Theoretical and experimental study of the frost heaving characteristics of the saturated sandstone under low temperature

Cold Regions Science and Technology 174 (2020) 103036

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Theoretical and experimental study of the frost heaving characteristics of the saturated sandstone under low temperature

T

Shibing Huanga,b, Yuhang Yea, Xianze Cuic,⁎, Aiping Chenga, Guofeng Liub a

School of Resources and Environmental Engineering, Wuhan University of Science and Technology, Wuhan, China Key Laboratory for Bridge and Tunnel of Shaanxi Province, School of Highway, Chang’ an University, Xi’ an, Shaanxi, China c College of Hydraulic & Environmental Engineering, China Three Gorges University, Yichang, China b

ARTICLE INFO

ABSTRACT

Keywords: Saturated sandstone Unfrozen water film Pore size distribution Frost heaving pressure Disjoining pressure

The pore structure and unfrozen water film have a significant influence on the frost heave of saturated porous media. The freezing process and frost heaving characteristics of saturated sandstones have been studied in this research by theoretical and experimental approaches. A developed micromechanical model has been proposed, considering the distribution function of pore size, the effect of unfrozen water film, and the interfacial free energy. The distribution of pore size in sandstone can well satisfy a dual-pore structure model, including the thin pores and coarse pores. Besides, four typical thickness functions are used to investigate the influence of the thickness of unfrozen water film on the disjoining pressure and frost heaving strains. The exponential equation suggested by Fagerlund (1973) may be better to quantify the thickness of this film by comparing with the experimental results. In addition, the frost heaving strains are also very close to the experimental values if ignoring the unfrozen water film. This calculation results may give the reasons why a satisfactory results can also be obtained with the absence of the unfrozen water film in the previous models. This study provides a better understanding of the frost heaving mechanism of saturated porous materials.

1. Introduction Frost heave of rock is a main cause of damage of the rock engineering and stone buildings in cold regions (Feng et al., 2014; Luo et al., 2014; Xia et al., 2018; Huang et al., 2018a, 2019; Liu et al., 2019b). A huge frost heaving pressure in the saturated pore would arise under low temperature due to the 9% volumetric expansion of freezing water. The freeze-thaw damage of rock mass is primarily induced by the repeated generation and dissipation of this frost heaving pressure, and it should be responsible for the frost damage of rock engineering in cold regions (Hori and Morihiro, 1998; Tan et al., 2018; Liu et al., 2019a). This frost heaving pressure can cause pore expansion and frost heaving strain of the saturated rock. The frost heaving strain, as an important and accessible index, reflects the frost-resistance properties and potential freeze-thaw damage degree (Prick, 1995). A bigger irreversible residual frost heaving strain represents a more serious damage caused by the ice crystallization (Zhao et al., 2015; Wang et al., 2019). Therefore, it is of significance to study the frost heaving mechanism of saturated porous rocks under low temperature. The freezing process of pore water has been widely investigated and many representation models have been proposed in the previous

⁎

decades for cement-based materials, in order to prevent the frost damage of this artificial building materials (Coussy and Monteiro, 2007; Zeng et al., 2011; Yang et al., 2015). Similarly, because the frost damage phenomena usually occurs in rock engineering in cold regions, the frost heave of porous rocks has also attracted many researchers. Mellor (1970) suggested that the maximum volumetric freezing strain in rocks is about 9% of the porosity without confinement of the pore ice. Lv et al. (2019) conducted several frost heave experiments on rocks to investigate the difference of frost heaving strains under uniform and unidirectional freezing conditions. Based on the ideal frost heave function from Mellor (1970), Lv et al. (2019) proposed a reliable empirical frost heave model for the rock by introducing a constraint coefficient. Matsuoka (1990) measured the linear frost heaving strains of rocks and proposed that the rock with a large surface area per unit volume had a larger freezing expansion in open systems. Inada et al. (1997) found the porosity and temperature are crucial parameters influencing the frost deformation characteristics of rocks. Huang et al. (2018c) built the relationship between the frost heaving strain and unfrozen water content without considering the effect of unfrozen water film. In addition, there are many coupled thermal-hydro-mechanical models have been proposed to estimate the stresses and strains

Corresponding author. E-mail addresses: [email protected] (S. Huang), [email protected] (X. Cui), [email protected] (A. Cheng), [email protected] (G. Liu).

https://doi.org/10.1016/j.coldregions.2020.103036 Received 24 November 2019; Received in revised form 1 February 2020; Accepted 12 March 2020 Available online 13 March 2020 0165-232X/ © 2020 Elsevier B.V. All rights reserved.

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of rocks subject to freezing (Neaupane et al., 1999; Kang et al., 2013; Duca et al., 2015; Huang et al., 2018b). However, up to now, the roles of the pore size distribution, interfacial free energy and unfrozen water film for saturated rocks during freezing are not clear. There is a general theoretical consensus in the previous literature that the freezing process initiates from the big pores and gradually penetrates into the smaller pores as the temperature decreases (Scherer, 1999; Liu et al., 2011). Therefore, the pore size distribution has a great influence on the freezing process. Besides, the interfacial free energy always exists in the water-ice interface and in the ice-pore wall interface, which may induce a membrane stress between them (Yang et al., 2015). This membrane stress has an effect on the frost heaving pressure during freezing. The unfrozen water film also plays an important role in the freezing process (Zeng and Li, 2019). Both the disjoining pressure and frozen water content increase with decreasing the thickness of the unfrozen water film between the solid ice and pore wall in the frozen pores. Vlahou and Worster (2010) advocated that the disjoining pressure had the potential to fracture the voids of rocks. In recent years, there are two representative models used to describe the freezing process of the cement-based materials, namely poromechanics model (Coussy and Monteiro, 2008; Zeng et al., 2013) and micromechanics model (Yang et al., 2015). The poromechanics model is assumed that the frost heave is caused by water pressure in the unfrozen pores and ice pressure in the frozen pores. Yang et al. (2015) extended this model by considering the effect of the unfrozen water film between the ice and pore wall in frozen pores. A significant disjoining pressure may arise in this unfrozen water film. Another difference between these two models lies in the approach for determining the macroscopic poroelastic properties. However, the freezing process of the nature rocks are still lack of theoretical reference. In order to investigate the freezing process and frost heaving mechanism of saturated rocks, a developed micromechanical model for the freezing rock is proposed in this study. The ice crystallization in pores and the contribution of unfrozen water film to the freezing process are investigated in Section 2. The frost heaving pressure and strain during freezing is detailedly deduced within the micromechanical framework in Section 4. In Section 5, this developed micromechanical model is validated by conducting a frost heaving experiment of the sandstone, and the reasonable thickness equation of the unfrozen water film is also obtained. Some limitations of this study are discussed and several significant conclusions are drawn in Section 5.

freezing point of bulk water. Tf is the freezing temperature, which should be smaller than Tm. pm is the initial water pressure before freezing. ρl and ρi are the densities of water and ice, respectively. pi and pl are the ice pressure and water pressure after freezing, respectively. Based on the Taylor series expansion, the Gibbs-Duhem relationship can be derived from Eq. (4) (Wettlaufer and Worster, 2006):

pi

pi

(pl

Si ) dT =

l dpl

pl =

i

ln

Tf Tm

(pl

(pl

il

(

pm ) 1

i l

)

+h (7)

i

5%

l

T Tm

i

(8) (9)

Parameters

values

Unit

Reference

Kl Kf Kc αl αf αi

1790 1790 7810 22.9–8.244ΔT 22.9–8.244ΔT 51.67 917

MPa MPa MPa 10−6 × °C−1 10−6 × °C−1 10−6 × °C−1 kg/m3

Lide (2004)

1000

kg/m3

0 334.88 × 103 300 0.3 2 2 0.029 0.35

°C m2·s−2 MPa nm MPa nm N/m N/m

Huang et al. (2018c)

Tm ℓ k1 λ1 k2 λ2 γil γwl

i

pl

2 T Tm i

1.475pl (MPa 1°C)

0 c 0 l

(3)

pm ) 1

(6)

Table 1 Values of the main parameters used in the calculation model.

Integrate Eq. (3), there is

pi

(5)

l

When the true water pressure induced by ice crystallization exceeds the tensile strength of rock matrix, the pore will be fractured and propagated. Although the water pressure inside the freezing porous media may reach several hundred megapascals from the elastic poromechanics model and micromechanics model (Zeng et al., 2013; Yang et al., 2015). The true values may be not so big due to the limitation of the tensile strength of the rock. Bridgman (1912) has conducted a famous Pinch-off-test to investigate the tensile strength caused by pore water pressure without apply any mechanical force. The test results

(2)

i dpi

i

2 il ri

pm ) 1

Tf

Where j = i, l, representing the ice and liquid, respectively. S is the entropy; T is the temperature; ν is the specific volume and p is the pressure. Substituting Eq. (2) into Eq. (1), yields

(Sl

pm ) 1

Substituting the values of parameters in Table 1 into Eq. (8), yields.

According to the definition of Gibbs free energy, it can be expressed j dp

(pl

Where h = rc − ri is the thickness of the unfrozen water film between the ice and pore wall. With the freezing of liquid, the water pressure will increase gradually. rc is the critical radius of the pore and ΔT = Tm − Tf. The influence of the difference in densities between water and ice is usually ignored in the previous literature in order to eliminate the second item in the denominator of Eq.(7) (Coussy and Monteiro, 2008; Huang et al., 2018c). It is assumed that this item can be negligible when the value of this item is less than 5% of the value of the phase item:

(1)

Sj dT +

pl =

rc =

Considering the phase balance of water-ice system, the specific Gibbs free energy of the liquid should be equal to that of the ice.

dGj =

Tf Tm

Where γil is the ice-liquid interfacial free energy, and ri is the curvature radius of the ice. Substituting Eq. (6) into Eq. (5) gives the critical radius rc of the freezing pores corresponding to the freezing temperature Tf:

2.1. Ice crystallization in pores

as

i

The stress equilibrium in the solid-liquid interface of the spherical pore can also be expressed by the Young-Laplace equation as follows:

2. Thermodynamic framework

dGi = dGl

Tm

pl =

(4)

Where ℓ is the latent heat of per unit mass of water. Tm is the 2

Yang et al. (2015)

Churaev and Derjaguin (1985)

Watanabe and Mizoguchi (2002) Parks (1984)

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0 -5 -10 (10,

Tf

-15 -20

Fig. 2. Water and ice in partial frozen porous medium.

-25 -30 0

5

10

15

20

r

Fig. 1. The critical freezing temperature Tf versus the pore water pressure pl.

r max

show that an enough water pressure can break the rock, and this water pressure is very close to the tensile strength of the rock. This point has been further confirmed by Pang et al. (2015). It implies that most of the true pore water pressure in freezing rock should be smaller than the tensile strength of the rock regardless of the huge nominal water pressure from the theoretical models. Generally, the tensile strength of the sandstone is below 10 MPa (about ten percents of the compressive strength). If 10 MPa is chosen as the extreme true pore water pressure, when the freezing temperature is smaller than −14.8 °C, the pressure

r max

(

item (pl

pm ) 1

i l

2 i

il

Vj = 1

(13) (14)

(15)

m1

(16)

2 il Tm +h T i

and M2 =

m2

. 2 il Tm +h T i

m1 and m2 are the

characteristic size of coarse pores and thin pores, respectively (Appendix A). V1 and V2 are the volume fractions of the coarse and thin pores, respectively. V1 + V2 = 1. m1, m2, V1 and V2 can be determined by the cumulative distribution curve of the pore size. Eq. (16) is of practical significance when Tf ≤ Tm, or else Sl = 1 above the freezing point of bulk water. The volume fraction of the unfrozen water film and ice content in frozen pores can be written as

Sf =

rc rmax

Sc = F (rc )

3h (r ) dr r

(17) (18)

Sf

Therefore, substituting Eq. (12) into Eq. (17), the volume fraction of the unfrozen water film is (Appendix B)

Sf = 3h

V1 V + 2 m1 m2

3h

V1 V + 1 e( rc m1

m1 rc

)+

V2 V + 2 e( rc m2

m2 rc

) (19)

2.3. The thickness of the unfrozen water film

n j=1

(12)

F (rc )

Where M1 =

(10)

Vg exp ( mg / r ),

m2 r

Sl = V1 exp (M1 T ) + V2 exp (M2 T )

Huang et al. (2018c) proposed that the exponential function could well describe the cumulative distribution of pore size in the sandstone. By extension, a more general exponential form is used in this study to express the cumulative distribution function of pore size:

g=1

m2 exp r2

Where Sl is the total unfrozen water content in the unfrozen pores. Substituting the critical pore radius in Eq. (11) into Eq.(15) using a dual-porosity model, the unfrozen water content can be obtained as

2.2. Unfrozen water content function

F (r ) = 1

V2

(r ) dr = 1

Sl = 1

The pore size distribution is of significant importance to simulate the freezing process of pore water, because the water in pores smaller than rc will not freeze. It means that the ice crystals can penetrate all the pores bigger than rf corresponding to the freezing temperature Tf. With decreasing the freezing temperature, the critical pore radius rc decreases, and more liquid water freezes gradually.

n

m1 r

The total volume of the unfrozen water includes the liquid water in unfrozen pores and the unfrozen water film in frozen pores (Fig. 2). Therefore, the unfrozen water content in unfrozen pores is approximate the volume fraction of pores smaller than rc:

) must be less than 5% of the temperature item

Tm +h T

m1 exp r2

(r ) dr = F (r )

r min

in Eq. (7). Therefore, the influence of water pressure on the critical pore radius can be ignored when the freezing temperature is smaller than −14.8 °C. From Eq. (9), the critical freezing temperature against the pore water pressure is plotted in Fig. 1. Therefore, it is assumed that the influence of water pressure on the freezing point can be ignored. The critical pore radius can be derived as T Tm i

V1

According to the definition of the distribution function of pore size, the following equations are automatically satisfied:

25

pl

rc =

dF (r ) = dr

(r ) =

The thickness of the unfrozen water film between the ice and pore wall in frozen pores decreases with decreasing the freezing temperature. However, the effect of this unfrozen water film on the freezing process of pore water is not fully understood. Zeng et al. (2019) discussed the thickness of the unfrozen water film and its influence on the frost heaving strains, which has provided a better understanding of the role of this film in the confined freezing process of porous media. There are four typical functions usually used to determine the thickness of the unfrozen water film. These four thickness functions can cover the lower

(11)

where Vg = volume fraction of pore phase g, and mg = characteristic parameter of pore phase g. This equation has considered the multiply-pore structures of rocks. For the sandstone, a dual-porosity model may be sufficient to distinguish the coarse and thin pores. According to Eq. (11), the distribution of pore size is 3

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Table 2 The widely used calculation functions of the thickness of the unfrozen water film. References

Value (nm)

Description

Abbreviation

Dash et al. (2006) Fagerlund (1973) Doppenschmidt and Butt (2000) Golecki and Jaccard (1978)

h h h h

The lower bound Close to the lower bound The medium value The upper bound

Dash's function Fagerlund's function Doppenschmidt's function Golecki's function

= = = =

0.3 1.97(−ΔT)(−1/3) 32 − 21 lg (−ΔT) 95 − 54 lg (−ΔT)

the pore water pressure due to the 9% volumetric expansion of freezing water when there is not enough empty pore space to store the extruded water. Therefore, the water pressure is supposed to occur in saturated porous media (Coussy and Monteiro, 2007). The cryo-suction is the supplement of the water pressure. It is used to explain the continual expansion without air voids and shrinkage with air void. The crystallization pressure is the ice pressure to resist the water pressure and the interface tension between the ice and the unfrozen water, and it can be derived by the Laplace's equation (Scherer, 1999). The disjoining pressure produces in the unfrozen water film between the rock and the ice. With the decreasing of freezing temperature, the thickness of this unfrozen water film decreases while the disjoining pressure increases (De Gennes, 1985; Rempel et al., 2001). In this study, assume there are narrow channels connecting these spherical pores for water flow. The pore ice penetrates from the big pore to the small one when the freezing temperature decreases. It should be assumed that the water pressure in unfrozen pores and channels is equal to that in the unfrozen water film in order to keep balance between them. Besides, the unfrozen water film is just dependent of freezing temperature as shown in Table 2. It notes that the flow of this unfrozen water film is negligible because the thickness of this film is the same around the pore wall. Therefore, the ice pressures on the internal interface are equal. However, the ice pressure in the entry interface may be different, because disjoining pressure may not arise here with the absence of the unfrozen water film (Fig. 4). Considering the difference in ice pressure between the internal interface and the entry interface. After a pore freezing, the entrance of this pore should satisfy the following equation:

100

80

h

60

40

20

0 -30

-25

-20

-15

-10

-5

0

Fig. 3. The thickness of the unfrozen water film versus freezing temperature.

bound and upper bound of the film thickness as shown in Table. 2. As shown in Fig. 3, the thickness of the unfrozen water film calculated by the Golecki's function is the largest. When the temperature reaches −30 °C, it is still larger than 15 nm. The thicknesses from the Fagerlund's function and Dash's function are very small and close to 0 at −30 °C. The Doppenschmidt's function gives the thickness of the unfrozen water film among them.

pie = pl +

2.4. The state equation of pore

=

peq N

3

p

T

il

(21)

h

Where pie is the ice pressure in the entry interface. Similarly, in frozen pores, the relation between the ice pressure and water pressure in the internal interface can be expressed as

During the freezing process, the porosity of freezing sandstone samples will increase with increasing ice fraction. As the water gradually freezes, the pores will expand under frost heaving pressure. However, the rock matrix will shrink because of the decrease in temperature. Therefore, the freezing samples may be subjected to the thermal strain load and the frost heaving pressure in pores. Therefore, the change of porosity can be expressed as (Yang et al., 2015) 0

2 r

pi p = pl +

(h ) +

2

il

r

h

(22)

pip

Where is the ice pressure in the internal interface of frozen pores. Because the ice in the frozen pores is connected with the liquid water from the unfrozen pores at the entrance, there should be no unfrozen water film here. However, due to the existing of the unfrozen water film in the internal interface between the solid ice and pore wall, the disjoining pressure is taken into account as shown in Eq. (22). The Eqs. (21) and (22) in this study are only used to explain the mechanical balance between the water pressure and ice pressure as a supplement for the Eq. (25) given by Yang et al. (2015). However, the difference in ice pressure around the solid ice between the internal interface and the entrance interface is ignored when calculating the macroscopic equivalent pressure because the entrance area is very smaller than the internal area of the pore. Considering the disjoining pressure, the relation between the pore wall pressure and the water pressure in frozen pores is

(20)

Where ϕ and ϕ0 are the current porosity and the initial porosity, respectively. peq is the equivalent frost heaving pressure. N is the Biot modulus. αp is the linear thermal dilation coefficient of the porous material. δT = Tf − T0, and T0 is the initial temperature. 3. Frost heaving of rocks 3.1. Frost heaving pressure in pores Yang (2013) argued that there are four typical hypotheses to describe the frost heaving pressure in pores, including the pore water pressure, the cryo-suction, the crystallization pressure and the disjoining pressure hypotheses. The pore water pressure hypothesis emphasizes that the frost expansion of saturated rock is mainly caused by

pw = pl +

(h )

2

wl

r

(23)

Where pw is the frost heaving pressure applied on the pore wall. γwl is the free energy of the pore wall-liquid interface. 4

Cold Regions Science and Technology 174 (2020) 103036

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Fig. 4. Pressure balance relationship in the pores.

In unfrozen pores, the pore wall pressure is

2

pw = pl

150

Nonfilm

wl

(24)

r

120

Considering the effect of disjoining pressure in the unfrozen water film, Yang et al. (2015) used a macroscopic equivalent pressure to replace the frost heaving pressure in pores:

peq =

rmin rmax

pw (r ) dr

90

(25) 60

Substituting Eqs. (23) and (24) into Eq. (25), yields

peq =

rc

pl +

rmax

(h )

2

wl

(r ) dr +

r

rmin rc

2

pl

wl

r

(r ) dr

30

(26) It can be simplified as rmin

peq = pl

rmax

2

wl

r

0

(r ) dr +

rc rmax

(h) (r ) dr

(27)

-22

Eq. (27) shows that the proposed equivalent pressure includes the contribution of pore water pressure, the membrane stress induced by surface tension effect and the disjoining pressure in the unfrozen water film. Substituting the Eq. (12) into the Eq. (27), yields

peq = pl

2

wl

V1 V + 1 e rc m1

m1 rc

+

V2 V + 2 e rc m2

m2 rc

+

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

Tf Fig. 5. The disjoining pressure versus freezing temperature.

is usually deduced according to the conservation rule of water-ice mass (Liu et al., 2018). Because the water-ice medium will not escape from the sandstone sample under undrained condition. Therefore, the total mass of water-ice medium in the freezing rock is equal to the mass of water in the rock before freezing:

(h) F (rc ) (28)

The disjoining pressure is a thermodynamic characteristic of unfrozen water film. The disjoining pressure between the pore walls and ice can be expressed by using an exponential function (Churaev and Sobolev, 2002):

(h) = k1 exp( h/ 1) + k2 exp( h/ 2)

-20

i i

+

l l

+

f

f

0 l 0

=

(30)

0

Where ρl is the density of the bulk water before freezing. ρl and ρc are the current densities of water and ice in the freezing rock, respectively. ρf is the current density of the unfrozen water film. ϕi, ϕl and ϕf are the volume fractions of solid ice, liquid water and unfrozen water film, respectively. The linear form can be employed to express the relation between the densities and the pressure for the water and ice as below (Coussy and Monteiro, 2007):

(29)

Where k1 and k2 are the force parameters; λ1 and λ2 characterize the range of structural forces action. The disjoining pressure is from 0 to about 110.8 MPa by adopting different thickness functions as shown in Fig. 5. This illustrates that an appropriate thickness function should be determined in order to derive a believable frost heaving pressure and deformation. The pore water pressure in freezing cement-based material and rock

j 0 j

5

=1+

pj Kj

3

j

T

(31)

Cold Regions Science and Technology 174 (2020) 103036

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Where Kj is the bulk modulus of the jth phase. αj is the linear thermal dilation coefficient of the jth phase; j = i, l and f. Substituting Eqs. (20) (28) and (31) into Eq. (30), the water pressure containing the effect of unfrozen water film in pores can be derived as

pl =

3

1 M

T+

p

f

(h )

+

Kf

Where M =

T+

l l

{

l

Kl

F (rc ) +2 N

+

c

c

0

wl

+

1 N

0 i 0 l

i

V1 V + 1 e rc m1

+

Kf

T

f

f

1 N

f

+

K c l0

0 i 0 l

T

i i

m1 rc

+

1 +

i

Sm T Ki

V2 V + 2 e rc m2

heaving strain is

=

m2 rc

is the fusion entropy

i

=

m

Tm

per unit volume of water freezing. If the membrane stress is ignored in Eq. (32), the water pressure is as the same as that in micromechanics model given by Yang et al. (2015):

3

T+

1 pl = M

0 i 0 l

i

T+

l l

1

i

i i

0 i 0 l

T

+

0 i 0 l

Sm T Ki

f

f

(h )

Kp = b=1

f

F (rc ) N

+

T+

T+

l l

c

c

T

c l

0 c

0

c l

0

S T 1 + c m Kc

0

c

sp

T+

Where Mp =

{

l l

l

Kl

+

T+

c

Kc

c

+

c

1 Np

T)

c

c

}; N = p

l

0

1 +

0

c

Kc

+

l

0

Ks

bc

c

Ks

Sm T

is the Biot modulus in por-

omechanics model; b is the Biot coefficient of porous material. The expression of water pressure in Eq. (34) from this developed model is very similar to that of Eq. (35) given in the poromechanics model (Coussy and Monteiro, 2008; Zeng et al., 2013). The little difference between Eq. (34) and Eq. (35) is caused by the state equation of pore under freezing. In poromechanics model, the variation of porosity is described by the water pressure and ice pressure with respect to the Biot modulus. However, in micromechanics model, the variation of porosity is expressed by an equivalent pressure, see Eq. (20). Besides, there are also some differences in the poroelastic parameters. Anyhow, the basic ideas of these models used to describe the ice crystallization process are from the poroelastic theory. 3.2. Frost heaving strains Generally, the total frost heaving strain mainly includes the thermal shrinkage strain caused by the decreasing of temperature and the frost expansion strain caused by frost heaving pressure in pores.

=

T

+

f

(36)

Where T

=

f

=b

m

T

(37)

peq 3Kp

0 Km

0)

+ 4Gm

Kp Km (3Km + 4Gm

0) 0 0)

(40)

The approximately homogenous red sandstones containing clay minerals are selected from Yichang, Hubei Province. Two saturated cylindrical samples with diamater of 50 mm and height of 100 mm had been prepared in this experiment. The mineral composition of these fresh red sandstones has been identified through XRD test (Fig. 6). The main mineral components of red sandstone are feldspar (43.25%), quartz (36.07%), calcite (9.44%), illite (7.24%) and chlorite (4%) (Liu et al., 2019). The mercury intrusion porosimetry (MIP) companied with gas adsorption by BET method were performed on the sandstone samples to characterizing the pore size distribution of sandstone, which has a significant influence on the freezing process of pore water. The freeze-resistant strain gauges BX120-20AA were used to continuously measure the frost heaving strains of the samples with a strain collection device XL2101G. A couple of strain gauges were attached on the surface of a sandstone specimen during freezing. One strain gauge was parallel to the axial direction of the specimen and the other one was along the circumferential direction of the specimen. In order to ensure that the strain gauges are closely bonded on the saturated sandstone. An thin dampproof layer of 704 glue was attached to the surface of the sandstone first, followed by the special adhesive glue and strain gauges. Then, an external dampproof layer of 704 glue was attached to the strain gauges again to prevent the external moisture from the atmosphere. A temperature sensor PT-100 was placed inside the specimen to monitor the internal temperature of the specimens. To eliminate the temperature effect on the strain gauge sensitive grid and glue, a piece of quartz glass with stable thermal dilation coefficient is used as a contrast (Fig. 7). The sandstone samples were put in an airconditioned room first until the temperature of the samples reaches about 20 °C. Then they were gradually freezing in a freeze chamber at about −20 °C for 6 h. Besides, The sandstones were sealed by a plastic film during freezing to avoid water evaporation. The main experimental devices are shown in Fig. 8 and the internal temperature of the saturated samples against the freezing time is shown in Fig. 9.

0

(35) b

3

4.1. Experimental procedure

0

The detailed expression of the water pressure in poromechanics model is (Coussy and Monteiro, 2008; Zeng et al., 2013): 1 3( Mp

4Km Gm (1

4. Validation by a frost heaving experiment

(34)

pl =

(39)

(33)

Moreover, if the disjoining pressure is not considered as well as the membrane stress in Eq. (32), the water pressure can be simplified as

1 pl = 3 M

peq 3K p

1 = N 4Km Gm (1 m = (1 0) p

T

Kf

T+b

Combining with Eq. (28), Eq. (29) and Eq. (32), the frost heaving strain can be calculated by Eq. (39). In addition, the poroelastic parameters should be characterized by the commonly used physico-mechanical parameters. The poroelastic parameters includes the Biot coefficient b, the Biot modulus N and the linear thermal dilation coefficient of rock matrix αm. According to the theory of microporoelasticity, the related poroelastic parameters are (Appendix C):

0 i 0 l

(32)

}, S

m

4.2. Determination of the critical parameters

(38)

First, the basic mechanical parameters of the sandstone are derived by conducting the uniaxial compression test on the WAW-300 electrohydraulic servo testing machine, including the elasticity modulus and

Where Kp is the bulk modulus of porous material. εT is the thermal strain and εf is the frost strain caused by frost heaving pressure. Substituting Eq. (37) and Eq. (38) into Eq. (36), the total frost 6

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4.3. Comparison between the theoretical and experimental results

30

25

1-Feldspar 2-Quantz 3-Calcite 4-Illite 5-Chlorite

2-Quantz 36.07%

3-Calcite 9.44%

1

20

3 4-Illite 7.24%

1- Feldspar 43.25%

15

Intensity(

Substituting the values of common parameters in Table 1 and the parameter values of the sandstone in Table 3 into Eq. (16) and Eq. (19), the total unfrozen water content can be derived as shown in Fig. 12. It is obvious that there is a big difference in the total unfrozen water content when using different thickness function in Table 2. However, the change of the total unfrozen water content calculated by the Dash's function and the Fagerlund's function are very close to the value without considering the unfrozen water film. The unfrozen water contents are approximately 2%, 4.8% and 0% by using Dash's function, Fagerlund's function and nonfilm function (h = 0) when the temperature falls to −20 °C. However, there are still 25.8% and 51.2% of the initial pore water keeps liquid at −20 °C using the Doppenschmidt's function and Golecki's function, respectively. Therefore, the thickness function of the unfrozen water film has a great influence on the unfrozen water content under freezing. Substituting the values of parameters in Table 1 and Table 3 into the Eq. (32), the frost heaving water pressure is derived in Fig. 13. It is shown that with the decrease of the thickness of the unfrozen water film, the pore water pressure increases. Especially, the pore water pressure has a remarkably increase when the temperature falls just below the freezing point of the bulk water, because of the fast freezing of pore water. The equivalent frost heaving pressure obeys the same change rule as shown in Fig. 14. However, the equivalent frost heaving pressure combining with the Dash's function is the largest. Because the disjoining pressure using Dash's function is much larger than the values from the other thickness functions (Fig. 5). Moreover, the values of the equivalent frost heaving pressure from the Fagerlund's function are very close to that without unfrozen water film under any freezing temperature. Substituting the equivalent frost heaving pressure in Fig. 15 into Eq. (39), the frost heaving strains are derived using the above thickness functions as shown in Fig. 15. The frost heaving strains using Fagerlund's function agree well with the experimental results. In addition, if the influence of unfrozen water film is not considered, the calculated frost heaving strains are also acceptable and consistent with the experimental results. This results may give the positive evidence for the poromechanics model (Coussy and Monteiro, 2008; Zeng et al., 2013), in which a satisfactory frost heaving strain can also be derived without considering the influence of unfrozen water film. Therefore, it can be suggested that the Fagerlund's function is more suitable to characterize the thickness of the unfrozen water film in the sandstone. Besides, according to the experimental and theoretical results, the frost heave of this saturated sandstone can be divided into three stages: ① When the

5-Chlorite 4%

10 2

5

1 43

5 3

1

4

1

1 11 1 2

0 10

20

30

40

50

1 5

1

1 2

2

60

70

80

90

2 Fig. 6. X-ray diffraction pattern of the red sandstone. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Poisson's ratio of sandstone. The porosity is derived from the MIP and gas adsorption test. By fitting the experimental results of the pore size distribution using Eq. (11), the pore distribution parameters for the red sandstone, including characteristic size of coarse pores and thin pores and their volume fractions, can be determined as shown in Fig. 10. Therefore, the total physico-mechanical parameters of sandstone are listed in Table. 3. From the cumulative distribution function of pore volume fraction, the commonly used differential scheme is derived as follows:

| F (r )/ log(r )| = 2.303

m1 V1e r

m1/ r

+

m2 V2 e r

m2 / r

(41)

Fig. 11 shows that there are two peak points on the differential distribution curves. The pore radius of these two peak points are approximately m1 and m2 (Appendix A). Therefore, m1 and m2 represent the characteristic values of the thin pores and coarse pores, respectively. Besides, the volume fraction of coarse pores is bigger than that of thin pores in this sandstone. They are 61% and 39%, respectively. Accordingly, the thin pores and coarse pores can be well distinguished by the proposed dual-pore size distribution model, see Eqs. (11)~(14).

Fig. 7. The sandstone samples and quartz glass. 7

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S. Huang, et al.

Fig. 8. The main experimental devices.

30 Freeze box

saturated sample

Experiment Calculation

1.0

20

0.8 1e

10 F(r

0.6

0

0.4

-10

0.2

-20

0.0

0

1

2

3

4

5

-5

1

10-6

-30

1 -4

m1 m2

10-5

10-4

6

10-3

10-2

10-1

100

r Fig. 10. Cumulative pore volume fraction of the red sandstone. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. The temperature inside the saturated sample during freezing.

temperature decreases to about 0 °C, the sandstone keeps shrinkage due to the thermal contraction of rock matrix. ② With the continuous decrease of temperature, the saturated sandstone begins to expand caused by the expansion of freezing pore water. Almost 90% of the pore water has frozen at −5 °C and the frost heaving strain reaches the maximum value. The calculated maximum strain is about 2500 × 10−6 using Fagerlund's function. The experimental values is about from 2000 × 10−6 to 2600 × 10−6, which are very close to the theoretical results using Fagerlund's function. ③ Afterwards, there will be a stable strain arising because the thermal contraction of rock matrix is very close to the pore expansion caused by the freezing of the residual liquid water, both of which are very small. It should be noted that the present model is based on the assumption of elastic spherical pore. The freezing of pore water initiates from the big pores and penetrates into the small pores as the temperature decreases, according to the phase balance theory of water-ice system. Besides, although the disjoining pressure increases quickly to be about 28 MPa using Fagerlund's function, the unfrozen water content is less than 5% at −20 °C. Eq.(28) shows that the contribution of the

disjoining pressure to the macroscopic equivalent pressure must be very small when the volume fraction of the unfrozen water is less than 5%. Therefore, the calculated frost heaving strain with nonfilm are very close to the theoretical value using Fagerlund's function in this sandstone. Anyhow, This developed model has provided a comprehensive understanding of the freezing process and the related frost heaving characteristics of the sandstone. 5. Discussions and conclusions In this study, to make accurate estimation of the freezing strain of the sandstone, a developed micromechanic model for the freezing sandstone has been proposed. A frost heaving experiment had also been carefully conducted in order to validate this model and determine the appropriate thickness function of the unfrozen water film. The freezing process of pore water and the frost heaving mechanism of saturated sandstone are derived. Different from the previous models, the prevent model in this study has considered all the possible micromechanical 8

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Table 3 The physico-mechanical parameters of the sandstone. σc (MPa)

vp (km/s)

Ep (GPa)

μp

ϕ0 (%)

m1 (mm)

m2 (mm)

V1

αp (°C−1)

52.7

2.46

9.54

0.25

11.9

5.46 × 10−4

1.87 × 10−5

0.61

9.4 × 10−6

360

1.0 0.9 0.8

300 270 240

0.6

210

0.5

180

pl

log(r

0.7

F(r

Nonfilm

330

Coarse pore Thin pore All by the model

0.4

150 120

0.3

90

0.2

60

0.1

30

0.0 1E-6

1E-5

1E-4

0.001

0.01

0.1

0 -22

1

-20

-18

-16

-14

-12

Fig. 11. Differential pore-size distribution of the red sandstone. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

1.0

-6

-4

-2

0

2

450

Nonfilm

Nonfilm

420 390

0.8

360 330

0.7

300

0.6

270 240

0.5

pe

Sl+Sf

-8

Fig. 13. The water pressure in the red sandstone during freezing. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.9

0.4

210 180 150

0.3

120

0.2

90 60

0.1 0.0 -22

-10

Tf

r

30

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0 -22

2

Tf

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

Tf

Fig. 12. The unfrozen water contents using different thickness functions.

Fig. 14. The equivalent frost heaving pressure in the red sandstone during freezing. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

action, including the membrane stress, disjoining pressure and water pressure during freezing. The contribution of the pore size distribution and thickness of unfrozen water film to the unfrozen water content has also been investigated. Because the unfrozen water content has a great influence on the frost heaving pressure as shown in Eq. (32). However, the frost heave is assumed to be an elastic deformation in this model although the equivalent frost heaving pressure may be more than 200 MPa. Actually, the residual frost heaving strain will increase with increasing the number of freeze-thaw cycles (Wang et al., 2019). It illustrates that the plastic strain arises inside the rock and the rock matrix has gradually entered into the elastic-plastic state with the

freezing of pore water. Therefore, an elastic-plastic micromechanical model should be built when researching the long-term frost heaving characteristics of rocks under freeze-thaw. In one freeze-thaw cycle, the damage of the rock caused by ice crystallization may be ignored. Hence, the poroelastic theory is used in this research as a preliminary study. From this study, the following conclusions can be drawn: 1. The size distribution of pores in the sandstone can well satisfy the dual-pore structure function, including thin pores and coarse pores. The characteristic parameters m1 and m2 in this function are 9

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S. Huang, et al.

5

characterize the thickness of the unfrozen water film in the sandstone. The frost heaving strain from the experiment are in good agreement with the calculated results from the developed micromechanic model combined with the Fagerlund's function. 3. A satisfactory frost heaving strain for sandstone can be also derived if the unfrozen water film is not considered. Because the unfrozen water content is less than 5% at −20 °C when using Fagerlund's function. As a result, the contribution of the disjoining pressure to the macroscopic equivalent pressure is very small in this sandstone. 4. Both the theoretical and experimental results show that the maximum frost heaving strain of the saturated sandstone exceeds 2000 × 10−6 under low temperature. The frost heave of this saturated sandstone has experienced three typical stages, including shrinkage stage caused by contraction of rock matrix, quick expansion stage caused by pore water freezing and stable strain stage.

Nonfilm Experimental values

4

//

//

-3

3

p

2 1 0 -1 -30

-20

-10

0

10

20

Declaration of Competing Interest

30

Tf

The authors declared that they have no conflicts of interest to this work.

Fig. 15. The frost heaving strains of the sandstone considering the influence of unfrozen water film. (‘ // ’ and ‘⊥’ represent the directions parallel and vertical to the axis of the cylinderical samples, respectively).

Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No. 41702291, 41702254), the Open Fund of Key Laboratory for Bridge and Tunnel of Shaanxi Province (Grant No. 300102219529) and the Natural Science Foundation of Hubei Province (Grant No. 2018CFB613).

approximately the two peak points of the differential distribution curves. The influence of the pore size distribution on the freezing process is comprehensively investigated in this study. 2. The contribution of the unfrozen water film has detailedly introduced in this research. The Fagerlund's function is suggested to Appendix A. Appendix

The values of the two peak points on the differential distribution curves are the solutions of the following derivation function (A-1)

| [ F (r )/ log(r )]/ r| = 0 Substituting the expression of F(r) into Eq.(A-1), yields

m1 m1 r2 r

1 V1e

m1/ r

+

m2 m2 r2 r

m2 / r

1 V2 e

=0

(A-2)

It can be simplified as

m1 (m1

r)

V1 e V2

m1 m2 r

+ m2 (m2

r) = 0

(A-3)

It can be proved that m1 and m2 are the approximate solution of (A-3). Appendix B. Appendix According to the definition of the differential distribution function,

dF (r ) = dr

(r ) =

V1

m1 exp r2

m1 r

V2

m2 exp r2

m2 r

(B-1)

There is r min

(r ) dr = 1

r max

(B-2)

Substituting Eq. (B-1) into Eq. (12), yields

Sf =

3h r

rc rmax

V1

m1 exp r2

m1 r

V2

m2 exp r2

m2 r

dr

(B-3)

Using step-by-step integration:

Sf = 3h

Sf =

rc rmax

3hV1

V1 1 ( e r

1 d exp r

m1 r

)

rc

rmax

m1 r rc rmax

+ e(

m1 r

rc rmax

) d1 r

V2

1 d exp r 3hV2

1 ( e r

m2 r m2 r

)

(B-4) rc

rmax

rc rmax

10

e(

m2 r

) d1 r

(B-5)

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1 3hV1 e ( r

Sf =

m1 r

rc

)

1 3hV2 e ( r

rmax

m2 r

rc

)

3h V1

rmax

1 ( e r2

rc rmax

m1 r

) dr + V2

rc rmax

1 ( e r2

m2 r

) dr (B-6)

The final fraction of the unfrozen water film is

Sf = 3h

V1 V + 2 m1 m2

V1 V + 1 e( rc m1

3h

m1 rc

)+

V2 V + 2 e( rc m2

m2 rc

)

(B-7)

Appendix C The spherical pore expansion model subjected to the external and internal pressures is shown in Fig. 16. Assume the mechanical interaction between pores is negligible. When an external pressure is applied on the external surface of a hollow sphere, the displacement field solution is

ure = pe

a13a23 1 a23r + , 3 3 2 4Gs (a2 a1 ) r 3Ks (a23 a13)

r

(a2) = pe

(C-1)

The volume fraction of void corresponding to the initial porosity is: 0

=

a13 a23

(C-2)

Therefore, the displacement of the external surface is.

a2 = a2 pe

0

4Gs (1

0)

+

1 3Ks (1

0)

,

r

(a2) = pe

(C-3)

The bulk strain of this hollow sphere under external pressure can be expressed as. e p

=

(a2 + a2 )3 a2 3

a23

3

a2 , a2

r

(a2) = pe

(C-4)

Then, it is e p

= 3pe

0

4Gs (1

0)

+

1 3Ks (1

0)

,

r

(a2) = pe

(C-5)

According to the definition of the bulk modulus of porous material, it can be defined as

Kp =

pe (C-6)

p

Substituting Eq. (B-3) and (B-5) into (B-6), the bulk modulus of porous material yields

Kp =

4Km Gm (1 0) 3 0 Km + 4Gm

(C-7)

When the internal frost heaving pressure is applied on the internal surface of a spherical pore, the displacement field solution under this boundary condition is.

uri = pin

a13a23 1 a13r + , 3 3 2 4Gs (a2 a1 ) r 3Ks (a23 a13)

r

(a1) = pin

(C-8)

The change of the external radius is

a2 = pin

0

4Gs (1

0)

a2 +

0

3Ks (1

0)

a2 ,

r

(a1) = pin

(C-9)

External pressure

(b nternal pressure

Fig. 16. The expansion deformation of pore under external and internal pressure. (a) External pressure (b) Internal pressure. 11

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S. Huang, et al.

Similar, the bulk strain of this hollow sphere under external pressure can be expressed as. in p

0

= 3pin

4Gs (1

0)

+

0

3Ks (1

0)

,

r

(a1) = pin

(C-10)

If the volumetric deformation of this hollow sphere is equal under internal pressure and external pressure: e p

=

in p

(C-11)

Substituting Eq. (B-5) and Eq. (B-10) into Eq. (B-11), there is 0 (3K s + 4Gs ) p 3Ks 0 + 4Gs in

pe =

(C-12)

According to the definition of Biot coefficient, it can be written as 0 (3K s + 4Gs ) 3Ks 0 + 4Gs

b=

(C-13)

Or

Ks Kp

b=1

(C-14)

Under inner pressure, the change of porosity can be expressed as 0

a1 a1

=3

Where

a1 a1

3

0

p

T

1 4Gs (1

= peq

(C-15) 0)

+

0

3K s (1

0)

.

It can be also written using Boit modulus as 0

=

peq N

3

p

T

(C-16)

Therefore, the Boit modulus is

1 1 =3 N 4Gm (1

0)

+

0

3Km (1

0)

0

=

(3Km + 4Gm

0) 0

4Km Gm (1

0)

(C-17)

uniaxial compression. Cold Reg. Sci. Technol. 162, 1–10. Inada, Y., Kinoshita, N., Ebisawa, A., Gomi, S., 1997. Strength and deformation characteristics of rocks after undergoing thermal hysteresis of high and low temperatures. Int J Rock Mech Min 34 (3–4) 140.e1-140.e14. Kang, Y.S., Liu, Q.S., Huang, S.B., 2013. A fully coupled thermo-hydro-mechanical model for rock mass under freezing/thawing condition. Cold Reg. Sci. Technol. 95, 19–26. Lide, D.R. (Ed.), 2004. CRC handbook of chemistry and physics. Vol. 85 CRC press. Liu, L., Ye, G., Schlangen, E., Chen, H.S., Qian, Z.W., Sun, W., Van Breugel, K., 2011. Modeling of the internal damage of saturated cement paste due to ice crystallization pressure during freezing. Cement Concrete Comp 33 (5), 562–571. Liu, L., Qin, S.N., Wang, X.C., 2018. Poro-elastic–plastic model for cement-based materials subjected to freeze–thaw cycles. Constr. Build. Mater. 184, 87–99. Liu, Y., Cai, Y., Huang, S., Guo, Y., Liu, G., 2019. Effect of water saturation on uniaxial compressive strength and damage degree of clay-bearing sandstone under freezethaw. Bull. Eng. Geol. Environ. 1–16. https://doi.org/10.1007/s10064-019-01686-w. Liu, B., Ma, Y.J., Liu, N., Han, U.H., Li, D.Y., Deng, H.L., 2019a. Investigation of pore structure changes in Mesozoic water-rich sandstone induced by freeze-thaw process under different confining pressures using digital rock technology. Cold Reg. Sci. Technol. 161, 137–149. Liu, H.Y., Yuan, X.P., Xie, T.C., 2019b. A damage model for frost heaving pressure in circular rock tunnel under freezing-thawing cycles. Tunn Undergr Sp Tech 83, 401–408. Luo, X.D., Jiang, N., Zuo, C.Q., Dai, Z.W., Yan, S.T., 2014. Damage characteristics of altered and unaltered diabases subjected to extremely cold freeze–thaw cycles. Rock Mech. Rock. Eng. 47 (6), 1997–2004. Lv, Z.T., Xia, C.C., Li, Q., Si, Z.P., 2019. Empirical frost heave model for saturated rock under uniform and unidirectional freezing conditions. Rock Mech. Rock. Eng. 52 (3), 955–963. Matsuoka, N., 1990. Mechanisms of rock breakdown by frost action: an experimental approach. Cold Reg. Sci. Technol. 17 (3), 253–270. Mellor, M., 1970. Phase composition of pore water in cold rocks (No. CRREL-RR-292). In: Cold regions Research and Engineering Lab Hanover NH. Neaupane, K.M., Yamabe, T., Yoshinaka, R., 1999. Simulation of a fully coupled thermo–hydro–mechanical system in freezing and thawing rock. Int J Rock Mech Min 36 (5), 563–580. Pang, X.Y., Meyer, C., Funkhouser, G.P., Darbe, R., 2015. An innovative test apparatus for oil well cement: In-situ measurement of chemical shrinkage and tensile strength. Constr. Build. Mater. 74, 93–101. Parks, G.A., 1984. Surface and interfacial free energies of quartz. J Geophys Res- Sol Ea 89 (B6), 3997–4008.

References Bridgman, P.W., 1912. Breaking tests under hydrostatic pressure and conditions of rupture. The London, Edinburgh, and Dublin Philos. Mag. J. Sci. 24 (139), 63–80. Churaev, N.V., Derjaguin, B.V., 1985. Inclusion of structural forces in the theory of stability of colloids and films. J. Colloid Interface Sci. 103 (2), 542–553. Churaev, N.V., Sobolev, V.D., 2002. Disjoining pressure of thin unfreezing water layers between the pore walls and ice in porous bodies. Colloid J 64 (4), 508–511. Coussy, O., Monteiro, P., 2007. Unsaturated poroelasticity for crystallization in pores. Comput. Geotech. 34 (4), 279–290. Coussy, O., Monteiro, P.J., 2008. Poroelastic model for concrete exposed to freezing temperatures. Cem. Concr. Res. 38 (1), 40–48. Dash, J.G., Rempel, A.W., Wettlaufer, J.S., 2006. The physics of premelted ice and its geophysical consequences. Rev. Mod. Phys. 78, 695–741. De Gennes, P.G., 1985. Wetting: Statics and dynamics. Rev. Mod. Phys. 57 (3), 827–863. Doppenschmidt, A., Butt, H.J., 2000. Measuring the thickness of the liquid-like layer on ice surfaces with atomic force microscopy. Langmuir 16, 6709–6714. Duca, S., Alonso, E.E., Scavia, C., 2015. A permafrost test on intact gneiss rock. Int J Rock Mech Min 77, 142–151. Fagerlund, G., 1973. Determination of pore-size distribution from freezing-point depression. Mater. Struct. 6, 215–225. Feng, Q., Jiang, B.S., Zhang, Q., Wan, L.P., 2014. Analytical elasto-plastic solution for stress and deformation of surrounding rock in cold region tunnels. Cold Reg. Sci. Technol. 108, 59–68. Golecki, I., Jaccard, C., 1978. Intrinsic surface disorder in ice near the melting point. J. Phys. C Solid State Phys. 11, 4229–4237. Hori, M., Morihiro, H., 1998. Micromechanical analysis on deterioration due to freezing and thawing in porous brittle materials. Int. J. Eng. Sci. 36 (4), 511–522. Huang, S.B., Liu, Q.S., Cheng, A.P., Liu, Y.Z., 2018a. A statistical damage constitutive model under freeze-thaw and loading for rock and its engineering application. Cold Reg. Sci. Technol. 145, 142–150. Huang, S.B., Liu, Q.S., Cheng, A.P., Liu, Y.Z., Liu, G.F., 2018b. A fully coupled thermohydro-mechanical model including the determination of coupling parameters for freezing rock. Int J Rock Mech Min 103, 205–214. Huang, S.B., Liu, Q.S., Liu, Y.Z., Ye, Z.Y., Cheng, A.P., 2018c. Freezing strain model for estimating the unfrozen water content of saturated rock under low temperature. Int J Geomech 18 (2), 04017137. Huang, S.B., Liu, Y.Z., Guo, Y.L., Zhang, Z.L., Cai, Y.T., 2019. Strength and failure characteristics of rock-like material containing single crack under freeze-thaw and

12

Cold Regions Science and Technology 174 (2020) 103036

S. Huang, et al. Prick, A., 1995. Dilatometrical behaviour of porous calcareous rock samples subjected to freeze-thaw cycles. Catena 25 (1–4), 7–20. Rempel, A.W., Wettlaufer, J.S., Worster, M.G., 2001. Interfacial premelting and the thermomolecular force: Thermodynamic buoyancy. Phys. Rev. Lett. 87 (8), 088501. Scherer, G.W., 1999. Crystallization in pores. Cem. Concr. Res. 29 (8), 1347–1358. Tan, X.J., Chen, W.Z., Liu, H.Y., Wang, L.Y., Ma, W., Chan, A.H.C., 2018. A unified model for frost heave pressure in the rock with a penny-shaped fracture during freezing. Cold Reg. Sci. Technol. 153, 1–9. Vlahou, I., Worster, M.G., 2010. Ice growth in a spherical cavity of a porous medium. J. Glaciol. 56 (196), 271–277. Wang, W.J., Yang, X.L., Huang, S.B., Yin, D., Liu, G.F., 2019. Experimental study on the shear behavior of the bonding interface between sandstone and cement mortar under freeze-thaw. Rock Mech. Rock. Eng. 1–27. Watanabe, K., Mizoguchi, M., 2002. Amount of unfrozen water in frozen porous media saturated with solution. Cold Reg. Sci. Technol. 34 (2), 103–110. Wettlaufer, J.S., Worster, M.G., 2006. Premelting dynamics. Annu. Rev. Fluid Mech. 38, 427–452.

Xia, C.C., Lv, Z.T., Li, Q., Huang, J.H., Bai, X.Y., 2018. Transversely isotropic frost heave of saturated rock under unidirectional freezing condition and induced frost heaving force in cold region tunnels. Cold Reg. Sci. Technol. 152, 48–58. Yang, R.W., 2013. Contributions to Micromechanical Modelling of Transport and Freezing Phenomena within Unsaturated Porous Media. PhD thesis. Université Paris-Est. Yang, R.W., Lemarchand, E., Fen-Chong, T., Azouni, A., 2015. A micromechanics model for partial freezing in porous media. Int. J. Solids Struct. 75, 109–121. Zeng, Q., Li, K.F., 2019. Quasi-liquid layer on ice and its effect on the confined freezing of porous materials. Crystals 9 (5), 250. Zeng, Q., Fen-Chong, T., Dangla, P., Li, K.F., 2011. A study of freezing behavior of cementitious materials by poromechanical approach. Int. J. Solids Struct. 48 (22−23), 3267–3273. Zeng, Q., Fen-Chong, T., Li, K.F., 2013. Elastic behavior of saturated porous materials under undrained freezing. Acta Mech Sinica-Prc 29 (6), 827–835. Zhao, X.F., Lv, X.J., Wang, L., Zhu, Y.F., Dong, H., Chen, W., Li, J.K., Ji, B., Ding, Y.B., 2015. Research of concrete residual strains monitoring based on WLI and FBG following exposure to freeze–thaw tests. Cold Reg. Sci. Technol. 116, 40–48.

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