2. School of Civil Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150090, China;
3. State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou, Gansu 730000, China
In regions with seasonally frozen ground or permafrost, frost heave is an uneven upward movement of the ground surface during extended exposure to subfreezing temperatures (Cheng and He, 2001; Xu et al., 2001 ). Experimental observations indicate that when finegrained soils, usually called frostsusceptible soils, are subjected to subfreezing temperatures, substantial frost heave may be observed in the field or laboratory (Konrad and Morgenstern, 1980; Sheng et al., 2013 ). Frost heave is attributed to the solidification of a portion of the water in soil pores and is associated with water migration, particularly in the presence of adequate water supply (Zhao et al., 2009 ; Zhou and Li, 2012).
In practice, the detrimental effect of frost action in frostsusceptible soils is an inevitable and significant concern in geotechnical engineering (Penner and Ueda, 1977; Peppin et al., 2011 ). In general, frost heave in cold regions is a primary cause of damage that can lead to the failure of various types of infrastructure such as roadways, bridges, pipelines, and foundations. Failures primarily occur when footings are located in frostsusceptible soils above the final frostpenetration line (Everett, 1961; Wang and Hu, 2004; Abzhalimov and Golovko, 2009).
For this reason, numerous efforts have been made over decades to explain the frost heave mechanism of a variety of soils; these efforts have included laboratory tests on small specimens, model experiments, and field investigation (Taber, 1930; Singh and Chaudhary, 1995; Ma and Wang, 2012; Zheng and Kanie, 2015). Recently in China, there has been renewed interest in researching the frostheave mechanism and its adverse effects on engineering in frostsusceptible soils (Zhou and Zhou, 2012; Lai et al., 2014 ; Ma et al., 2015 ).
In this regard, typical methods to estimate the amount of frost heave are based on assumptions about saturated soils; proposed methods include the capillary model, the rigidice model, the segregationpotential model, thermomechanical models, and the semiempirical model (O'Neil and Miller, 1985; Fremond and Mikkola, 1991; Nixon, 1991; Black, 1995; Michalowski and Zhu, 2006). Most of these approaches, requiring the use of many parameters, virtually explain the complicated phenomenon of saturated soils to predict accurately the amount of frost heave. However, these models may inherently show reduced accuracy when applied to the engineering problem of predicting frostheave amounts of unsaturated soils. In this work, a series of onesided, multistage freezing experiments involving an unsaturated silty clay column in an open system were conducted. Here, experimental results are presented and analyzed. On this basis, a mathematical procedure is proposed to predict the maximum frost heave with satisfactory accuracy and simplicity. Finally, some conclusions are drawn and future research directions proposed.
2 Test and materials 2.1 Sample preparationsFrostsusceptible silt clay obtained from the city of Harbin in Northeast China was used to prepare all reconstituted samples. A grainsize distribution of this clay is shown in Figure 1; and the clay exhibits the following grainsize characteristics: curvature coefficient, C_{c}=1.25; soil nonuniformity index, 10.04; and plasticity index, 11.57%. All circular solid samples were prepared with a diameter of 10 cm and height of 15 cm. These soil samples were compacted using slight vibration at an optimal moisture content of approximately 14% to achieve a density of about 1,670 kg/m^{3}. The degree of saturation of the soil samples was 59%.
Figure 2 shows the experiment setup. The apparatus includes four main components: temperature control, water supply, loading, and data acquisition (Figure 3). The surface of the inner container was covered with Vaseline to minimize side friction and adhesion between the sample and the container. Before testing, the sample was placed into a plexiglas container with inner dimensions of 20 cm in height and 10 cm in diameter. To enhance the cooling efficiency, the container circumference was insulated with a 50mmthick layer of insulating Styrofoam. All prepared specimens in the container were placed in a temperature chamber to avoid heat loss and kept at a constant environmental temperature of −2 °C.
Alcohol, at temperatures ranging from −40 °C to +30 °C with an accuracy of ±0.1 °C, was used as the controlled liquid circulating through the plate ( Figure 3). Throughout the test, the tops of the samples were the cooling end, subjected to subzero temperatures (T_{c}) through a copper plate; the bottoms of the samples were the warm end, subjected to the abovezero temperature (T_{w}), 2 °C, through another copper plate.
Additionally, the watersupplement system had a Mariotte flash connected to the bottom plate through a plastic tube. In all experiments, an unpressurized watersupplement process was employed.
2.3 Determination of freezing duration and test proceduresThere were six freezingtest cases (i.e., Cases FT0 to FT5) conducted in this study (Table 1). Various overburdenstress conditions were enforced on the top of the soil column. All test specimens began to freeze after standing for 24 hours under corresponding load with no water supplement. A displacement transducer was installed at the plate to continuously monitor the accumulated frost heave at the specimen surface. The side of the container included four holes, accommodating four thermistors for each sample. These thermistors were inserted laterally through the holes, approximately 3 cm into the sample, at 3cm intervals to measure internal soil temperature (T_{soil}); two additional thermistors were placed at the top and bottom of each sample.
Usually, evaluating maximum frost heave requires the maximum possible time duration at a given T_{c}. To minimize and reasonably determine freezing duration, the top of the sample in Case FT0 was successively cooled to a T_{c} of −5 °C over a 168hour period. The frost heave and frostheave rate with time elapsed in Case FT0 are presented in Figure 4. For ease of presentation, experimental data for frost heave is marked as S(t), where t is the freezing duration. From Figure 4, frost heave, S_{FT0}(t), accumulated with t and approached a maximum value as the t approached 168 hours of freezing. Nevertheless, the frostheave rate, dS(t)/dt, gradually reduced as t lapsed; and it tended toward zero by the end of the freezing period (i.e., t=168 hours). Using data of S_{FT0}(t), a straight line is obtained by plotting 1/S(t) versus 1/t in Figure 5. Thus, the relationship between S(t) and t through the freezing period could be approximately expressed using a hyperbolic form.
Further, the correlation between frost heave and freezing time for various ranges of elapsed time is shown in Figures 6a to 6g. From these figures, we see a correlation by using just the data from 0 to 72 hours; thus, the validity of the function and parameters used is confirmed. For this purpose, the 72hour freezing duration for every stage in the following multistage freezing test cases was determined.
In Cases FT1 to FT5, identical thermal conditions, i.e., Stage 1 to Stage 4 (Figure 7), were applied to the samples. The top of each sample was successively cooled through one copper plate to various freezing temperatures of 0 °C, −5 °C, −10 °C, −15 °C, and −20 °C, stage by stage; and each freezing stage had a 72hour duration.
Figure 8 comprises the frostheave time histories (i.e., S_{FT0}(t) and S_{FT1}(t)) in cases FT0 and FT1. It was found that the maximum frost heave was not attained by the end of every freezing stage in Case FT1. The recorded frost heave, S(t), versus time for Cases FT1 to FT5 is illustrated in Figure 9, in which each freezing interval is 72 hours. For each freezing stage, a similar trend of the frost heave under various overburden stresses was observed, at first accumulating rapidly and then more slowly until freezing ended. As T_{c} was decreased from one stage to the next, frost heave demonstrated a slight but abrupt increase.
Frostheave amounts under overburden stresses (σ_{v}) of 0.000 MPa, 0.015 MPa, 0.028 MPa, 0.041 MPa, and 0.054 MPa attained 18.1 mm, 15.1 mm, 13.0 mm, 11.4 mm, and 10.0 mm, respectively, by the end of freezing (i.e., t=288 hours). Less frost heave occurred as the overburden stress (σ_{v}) increased, indicating that the reduction in frost heave maybe due to increasing σ_{v} at the sample surface. It may be that water was inhibited from entering into the soil due to the increase of σ_{v}.
3.2 Formulation of equation for frostheave increment due to change of T_{c}Maximum frost heave is a critical parameter for the design of foundations in frozen regions (Singh and Chaudhary, 1995; Peppin et al., 2011 ; Sheng et al., 2013 ). A new approach is firstly proposed to evaluate maximum frost heave by determining the appropriate equation for the relationship between incremental frost heave (ΔS) induced by the change of T_{c} and t, utilizing information obtained during the each stage.
From Figure 8, variations of ΔS_{1,2}(t) (i.e., ΔS_{1,2}(t)=S_{FT0}(t)−S_{FT1}(t)) and the associated frostheave increment rate (i.e., dΔS_{1,2}(t)/dt) are obtained. These variations are presented in Figure 10, where the t varies in the range of 72~144 hours. Using data for ΔS_{1,2}(t) (Figure 10), a straight line is obtained by plotting 1/ΔS_{1,2}(t_{1}) versus 1/t_{1} (i.e., t_{1}=t−72, t varies over the range of 72~144 hours), as presented in Figure 11. We have
$\frac{1}{{\Delta {S_{1,2}}({t_1})}} = a + b\frac{1}{{{t_1}}}$

(1) 
Which can be expressed as
$\Delta {S_{1,2}}({t_1}) = \frac{{{t_1}}}{{a + b{t_1}}}$

(2) 
where the parameters a and b are undetermined coefficients. The values of a and b can be obtained from the slope and intercept of the straight line. Even with sufficient time, the ΔS_{1,2}(t_{1}) induced by the change of T_{c} from −5 °C to −10 °C can be fitted with the hyperbolic curve.
3.3 Estimation method for maximum frost heaveIn this section, the proposed method for estimation of maximum frost heave is outlined. The above investigations have clarified that a hyperbolic curve can be used to fit the frostheave increment (Figures 6c and 10) induced by the change of T_{c}.
It should be noted that ΔS_{0,1}(t) (ΔS_{0,1}(t)=S(t)−S(0), Figure 12a) is a known quantity, where t varies from 0 to 72 hours. After careful inspection, we have
$\Delta {f_0}(t) = \frac{t}{{{a_0} + {b_0}t}}$

(3) 
where the parameters a_{0} and b_{0} are undetermined coefficients. A straight line can be obtained by plotting 1/Δf_{0}(t) versus 1/t. The value of a_{0} and b_{0} can be obtained from the slope and intercept of the straight line. This approach may also be used to calculate the undetermined coefficients in the hyperbolic curves' equations defining the other stages.
Theoretically, as time t→∞, the maximum frost heave, S_{max,1}, approaches that attained at T_{c} of −5 °C. Thus, it could be assessed from Equation (3) as
${S_{\max ,1}} = \mathop {\lim }\limits_{t \to \infty } \Delta {f_0}(t) = \frac{1}{{{b_0}}}$

(4) 
The predicted frost heave continues to grow at the T_{c} of −5 °C where t exceeds 72 hours, and the contribution of relevant frost heave can be extracted using Equation (3). Therefore, to remove this contribution, the ΔS_{1,2}(t) occurring during the change of T_{c} from −5 °C to −10 °C can be obtained from Equation (5) as follows:
$\Delta {S_{1,2}}(t) = S(t)  \Delta {f_0}(t)$

(5) 
where t varies in the range 72~144 hours. Further, ΔS_{1,2}(t) is a known quantity (Figure 12a).
Therefore, a similar hyperbolic equation is utilized to fit the discrete data of ΔS_{1,2}(t) below:
$\Delta {f_1}({t_1}) = \frac{{{t_1}}}{{{a_1} + {b_1}{t_1}}}$

(6) 
where t_{1} (i.e., t_{1}=t−72) is t at the beginning of Stage 2, and parameters a_{1} and b_{1} are undetermined coefficients.
Combined with Equations (3) and (6), maximum frost heave, S_{max,2}, at T_{c} of −10 °C is described in the following form:
${S_{\max ,2}} = \mathop {\lim }\limits_{t \to \infty } \Delta {f_0}(t) + \mathop {\lim }\limits_{t \to \infty } \Delta {f_1}(t) = \sum\limits_{j = 0}^1 {\frac{1}{{{b_j}}}} $

(7) 
Maximum frost heave, S_{max,3} and S_{max,4} at T_{c} of −15 °C and −20 °C, respectively, can be similarly calculated.
To remove the overall effect of the change of T_{c} on the frost heave, we can separate out the effect of the change in T_{c} during Stage 1 on the increase of frostheave amount by extrapolating the curve obtained from Stage 1.
Similarly, the effect of Stage 2 is extrapolated to isolate the effect of the change in T_{c} on the next freezing stage. In this analysis, in addition to the assumptions made in the extrapolation, we have also supposed that the effect of T_{c} on frost heave is additive. On this basis, Figures 12a to 12e, present the recorded and predicted frostheave increment at each freezing stage under various overburden stresses.
3.4 Empirical equation between S_{max,n} and T_{c}Finally, frost heave can be assumed as a function of σ_{v} and T_{c}. By using the abovementioned mathematical procedure, data for S_{max,n} (n=1, 2, 3, and 4) can be obtained; and eventually, the relationship between S_{max,n} and T_{c} under various overburden stresses, as shown in Figure 13, may be derived. The hyperbolic curve is still chosen to fit data for S_{max,n} during each freezing stage in this uniform form:
${S_{\max }}({T_{\rm c}}) = \frac{{{T_{\rm c}}}}{{m + n{T_{\rm c}}}}$

(8) 
where S_{max}(T_{c}) is the fitting formula and represents the maximum frost heave at T_{c}. Parameters m and n are undetermined coefficients, which as the function of the σ_{v}, are expressed by Equations (9) and (10) as follows:
$m =  {\rm{2}}{\rm{.05}}{{\rm{e}}^{{\rm{0}}{\rm{.02}}{\sigma _v}}}$

(9) 
$n = {\rm{0}}{\rm{.42 + }}\frac{{{\sigma _v}}}{{{\rm{205}}{\rm{.4 + 0}}{\rm{.5}}9{\sigma _v}}}$

(10) 
A series of laboratory tests was conducted to investigate the frost heave of Harbin silty clay with free access to an external water source under onesided, multistage freezing conditions. The results and conclusions are summarized as follows.
(1) Frost heave at the ground surface decreases due to increased overburden pressure, which may act as a constraint on frost heave. Frost heave can have more abrupt increases from one stage to the next. Overall, the evolution of frostheave amount goes through two phases: rapid accumulation at the beginning of the freezing stage and then slow increase during the remainder of it.
(2) Maximum frost heave is not attained under test conditions with a 72hour freezing period for each stage. Sufficient freezing time is required to reach maximum frost heave under a given thermal boundary setting. Further, a hyperbolic equation can be used to approximate the relations between incremental frost heave and freezing time, as well as those between frostheave amount and freezing time at each freezing stage.
(3) Based on the assumption of the effect of the freezing temperature at the cooling end and its change on frost heave being an isolated additive, a simple method considering the effect of overburden pressure is proposed to estimate maximum frost heave. Meanwhile, the magnitude of maximum surface frost heave is found to be a function of cooling temperature and overburden stresses.
(4) This investigation provides a practical method to estimate maximum frost heave with accuracy sufficient for engineering. Future studies should address other important factors and introduce them into the theoretical model and framework. Including these additional factors can achieve a more general understanding of the frostheave mechanism in unsaturated frostsusceptible soils.
Acknowledgments:We gratefully acknowledge support for this research from the State Key Program of National Natural Science of China (Grant No. 41430634), the National Natural Science Foundation of China (Grant Nos. 41702382, 51578195, 51378161, and 51308547), the Foundation Project Program 973 of China (No. 2012CB026104), and the State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining & Technology (Grant No. SKLGDUEK1209).
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