J. Cent. South Univ. (2021) 28: 2514-2526

DOI: https://doi.org/10.1007/s11771-021-4783-x

Primary and secondary consolidation compression for saturated soil considering coupling effect of loading and heating

DENG Yue-bao(邓岳保)¹, MAO Wei-yun(毛伟赟)¹, KONG Gang-qiang(孔纲强)², HAN Yi-dong(韩逸冬)¹

1. Institute of Geotechnical Engineering, Ningbo University, Ningbo 315211, China;

2. Key Laboratory of Geomechanics and Embankment Engineering (Ministry of Education),Hohai University, Nanjing 210098, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: Geotechnical engineering that relates to the energy and environmental problem is receiving more and more attention worldwide. It is of great theoretical and practical value to study the properties of soil under thermal mechanical coupling and its mathematical description. Firstly, based on the general function, a unified primary and secondary consolidation model of saturated soil considering heating temperature is deduced. Combining the existing research achievements, a practical model is obtained which comprehensively reflects the effective stress change, creep and heating effects. After that, a series of thermo-consolidation tests are carried out using a temperature controlled consolidation instrument to study the effects of effective stress, temperature and consolidation duration on saturated soils. The corresponding functional formulas and parameters are obtained thusly. On this basis, the calculation and analysis are carried out to check the reliability and applicability of the newly proposed model. The new model is simple and practical and the parameters are easy to be obtained. And it describes the main law of consolidation compression of saturated soils under the thermal mechanical coupling effect. Therefore, it is suggested for theoretical analysis of thermal geotechnical engineering problems.

Key words: soft soil; primary consolidation; secondary consolidation; temperature effect; consolidation compression test

Cite this article as: DENG Yue-bao, MAO Wei-yun, KONG Gang-qiang, HAN Yi-dong. Primary and secondary consolidation compression for saturated soil considering coupling effect of loading and heating [J]. Journal of Central South University, 2021, 28(8): 2514-2526. DOI: https://doi.org/10.1007/s11771-021-4783-x.

1 Introduction

The phenomenon of deformation with time for saturated soil under load can be described by primary consolidation and secondary consolidation in soil mechanics. The primary consolidation settlement is the deformation caused by the increasing of effective stress, while the secondary consolidation settlement is the deformation that occurs when the effective stress is constant. In 1920s, TERZAGHI et al [1] assumed a simple linear relationship between the change of pore ratio and the effective stress, and described the consolidation compression of saturated soil for the first time. In 1940 s, TAYLOR and MERCHANT found that the secondary consolidation stage after the completion of the primary consolidation of soft soil, and they used a constant secondary consolidation coefficient C_α to describe the size of secondary consolidation. This finding considered an important part of classical soil mechanics. In 1977,MESRI [2] found that there is a certain correlation between the compression characteristics of primary and secondary consolidation for a certain soil of which ratio C_α/C_c is a constant value, which promotes the application of secondary consolidation theory in practical engineering. As the research progresses, the researchers gradually realized that the soil secondary consolidation showed some nonlinearity, and it could be affected by consolidation time, vertical pressure and temperature and many other factors [2-5].

The effect of temperature change on soil or soft ground is negligible for most geotechnical engineering issues. However, in energy and environmental geotechnical problems such as nuclear waste burial, geothermal resource development, energy underground structure, oil and gas pipeline, heating pipeline and landfill, the influence of temperature change on soil layer of site cannot be ignored [6-10]. Additionally, the raise and discussion on the technology of heat drainage consolidation foundation treatment also promotes the study of saturated soil temperature effect by scholars [11, 12]. Under this background, the study on soil physical and mechanical properties under load temperature coupling has great theoretical and practical value.

Actually, a symposium on the impact of thermal effects on soil properties was held in Washington D.C. in the 1960s. After that the scholars paid more and more attention on the research of soil’s temperature effect. Over 50 years, different scholars have carried out a series of studies on the temperature effect of soil, such as the effect of temperature changing on the physical and mechanical properties of soil [7, 13-16] and the study on constitutive model considering the coupled effect of load and temperature. Among these studies, HUECKEL and BORSETTO [16] improved Cambridge Model by thermal evolution of yield limit under constant plastic strain, and developed the first thermal constitutive model of soil. Since then, scholars from worldwide have made a series of improvements [17-21]. Based on the theory of particle fluid dynamics and the concept of entropy, CHENG et al [22-24] established the thermodynamic model of Tsinghua University, considering the multiphase and complex stress conditions of saturated soil. SONG et al [25] studied the influences of thermal gradient on the fluid flux (thermo-osmosis effect) in the thermo-hydro-mechanical analysis of soil, and found that the thermo-osmosis effect will cause negative pore pressure and increase the effective stress in porous medium. It should be noted that, the above thermal constitutive model which is based on the model of adjacent state and the theory of thermo-dynamics is theoretically complete, but it is very complicated.

Consolidation compression test is an important method to study the mechanical properties of saturated soil, and it has strong utility, so many scholars carry out consolidation compression tests to study the temperature effect of soil. In 1967, PASSWELL found that the change of saturated soil volume during the heating process was basically consistent with the curve obtained by standard consolidation test, and put forward the concept of thermal consolidation [12]. ERIKSSON [26] targeted graded loading and constant strain rate consolidation tests for different clays, and found that the effect of temperature on pre-consolidation pressure was obvious, and the effect on soil compression index was not obvious. MORITZ carried out CRS consolidation compression experiments on Swedish clay at different temperatures, also found that the pre-consolidation pressure decreases with increasing temperature, and gave the double logarithmic empirical formula considering the influence of temperature on the pre-consolidation pressure of soil [7]. LALOUI et al [27] carried out the thermal consolidation test on kaolin, combining with the previous test data on temperature and pre-consolidation pressure, proposed a simple logarithmic empirical formula for temperature and pre-consolidation pressure. WANG [7] obtained the conversion relation between the above two expressions based on theoretical derivation. It should be noted that none of the above studies deals with the secondary consolidation of soil, and relatively few studies have been conducted on the effect of temperature on the secondary consolidation of soil. PLUM et al [28] carried out consolidation compression at room temperature for 3 d and then heated to 50 °C for 5 d; it was found that compared with normal temperature, the secondary consolidation coefficient C_α increased by 10%. GREEN [29] carried out one-dimensional consolidation creep tests at different temperatures (5 to 50 °C); the results showed that the secondary consolidation coefficient was related to the effective stress and temperature, and the effect of temperature on C_α was more obvious under low effective stress. In Ref. [30] tests on marine soft soil found that the secondary consolidation coefficient increased linearly with temperature increasing. COCCIA et al [31] combed the temperature effect of secondary consolidation of soil. WANG et al [32] and ZEINALI et al [33] studied the thermal consolidation considering the creep effect. In above paper, the temperature effect of primary and secondary consolidation compression is studied qualitatively, or a mathematical description is given for a mechanical property of primary and secondary consolidation compression. However, the unified constitutive model for primary and secondary consolidation compression of soil is still lacking, and the consolidation compression model which can fully describe the effective stress change, temperature change and deformation time effect has not been reported.

In response, a comprehensive consolidation compression model considering the effective stress variation, soil creep properties and thermal expansion and contraction properties is derived from the general functional relationship between soil void ratio, effective stress, temperature and time. Then, combined with the research results of classical soil mechanics and existing soil thermal consolidation characteristics, a practical compression model of primary and secondary consolidation under thermal mechanical coupling is established. A series of temperature controlled consolidation compression tests under different stress temperature paths were carried out based on the advanced temperature controlled consolidation compressor, the corresponding parameters and their value range are obtained, and the calculation and analysis are carried out to provide a certain theoretical basis and parameter value basis for the thermal geotechnical engineering problems.

2 Primary and secondary consolidation under loading and heating

2.1 Unified model

The causes of saturated soil’s consolidation and compression can be summarized as three aspects: effective stress change (load change), time attribute of soil deformation (creep) and environmental change, such as the temperature change. For the general thermal geotechnical engineering, the temperature of saturated soil (below groundwater level) does not change much, so the change of dry and wet state of soil caused by heating is ignored.

Thus, set that the void ratio (or strain) of saturated soil during consolidation and compression is a function of effective stress, time and temperature:

e=e(σ'_v, t, T)                             (1)

where the effective stress σ'_v and temperature T are both functions of time.

According to the general mechanics of materials, the following two assumptions are made. Firstly, the partial derivative relationship between the porosity ratio of soil and the effective stress is the compressibility index of soil; this index varies with the change of effective stress and temperature. Secondly, the partial derivative relationship between the pore ratio of soil and temperature is the thermal expansion and contraction property of soil, which is independent of time and temperature level.

Thus, according to formula (1), the rate of change of pore ratio can be obtained:

     (2)

In the formula: ① the first term on the right represents the change in the porosity ratio caused by the effective stress change, describing the elastic-plastic deformation of the soil. The expression /σ'_v is related to the effective stress and temperature T and is determined by conducting thermal consolidation tests at different stress levels and temperatures. ② The second term indicates the change of soil pore ratio with time under the condition of effective stress invariant, describing the clay deformation (creep) of soil. The clay deformation of soil is affected by both stress level and temperature, and its variation law can be obtained by consolidation creep test at different stress levels and different temperatures. ③ The third term indicates the change of pore ratio caused by thermal expansion and contraction. Expression e/T is related to the effective stress (or stress level) and can be determined by variable temperature compression tests at different stress levels.

 (3a)

According to the integral of formula (2), the change of soil porosity ratio can be obtained at any time under the action of thermal-mechanical coupling is:

 (3b)

Based on the results of classical soil mechanics research [1], define the boundary point of main consolidation and secondary consolidation as t_p. The change of effective stress from 0 to t_p is the main consolidation stage; while from t_p to time t, the effective stress is constant, and the pore ratio changes with time, which is the secondary consolidation stage. Moreover, set the process of temperature variation from the initial temperature T₀ at the time t_T0; then the temperature rises linearly and reaches a stable value T_f at time t_Tf; After that temperature keeps constant. On this basis, the following expression can be obtained:

        (4)

Formula (4) is the unified mathematical model of consolidation and compression of saturated soil under thermal-mechanical coupling. In the formula the first term on the right represents the change of the primary consolidation pore ratio; the second term represents the change of the secondary consolidation pore ratio; and the third term represents the effect of the thermal expansion and contraction properties of the soil on the pore ratio. For example, the compressibility index of soil in primary consolidation stage that is not the main consolidation stage is not required to be constant (material nonlinearity); the creep property of secondary consolidation as soil material is always existing (according to the assumption of creep theory B); the creep deformation in the main consolidation stage is included in the primary consolidation deformation [2]. The parameters of the above expressions can be got by carrying out thermal consolidation test at different temperatures and the variable temperature test was at different stress levels.

2.2 Practical model

Considering the complex properties and its remarkable regional nature, it is difficult to obtain a unified detailed mathematical expression. Based on the existing research results and some simplifications, a practical model of saturated soil for the consolidation and compression reflecting multiple factors is proposed here.

First, expression (4) can be rewritten as follows:

                        (5)

where De_p, De_c and De_T are the pore ratio change caused by primary consolidation, secondary consolidation and thermal expansion, respectively; among them, De_c is related to the time.

1) Pore ratio change by primary consolidation. According to formula (4) and formula (5), the change of pore ratio in the primary consolidation stage of saturated soil can be expressed as follows:

          (6)

Around the soil temperature effect in the primary consolidation stage, scholars at home and abroad have carried out a series of studies, and have reached the following two consensuses. On one hand, the effect of heating on soil compression index C_c is very small and it can be ignored; on the other hand, the effect of heating on soil compressibility is mainly manifested in the effect of pre-consolidation pressure, that is, the pre- consolidation pressure decreases with the increase of temperature. On the above two points of consensus, the following additions will be made.

① The effect of heating on soil compression is related to thermal expansion and contraction properties, thermal elasto-plasticity, thermal viscosity and strength properties of soil materials. The compression of soil after heating may increase (thermal settlement) or decrease (thermal expansion), which is closely related to soil properties and their state. For strongly over-consolidated soil, heating causes soil expansion; while for normal consolidated soil, under-consolidated soil and slightly over-consolidated soil, the soil is characterized by thermal settlement during heating.

② Considering the objective property of the pre-consolidation pressure (the maximum pressure in history), thus the effect of temperature effect is described by the apparent pre-consolidation pressure in the following. The effect of temperature on the apparent pre-consolidation pressure of saturated soil is essentially the effect of temperature on the yield surface material (or the influence of range from elasticity to plasticity). With the increase of temperature, the structure of saturated soil is impacted, and its elastic deformation range is reduced, and thus the saturated soil is easier to yield.

③ When the soil is heated, the porosity of the soil increases due to the thermal expansion, which makes the instantaneous settlement of the soil increase under load, and the compressibility of the initial section of the e-lg p curve increases, which means the slope of the initial section of e-lg p (or C_e) increase.

As to the relationship between apparent pre-consolidation pressure and temperature, the empirical formula proposed by LALOUI and CEKEREVAC [27] is widely accepted, and it can be expressed as follows:

                   (7)

where △T is the temperature variation, and △T=T_f-T₀; γ is the coefficient of temperature influence on the apparent pre-consolidation pressure, and γ=0.3-0.4; p_c is the pre-consolidation pressure of the soil under room temperature condition; △p_cT is the change of the apparent pre-consolidation pressure with temperature. Then, the apparent pre-consolidation pressures at different temperatures can be calculated by:

p_cT=p_c+△p_cT                                             (8)

As to the influence of heating on the compressibility of soil before yielding, based on the ERIKSSON’s [26] results, the following formula is given:

                     (9)

where p₀ is the current vertical pressure of the soil; for the normal consolidated soil, there is p₀=p_c; C_e is the compression index (or rebound re- compression index) of the initial section of the soil at normal temperature; and C_eT is the initial section compression index of the soil at temperature T after heating.

Next, the e-lgp curve method considering the soil stress history can be used to calculate the primary consolidation compressibility under thermal-mechanical coupling:

① When p₀+p_cT,

                   (10a)

② When p₀+p≥p_cT,

        (10b)

where Dp is the increment of vertical pressure; C_cT=C_c, which means the effect of heating on soil compression index is ignored.

Formulae (6)-(10) calculate the total amount of pore ratio change in the primary consolidation stage. And the calculation process of the pore ratio change with time (e_pt) during the primary consolidation stage is as follows.

The formula for calculating the consolidation degree according to the classical consolidation theory is introduced here,

                 (11)

where C_vT is the coefficient of consolidation of soil under temperature T; H is the longest drainage distance. It is worth noting that parameter C_vT will increase with temperature rising, and the consolidation rate will be accelerated under heating.

Then the change of pore ratio at any time in the primary consolidation stage e_pt is the product of U_t and e_p, that is:

           (12)

2) Pore ratio change by secondary consolidation

Based on the relative research results at room temperature, the formula of the secondary consolidation calculation for thermal-mechanical coupling is as follows:

               (13)

where C_αT is the coefficient of secondary consolidation under different temperatures.

At present, there are few reports on the influence of heating on soil secondary consolidation coefficient. PLUM et al [28] found that the secondary consolidation coefficient was 0.0048 at room temperature and it changed to 0.0053 after heating, and the slope of the secondary consolidation curve increased by 10% due to heating. HOUSTON et al [30] carried out creep tests at constant stress level for deep sea sediment, and the temperature range of 4-200 °C and found that the secondary consolidation coefficient increased approximately linearly with temperature. BURGHIGNOLI et al [13] carried out the study on heat transfer and soil creep during thermal consolidation of clay soil, and found that the temperature cycle can lead to soil hardening under the drainage condition and the soil creep affects the thermal consolidation process, and it is related to the stress level. To sum up, the secondary consolidation coefficient C_αT is related to temperature T and stress level p, but the corresponding empirical formula is not reported so far.

Based on the experimental studies, the empirical formula is proposed:

                   (14)

where η is a empirical parameter related to the temperature; b is a empirical parameter related to the stress level. Both parameters η and b can be determined by the consolidation creep test.

3) Pore ratio change by thermal expansion

The properties of thermal expansion and contraction of soil also cause the change of pore ratio of soil. Previous studies [17, 27] have shown that, the thermal expansion and contraction of soil is thermal elasticity behavior. Thus, the following formula is used to describe the deformation of soil caused by temperature change:

      (15)

In the formula, α_T is the coefficient of thermal expansion of saturated soil under drainage conditions. Existing studies have shown that the coefficient α_T is related to the consolidation pressure. ABUEL-NAGA et al [34] carried out tests for Boom clay, MC clay and Bangkok clay, and found that the thermal expansion coefficient of soil decreased with the increase of vertical pressure under drainage conditions, as shown in Figure 1.

Figure 1 Variation of thermal expansion coefficient with stress level for different soils

Define α_T100 as the thermal expansion coefficient at vertical pressure 100 kPa; then, according to Figure 1, the following formula can be used to describe the relationship between the secondary consolidation coefficient α_T and the vertical pressure p:

                         (16)

where coefficient α_T100 is dependent on soil type and its microstructure.

3) Comprehensive calculation formula

Combining formulae (5) to (16), and introducing variable ψ, then a concretized formula (4) can be got as:

  (17)

where p=p₀+Dp; and variable ψ is

                          (18)

Formula (17) reflects the elasticity (first item), plasticity (second item), viscosity (third item) and thermal expansion (fourth item) of soil during consolidation compression. Among them, the elastic deformation parameter is C_eT; the plastic deformation parameters are C_cT and p_cT; the viscous deformation parameters are η and b; parameters of thermal expansion are α_T100 and a. All the above parameters can be obtained by thermo consolidation compression test.

3 Test and parameters

3.1 Thermal consolidation compression test

3.1.1 Soil samples

The tested soil samples were taken from a foundation pit project in coastal area with depth of 3-4 m, which is a typical gray color silty clay in Ningbo city. This saturated soft soil is homogeneous and is slightly disturbed due to excavation with unloading effect. The basic physical property indexes of the test soil are shown in Table 1.

Table 1 Basic physical properties of test soil

3.1.2 Test instrument

Experimental instrument is advanced temperature control consolidation instrument KTL,as shown in Figure 2. The hardware part of the test system includes standard static loading frame with a maximum load of 10 kN, consolidation pressure chamber with high precision pore pressure sensor, standard volume pressure controller, temperature control system (temperature control range is room temperature to 65 °C), and multi-channel data acquisition device. Among them, the temperature control system includes circulation pipe, heat preservation device, and pipe joint, fluid circulation convection pump and temperature controller. The temperature control system uses the inlet and outlet to fill and drain the cavity of the consolidated pressure chamber, and then the water in the cavity is heated and controlled by the heating device, and thus the heating of the soil sample is realized. A variety of consolidation compression tests can be carried out with GeoSmartLab software. The test instrument is completely controlled by computer and collects data. It can measure variables such as axial stress, back pressure, pore pressure, axial displacement, and temperature. It has the functions of soil saturation, B value detection, graded loading, equal strain rate loading, and equal loading rate consolidation compression and so on. The specimen size used in the consolidation apparatus is standard consolidation specimen size.

Figure 2 Advanced soil test system with temperature controlling device:

3.1.3 Test steps and test program

The test is divided into three stages. First, the soil sample is prepared; then the bottom water permeable plate, filter paper and protective ring are placed on the base of the consolidation container in turn; the ring knife containing the soil sample is put into the protective ring; the upper cover is covered and tightly sealed, and the soil sample installation is completed. After that, the pore water pressure at the bottom of the consolidation vessel is opened and filled with degassing water to remove the remaining bubbles in the bottom and pipeline. The axial pressure and back pressure (with pressure difference is 5 kPa) are set by the software system to saturate the specimen; when B value reaches 0.95, the following tests with different temperature-stress paths were carried out.

Table 2 shows the test program including three test series. Series 1 is graded loading and unloading tests at different temperatures, to determine parameters C_c, C_e and p_c, and to assess the effect of temperature on these parameters. The temperature range is room temperature (26 °C) to 56 °C; the vertical pressure is 12.5 to 800 kPa; the loading time for all pressure levels is 24 h. Series 2 is consolidation creep tests at different temperatures to determine coefficients C_αT and to study the effect of vertical pressure and temperature on C_αT. The temperature range is the same as Series 1, and vertical pressure is 25, 50, 100, 200 kPa; the consolidation periods for each loading step is 3 d. Series 3 is variable temperature test under different load levels, that is 50 and 100 kPa, respectively. The temperatures are room temperature 26, 36, 46, 56 °C and room temperature 26 °C, single stage temperature is maintained for 1 h. Note that the process of each stage heating from initial temperature to the target temperature in the test is about 10 min.

Table 2 Test series with different temperature and loads

3.2 Test results and analysis

1) Effect of heating on primary consolidation

Figure 3 shows the e-lgp curves obtained from Series 1. It can be seen that at the beginning of heating, the soil has a small rebound, and then, with the increase of temperature, the compression curve moves down, which is consistent with previous study [26]. The C_c, C_e and p_c at different temperatures are shown in Table 3. It can be concluded that the effect of heating on compression indexes C_c and C_e is not obvious, and the apparent pre-consolidation pressure decreases gradually with the increase of temperature. Figure 4 is got from Table 3 and Formula (7). The figure shows that Formula (7) can better describe the relationship between the pre-consolidation pressure and the dimensionless temperature T/T₀ of the soil in this test, and γ=0.382.

Figure 3 e-p curves under different temperatures

Table 3 Temperature effects on C_c, C_e and p_c

Figure 4 Variation of p_c with temperature T

2) Effect of heating on secondary consolidation compression

A series of e-lgt curves at different temperatures were obtained for Series 2. It found that the deformation of soil samples with time showed a straight line on the e-lgt curves. Based on this, the secondary consolidation coefficients at different temperatures (C_αT) are obtained. Table 4 shows the secondary consolidation coefficient results. Figure 5 shows the scatter plot of the effect of temperature and vertical pressure on the secondary consolidation coefficient. Both Table 4 and Figure 5 show that the C_αT is mainly varied from 0.003 to 0.005; with the increase of temperature, the secondary consolidation coefficient increases, and this trend develops rapidly in the early stage and gradually slows down in the later stage; the vertical pressure has a certain effect on the secondary consolidation coefficient, and this effect becomes smaller with the increase of vertical pressure, but the law is not obvious.

Table 4 Effect of temperature and stress on secondary condition coefficient

Figure 5 Effects of temperature and vertical pressure on secondary consolidation coefficient:

According to Figure 5(a) and combining with Formula (14), it can be obtained η=0.0021, and b=0.003-0.004.

3) Effect of heating temperature on consolidation compression

Figure 1 shows the rebound of test soil unit at different pressures under drainage conditions. It can be seen from that the greater the effective stress (or stress level) is, the smaller the coefficient of thermal expansion will be. According to Formula (16), the related parameter values of Ningbo soft clay in this test are obtained as a=0.28, and α_T100=0.00015.

4 Calculation and analysis

4.1 Bangkok soil sample test

ABUEL-NAGA et al [34] carried out consolidation compression tests for Bangkok clay under normal temperature (25 °C) and high temperature (70 °C), respectively, i.e., T₀=25 °C, T_f=70 °C, DT=45. Soil pre-consolidation pressure p_c=200 kPa, and pore ratio of soil before heating e₀=1.65. The computation process is as follows.

First, the effect of heating on the pore ratio is calculated from Formula (16). The Bangkok clay parameters a=1.28 and α_T100=0.000707, and then parameter α_T can be calculated. Parameter e_T can be obtained according to Formula (15). The primary consolidation compressibility is calculated according to Formulae (7)-(10). p₀=20 kPa, C_cT=0.435, C_e=0.058; according to the common range of values, take the parameter γ=0.4. Table 5 shows parameters and the intermediate calculation results.

Figure 6 shows the comparison between the theoretical calculations about the measured e-lgp of Bangkok clay (without considering the secondary consolidation). It can be seen from the figure that the law of compression curve obtained by theoretical calculation is close to the measured values. The theoretical calculation value reflects the rebound effect of soil under heating condition. And, the downward shift of compression curve means the increase of total compressibility of soil, which also reflects the decrease of apparent pre-consolidation pressure of soil with temperature rise.

Table 5 Parameters for calculation of Bangkok clay

Figure 6 Comparison of theoretical results with measured results (Bangkok clay)

4.2 Ningbo soil

Table 6 presents the calculation formula and parameter value for consolidation compression computation of Ningbo soil. Firstly, calculate △e_p with the assumption that temperature T=26, 36, 46, 56, 66 °C, respectively. The computation results are shown in Figure 7. It can be seen that the compression amount increases with the increase of temperature.

Table 6 Parameters of Ningbo saturated soft soil

Secondly, calculate Δe_c. Figure 8 shows the secondary consolidation compression variation with compression time and temperature. It can be seen from that the amount of compression increases with time extension; the higher the temperature is, the greater the amount of compression will be; the effect of heating during early consolidation process is relatively obvious, and then the effect of heating gradually slows down.

Then, calculate △e_T. Figure 9 shows the thermal expansion deformations at different temperatures. It can be seen that with the increase of vertical pressure, the rebound of soil unit decreases; the higher the temperature is, the more obvious the rebound will be.

Figure 7 Primary consolidation compressions under different temperature conditions

Figure 8 Variation of secondary compression with T

Figure 9 Thermal elastic deformation variation with vertical pressure under different temperatures

Finally, the effects of primary consolidation, secondary consolidation compression and thermal expansion deformation on soil deformation are comprehensively analyzed as shown in Figure 10.

Figure 10 Comparison of e-p curves under different conditions:

Figure 10(a) shows the e-p curves with different consolidation compression durations at 46 °C. It can be seen that the longer the time of secondary consolidation is, the smaller the amount of deformation increase will be. Figure 10(b) shows e-p curves at different temperatures with t_f/t_p=100. It can be seen that the higher the temperature is, the greater the secondary consolidation deformation will be; in the rebound and recompression stage, the soil deformation behaves as rebound under heating condition; whereas in the normal consolidation compression stage, the soil deformation is thermal settlement. Totally, the soil compressibility increases with temperature rise in this calculation case.

According to the previous calculation, the effect of heating on the final deformation of soil changes with the soil over-consolidation ratio. Under large over-consolidation ratio condition (or soil structure is significant), or low load level condition (mainly elastic deformation of soil), the consolidation compression of soil during heating shows as rebound with temperature rise. For normal consolidation soil and weak consolidated soils (soil structure is none or weak), the effect of heating on the final deformation of soil is that the consolidation compression of soil under thermal coupling is greater than that under normal temperature.

5 Conclusions

1) According to the theoretical analysis of the primary and secondary consolidation compression, a unified expression is obtained for saturated soil compression under the coupling effect of loading and heating. Then, based on the existing research, a simple and practical thermal-mechanical coupled primary and secondary consolidation compression model is proposed, which takes into account the variation of apparent pre-consolidation pressure and secondary consolidation coefficient with temperature, as well as the influence of heating on the compressibility of the rebound and re- compression of soil.

2) The newly proposed model comprehensively reflects the effect of effective stress change, creep or viscosity behavior, and temperature change (thermal expansion) on the consolidation compression of soil. It can explain not only the rebound phenomenon of some soil (such as strong over consolidated soil) after heating, but also the thermal settlement phenomenon of some soil (such as normal consolidated soil and weak over consolidated soil) after heating.

3) A series of thermal-coupled consolidation compression experiments for a typical soft soil were carried out by using an advanced temperature- controlled consolidation instrument. Accordingly, the model parameters and their value ranges were obtained. It is found that the rebound amount of soil by heating is related to the vertical stress; the larger the vertical pressure is, the smaller the rebound amount will be. With the increase of temperature, the apparent pre-consolidation pressure of soil decreases. The secondary consolidation coefficient is related to both heating temperature and stress level; and the secondary consolidation coefficient is related to the change of temperature by natural logarithmic function.

4) The present thermal-mechanical coupled consolidation compression model is simple, and the parameters are easy to obtain, and can describe the main law of consolidation compression of saturated soil under the action of thermal-mechanical coupling. Therefore, it is suggested to be applied for the theoretical analysis of thermal-related geotechnical engineering.

Contributors

DENG Yue-bao developed the overarching research goals, established the mechanical model and edited the manuscript draft. MAO Wei-yun carried out the soil element tests and analyzed the measured data. KONG Gang-qiang revised the draft manuscript and suggested amendments. HAN Yi-dong edited and typeset the manuscript.

Conflict of interest

DENG Yue-bao, MAO Wei-yun, KONG Gang-qiang, and HAN Yi-dong declare that they have no conflict of interest.

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(Edited by YANG Hua)

中文导读

热力耦合作用下饱和土的主次固结压缩

摘要：能源与环境岩土工程当前受到越来越多的重视，研究热力耦合作用下土体力学特性及其数学描述具有重要的理论与实际价值。基于土体孔隙比-有效应力-温度-时间之间的泛函关系，推导了热力耦合作用饱和土的主次固结压缩统一模型。结合已有研究基础对其进行简化，得到能综合反映有效应力变化、蠕变以及加热影响的实用模型。使用温控固结仪进行一系列热力耦合的固结压缩试验，研究竖向压力、温度、压缩时间等对饱和土固结压缩的影响，得到相应的数学描述及其参数取值。在此基础上开展计算与分析，检验模型的可靠性与适用性。新模型简单实用，参数容易获得，且能较好地描述热力耦合作用下饱和土体固结压缩的主要规律，因此可供涉热岩土工程问题理论分析时参考。

关键词：饱和土；主固结；次固结；温度效应；固结压缩试验

Foundation item: Project(51608281) supported by the National Natural Science Foundation of China; Project(LGG21E080005) supported by the Provincial Natural Science Foundation of Zhejiang Province, China

Received date: 2020-06-01; Accepted date: 2020-12-20

Corresponding author: DENG Yue-bao, PhD, Associate Professor; Tel: +86-15058428762; E-mail: dengyuebao@nbu.edu.cn; ORCID: https://orcid.org/0000-0001-7512-8517