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A method for obtaining radial stress in the freezing direction in frozen soil samples
Journal Pre-proof A method for obtaining radial stress in the freezing direction in frozen soil samples
Hao Zheng, Shunji Kanie, Satoshi Akagawa PII:
S0165-232X(19)30079-5
DOI:
https://doi.org/10.1016/j.coldregions.2019.102899
Reference:
COLTEC 102899
To appear in:
Cold Regions Science and Technology
Received date:
7 February 2019
Revised date:
11 September 2019
Accepted date:
19 September 2019
Please cite this article as: H. Zheng, S. Kanie and S. Akagawa, A method for obtaining radial stress in the freezing direction in frozen soil samples, Cold Regions Science and Technology(2019), https://doi.org/10.1016/j.coldregions.2019.102899
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© 2019 Published by Elsevier.
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A method for obtaining radial stress in the freezing direction in frozen soil samples
Hokkaido
University,
2 Cryosphere Engineering Laboratory, Tokyo, Japan
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Hao Zheng1*, Shunji Kanie1, Satoshi Akagawa2
1
Faculty
of
Engineering,
Japan,
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[email protected]
Sapporo,
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Abstract:
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In this study, we design an innovative frost heave cell to investigate the characteristics
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of stress in the radial direction during the freezing of a soil sample. The frost heave cell
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is divided into several independent rings connected by silicone sheets, which aims to reduce the deformation influence of the frozen part on the unfrozen part. By installing mini-sized water pressure gauges, strain gauges, and thermal sensors in the frost heave cell, the circumferential strain measured by strain gauges on the surface of frost heave cell can be effectively evaluated. In order to transfer this circumferential strain into stress in the radial direction, we adopt the thick-walled cylinder theory. Then, we conducted one freezing test with Dotan to investigate the characteristics of radial stress during freezing. The results suggest that the radial stress caused by frost heave increases
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rapidly when the temperature drops from 0 ºC to -2 ºC. However, as the temperature decreases below -2 ºC, the radial stress gradually increases. When the temperature approaches -10 ºC, the radial stress plateaus and does not increase further. These findings indicate the unique characteristics of radial stress perpendicular to the freezing
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direction and suggest the need for further research on this topic. As more complicated
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applications of artificial freezing method in underground space of urban area are
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expected, the influence on the surrounding structures brought by three-dimensional
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freezing shows its significance. Our research is aiming to investigate the mechanism of
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freezing phenomenon of soil in three-dimensional space and then promote the
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application of artificial freezing method in a wide area.
Keywords: frost heave; radial stress; freezing direction; thick-walled cylinder theory
1. Introduction As widely observed in nature, soil freezing process can lead to significant volumetric expansion due to the phase change of water and the water migration towards the freezing front. This volume expansion is called frost heave. With continuous water migration to the freezing front, this volume expansion could be much larger than that
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caused simply by the phase change from water to ice, which is approximately 9%. For example, palsas and pingos, which appear in permafrost areas, contain a large amount of ice lenses and can reach up to 70 m in height in the case of pingos (Mackay, 1973) (Pissart, 2002). Frost heave can cause significant damages to infrastructures such as
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highways, railways, bridges, and pipelines. It occurs not only on Earth but also on Mars
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(Zent, Sizemore, & Remple, 2011). However, frost heave research is relatively recent,
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even on Earth. In fact, the role of water migration in the freezing process was first
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recognized as late as 1929 (Taber, 1929) (Taber, 1930). The driving force of water
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migration arises from the suction pressure produced during freezing process.
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Continuous water migration under suction pressure results in the formation of a
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horizontal ice layer known as a segregated ice lens. This horizontal ice layer mainly grows in the freezing direction; therefore, major frost deformation appears in the freezing direction (Rempel, 2007) (Lai, Pei, Zhang, & Zhou, 2014). Due to this characteristic, the majority of research concerning frost heave phenomenon has been focused on the freezing direction.
The first wave of research on frost heave came in the 1970s due to the oil crisis and because of the large amount of oil and gas buried in cold regions. For example, the
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Arctic holds an estimated 22% of Earth's oil and natural gas resources (Budzik, 2009). At this period, the most important issue of frost heave for researchers and engineers is the precise prediction of the amount of frost heave during complex thermal-hydraulic and mechanical processes. To date, various theories and experimental equations have
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been proposed to explain the frost heave mechanism and predict the amount of frost
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heave. For example, the Generalized Clausius-Clapeyron model (Edlefsen & Anderson,
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1943), the rigid ice model (Miller, 1978) (O’Neill & Miller, 1985), Takashi’s model
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(Takashi, Yamamoto, Ohrai, & Masuda, 1978) (Tsutomu, Ohrai, Yamamoto, &
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Okamoto, 1981), the segregation potential model (Konrad & Morgenstern, 1981)
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(Konrad, 1994), and the porosity rate model (Michalowski, 1993) (Michalowski & Zhu,
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2006). These models have been used to calculate the deformation and stress caused by frost heave in order to guarantee safe construction (Zheng & Kanie, 2014). For example, the interaction between soil and a buried pipeline caused by frost heave has been analyzed using a numerical method and compared with a full-scale experiment conducted in Cane, France (Razaqpur & Wang, 1996).
The majority of previous frost heave research has explored the characteristics of frost heave in the freezing direction because most of the deformation occurs in this direction.
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However, recent laboratory investigations have demonstrated that the formation of ice lenses is not a simple one-dimensional process. Vertical ice lens structures were also observed (Arenson, Azmatch, Sego, & Biggar, 2008). As a consequence, frost heave and stress in the radial direction of freezing, which has largely been ignored in previous
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studies, should be carefully considered. Indeed, due to the wide application of artificial
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freezing technology in civil engineering, stress and deformation in the radial direction
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of freezing have begun to garner increasing attention because of the complicated
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freezing mechanism, especially in underground urban areas where artificial freezing
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may affect the foundations of surrounding structures. When numerous freezing pipes are
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installed in a construction site, segregated ice lenses produced around the freezing pipe
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scatter in three-dimensional space. Thus, the influence of frost heave is not limited to the direction of freezing (Afshani & Akagi, 2015). Accordingly, understanding the deformation and stress in the radial direction of freezing is necessary and urgent for the safe application of artificial freezing.
Ueda et al. proposed an empirical formula to express the strains in the freezing direction and its radial direction based on the stress relationship in each direction (Ueda, Ohrai, & Tamura, 2007). This empirical equation is derived from three-dimensional frost heave
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tests without horizontal restriction. This is a pioneering attempt to predict the frost heave ratio in three-dimensional space. In their frost heave test, horizontal deformation is allowed because they used a rubber sleeve to cover the soil sample. However, this unconstrained condition for the soil sample in the radial direction may not be effective
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when the segregated ice lens develops between the soil sample and the rubber sleeve.
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This can cause uneven deformation in the radial direction. Therefore, to solve this issue,
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in our study, we propose an improved experimental method that can prevent uneven
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deformation in the radial direction of freezing. This proposed method provides a strong
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constraint instead of the rubber sleeve in the radial direction so that we can avoid the
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occurrence of ice lenses between soil sample and rubber sleeve.
2. Novel frost heave cell
We propose an innovative structure of frost heave cell to provide a strong constraint in the radial direction of soil sample. Acrylic material is adopted for the frost heave cell. The thickness of the acrylic mold is 15 mm, which is thick enough to constrain radial deformation and effectively obtain the strain. This frost heave cell can prevent the occurrence of segregated ice lenses on the lateral surface of the frozen soil sample due to its rigid constraints. In addition, we separate the entire frost heave cell into 10 parts
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that are then connected by silicone sheets to reduce the influence of deformation and stress caused by volume expansion of the frozen section on the unfrozen section. These 10 acrylic rings are stacked to form the entire frost heave cell. The height of the acrylic rings (No. 1 to No. 8) containing the soil sample is 10 mm with a thickness of 15 mm.
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The heights of the top and bottom rings are 25 mm and 40 mm, respectively. We set
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strain gauges and thermal sensors in rings No. 1 to No. 8 to obtain the circumferential
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strain and thermal distribution along the freezing soil sample. Mini-sized water pressure
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gauges are set in rings No. 2, 4, and 6, respectively. Figure 1 illustrates the schematic
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this study.
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diagram of the experimental system and the details of this new frost heave cell used in
Figure 1 Schematic diagram of frost heave testing system and novel frost heave cell
The frost heave cell shown in the right of Figure 1 is positioned between the bottom 7
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pedestal and the upper pedestal which could move vertically as a piston. In addition, the pedestal surface consists of porous stone which allows water supply and drainage efficiently between soil sample and water supply system. Vertical confining pressure is applied by the load placed on the top plate of the rod. A double tube burette is used to
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supply distilled and de-aired water to the freezing sample from the bottom pedestal.
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Water supply is measured by a differential pressure transducer which can measure the
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pressure difference between the base of the water column in the double tube burette and
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the air. The frost heave is measured with a laser displacement sensor. Concerning this
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frost heave testing system, more details can be found in research from Niu et al. (Niu,
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Zheng, & Li, 2018). Figures 2 and 3 show the installation methods of the mini-sized
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water pressure gauge and the strain gauges in the ring of the frost heave cell. In this novel frost heave cell, thermal sensors are inserted into the acrylic rings so that we can observe the temperature distribution along the soil sample in real-time. In addition, it is assumed that the influence from neighboring frozen parts on unfrozen parts is much smaller when we connect the independent rings using a silicone sheet than in a completely rigid frost heave cell.
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Figure 2 Installation of mini-sized water pressure gauge
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2.1 Thick-walled cylinder theory
Figure 3 Installation of strain gauges
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With this frost heave cell, we can measure the circumferential strain (b) on the outer
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surface of the frost heave cell using strain gauges. However, our aim in this study is to
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determine the stress or strain in the radial direction of the frozen soil ( r (a) or 𝜀𝑟 (𝑎)).
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Therefore, we need to calculate the radial stress inside the frost heave cell based on the measured outer circumferential strain (b) . Figure 4 illustrates the relative location of the measured strain ( (b) ) and the objective strain (𝜀𝑟 (𝑎)). a and b are the radii of the inner and outer surface of the frost heave cell. As shown in Figure 2, the ring of the frost heave cell is an ideal thick-walled cylinder. Therefore, we adopt the thick-walled cylinder theory to calculate the radial stress produced by the frozen soil.
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Figure 4 Diagram of measured strain and objective strain on the frost heave cell
equation:
1 2 a b2
b2 a ( 1 2 ) a
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r (a)
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The radial stress acting on the inside of the frost heave cell is obtained by the following
u a 2(1 )b t u b
(1)
Here, is Poisson’s ratio. u a and u b are the displacements in the radial direction at a and b, respectively. t is expansion strain due to temperature change and
E . E is Young’s Modulus. Similarly, the radial stress acting on the (1 )(1 2 )
outside of the frost heave cell can be calculated as follows:
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r (b)
1 2(1 )a a b 2 2
b (1 2 )
a 2 u a t b u b
(2)
(3)
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u a r a 1 K t u b 1 r (b)
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Based on Equations (1) and (2), the following matrix equation can be obtained:
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And,
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a (1 2 ) b 2 a 1 a 2 b2 2(1 )a
2(1 )b 2 b (1 2 ) a b
(4)
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K
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In this study, we measure the circumferential strain b on the outer surface of the acrylic cell using strain gauges. Based on the measured b , the displacement in the radial direction can be calculated as u b b b . Then, considering the boundary conditions on the outer surface of the acrylic cell, we know that r b 0 . We ignore the expansion strain t of frost heave cell due to temperature change in the thick-walled cylinder theory. Instead, we compensate the thermal influence on the whole experimental system comprehensively by the data obtained from validation test for
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strain gauges. Therefore, by substituting u b b b and r b 0 into Equation (2), we can obtain the displacement u a . Consequently, we can substitute u a and
u b into Equation (1) to obtain the radial stress inside the acrylic cell r a . The
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details concerning this derivation can be found in Appendices A.
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2.2 Equipment verification
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Before initiating the frost heave test and evaluating the stress produced by frost heave in
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the radial direction of freezing, it is important to verify the accuracy of the sensors used
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in this frost heave cell. Therefore, a series of verification tests have been conducted to
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confirm the reliability of the sensors. First, we conduct a pressure test with pure water to
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check whether the test instrument can accurately measure the pressure or if some correction is required. As the experimental result shown in Figure 5, we fill the frost heave cell with pure water, increase the water pressure from 0 kPa to 500 kPa, then reduce the water pressure back to 0 kPa. The pressure value of the strain gauge is calculated using the thick-walled cylinder theory. Figure 5 verifies the effectiveness of both the mini-sized water pressure gauges and strain gauges, which respond identically to the change in water pressure. Therefore, we can conclude that the instruments are reliable for measuring pressure changes accurately.
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After confirming the working status of the sensors under various pressures, we then validate the temperature influence on the mini-sized water pressure gauges and strain gauges. Because during the frost heave experiment, the temperature will change with
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time. If the value of the measuring instrument fluctuates with temperature, which can
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lead to thermal expansion, we will examine the measured value to verify whether a
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correction is required. We fill the frost heave cell with pure water again, decrease the
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water temperature over 20 h from 1 ºC to -1 ºC, then compare the pressures obtained by
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the strain gauges with those obtained by mini-sized water pressure gauges. The
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experimental result is shown in Figure 6. As the water pressure gauge measures the
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water pressure directly, the observed values are not affected by the change of water temperature. In contrast, the pressure obtained by the strain gauge may include thermal influence on frost heave cell and other parts of the system. It is noted that the values obtained from strain gauges exhibit slight drift with an increase of temperature. Assuming the relationship between temperature and pressure obtained by strain gauge is linear, we introduce a correction coefficient derived from a regression formula. Based on the experimental data, the thermal influence on the stress obtained by strain gauge is 7 kPa/ºC. Therefore, we compensate the radial stress by this value as a comprehensive
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correction for thermal influence. In this study, all the radial stresses obtained by strain gauges are corrected by considering the thermal influence when temperature changes.
600
WPG SG
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500
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300
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200
100
0 2
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0
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Pressure (kPa)
400
4
6
8
10
Time (hr)
Figure 5 Pressure verification of the mini-sized water pressure gauge (WPG) and strain gauge (SG)
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30
SG WPG
20
2
10
0
-10
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Pressure (kPa)
R =0.92
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-20
-30 -1.0
-0.5
0.0
0.5
1.0
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Temperature (C)
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Figure 6 Temperature verification of the water pressure gauge (WPG) and strain gauge
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(SG)
3. Frost heave experiment
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3.1 Experimental conditions
After confirming the performance of the proposed frost heave cell, we conduct a frost heave test using typical diluvial silt known as Dotan. The physical properties of Dotan are shown in Table 1. The purpose of this experiment is to show the applicability of the newly designed frost heave cell and thick-walled cylinder theory. Further, it can be used as a preliminary study of the stress produced by frost heave in the radial direction of freezing. Before the experiment, Dotan is dried and powdered to fine particles. Then, it is stirred into a slurry with lots of water. This slurry is then placed into a vacuum 15
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container for de-airing processing. After that, this de-aired slurry is poured into the frost heave cell and consolidated by the application of a load on top of the upper pedestal, which is 100 kPa in this study. This experiment is conducted in a low-temperature room where room temperature is set to approximately 1 ºC. We adopt the ramped freezing
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condition for this experiment. This freezing condition changes the temperatures of the
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upper and the bottom of the soil sample with time. But the temperature difference
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between upper and bottom of the soil sample keeps constant in order to guarantee a
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constant freezing rate (Konrad J.-M. , 1989), which is 1.0 mm/hr in this study. The
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initial height of the sample is 92 mm. Based on the designed thermal gradient (0.1 ºC
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/mm), we set the temperature difference between the cold side and warm side as 9.2 ºC.
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We need about 92 hours to freeze the whole soil sample considering the designed freezing rate. Experimental conditions are summarized in Table 2. During the experiment, the following values are measured: frost heave amount in freezing direction, absorbed water amount, temperature of upper and bottom pedestals, temperature of each ring, and strains obtained by the strain gauges. Each measured value is recorded by the data logger every 5 min.
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Table 1 Soil properties of Dotan 2700 kg/m3 62.3% 41.9%
Density of soil particle Liquid limit Plastic limit Grain size content 2 - 75 mm 0.075 - 2 mm 0.005 – 0.075 mm
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0% 24% 48% 28%
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Smaller than 0.005 mm
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Table 2 Experimental conditions Frost-susceptible material
Reconstructed sample of Dotan (saturated) 100 kPa
Height of sample
92 mm 1.0 mm/hr 0.1 ºC /mm
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Thermal gradient
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Freezing rate
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Constraining stress 1 in freezing direction
3.2 Experimental results
In this section, we present the results of the frost heave test by Dotan in order to illustrate the characteristics of radial stress caused by frost heave. The displacement in the freezing direction over time, the absorbed water, and the temperatures of upper and bottom pedestals (Tc and Tw, respectively) are shown in Figure 7. Here Tc represents the temperature of upper pedestal and Tw means the temperature of bottom pedestal. Temperatures of these two pedestals are changed with time to realize the ramped 17
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freezing condition. For the sake of clarity, note that “Absorbed water” in Figure 7 is in unit of millimeter, not milliliter. In Figure 7, “Absorbed water” is the real water amount absorbed to soil sample divided by the cross area of the soil sample. Therefore, it is given in millimeter and can be regarded as the height when it is in a cylinder as the frost
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heave cell (height=volume/area). This height is used for direct comparison with the
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“Displacement” and they exhibit clear similarities in their trends. This is reasonable
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because frost heave is mainly produced by the absorbed water to the soil sample. This
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absorbed water is driven by the suction pressure and flows to a freezing front where ice
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lenses occur (Konrad & Shen, 1996). Figure 7 verifies that the temperature control
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system in the upper and bottom pedestals is effective during the experiment, and the
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experiment can be performed properly according to the setting conditions of the freezing rate and thermal gradient. After the freezing period, the total frost heave is approximately 15 mm. Given the original height of the Dotan sample (92 mm), we infer a frost heave ratio of approximately 16.3%.
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Displacement Absorbed water Tw Tc
25 20
5
15
0
10
-5
5
-10
0
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10
-15 -20 20
40
60
-5
80
-10 100
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0
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15
30
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Time (hr)
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Figure 7 Frost heave in the freezing direction and absorbed water (Dotan, 1 =100 kPa)
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The increase of radial stress with time obtained from the strain gauges attached to the outer surface of each ring is shown in Figure 8. We note that the radial stress increases significantly in about every 10 hours from SG 1 to SG 8 in turn (SG: strain gauge). This is identical to the freezing rate of our experimental conditions. As the height of each ring is 10 mm and the freezing rate is 1 mm/h, 10 hours imply the freezing time for each ring. After the rapid increase, the radial stress increases at a slower rate until the experiment ends. Finally, approximately 90 h after freezing began, the radial stress of each ring becomes stable and does not increase further. In addition, we find that before
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Temperature ( C)
20
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Displacement and absorbed water (mm)
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the increase of radial stress, there is a stage of decrease and the radial stress becomes negative. In our opinion, this negative radial stress is caused by drainage consolidation when the soil sample is freezing. This consolidation during freezing is the result of suction pressure that drives water to the freezing front. This suction pressure causes an
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increase of effective stress and therefore further consolidation occurs (Chamberlain,
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1981) (Konrad & Samson, 2000). And this negative radial stress is different from the
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frost shrinkage reported by Liu et al. (Liu, Liu, Li, & Fang, 2019). In their paper, they
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report that frost shrinkage will occur in unsaturated soil with a low degree of saturation.
700
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SG-1 SG-2 SG-3 SG-4 SG-5 SG-6 SG-7 SG-8
600 500
Radial stress (kPa)
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However, the soil sample used in our experiment is in a saturated condition.
400 300 200 100 0 -100 0
20
40
60
Time (hr)
20
80
100
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Figure 8 Radial stress obtained by the strain gauges (SG) against time (Dotan, 1 =100 kPa)
Furthermore, the radial stress varies with the change of temperature in each ring (Figure
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9), as revealed by thermal sensors inserted into the frost heave cell. We note that the
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stress increases drastically when the temperature is between 0 ºC and -2 ºC. Before this
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sudden increase, the radial stress does not increase until 0 ºC, which demonstrates that
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the rings of the frost heave cell, which are connected by silicone sheets, do not affect
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each other, even though some experience early expansion due to frost heave. When the
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temperature of the ring drops below -2 ºC, the radial stress gently increases. The
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increase of radial stress shows a bi-linear trend (red lines in Figure 9). This indicates that the formation of significant ice lenses has finished when the temperature reaches -2 ºC. For the second linear increase of radial stress which has a slower increasing rate, we think that it is not produced by the significant ice lens. The slow increase of radial stress may be caused by the reduction of unfrozen water content in the frozen soil. As the temperature becomes colder, the unfrozen water between the soil particles is frozen into ice and some volume expansion occurs (Watanabe & Wake, 2009), resulting in an increase of radial stress. And this transformation of unfrozen water does not produce
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significant ice lens. When the temperature closes to -10 ºC, the radial stresses in all eight rings do not increase further. For the radial stress, at the first red line, it increases from 0 to about 300 kPa and at the second red line, it increases from 300 kPa to 500 kPa. If we assume the increase of radial stress at the first red line from 0 to about 300 kPa
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results from the significant ice lenses and the increase at the second red line from 300
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kPa to 500 kPa is a result of the reduction of unfrozen water content, it seems that the
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significant ice lens does not produce much more radial stress compared to the unfrozen
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water reduction in the second red line. This is because the ice lens grows in freezing
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direction and most of the volume expansion is also distributed to freezing direction,
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resulting in a limited increase of radial stress. The increasing trend is very similar for all
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rings of frost heave cell. This bi-linear increasing trend of radial stress produced by frost heave is quite different from the stress in the freezing direction (Wang & Zhou, 2018). In addition, in this study, the local water migration within each ring is not considered because we freeze the whole surface of the soil sample simultaneously. Therefore, water migration in the horizontal direction can be neglected. Furthermore, for the vertical water migration, frozen soil has low permeability and as the temperature of frozen soil decreases, the amount of unfrozen water becomes limited as well. Consequently, it is difficult to absorb water from the frozen soil to the ice lens. The most possible path for
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water migration in vertical direction is from the unfrozen soil to the ice lens which is just the direction of water supply in this experiment.
700
400
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300 200
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Radial stress (kPa)
500
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600
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SG-1 SG-2 SG-3 SG-4 SG-5 SG-6 SG-7 SG-8
0 -100 0
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2
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100
-2
-4
-6
-8
-10
Temperature (C)
Figure 9 Radial stress obtained by the strain gauges (SG) against temperature (Dotan, 1 =100 kPa)
In order to confirm the reduction of unfrozen water content in the frozen Dotan, we adopt the pulsed nuclear magnetic resonance (NMR) method. This method employs a 90° pulse to acquire the free induction decay (FID) curve, which consists of FID components from pore ice and unfrozen water. The FID of pore ice has a short decay
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time in the range of tens of microseconds, whereas the FID of unfrozen water has a much longer decay time. Accordingly, the FID of unfrozen water can be determined (Akagawa, Iwahana, Watanabe, Chuvilin, & Istomin, 2012). The preparing method for the sample used for the unfrozen water content test is the same as that of the freezing
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test. Then we cut this soil sample into small pieces and measure the unfrozen water
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content under different temperatures. The details of pulsed NMR test can be found in
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the research from Akagawa et al. (Akagawa, Iwahana, Watanabe, Chuvilin, & Istomin,
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2012).
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Figure 10 shows that as temperature drops, the unfrozen water content decreases as well.
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When the temperature approaches -10 ºC, the unfrozen water content comes to stable and is no longer reduced. Similarly, the radial stress shown in Figure 9 also reaches a stable value when the temperature approaches -10 ºC. This agreement supports our explanation that the increasing rate of radial stress is clearly affected by the forming rate of ice. When the temperature decreases from 0 ºC to -2 ºC, segregated ice lenses appear quickly and produce significant volume expansion. This leads to a rapid increase of the radial stress perpendicular to the freezing direction. However, when temperature drops from -2 ºC to -10 ºC, no further segregated ice lenses occur. During this stage, as
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temperature decreases, the amount of unfrozen water between soil particles decreases as well. This reduction of unfrozen water content also leads to volume expansion because unfrozen water becomes ice. However, the amount of volume expansion produced at this stage (from -2 ºC to -10 ºC) is relatively smaller and slower compared with that
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produced by significant ice lenses (from 0 ºC to -2 ºC), but also contributes to the
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increase of radial stress.
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40
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30 25 20
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Unfrozen water content (%)
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15 10 5 0 -10
-8
-6
-4
-2
0
Temperature (C) Figure 10 Unfrozen water content of Dotan under different temperatures measured by pulsed NMR test
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4. Conclusions This research is aimed to further our understanding of stress in the radial direction of freezing by designing a novel frost heave cell with various sensors to measure the circumferential strain caused by frost heave. We then develop a corresponding
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calculation method for stress in the radial direction, which is perpendicular to the
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freezing direction. The applicability and accuracy of this novel frost heave cell are
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successfully evaluated through verification tests. We then conduct a frost heave test
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using Dotan to show the potential applications of this method and clarify the increasing
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findings.
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trend of stress in the radial direction of freezing. As a result, we obtain the following
1. The newly designed frost heave cell and thick-walled cylinder theory are successfully applied to evaluating the radial stress produced by frost heave. 2. In the experiment using Dotan, a bi-linear increasing trend of radial stress is observed. The turning point of this increasing trend is found to be -2 ºC, after which the rate of radial stress increase is reduced. 3. The radial stress in the first linear increase from 0 ºC to -2 ºC is produced by significant ice lens. The radial stress in the second linear increase from -2 ºC to -8
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ºC is produced by the reduction of unfrozen water content in the frozen soil. The significant ice lens does not produce much more radial stress compared with the reduction of unfrozen water content because significant ice lens grows in freezing direction and most of the volume expansion is distributed to freezing direction,
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resulting in a limited increase of radial stress.
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4. The unfrozen water content under various temperatures obtained by the pulsed NMR
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test confirms that the increasing rate of radial stress is dominated by the formation
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of significant ice lenses and the reduction of unfrozen water content.
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In conclusion, the evaluation of three-dimensional stress conditions during freezing
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should be seriously considered when applying artificial freezing technology to large projects in urban areas. Substantial further research is required because the increasing trend of radial stress is complicated. In future, we are going to conduct a series of frost heave experiments under different conditions with different materials in order to investigate the characteristics and mechanism of the radial stress caused by freezing soil. We hope we can distinguish the contributions from ice lens and the reduction of unfrozen water content. It is expected that through our research concerning three-dimensional frost heave, frost heave in three-dimensional space and its
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corresponding stress can be predicted properly to alleviate the turbulence to surrounding structures. Furthermore, we also hope our research can contribute to the development in cold regions.
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Acknowledgments
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This work was partly supported by Ueda Memorial Foundation Research Grant [ grant
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numbers: 09Rm01-01]. In addition, we appreciate the anonymous reviewers and the
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Appendices A
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editors for their helpful comments on this manuscript.
In this section, we illustrate the details of the thick-walled cylinder theory to calculate the radial stress produced by freezing soil. The equilibrium equation of this frost heave cell in the polar coordinate system is derived first. The polar coordinate system and microelement of frost heave cell are illustrated in Figure A-1. The components of stress with respect to the small angle is shown in Figure A-2.
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Figure A-1 The balance of microelement in the polar coordinate system
Figure A-2 Components of stress acting on the surface of the microelement
According to Figure A-1, the radial component of force due to the radial normal stress is
r dr r dr d r rd r r 29
(A-1)
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According to Figure A-2, the radial component of force due to the circumferential normal stress is
(A-2)
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d d d dr sin dr sin 2 2
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According to Figure A-2, the radial component of force due to the shear stress is
(A-3)
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d d r dr cos r r d dr cos 2 2
R is the body force in the radial direction acting on the per unit area. Thus, considering the balance of forces in the radial direction, we can obtain the following equation:
r r 2 drd dr d r rd r r d d d dr sin drd sin dr sin 2 2 2 d r d d r dr cos drd cos r dr cos 2 2 2 Rdrd 0
r rd r drd r
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(A-4)
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Then, we reorganize the Equation (A-4) as follows:
r d drd r dr 2 d 2 dr sin r r 2 d r d drd sin drd cos Rdrd 0 2 2
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r drd r
sin
d d , 2 2
d 1 .We can ignore the high-order of these small terms. Accordingly, Equation 2
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cos
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Here, because d is infinitesimal, we can approximate that
(A-5)
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r drd drd r drd Rdrd 0 r
(A-6)
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r drd r
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(A-5) can be further simplified:
Dividing both sides of Equation (A-6) by rdrd , the above equation can be rewritten as:
r 1 r r R 0 r r r r
(A-7)
Equation (A-7) is, therefore, the equilibrium equation in the radial direction of this frost heave cell.
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As the problem being considered is an axisymmetric problem in which isotropy is maintained, there is no change in stress along the circumferential direction; i.e., r 0 . Moreover, if we do not consider the body force (R=0), Equation (A-7) can be
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(A-8)
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d r 1 r 0 dr r
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transformed as follows:
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On the other hand, the stress and strain relationship, in this case, can be considered as
1 (1 ) r t E
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the plane strain state when expansion strain ( t ) due to freezing is considered.
Here,
1 r (1 ) t E
(A-9)
(A-10)
r and are the strains in the radial and circumferential directions,
respectively. E is Young’s Modulus. is Poisson’s ratio. Strain can be expressed by displacement as follows:
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r
du dr
u r
(A-11)
Here, u is the displacement in the radial direction and r is the radius.
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1 ( r ) E
(A-12)
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Subtracting Equation (A-10) from Equation (A-9), the following equation is established:
E E ( r ) 1 1
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Then, by substituting Equation (A-11) into Equation (A-12), we obtain:
du u dr r
(A-13)
According to the relationship between stress and strain in the plane strain condition, we can obtain the following equation:
r
E du (1 ) (1 )(1 2 ) dr
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u r
(A-14)
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We then differentiate both sides of Equation (A-14) with respect to r,
d r E d 2u u 1 du (1 ) 2 2 dr (1 )(1 2 ) dr r dr r
(A-15)
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Substituting Equation (A-13) and Equation (A-15) into Equation (A-8) and reorganizing
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it produces the following equation:
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E d 2u u 1 du 1 E ( 1 ) 2 2 (1 )(1 2 ) dr r dr r 1 r
du u 0 dr r
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1 d 2u u 1 du 1 du u 0 (1 ) 2 2 1 2 dr r dr r dr r 2 r d 2u 1 2 du du u u ( 1 ) (1 2 ) 2 2 0 2 dr r dr r dr r r 1 d 2u 1 du u 2 0 (1 ) 2 (1 ) 1 2 dr r dr r
(A-16)
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1 1 2
d 2u 1 du u 0 dr 2 r dr r 2
The above equation can be transformed as follows:
d 1 d (ru) 0 dr r dr
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(A-17)
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By integrating Equation (A-17) twice, the displacement can be expressed by the following equation including two unknown coefficients C1 and C2.
1 1 C1r C2 2 r
(A-18)
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u
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According to Equation (A-18), the displacements u a and ub can be expressed as the
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following equation. Here, a is the inner radius and b is the outer radius.
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u (a) a 2 1 a C1 u (b) b 2 1 b C2
r u
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According to Equation (A-11), we can obtain the following relationship:
r 1 du d (r ) C1 C2 r 2 r dr dr 1 1 1 1 C1 2 C2 r C1 2 C2 2 r 2 r r
The above equations can be expressed in matrix form as:
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(A-19)
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r (r ) 1 2 r 1 2
1 r 2 C1 1 r 2 C2
(A-20)
Then, we also combine Equation (A-9) and (A-10) into a matrix form:
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r (r ) 1 t 1 (r ) 1
(A-21)
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r (r ) 1 1 E r
The above equation can also be expressed as follows:
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r (r ) 1 E t 1 (r ) (1 )(1 2 ) 1
(A-22)
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r (r ) 1 E r (1 )(1 2 )
r (r )
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If we extract the radial direction (r direction) from Equation (A-22), we obtain
(r ) 1 E E 1 r t (1 )(1 2 ) (r ) (1 )(1 2 ) 1
(A-23)
When we focus on the stress at the inner side of frost heave cell a, it becomes:
r (a)
E 1 (1 )(1 2 )
(a) 1 E r t (a) (1 )(1 2 ) 1
(A-24)
Based on Equation (A-19) and (A-20), the strain can be expressed by displacement, 36
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r (a) 1 2 a 1 2
1
1 a 2 a 2 1 / a u (a) 1 a 2 b 2 1 b u (b)
(A-25)
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Then, substituting Equation (A-25) into Equation (A-24), we obtain:
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1 2 1 a 2 a 2 1 a u a E 1 r (a) 2 (1 )(1 2 ) 1 2 1 a b 2 1 b u b
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E t (1 )(1 2 )
(A-26)
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1
1 2 1 2
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Here,
1 a 2 a 2 1 a 1 a 2 b 2 1 b
1
a b 2 a 1 a 2 b 2 a b 2 a
2b 0
(A-27)
Therefore, by substituting Equation (A-27) into Equation (A-26), the following equation can be obtained:
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r (a)
a b 2 a 2b u a E 1 E 1 t 2 2 2 (1 )(1 2 ) a b 0 u b (1 )(1 2 ) a b a
r (a)
1 2 a b2
u a b2 b2 2(1 )b ( 1 ) a a t u b a a
r (a)
1 2 a b2
b2 a ( 1 2 ) a
Here
E . From the above equation, the radial stress acting on the inside (1 )(1 2 )
(A-28)
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of
u a 2(1 )b t u b
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of the microelement is obtained (Equation (A-28)). Similarly, the radial stress acting on
1 2(1 )a a b 2 2
b (1 2 )
a 2 u a t b u b
(A-29)
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r (b)
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the outside of the microelement can be calculated as follows:
From Equations (A-28) and (A-29), the following matrix equation is obtained:
u a r a 1 K t u b 1 r (b)
And,
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(A-30)
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K 2 1 2 a b
a (1 2 ) b 2 a 2(1 )a
2(1 )b b (1 2 ) a 2 b
(A-31)
In this study, as explained in the previous section, we measure the circumferential strain
b on the outer surface of the acrylic cell using strain gauges. Based on the
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measured b , the displacement in the radial direction can be calculated as
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u b b b . Then, considering the boundary conditions on the outer surface of the
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acrylic cell, we know that r b 0 . We ignore the expansion strain t of frost heave
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cell due to temperature change in the thick-walled cylinder theory. Instead, we
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compensate the thermal influence on the whole experimental system comprehensively
and
r b 0 into Equation (A-29), we can obtain the
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u b b b
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by the data obtained from validation test for strain gauges. Therefore, by substituting
displacement u a .
u a =
b
b 2a 1 v
2
a 2 1 2v
(A-32)
Consequently, we can substitute u a and u b into Equation (A-28) to obtain the radial stress inside the acrylic cell r a . This is the final goal.
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References
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Afshani A., & Akagi H. (2015). Artificial Ground Freezing Application in Shield Tunneling. Japanese Geotechnical Society Special Publication, 3(2), pp. 71-75. doi:https://doi.org/10.3208/jgssp.v03.j01 Akagawa S., Iwahana G., Watanabe K., Chuvilin E., & Istomin V. (2012). Improvement of Pulse-NMR Technology for Determining the Unfrozen Water Content in Frozen Soils. Tenth International Conference on Permafrost, 2, pp. 21-26. Salekhard, Russia. Arenson U.L., Azmatch F.T., Sego D., & Biggar W.K. (2008). A New Hypothesis on Ice Lens Formation in Frost-Susceptible Soils. Proceedings of the Ninth International Conference on Permafrost. 1, pp. 59-64. Fairbanks, Alaska. Budzik P. (2009). Arctic Oil and Natural Gas Potential. U.S. Energy Information Administration, Office of Integrated Analysis and Forecasting, Oil and Gas Division. Chamberlain Edwin J. (1981). Overconsolidation effects of ground freezing. 40
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Engineering Geology, 18, 97-110. Edlefsen N., & Anderson A. (1943). Thermodynamics of Soil Moisture. Hilgardia, 15(2), pp. 31-298. Konrad, J. M. (1989). Influence of overconsolidation on the freezing characteristics of a clayey silt. Canadian Geotechnical Journal, 26(1), pp.9-21. Konrad J.M. (1994). Frost Heave in Soils: Concepts and Engineering. Canadian Geotechnical Journal, 31(2), pp. 223-245. doi:https://doi.org/10.1139/t94-028 Konrad J.M., & Morgenstern N.R. (1981). The Segregation Potential of a Freezing Soil. Canadian Geotechnical Journal, 18(4), 10. Konrad, J.-M., Shen M. (1996). 2-D frost action modeling using the segregation
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potential of soils. Cold Regions Science and Technology, 24, 263-278. Konrad Jean-Marie, Samson Martin. (2000). Hydraulic conductivity of kaolinite–silt mixtures subjected to closed-system freezing and thaw consolidation. Canadian Geotechnical Journal, 37(4), 857-869. Lai Y., Pei W., Zhang M., & Zhou J. (2014). Study on Theory Model of Hydro-Thermal–Mechanical Interaction Process in Saturated Freezing Silty Soil. International Journal of Heat and Mass Transfer, 78, pp. 805-819. Liu, Z., Liu, J., Li, X., & Fang, J. (2019). Experimental Study on the Volume and Strength Change of An Unsaturated Silty Clay upon Freezing. Cold Regions Science and Technology, 157, pp. 1-12.
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Mackay R.J. (1973). The Growth of Pingos, Western Arctic Coast, Canada. Canadian Journal of Earth Sciences, 10(6), 979-1004. doi:https://doi.org/10.1139/e73-086 Michalowski L.R. (1993). A Constitutive Model of Saturated Soils for Frost Heave Simulations. Cold Regions Science and Technology, 22(1), pp. 47-63. https://doi.org/10.1016/0165-232X(93)90045-A Michalowski L.R., & Zhu M. (2006). Frost Heave Modelling using Porosity Rate Function. International Journal for Numerical and Analytical Methods in Geomechanics, 30(8), pp. 703-722. doi:10.1002/nag.497 Miller D.R. (1978). Frost Heaving in Non-Colloidal Soils. Proceedings 3th International Conference on Permafrost, 1, pp. 708-713. Niu, F., Zheng, H., & Li, A. (2018). The study of frost heave mechanism of high-speed railway foundation by field-monitored data and indoor verification experiment. Acta Geotechnica, pp. 1-13. O’Neill K., & Miller D.R. (1985). Exploration of a Rigid-Ice Model of Frost Heave. Water Resources Research, 21(3), pp. 281–296. Pissart A. (2002). Palsas, Lithalsas and Remnants of These Periglacial Mounds. A 41
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Progress Report. Progress in Physical Geography, 26(4), pp. 605-621. Razaqpur G.A., & Wang D. (1996). Frost-induced Deformations and Stresses in Pipelines. International Journal of Pressure Vessels and Piping, 69(2), pp. 105-118. Rempel W.A. (2007). Formation of Ice Lenses and Frost Heave. Journal of Geophysical Research, 112(F2). doi:https://doi.org/10.1029/2006JF000525 Taber S. (1929). Frost Heaving. The Journal of Geology, 37, pp. 428-461. Taber S. (1930). The Mechanics of Frost Heaving. The Journal of Geology, 38(4), pp. 303-317. Takashi, T., Yamamoto, H., Ohrai, T., & Masuda, M. (1978). Effect of Penetration Rate of Freezing and Confining Stress on The Frost Heave Ratio of Soil. Third
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International Conference on Permafrost, pp. 737-742. Tsutomu T., Ohrai T., Yamamoto H., & Okamoto J. (1981). Upper Limit of Heaving Pressure Derived by Pore-Water Pressure Measurements of Partially Frozen Soil. Engineering Geology, 18(1), pp.245-257. Ueda Y., Ohrai T., & Tamura T. (2007). Three-dimensional Ground Deformation Analysis with Frost Heave Ratios of Soil. Doboku Gakkai Ronbunshuu C, 63, pp. 835-847. Wang, P., & Zhou, G. (2018). Frost-heaving pressure in geotechnical engineering materials during freezing process. International Journal of Mining Science and Technology, 28(2), pp. 287-296.
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Watanabe K., Wake T. (2009). Measurement of Unfrozen Water Content and Relative Permittivity of Frozen Unsaturated Soil using NMR and TDR. Cold Regions Science and Technology, 59(1), pp. 34-41. https://doi.org/10.1016/j.coldregions.2009.05.011 Zent P.A., Sizemore G.H., & Remple W.A. (2011). Ice Lens Formation and Frost Heave at the Phoenix Landing Site. 2011 International Conference: Exploring Mars Habitability. Lisbon, Portugal. https://ntrs.nasa.gov/search.jsp?R=20110014308 Hao Zheng and Shunji Kanie. (2014). Two-dimensional Frost Heave Evaluation by Numerical Model Considering Nonlinear Elasticity of Unfrozen Soil, Journal of Seppyo, 76(6), pp. 461-480. ISSN0373-1006, The Japanese society of Snow and Ice.
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The authors declare no conflict of interest.
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Highlights:
Designing a novel frost heave cell to avoid uneven radial deformation
Establishing a calculating method of the radial stress in freezing direction
Investigating the characteristics of radial stress during freezing of soil
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Hao Zheng, Shunji Kanie, Satoshi Akagawa PII:
S0165-232X(19)30079-5
DOI:
https://doi.org/10.1016/j.coldregions.2019.102899
Reference:
COLTEC 102899
To appear in:
Cold Regions Science and Technology
Received date:
7 February 2019
Revised date:
11 September 2019
Accepted date:
19 September 2019
Please cite this article as: H. Zheng, S. Kanie and S. Akagawa, A method for obtaining radial stress in the freezing direction in frozen soil samples, Cold Regions Science and Technology(2019), https://doi.org/10.1016/j.coldregions.2019.102899
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier.
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A method for obtaining radial stress in the freezing direction in frozen soil samples
Hokkaido
University,
2 Cryosphere Engineering Laboratory, Tokyo, Japan
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Hao Zheng1*, Shunji Kanie1, Satoshi Akagawa2
1
Faculty
of
Engineering,
Japan,
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[email protected]
Sapporo,
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Abstract:
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In this study, we design an innovative frost heave cell to investigate the characteristics
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of stress in the radial direction during the freezing of a soil sample. The frost heave cell
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is divided into several independent rings connected by silicone sheets, which aims to reduce the deformation influence of the frozen part on the unfrozen part. By installing mini-sized water pressure gauges, strain gauges, and thermal sensors in the frost heave cell, the circumferential strain measured by strain gauges on the surface of frost heave cell can be effectively evaluated. In order to transfer this circumferential strain into stress in the radial direction, we adopt the thick-walled cylinder theory. Then, we conducted one freezing test with Dotan to investigate the characteristics of radial stress during freezing. The results suggest that the radial stress caused by frost heave increases
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rapidly when the temperature drops from 0 ºC to -2 ºC. However, as the temperature decreases below -2 ºC, the radial stress gradually increases. When the temperature approaches -10 ºC, the radial stress plateaus and does not increase further. These findings indicate the unique characteristics of radial stress perpendicular to the freezing
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direction and suggest the need for further research on this topic. As more complicated
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applications of artificial freezing method in underground space of urban area are
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expected, the influence on the surrounding structures brought by three-dimensional
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freezing shows its significance. Our research is aiming to investigate the mechanism of
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freezing phenomenon of soil in three-dimensional space and then promote the
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application of artificial freezing method in a wide area.
Keywords: frost heave; radial stress; freezing direction; thick-walled cylinder theory
1. Introduction As widely observed in nature, soil freezing process can lead to significant volumetric expansion due to the phase change of water and the water migration towards the freezing front. This volume expansion is called frost heave. With continuous water migration to the freezing front, this volume expansion could be much larger than that
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caused simply by the phase change from water to ice, which is approximately 9%. For example, palsas and pingos, which appear in permafrost areas, contain a large amount of ice lenses and can reach up to 70 m in height in the case of pingos (Mackay, 1973) (Pissart, 2002). Frost heave can cause significant damages to infrastructures such as
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highways, railways, bridges, and pipelines. It occurs not only on Earth but also on Mars
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(Zent, Sizemore, & Remple, 2011). However, frost heave research is relatively recent,
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even on Earth. In fact, the role of water migration in the freezing process was first
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recognized as late as 1929 (Taber, 1929) (Taber, 1930). The driving force of water
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migration arises from the suction pressure produced during freezing process.
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Continuous water migration under suction pressure results in the formation of a
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horizontal ice layer known as a segregated ice lens. This horizontal ice layer mainly grows in the freezing direction; therefore, major frost deformation appears in the freezing direction (Rempel, 2007) (Lai, Pei, Zhang, & Zhou, 2014). Due to this characteristic, the majority of research concerning frost heave phenomenon has been focused on the freezing direction.
The first wave of research on frost heave came in the 1970s due to the oil crisis and because of the large amount of oil and gas buried in cold regions. For example, the
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Arctic holds an estimated 22% of Earth's oil and natural gas resources (Budzik, 2009). At this period, the most important issue of frost heave for researchers and engineers is the precise prediction of the amount of frost heave during complex thermal-hydraulic and mechanical processes. To date, various theories and experimental equations have
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been proposed to explain the frost heave mechanism and predict the amount of frost
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heave. For example, the Generalized Clausius-Clapeyron model (Edlefsen & Anderson,
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1943), the rigid ice model (Miller, 1978) (O’Neill & Miller, 1985), Takashi’s model
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(Takashi, Yamamoto, Ohrai, & Masuda, 1978) (Tsutomu, Ohrai, Yamamoto, &
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Okamoto, 1981), the segregation potential model (Konrad & Morgenstern, 1981)
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(Konrad, 1994), and the porosity rate model (Michalowski, 1993) (Michalowski & Zhu,
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2006). These models have been used to calculate the deformation and stress caused by frost heave in order to guarantee safe construction (Zheng & Kanie, 2014). For example, the interaction between soil and a buried pipeline caused by frost heave has been analyzed using a numerical method and compared with a full-scale experiment conducted in Cane, France (Razaqpur & Wang, 1996).
The majority of previous frost heave research has explored the characteristics of frost heave in the freezing direction because most of the deformation occurs in this direction.
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However, recent laboratory investigations have demonstrated that the formation of ice lenses is not a simple one-dimensional process. Vertical ice lens structures were also observed (Arenson, Azmatch, Sego, & Biggar, 2008). As a consequence, frost heave and stress in the radial direction of freezing, which has largely been ignored in previous
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studies, should be carefully considered. Indeed, due to the wide application of artificial
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freezing technology in civil engineering, stress and deformation in the radial direction
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of freezing have begun to garner increasing attention because of the complicated
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freezing mechanism, especially in underground urban areas where artificial freezing
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may affect the foundations of surrounding structures. When numerous freezing pipes are
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installed in a construction site, segregated ice lenses produced around the freezing pipe
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scatter in three-dimensional space. Thus, the influence of frost heave is not limited to the direction of freezing (Afshani & Akagi, 2015). Accordingly, understanding the deformation and stress in the radial direction of freezing is necessary and urgent for the safe application of artificial freezing.
Ueda et al. proposed an empirical formula to express the strains in the freezing direction and its radial direction based on the stress relationship in each direction (Ueda, Ohrai, & Tamura, 2007). This empirical equation is derived from three-dimensional frost heave
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tests without horizontal restriction. This is a pioneering attempt to predict the frost heave ratio in three-dimensional space. In their frost heave test, horizontal deformation is allowed because they used a rubber sleeve to cover the soil sample. However, this unconstrained condition for the soil sample in the radial direction may not be effective
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when the segregated ice lens develops between the soil sample and the rubber sleeve.
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This can cause uneven deformation in the radial direction. Therefore, to solve this issue,
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in our study, we propose an improved experimental method that can prevent uneven
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deformation in the radial direction of freezing. This proposed method provides a strong
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constraint instead of the rubber sleeve in the radial direction so that we can avoid the
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occurrence of ice lenses between soil sample and rubber sleeve.
2. Novel frost heave cell
We propose an innovative structure of frost heave cell to provide a strong constraint in the radial direction of soil sample. Acrylic material is adopted for the frost heave cell. The thickness of the acrylic mold is 15 mm, which is thick enough to constrain radial deformation and effectively obtain the strain. This frost heave cell can prevent the occurrence of segregated ice lenses on the lateral surface of the frozen soil sample due to its rigid constraints. In addition, we separate the entire frost heave cell into 10 parts
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that are then connected by silicone sheets to reduce the influence of deformation and stress caused by volume expansion of the frozen section on the unfrozen section. These 10 acrylic rings are stacked to form the entire frost heave cell. The height of the acrylic rings (No. 1 to No. 8) containing the soil sample is 10 mm with a thickness of 15 mm.
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The heights of the top and bottom rings are 25 mm and 40 mm, respectively. We set
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strain gauges and thermal sensors in rings No. 1 to No. 8 to obtain the circumferential
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strain and thermal distribution along the freezing soil sample. Mini-sized water pressure
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gauges are set in rings No. 2, 4, and 6, respectively. Figure 1 illustrates the schematic
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this study.
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diagram of the experimental system and the details of this new frost heave cell used in
Figure 1 Schematic diagram of frost heave testing system and novel frost heave cell
The frost heave cell shown in the right of Figure 1 is positioned between the bottom 7
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pedestal and the upper pedestal which could move vertically as a piston. In addition, the pedestal surface consists of porous stone which allows water supply and drainage efficiently between soil sample and water supply system. Vertical confining pressure is applied by the load placed on the top plate of the rod. A double tube burette is used to
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supply distilled and de-aired water to the freezing sample from the bottom pedestal.
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Water supply is measured by a differential pressure transducer which can measure the
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pressure difference between the base of the water column in the double tube burette and
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the air. The frost heave is measured with a laser displacement sensor. Concerning this
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frost heave testing system, more details can be found in research from Niu et al. (Niu,
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Zheng, & Li, 2018). Figures 2 and 3 show the installation methods of the mini-sized
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water pressure gauge and the strain gauges in the ring of the frost heave cell. In this novel frost heave cell, thermal sensors are inserted into the acrylic rings so that we can observe the temperature distribution along the soil sample in real-time. In addition, it is assumed that the influence from neighboring frozen parts on unfrozen parts is much smaller when we connect the independent rings using a silicone sheet than in a completely rigid frost heave cell.
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Figure 2 Installation of mini-sized water pressure gauge
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2.1 Thick-walled cylinder theory
Figure 3 Installation of strain gauges
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With this frost heave cell, we can measure the circumferential strain (b) on the outer
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surface of the frost heave cell using strain gauges. However, our aim in this study is to
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determine the stress or strain in the radial direction of the frozen soil ( r (a) or 𝜀𝑟 (𝑎)).
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Therefore, we need to calculate the radial stress inside the frost heave cell based on the measured outer circumferential strain (b) . Figure 4 illustrates the relative location of the measured strain ( (b) ) and the objective strain (𝜀𝑟 (𝑎)). a and b are the radii of the inner and outer surface of the frost heave cell. As shown in Figure 2, the ring of the frost heave cell is an ideal thick-walled cylinder. Therefore, we adopt the thick-walled cylinder theory to calculate the radial stress produced by the frozen soil.
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Figure 4 Diagram of measured strain and objective strain on the frost heave cell
equation:
1 2 a b2
b2 a ( 1 2 ) a
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r (a)
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The radial stress acting on the inside of the frost heave cell is obtained by the following
u a 2(1 )b t u b
(1)
Here, is Poisson’s ratio. u a and u b are the displacements in the radial direction at a and b, respectively. t is expansion strain due to temperature change and
E . E is Young’s Modulus. Similarly, the radial stress acting on the (1 )(1 2 )
outside of the frost heave cell can be calculated as follows:
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r (b)
1 2(1 )a a b 2 2
b (1 2 )
a 2 u a t b u b
(2)
(3)
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u a r a 1 K t u b 1 r (b)
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Based on Equations (1) and (2), the following matrix equation can be obtained:
re
And,
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a (1 2 ) b 2 a 1 a 2 b2 2(1 )a
2(1 )b 2 b (1 2 ) a b
(4)
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K
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In this study, we measure the circumferential strain b on the outer surface of the acrylic cell using strain gauges. Based on the measured b , the displacement in the radial direction can be calculated as u b b b . Then, considering the boundary conditions on the outer surface of the acrylic cell, we know that r b 0 . We ignore the expansion strain t of frost heave cell due to temperature change in the thick-walled cylinder theory. Instead, we compensate the thermal influence on the whole experimental system comprehensively by the data obtained from validation test for
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strain gauges. Therefore, by substituting u b b b and r b 0 into Equation (2), we can obtain the displacement u a . Consequently, we can substitute u a and
u b into Equation (1) to obtain the radial stress inside the acrylic cell r a . The
of
details concerning this derivation can be found in Appendices A.
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2.2 Equipment verification
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Before initiating the frost heave test and evaluating the stress produced by frost heave in
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the radial direction of freezing, it is important to verify the accuracy of the sensors used
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in this frost heave cell. Therefore, a series of verification tests have been conducted to
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confirm the reliability of the sensors. First, we conduct a pressure test with pure water to
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check whether the test instrument can accurately measure the pressure or if some correction is required. As the experimental result shown in Figure 5, we fill the frost heave cell with pure water, increase the water pressure from 0 kPa to 500 kPa, then reduce the water pressure back to 0 kPa. The pressure value of the strain gauge is calculated using the thick-walled cylinder theory. Figure 5 verifies the effectiveness of both the mini-sized water pressure gauges and strain gauges, which respond identically to the change in water pressure. Therefore, we can conclude that the instruments are reliable for measuring pressure changes accurately.
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After confirming the working status of the sensors under various pressures, we then validate the temperature influence on the mini-sized water pressure gauges and strain gauges. Because during the frost heave experiment, the temperature will change with
of
time. If the value of the measuring instrument fluctuates with temperature, which can
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lead to thermal expansion, we will examine the measured value to verify whether a
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correction is required. We fill the frost heave cell with pure water again, decrease the
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water temperature over 20 h from 1 ºC to -1 ºC, then compare the pressures obtained by
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the strain gauges with those obtained by mini-sized water pressure gauges. The
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experimental result is shown in Figure 6. As the water pressure gauge measures the
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water pressure directly, the observed values are not affected by the change of water temperature. In contrast, the pressure obtained by the strain gauge may include thermal influence on frost heave cell and other parts of the system. It is noted that the values obtained from strain gauges exhibit slight drift with an increase of temperature. Assuming the relationship between temperature and pressure obtained by strain gauge is linear, we introduce a correction coefficient derived from a regression formula. Based on the experimental data, the thermal influence on the stress obtained by strain gauge is 7 kPa/ºC. Therefore, we compensate the radial stress by this value as a comprehensive
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correction for thermal influence. In this study, all the radial stresses obtained by strain gauges are corrected by considering the thermal influence when temperature changes.
600
WPG SG
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500
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300
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200
100
0 2
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0
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Pressure (kPa)
400
4
6
8
10
Time (hr)
Figure 5 Pressure verification of the mini-sized water pressure gauge (WPG) and strain gauge (SG)
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30
SG WPG
20
2
10
0
-10
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Pressure (kPa)
R =0.92
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-20
-30 -1.0
-0.5
0.0
0.5
1.0
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Temperature (C)
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Figure 6 Temperature verification of the water pressure gauge (WPG) and strain gauge
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(SG)
3. Frost heave experiment
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3.1 Experimental conditions
After confirming the performance of the proposed frost heave cell, we conduct a frost heave test using typical diluvial silt known as Dotan. The physical properties of Dotan are shown in Table 1. The purpose of this experiment is to show the applicability of the newly designed frost heave cell and thick-walled cylinder theory. Further, it can be used as a preliminary study of the stress produced by frost heave in the radial direction of freezing. Before the experiment, Dotan is dried and powdered to fine particles. Then, it is stirred into a slurry with lots of water. This slurry is then placed into a vacuum 15
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container for de-airing processing. After that, this de-aired slurry is poured into the frost heave cell and consolidated by the application of a load on top of the upper pedestal, which is 100 kPa in this study. This experiment is conducted in a low-temperature room where room temperature is set to approximately 1 ºC. We adopt the ramped freezing
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condition for this experiment. This freezing condition changes the temperatures of the
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upper and the bottom of the soil sample with time. But the temperature difference
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between upper and bottom of the soil sample keeps constant in order to guarantee a
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constant freezing rate (Konrad J.-M. , 1989), which is 1.0 mm/hr in this study. The
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initial height of the sample is 92 mm. Based on the designed thermal gradient (0.1 ºC
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/mm), we set the temperature difference between the cold side and warm side as 9.2 ºC.
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We need about 92 hours to freeze the whole soil sample considering the designed freezing rate. Experimental conditions are summarized in Table 2. During the experiment, the following values are measured: frost heave amount in freezing direction, absorbed water amount, temperature of upper and bottom pedestals, temperature of each ring, and strains obtained by the strain gauges. Each measured value is recorded by the data logger every 5 min.
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Table 1 Soil properties of Dotan 2700 kg/m3 62.3% 41.9%
Density of soil particle Liquid limit Plastic limit Grain size content 2 - 75 mm 0.075 - 2 mm 0.005 – 0.075 mm
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0% 24% 48% 28%
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Smaller than 0.005 mm
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Table 2 Experimental conditions Frost-susceptible material
Reconstructed sample of Dotan (saturated) 100 kPa
Height of sample
92 mm 1.0 mm/hr 0.1 ºC /mm
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Thermal gradient
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Freezing rate
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Constraining stress 1 in freezing direction
3.2 Experimental results
In this section, we present the results of the frost heave test by Dotan in order to illustrate the characteristics of radial stress caused by frost heave. The displacement in the freezing direction over time, the absorbed water, and the temperatures of upper and bottom pedestals (Tc and Tw, respectively) are shown in Figure 7. Here Tc represents the temperature of upper pedestal and Tw means the temperature of bottom pedestal. Temperatures of these two pedestals are changed with time to realize the ramped 17
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freezing condition. For the sake of clarity, note that “Absorbed water” in Figure 7 is in unit of millimeter, not milliliter. In Figure 7, “Absorbed water” is the real water amount absorbed to soil sample divided by the cross area of the soil sample. Therefore, it is given in millimeter and can be regarded as the height when it is in a cylinder as the frost
of
heave cell (height=volume/area). This height is used for direct comparison with the
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“Displacement” and they exhibit clear similarities in their trends. This is reasonable
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because frost heave is mainly produced by the absorbed water to the soil sample. This
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absorbed water is driven by the suction pressure and flows to a freezing front where ice
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lenses occur (Konrad & Shen, 1996). Figure 7 verifies that the temperature control
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system in the upper and bottom pedestals is effective during the experiment, and the
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experiment can be performed properly according to the setting conditions of the freezing rate and thermal gradient. After the freezing period, the total frost heave is approximately 15 mm. Given the original height of the Dotan sample (92 mm), we infer a frost heave ratio of approximately 16.3%.
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35
Displacement Absorbed water Tw Tc
25 20
5
15
0
10
-5
5
-10
0
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10
-15 -20 20
40
60
-5
80
-10 100
-p
0
o
15
30
re
Time (hr)
na
lP
Figure 7 Frost heave in the freezing direction and absorbed water (Dotan, 1 =100 kPa)
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The increase of radial stress with time obtained from the strain gauges attached to the outer surface of each ring is shown in Figure 8. We note that the radial stress increases significantly in about every 10 hours from SG 1 to SG 8 in turn (SG: strain gauge). This is identical to the freezing rate of our experimental conditions. As the height of each ring is 10 mm and the freezing rate is 1 mm/h, 10 hours imply the freezing time for each ring. After the rapid increase, the radial stress increases at a slower rate until the experiment ends. Finally, approximately 90 h after freezing began, the radial stress of each ring becomes stable and does not increase further. In addition, we find that before
19
Temperature ( C)
20
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Displacement and absorbed water (mm)
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the increase of radial stress, there is a stage of decrease and the radial stress becomes negative. In our opinion, this negative radial stress is caused by drainage consolidation when the soil sample is freezing. This consolidation during freezing is the result of suction pressure that drives water to the freezing front. This suction pressure causes an
of
increase of effective stress and therefore further consolidation occurs (Chamberlain,
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1981) (Konrad & Samson, 2000). And this negative radial stress is different from the
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frost shrinkage reported by Liu et al. (Liu, Liu, Li, & Fang, 2019). In their paper, they
re
report that frost shrinkage will occur in unsaturated soil with a low degree of saturation.
700
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SG-1 SG-2 SG-3 SG-4 SG-5 SG-6 SG-7 SG-8
600 500
Radial stress (kPa)
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However, the soil sample used in our experiment is in a saturated condition.
400 300 200 100 0 -100 0
20
40
60
Time (hr)
20
80
100
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Figure 8 Radial stress obtained by the strain gauges (SG) against time (Dotan, 1 =100 kPa)
Furthermore, the radial stress varies with the change of temperature in each ring (Figure
of
9), as revealed by thermal sensors inserted into the frost heave cell. We note that the
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stress increases drastically when the temperature is between 0 ºC and -2 ºC. Before this
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sudden increase, the radial stress does not increase until 0 ºC, which demonstrates that
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the rings of the frost heave cell, which are connected by silicone sheets, do not affect
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each other, even though some experience early expansion due to frost heave. When the
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temperature of the ring drops below -2 ºC, the radial stress gently increases. The
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increase of radial stress shows a bi-linear trend (red lines in Figure 9). This indicates that the formation of significant ice lenses has finished when the temperature reaches -2 ºC. For the second linear increase of radial stress which has a slower increasing rate, we think that it is not produced by the significant ice lens. The slow increase of radial stress may be caused by the reduction of unfrozen water content in the frozen soil. As the temperature becomes colder, the unfrozen water between the soil particles is frozen into ice and some volume expansion occurs (Watanabe & Wake, 2009), resulting in an increase of radial stress. And this transformation of unfrozen water does not produce
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significant ice lens. When the temperature closes to -10 ºC, the radial stresses in all eight rings do not increase further. For the radial stress, at the first red line, it increases from 0 to about 300 kPa and at the second red line, it increases from 300 kPa to 500 kPa. If we assume the increase of radial stress at the first red line from 0 to about 300 kPa
of
results from the significant ice lenses and the increase at the second red line from 300
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kPa to 500 kPa is a result of the reduction of unfrozen water content, it seems that the
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significant ice lens does not produce much more radial stress compared to the unfrozen
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water reduction in the second red line. This is because the ice lens grows in freezing
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direction and most of the volume expansion is also distributed to freezing direction,
na
resulting in a limited increase of radial stress. The increasing trend is very similar for all
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rings of frost heave cell. This bi-linear increasing trend of radial stress produced by frost heave is quite different from the stress in the freezing direction (Wang & Zhou, 2018). In addition, in this study, the local water migration within each ring is not considered because we freeze the whole surface of the soil sample simultaneously. Therefore, water migration in the horizontal direction can be neglected. Furthermore, for the vertical water migration, frozen soil has low permeability and as the temperature of frozen soil decreases, the amount of unfrozen water becomes limited as well. Consequently, it is difficult to absorb water from the frozen soil to the ice lens. The most possible path for
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water migration in vertical direction is from the unfrozen soil to the ice lens which is just the direction of water supply in this experiment.
700
400
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300 200
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Radial stress (kPa)
500
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600
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SG-1 SG-2 SG-3 SG-4 SG-5 SG-6 SG-7 SG-8
0 -100 0
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2
na
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100
-2
-4
-6
-8
-10
Temperature (C)
Figure 9 Radial stress obtained by the strain gauges (SG) against temperature (Dotan, 1 =100 kPa)
In order to confirm the reduction of unfrozen water content in the frozen Dotan, we adopt the pulsed nuclear magnetic resonance (NMR) method. This method employs a 90° pulse to acquire the free induction decay (FID) curve, which consists of FID components from pore ice and unfrozen water. The FID of pore ice has a short decay
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time in the range of tens of microseconds, whereas the FID of unfrozen water has a much longer decay time. Accordingly, the FID of unfrozen water can be determined (Akagawa, Iwahana, Watanabe, Chuvilin, & Istomin, 2012). The preparing method for the sample used for the unfrozen water content test is the same as that of the freezing
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test. Then we cut this soil sample into small pieces and measure the unfrozen water
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content under different temperatures. The details of pulsed NMR test can be found in
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the research from Akagawa et al. (Akagawa, Iwahana, Watanabe, Chuvilin, & Istomin,
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2012).
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Figure 10 shows that as temperature drops, the unfrozen water content decreases as well.
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When the temperature approaches -10 ºC, the unfrozen water content comes to stable and is no longer reduced. Similarly, the radial stress shown in Figure 9 also reaches a stable value when the temperature approaches -10 ºC. This agreement supports our explanation that the increasing rate of radial stress is clearly affected by the forming rate of ice. When the temperature decreases from 0 ºC to -2 ºC, segregated ice lenses appear quickly and produce significant volume expansion. This leads to a rapid increase of the radial stress perpendicular to the freezing direction. However, when temperature drops from -2 ºC to -10 ºC, no further segregated ice lenses occur. During this stage, as
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temperature decreases, the amount of unfrozen water between soil particles decreases as well. This reduction of unfrozen water content also leads to volume expansion because unfrozen water becomes ice. However, the amount of volume expansion produced at this stage (from -2 ºC to -10 ºC) is relatively smaller and slower compared with that
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produced by significant ice lenses (from 0 ºC to -2 ºC), but also contributes to the
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increase of radial stress.
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40
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30 25 20
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Unfrozen water content (%)
35
15 10 5 0 -10
-8
-6
-4
-2
0
Temperature (C) Figure 10 Unfrozen water content of Dotan under different temperatures measured by pulsed NMR test
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4. Conclusions This research is aimed to further our understanding of stress in the radial direction of freezing by designing a novel frost heave cell with various sensors to measure the circumferential strain caused by frost heave. We then develop a corresponding
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calculation method for stress in the radial direction, which is perpendicular to the
ro
freezing direction. The applicability and accuracy of this novel frost heave cell are
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successfully evaluated through verification tests. We then conduct a frost heave test
re
using Dotan to show the potential applications of this method and clarify the increasing
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findings.
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trend of stress in the radial direction of freezing. As a result, we obtain the following
1. The newly designed frost heave cell and thick-walled cylinder theory are successfully applied to evaluating the radial stress produced by frost heave. 2. In the experiment using Dotan, a bi-linear increasing trend of radial stress is observed. The turning point of this increasing trend is found to be -2 ºC, after which the rate of radial stress increase is reduced. 3. The radial stress in the first linear increase from 0 ºC to -2 ºC is produced by significant ice lens. The radial stress in the second linear increase from -2 ºC to -8
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ºC is produced by the reduction of unfrozen water content in the frozen soil. The significant ice lens does not produce much more radial stress compared with the reduction of unfrozen water content because significant ice lens grows in freezing direction and most of the volume expansion is distributed to freezing direction,
of
resulting in a limited increase of radial stress.
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4. The unfrozen water content under various temperatures obtained by the pulsed NMR
-p
test confirms that the increasing rate of radial stress is dominated by the formation
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of significant ice lenses and the reduction of unfrozen water content.
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In conclusion, the evaluation of three-dimensional stress conditions during freezing
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should be seriously considered when applying artificial freezing technology to large projects in urban areas. Substantial further research is required because the increasing trend of radial stress is complicated. In future, we are going to conduct a series of frost heave experiments under different conditions with different materials in order to investigate the characteristics and mechanism of the radial stress caused by freezing soil. We hope we can distinguish the contributions from ice lens and the reduction of unfrozen water content. It is expected that through our research concerning three-dimensional frost heave, frost heave in three-dimensional space and its
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corresponding stress can be predicted properly to alleviate the turbulence to surrounding structures. Furthermore, we also hope our research can contribute to the development in cold regions.
of
Acknowledgments
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This work was partly supported by Ueda Memorial Foundation Research Grant [ grant
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numbers: 09Rm01-01]. In addition, we appreciate the anonymous reviewers and the
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Appendices A
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editors for their helpful comments on this manuscript.
In this section, we illustrate the details of the thick-walled cylinder theory to calculate the radial stress produced by freezing soil. The equilibrium equation of this frost heave cell in the polar coordinate system is derived first. The polar coordinate system and microelement of frost heave cell are illustrated in Figure A-1. The components of stress with respect to the small angle is shown in Figure A-2.
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Figure A-1 The balance of microelement in the polar coordinate system
Figure A-2 Components of stress acting on the surface of the microelement
According to Figure A-1, the radial component of force due to the radial normal stress is
r dr r dr d r rd r r 29
(A-1)
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According to Figure A-2, the radial component of force due to the circumferential normal stress is
(A-2)
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d d d dr sin dr sin 2 2
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According to Figure A-2, the radial component of force due to the shear stress is
(A-3)
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na
d d r dr cos r r d dr cos 2 2
R is the body force in the radial direction acting on the per unit area. Thus, considering the balance of forces in the radial direction, we can obtain the following equation:
r r 2 drd dr d r rd r r d d d dr sin drd sin dr sin 2 2 2 d r d d r dr cos drd cos r dr cos 2 2 2 Rdrd 0
r rd r drd r
30
(A-4)
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Then, we reorganize the Equation (A-4) as follows:
r d drd r dr 2 d 2 dr sin r r 2 d r d drd sin drd cos Rdrd 0 2 2
of
r drd r
sin
d d , 2 2
d 1 .We can ignore the high-order of these small terms. Accordingly, Equation 2
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cos
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Here, because d is infinitesimal, we can approximate that
(A-5)
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r drd drd r drd Rdrd 0 r
(A-6)
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r drd r
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(A-5) can be further simplified:
Dividing both sides of Equation (A-6) by rdrd , the above equation can be rewritten as:
r 1 r r R 0 r r r r
(A-7)
Equation (A-7) is, therefore, the equilibrium equation in the radial direction of this frost heave cell.
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As the problem being considered is an axisymmetric problem in which isotropy is maintained, there is no change in stress along the circumferential direction; i.e., r 0 . Moreover, if we do not consider the body force (R=0), Equation (A-7) can be
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(A-8)
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d r 1 r 0 dr r
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transformed as follows:
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On the other hand, the stress and strain relationship, in this case, can be considered as
1 (1 ) r t E
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r
na
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the plane strain state when expansion strain ( t ) due to freezing is considered.
Here,
1 r (1 ) t E
(A-9)
(A-10)
r and are the strains in the radial and circumferential directions,
respectively. E is Young’s Modulus. is Poisson’s ratio. Strain can be expressed by displacement as follows:
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r
du dr
u r
(A-11)
Here, u is the displacement in the radial direction and r is the radius.
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1 ( r ) E
(A-12)
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r
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Subtracting Equation (A-10) from Equation (A-9), the following equation is established:
E E ( r ) 1 1
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r
na
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Then, by substituting Equation (A-11) into Equation (A-12), we obtain:
du u dr r
(A-13)
According to the relationship between stress and strain in the plane strain condition, we can obtain the following equation:
r
E du (1 ) (1 )(1 2 ) dr
33
u r
(A-14)
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We then differentiate both sides of Equation (A-14) with respect to r,
d r E d 2u u 1 du (1 ) 2 2 dr (1 )(1 2 ) dr r dr r
(A-15)
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Substituting Equation (A-13) and Equation (A-15) into Equation (A-8) and reorganizing
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it produces the following equation:
re
E d 2u u 1 du 1 E ( 1 ) 2 2 (1 )(1 2 ) dr r dr r 1 r
du u 0 dr r
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1 d 2u u 1 du 1 du u 0 (1 ) 2 2 1 2 dr r dr r dr r 2 r d 2u 1 2 du du u u ( 1 ) (1 2 ) 2 2 0 2 dr r dr r dr r r 1 d 2u 1 du u 2 0 (1 ) 2 (1 ) 1 2 dr r dr r
(A-16)
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1 1 2
d 2u 1 du u 0 dr 2 r dr r 2
The above equation can be transformed as follows:
d 1 d (ru) 0 dr r dr
34
(A-17)
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By integrating Equation (A-17) twice, the displacement can be expressed by the following equation including two unknown coefficients C1 and C2.
1 1 C1r C2 2 r
(A-18)
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u
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According to Equation (A-18), the displacements u a and ub can be expressed as the
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following equation. Here, a is the inner radius and b is the outer radius.
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u (a) a 2 1 a C1 u (b) b 2 1 b C2
r u
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According to Equation (A-11), we can obtain the following relationship:
r 1 du d (r ) C1 C2 r 2 r dr dr 1 1 1 1 C1 2 C2 r C1 2 C2 2 r 2 r r
The above equations can be expressed in matrix form as:
35
(A-19)
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r (r ) 1 2 r 1 2
1 r 2 C1 1 r 2 C2
(A-20)
Then, we also combine Equation (A-9) and (A-10) into a matrix form:
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r (r ) 1 t 1 (r ) 1
(A-21)
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r (r ) 1 1 E r
The above equation can also be expressed as follows:
re
r (r ) 1 E t 1 (r ) (1 )(1 2 ) 1
(A-22)
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r (r ) 1 E r (1 )(1 2 )
r (r )
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If we extract the radial direction (r direction) from Equation (A-22), we obtain
(r ) 1 E E 1 r t (1 )(1 2 ) (r ) (1 )(1 2 ) 1
(A-23)
When we focus on the stress at the inner side of frost heave cell a, it becomes:
r (a)
E 1 (1 )(1 2 )
(a) 1 E r t (a) (1 )(1 2 ) 1
(A-24)
Based on Equation (A-19) and (A-20), the strain can be expressed by displacement, 36
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r (a) 1 2 a 1 2
1
1 a 2 a 2 1 / a u (a) 1 a 2 b 2 1 b u (b)
(A-25)
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Then, substituting Equation (A-25) into Equation (A-24), we obtain:
ro
1 2 1 a 2 a 2 1 a u a E 1 r (a) 2 (1 )(1 2 ) 1 2 1 a b 2 1 b u b
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E t (1 )(1 2 )
(A-26)
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1
1 2 1 2
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Here,
1 a 2 a 2 1 a 1 a 2 b 2 1 b
1
a b 2 a 1 a 2 b 2 a b 2 a
2b 0
(A-27)
Therefore, by substituting Equation (A-27) into Equation (A-26), the following equation can be obtained:
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r (a)
a b 2 a 2b u a E 1 E 1 t 2 2 2 (1 )(1 2 ) a b 0 u b (1 )(1 2 ) a b a
r (a)
1 2 a b2
u a b2 b2 2(1 )b ( 1 ) a a t u b a a
r (a)
1 2 a b2
b2 a ( 1 2 ) a
Here
E . From the above equation, the radial stress acting on the inside (1 )(1 2 )
(A-28)
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of
u a 2(1 )b t u b
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of the microelement is obtained (Equation (A-28)). Similarly, the radial stress acting on
1 2(1 )a a b 2 2
b (1 2 )
a 2 u a t b u b
(A-29)
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r (b)
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the outside of the microelement can be calculated as follows:
From Equations (A-28) and (A-29), the following matrix equation is obtained:
u a r a 1 K t u b 1 r (b)
And,
38
(A-30)
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K 2 1 2 a b
a (1 2 ) b 2 a 2(1 )a
2(1 )b b (1 2 ) a 2 b
(A-31)
In this study, as explained in the previous section, we measure the circumferential strain
b on the outer surface of the acrylic cell using strain gauges. Based on the
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measured b , the displacement in the radial direction can be calculated as
ro
u b b b . Then, considering the boundary conditions on the outer surface of the
-p
acrylic cell, we know that r b 0 . We ignore the expansion strain t of frost heave
re
cell due to temperature change in the thick-walled cylinder theory. Instead, we
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compensate the thermal influence on the whole experimental system comprehensively
and
r b 0 into Equation (A-29), we can obtain the
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u b b b
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by the data obtained from validation test for strain gauges. Therefore, by substituting
displacement u a .
u a =
b
b 2a 1 v
2
a 2 1 2v
(A-32)
Consequently, we can substitute u a and u b into Equation (A-28) to obtain the radial stress inside the acrylic cell r a . This is the final goal.
39
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References
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Watanabe K., Wake T. (2009). Measurement of Unfrozen Water Content and Relative Permittivity of Frozen Unsaturated Soil using NMR and TDR. Cold Regions Science and Technology, 59(1), pp. 34-41. https://doi.org/10.1016/j.coldregions.2009.05.011 Zent P.A., Sizemore G.H., & Remple W.A. (2011). Ice Lens Formation and Frost Heave at the Phoenix Landing Site. 2011 International Conference: Exploring Mars Habitability. Lisbon, Portugal. https://ntrs.nasa.gov/search.jsp?R=20110014308 Hao Zheng and Shunji Kanie. (2014). Two-dimensional Frost Heave Evaluation by Numerical Model Considering Nonlinear Elasticity of Unfrozen Soil, Journal of Seppyo, 76(6), pp. 461-480. ISSN0373-1006, The Japanese society of Snow and Ice.
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The authors declare no conflict of interest.
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Highlights:
Designing a novel frost heave cell to avoid uneven radial deformation
Establishing a calculating method of the radial stress in freezing direction
Investigating the characteristics of radial stress during freezing of soil
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