system

3.1. Influence of loading frequency on dynamic deformation

(1) Characteristics of stress-strain hysteresis curves.

The stress-strain hysteresis curves contain a lot of information which could reflect kinetic parameters such as energy loss, dynamic elastic modulus, and damping ratio. The typical stress-strain hysteresis curves are shown in Fig. 2. Each stress-strain hysteresis loop is divided into two phases-loading and unloading. The length of the $x$-axis represents the plastic strain generates during the loading and unloading phase.

Fig. 2Typical hysteresis curves under different frequency loading: (Dynamic stress amplitude σd= 65 kPa, effective stress σ1'=σ2'= 250 kPa, σ3'= 200 kPa)

As shown in Fig. 2, the development patterns of muck soil hysteresis curves are similar under the same consolidation pressure, cyclic stress amplitude and different loading frequency cyclic loading. The initial stress-strain hysteresis curve of the muck under the low-frequency cyclic loading developed rapidly. Its curve shape change from sparse to tight and then change to slightly sparse. The higher the frequency developed, the earlier the tight phase would be and the hysteresis loop would be closer. Results show that the shorter the time of the load under the high frequency loading, and the plastic strain development and the loss of energy were limited. Therefore, the area of a single hysteresis loop with a high frequency was relatively small and the degree of closure was higher than the low frequency condition. The energy of the loss increased as the stress amplitude increased, so that the area of the hysteresis loop and the length of the open section increased with the increasing of the stress amplitude under the same cycle frequency and number of times. When the consolidation stress was large, the arrangement of the particles was tight. The hysteresis loop entered the compact phase early, which means the soil entered the elastic phase earlier.

(2) Characteristics of cumulative plastic strain.

Fig. 3(a) and (b) present the relationship between cumulative plastic strain ${\epsilon }_d$ and number of cycles $N$ under the isotropic consolidation, different frequency and the same cyclic stress amplitude (${\sigma }_d=$ 20 kPa and 50 kPa). The typical curves of ${\epsilon }_d$-$N$ under the anisobaric consolidation are shown in Fig. 3(c).

As shown in Fig. 3, the following three characteristics are found from ${\epsilon }_d$-$N$ curves.

Fig. 3Curves of εd-N under different frequency loading

a)${\sigma }_1^{\text{'}}={\sigma }_2^{\text{'}}={\sigma }_3^{\text{'}}=$ 200 kPa, ${\sigma }_d=$ 20 kPa

b)${\sigma }_1^{\text{'}}={\sigma }_2^{\text{'}}={\sigma }_3^{\text{'}}=$ 200 kPa, ${\sigma }_d=$ 50 kPa

c)${\sigma }_1^{\text{'}}={\sigma }_2^{\text{'}}=$ 250 kPa, ${\sigma }_3^{\text{'}}=$ 200 kPa, ${\sigma }_d=$ 65 kPa

1) The characteristic of plastic strain: the cumulative plastic strain and plastic strain growth rate decrease as loading frequency increases under isotropic consolidation or anisotropic consolidation with high cyclic stress or low cyclic stress condition. As shown in the Fig. 3(a), the cumulative plastic strain corresponding to the vibration frequency of 0.5 Hz was reduced by 77 % compared with 0.1 Hz; the cumulative plastic strain corresponding to 1 Hz was reduced by 23 % compared with 0.5 Hz under the cyclic load amplitude of 20 kPa. Fig. 3 indicates that the longer the time acting on the soil at low loading frequency, the greater the energy absorbed by the soil skeleton, and the greater the plastic strain caused.

2) The mode of plastic strain: The plastic strain developed at a higher growth rate when the cumulative plastic strain with a vibration frequency of 0.1 Hz belonged to the development type, while 0.5 Hz and 1 Hz were stable type. As shown in Fig. 3(b), both 0.1 Hz and 0.25 Hz are both destructive under cyclic stress of 50 kPa, and the cumulative plastic strain developed leaping in the later stage of the cycle. The growth rate increased sharply until the soil was destroyed, but when the loading frequency raise up to 0.5 Hz, the plastic strain developed stably with a small amount of strain.

3) Frequency threshold: between 0.25 Hz and 0.5 Hz, there was a frequency threshold, which made the development mode of cumulative plastic strain developed from stability to failure. As shown in Fig. 3(c), there was also a threshold frequency for the deviator consolidation. Once the loading frequency was lower than the threshold frequency, the cumulative plastic strain development mode would be abrupt.

(3) Characteristics of dynamic modulus.

Fig. 4 presents the relationship between the dynamic elastic modulus $E_d$ and cumulative plastic strain ${\epsilon }_d$ under the same cyclic stress, but different frequencies, the isotropic consolidation (stress ${\sigma }_1^{\mathrm{\text{'}}}={\sigma }_2^{\mathrm{\text{'}}}={\sigma }_3^{\mathrm{\text{'}}}=$ 200 kPa). The dynamic elastic modulus decays rapidly at the beginning, slowing down at the later stage, and gradually stabilizing.

Fig. 4Curves of Ed-εd under different frequency loading (σ1'=σ2'=σ3'= 200 kPa, σd= 20 kPa)

3.2. Influence of stress amplitude on dynamic deformation

(1) Characteristics of cumulative plastic strain.

Fig. 5 presents the relationship between cumulative plastic strain ${\epsilon }_d$ and the number of cycles $N$ under the isotropic consolidation and the same frequency. As shown in Fig. 5, the cumulative plastic strain and the plastic strain rate increases cyclic stress increases during the loading frequency of 0.1 Hz to 1 Hz. The cumulative plastic strain increased by 75.88 % as the cyclic stress amplitude increased by 150 % (from 20 kPa to 50 kPa), and the cumulative plastic strain increased by 116 % as the cyclic stress amplitude increased by 30 % (from 50 kPa to 65 kPa) at the same vibration frequency. Results showed that the cumulative plastic strain of the muck soil increased nonlinearly with the dynamic stress and there was a critical dynamic cyclic stress. The strain increased sharply and the soil structure would be destroyed if the dynamic stress amplitude reached the critical value.

Fig. 5Curves of εd-N under different stress amplitude (σ1'=σ2'=σ3'= 200 kPa, f= 1 Hz)

(2) Characteristics of dynamic modulus.

As showed in Fig. 6, the dynamic modulus-plastic strain curves under different cyclic stress amplitudes have similar trend. With the increasing of plastic strain, the dynamic elastic modulus reduced rapidly at the initial stage of loading, and linearly decreased greatly with the cumulative plastic strain. As the amplitude of the stress increased, the dynamic elastic modulus reduced with an increasing trend. As the magnitude of the stress increased, the energy of the impact would increase, the more energy was absorbed by the soil specimen, the easier the soil skeleton was destroyed, and the faster the dynamic elastic modulus reduced. The difference of the dynamic elastic modulus caused by the stress amplitude was small at the frequency of 0.1 Hz, while the effect of the dynamic stress amplitude on the dynamic elastic mode was more obvious when the frequency was greater than 1 Hz.

Fig. 6Curves of Ed-εd under different loading amplitude (σ1'=σ2'= 250 kPa, σ3'= 200 kPa, f= 0.5 Hz)

3.3. Influence of solid pressure on dynamic deformation

(1) Characteristics of cumulative plastic strain.

Fig.7 presents the relationship between the cumulative plastic strain ${\epsilon }_d$ and the number of cycles $N$ under different intermediate principal stresses for the same loading frequency and stress amplitude.

Fig. 7Curves of εd-N (f= 0.1 Hz, σd= 20 kPa)

As shown in Fig. 7, the larger the intermediate principal stress developed, the smaller the final value of the cumulative plastic strain at the same cyclic stress and loading frequency would be. This was because the soil structure becomes tight and the stiffness of the muck increased as the consolidation stress increased, the ability to resist elastic deformation become strong. Results showed that under the low-frequency cyclic loading (0.1 Hz-1 Hz), the cyclic stress had no obvious plastic cumulative deformation of the muck, and the frequency influence was more significant.

(2) Characteristics of dynamic modulus.

Fig. 8 presents the relationship between dynamic elastic modulus $E_d$ and the cumulative plastic strain ${\epsilon }_d$ under different medium principal stresses. As shown in Fig. 8, the trend of the dynamic elastic modulus of the muck decreased with the increasing of the plastic strain was similar. The curve was divided into three stages: rapid decline stage (${\epsilon }_d$ ≤ 2 %), slow decline stage (2 % < ${\epsilon }_d$ < 3 %), and gradual stabilization stage (${\epsilon }_d$ > 3 %). The dynamic elastic modulus increased as the consolidation stress under certain plastic strain condition. In addition, the dynamic elastic modulus decreased more slowly in the isotopically consolidated soil than the deviator consolidated soil under the same consolidation pressure. It showed that the consolidation stress was an important factor affecting the dynamic elastic modulus.

Fig. 8The curves of Ed-εd (f= 0.5 Hz, dynamic stress σd= 65 kPa)