Energy absorption characteristics of cement-soil under confining pressure based on SHPB test

2.1. Effectiveness analysis of test data

The Ø50 mm split Hopkinson pressure bar (SHPB) apparatus is adopted in the present work, and due to relatively low wave impedance of cement-soil, the semiconductor strain gauge is pasted on the transmitted bar to record the weak signal of transmitted wave. Confining pressure device is shown in Fig. 1.

Fig. 1Confining pressure device

In order to clarify the effectiveness of test data, the propagation theory of one-dimensional wave and uniform stress hypothesis need to be satisfied. To solve the problem of uniform stress in SHPB tests, some measures were taken and original waveform collected in SHPB test is shown in Fig. 2. As clear in Fig. 2, the half-sine wave pattern loading is achieved and the waveform is smooth. It also can be found that the time of the rising edge of incident pulse is about 0.14 ms, and the time of duration is about 0.4 ms. So, the number of transmission-reflection cycles ($c_{t-f}$) for stress pulse in cement-soil specimen can be calculated by the following Eq. (1) as:

where $t_r$ is the time of the rising edge of incident pulse (about 0.14 ms), $t_{t-f}$ is the time of one transmission-reflection cycle for stress pulse in cement-soil specimen.

And, $t_{t-f}$ can be expressed as:

where $h$ is the length of specimen (0.025 m), $v$ is the propagation velocity of stress pulse in cement-soil (about 1450 m/s), so the $t_{t-f}$ is 0.0345 ms.

Finally, $c_{t-f}$ can be obtained by calculation as:

Thus, there is enough time to achieve uniform stress state for cement-soil specimen.

Fig. 2Original waveform (Cp= 2.0 MPa, Ip= 0.3 MPa). Cp – confining pressure; Ip – impact loading pressure (below)

The theory of one-dimensional wave requires $\lambda =ct>5d$[21], where $\lambda$ is the width of incident stress pules, $c$ is P-wave velocity (5190 m/s), $t$ is the propagation time of stress pulse (0.04 ms), $d$ is the diameter of specimen (0.05 m). It can be obtained by calculation as:

Therefore, the theory of one-dimensional wave can be clarified.

2.2. Materials

The silty clay tested in this investigation was obtained from foundation and its physical properties and grain size distribution are shown in Table 1 and Table 2, respectively. Ordinary Portland cement (P·O 42.5) was used as the binding agent in this work. Distilled water was used for specimen preparation.

2.3. Methods of energy calculation

Absorbed energy ($W_S$), which is the capacity to absorb energy of stress wave for cement-soil per unit volume, can be expressed as follow:

where $W_I$, $W_R$ and $W_T$ arethe energy of incident wave, reflected wave and transmitted wave, respectively, they can be calculated through Eq. (6):

where $E$, $C$, and $A$ are the Young’s modulus, P-wave velocity and the cross-sectional area, respectively; ${\epsilon }_I\left(t\right)$, ${\epsilon }_R\left(t\right)$ and ${\epsilon }_T\left(t\right)$ are the strain of incident wave, reflected wave, and transmitted wave, respectively; $t$ is the duration time of elastic wave.

3.1. Relationship between the absorbed energy and the incident energy

Through Eq. (5) and (6), the representative energy-time curve of cement-soil is obtained and shown in Fig. 3. In the initial loading, the incident energy increase rapidly with the propagation of stress wave, but the absorbed energy of cement-soil has a slight increase with the extension of stress wave. That is because most of incident energy is reflected back into incident bar, some is absorbed by cement-soil specimen, and only a small percentage of incident energy is transferred to transmitted bar [22]. It also can be seen that the time duration of the stress wave is about 0.35 ms, and the values of incident energy and absorbed energy maintain stable after reaching a certain time about 0.275 ms, which means the failure of cement-soil specimen.

Fig. 4 shows the relationship between the incident energy and the absorbed energy of cement-soil in SHPB test. From Fig. 4, the absorbed energy of cement-soil under different confining pressure increase with the increase of incident energy, and there is a good liner relationship between the incident energy and absorbed energy, which can be expressed as Eq. (7):

where $W_s$ and $W_I$ are the absorbed energy and incident energy, respectively; $a$ and $b$ are the parameters of fitting equation. The values of $a$, $b$ and $R^2$ are listed in Table 3 (for the range of incident energy is from 50 to 120 J).

Fig. 3Energy-time curve of the cement-soil (Cp= 2.0 MPa, Ip= 0.3 MPa)

Fig. 4Relationship between the incident energy and the absorbed energy

3.2. Absorbed energy of cement-soil

The absorbed energy under different confining pressure have been calculated and the relationship between the absorbed energy and confining pressure is shown in Fig. 5. It can be observed that absorbed energy is sensitive to the value of confining pressure, and the relationship between absorbed energy and confining pressure is also influenced by the change of strain rate.

As clear in Fig. 5, the absorbed energy increases firstly and then remains stable at a certain value with the increase of the confining pressure, which the turning point is 1.0 MPa of the confining pressure. For example, at the strain rate of 152 s^-1, the absorbed energy increase from 4.51 to 9.83 J with the confining pressures increasing from 0 to 1.0 MPa, after that, there is a relatively steady trend for the value of absorbed energy when the confining pressure is more than 1.0 MPa. In addition, the cement-soil specimens have fairly great improvement in the value of absorbed energy under confining pressure compared with that of uniaxial condition ($C_p=$0 MPa). It can be observed that the absorbed energy increase from 5.56 to 12.87 J with the confining pressures increasing from 0 to 1.0 MPa at strain rate of 160 s^-1, which causes an increase of 131.47 %.

The scatter diagram between absorbed energy and strain rate is shown in Fig. 6. All of groups show a similar trend that with the increase of strain rate, absorbed energy of cement-soil specimen increases. After data fitting, the relationship between the absorbed energy and strain rate can be assumed to be liner, and which can be expressed as Eq. (8):

where $W_s$ is the absorbed energy of cement-soil specimen, $\alpha$ and $\beta$ are the parameters of fitting equation. The values of $\alpha$, $\beta$ and $R^2$ are listed in Table 4 (for the range of strain rate is from 130 to 180 s^-1). Most of the values of $R^2$of fitting curves are more than 0.90, that is to say, Eq. (8) can well be described the strain rate sensitivity of the absorbed energy of cement-soil.

Fig. 5Absorbed energy versus confining pressure

Fig. 6Absorbed energy versus strain rate

3.3. Energy absorbency rate of cement-soil

In order to measure the absorption efficiency of cement-soil under confining pressure, the energy absorbency rate $\eta$ can be defined by Eq. (9):

where $W_s$ and $W_I$ are the absorbed energy and the incident energy, respectively.

Fig. 7 shows the comparison of the energy absorbency rate of cement-soil with different confining pressure and strain rate. It is can be seen that the confining pressure has an effect on the average value of energy absorbency rate. From Fig. 7(a), compared with uniaxial condition, the energy absorbency rate of cement-soil specimen has a significantly increase under confining pressure, but it shows no remarkable change when the value of confining pressure is from 1.0 to 2.0 MPa, which settles around a range of values about 12.0 %-14.0 %. By examining Fig. 7(b), there is no obvious regularity between the energy absorbency rate and strain rate.

Fig. 7Relationship among the confining pressure, strain rate and the energy absorbency rate

a)

b)

3.4. Discussion of energy mechanism for cement-soil

Test results both in this paper and of other studies [23-25] conclude that both the absorbed energy and the energy absorbency rate of the materials are influenced by the confining pressure. As for cement-soil tested in this work, its absorbed energy increases firstly and then decreases with the increase of the value of confining pressure, and so does the energy absorbency rate. The effect mechanism of confining pressure on energy absorption characteristics of cement-soil can be expressed as follow:

The critical state confining pressure, which is varies for different kinds of material, has been proposed in the previous literatures [26, 27]. There has a difference in the state of low confining pressure and high confining pressure for rock and soil as well as cement-soil. Specifically, both the absorbed energy and the energy absorbency rate of cement-soil increase with the increase of confining pressure when the values of confining pressure is lower than the critical state confining pressure, while the confining pressure show a negative effect on energy absorption when it is over the critical state confining pressure. In this paper, for cement-soil under confining pressure, a value of critical state confining pressure, about 1.0 MPa, is founded. Both the absorbed energy and the energy absorbency rate increase with the increase of confining pressure from 0 to 1.0 MPa, and then remain stable at a certain value with further increase of confining pressure. Such a similar result was presented by Jin [28]. Analyzing the causes, the counterforce during the lateral deformation of specimen under impact load increases with the value of confining pressure increasing, and then it is difficult for cracks inside of specimen to extend so that the absorbed energy of specimen keeps steady gradually [29]. On a macro level, the result of this behavior shows a smaller energy absorbency rate for cement-soil.

Compared with the uniaxial condition, the values of absorbed energy and the energy absorbency rate of cement-soil under confining pressure have a significant increase. Due to the energy absorption characteristics are closely related to the failure mode of specimen, so it can be explained through Fig. 8 and Fig. 9 that show the failure modes of cement-soil under different stress state. The cement-soil specimen under uniaxial state is broken into fragments, however, a better integrity for specimen is maintained under confining pressure state. Previous research showed that the initiation and propagation of fracture crack and deformation of specimen are limited under confining pressure state [30], so if the cement-soil specimen want to reach its failure state, more energy should be absorbed. That is to say, the confining pressure strengthens the capacity of energy absorption for cement-soil [31]. Thus, the absorbed energy and the energy absorbency rate increase naturally.

Fig. 8Failure modes of cement-soil under confining pressure

a)$C_p=$ 0.5 MPa, strain rate 130 s^-1

b)$C_p=$ 1.5 MPa, strain rate 152 s^-1

Fig. 9Failure modes of cement-soil under uniaxial condition

a) strain rate 152 s^-1

b) strain rate 160 s^-1

c) strain rate 172 s^-1

d) strain rate 180 s^-1

Many researches [17, 32] including this work show that the absorbed energy of cement-soil specimen is also related to the strain rate. In other words, the higher the strain rate, the more energy is absorbed for specimen. The strain rate effect on the absorbed energy of cement-soil is simply summarized in this paper.

In general, the value of strain rate has a positive correlation with impact loading pressure, so the higher the strain rate, the higher the impact loading pressure, which also means that incident energy increases gradually. Combining with the analysis in Section 3.1 and it can be found that the more energy is absorbed by cement-soil specimen with the increase of incident energy.

Mechanism of the strain rate effects on the energy absorption of cement-soil can also be explained through the failure modes of cement-soil specimen. As shown in Fig. 9, with the increasing of strain rate, the grain size of fragments becomes smaller and the number of fragments increases, which means the higher the damage degree of cement-soil specimen. From the perspective of micro-cracks and energy absorption of materials [33], the expansion of internal micro-crack and the generation of new fracture surface of cement-soil is a process of absorbing energy for cement-soil. The more serious the damage degree of cement-soil specimen is, the more energy of specimen is absorbed.