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A damage model for frost heaving pressure in circular rock tunnel under freezing-thawing cycles

T

⁎

Hongyan Liua, , Xiaoping Yuanb, Tiancheng Xiea a b

College of Engineering & Technology, China University of Geosciences (Beijing), Beijing 100083, China Helmholtz Centre Potsdam, German Research Centre for Geosciences (GFZ), Potsdam, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: Freezing-thawing cycles Circular tunnel Frost heaving pressure Rock damage elastic modulus Rock frost heaving ratio

The failure of rock tunnel lining in cold region involves the evaluation of frost heaving pressure in surrounding rock. Although many theoretical models of frost heaving pressure have been proposed, the combination eﬀect of reduction in rock elastic modulus and increase of rock void ratio due to freezing-thawing cycles has not been considered. For this issue, on basis of the elastic model for frost heaving pressure in the circular tunnel surrounding rock, this study establishes the relationship between the rock elastic modulus and the number of freezing-thawing cycles based on fracture mechanics and mesodamage theory. Thereafter, we propose the damage model for frost heaving pressure by considering the combination of rock elastic modulus and void ratio. It is assumed that frost heaving pressure mainly occurs when water in microcracks is frozen into ice. The calculation example shows that frost heaving pressure gradually increases and tends to be a constant with increasing the number of freezing-thawing cycles. It indicates that the proposed model is capable of calculating the variation of frost heaving pressure with the number of freezing-thawing cycles.

1. Introduction All over the world, the seasonal frozen and permafrost regions account for nearly 50% of the total land, and therefore many tunnels have been built or will be built in the cold region. Because the freezing fracture of the rock mass often occur in cold regions under cyclic freezing and thawing conditions (Duca et al., 2015), all the tunnel surrounding rock has diﬀerent levels of damage resulting from frost action (Mie, 1988; Mimouni et al., 2014; Lai et al., 2016). Because there are many microcracks in the rock, water may get into them, freeze into ice under low temperature, and produce large frost heaving pressure, which may cause damage to the tunnel surrounding rock and the lining (Rempel, 2007; Altindag et al., 2004; Hori and Morihiro, 1998). Thereafter, ice may be melted to water when temperature rises, and water may get into the newly formed cracks. Many freezing-thawing cycles will make the deterioration of rock mechanical properties, ﬁnally leading to the failure of the tunnel. Therefore, the proper calculation of the frost heaving pressure in the tunnel surrounding rock is signiﬁcant for the design and construction of the tunnel in the cold region. Many researchers have extensively made experimental and theoretical studies on the frost heaving pressure in the tunnel surrounding rock. First, Winkler (1968) found from experiments that the expansion ⁎

pressure of ice in void was 61.0, 133.0 and 211.5 MPa at the temperature of −5 °C, −10 °C and −20 °C, respectively, when the void volume was constant. Davidson and Nye (1985) obtained the variation of frost heaving pressure as function of rock crack depth and freezing time from the photoelastic tests. Asaad et al. (2015) conducted experiments on limestone samples subjected up to 50 freezing-thawing cycles under eight diﬀerent levels of saturations, and found that the key factor of aﬀecting the rock frost damage was not the number of freezing-thawing cycles but the critical level of water saturation. Zhang et al. (2004) studied the damage propagation of rock from a cold region tunnel under repeated freezing-thawing cycles with computerized tomography, and found that freezing-thawing cycles can make rock strength reduced and plasticity strengthened. Second, by assuming diﬀerent constitutive models of the tunnel surrounding rock, some calculation methods of frost heaving pressure are proposed. Lai et al. (1998) conducted a ﬁrst nonlinear analysis of coupling temperature, seepage and stress ﬁelds in cold regions tunnel. The ﬁnite element formula was derived based on Galerkin's method, and the results show that the inﬂuence of frost heaving pressure on the stress of tunnel lining is very large. Lai et al. (2000a,b) further proposed the analytical viscoelastic solution to examine frost heaving pressure in cold region tunnels using the Laplace transform method. They ﬁnd that frost heaving pressure leads to a remarkable expansion of the plastic zone in the

Corresponding author. E-mail address: [email protected] (H. Liu).

https://doi.org/10.1016/j.tust.2018.10.012 Received 18 June 2018; Received in revised form 21 October 2018; Accepted 23 October 2018 Available online 27 October 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.

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strain problem, because the length of tunnel is signiﬁcantly larger than the diameter of tunnel; and (5) after the tunnel is excavated, support structure is immediately constructed. The process of frost heaving is simultaneously disregarded. Assume that the frost heaving ratio of the frozen surrounding rock is α, and the volumetric expansion of the frozen surrounding rock is απ [c 2 − b2]. When the surrounding rock has been frozen, the corresponding total volume Vtotal of the model in Fig. 1 is the sum of its original volume πc 2 and the volumetric expansion of the frozen surrounding rock, namely

surrounding rock, and then the rock in the plastic zone experiences hardening after the Mohr–Coulomb yield criterion is reached. Feng et al. (2014) established a new elastoplastic calculation model for surrounding rock in cold region tunnels, where the entire surrounding rock is divided into four zones, namely the non-frozen elastic zone, the frozen elastic zone, the frozen plastic zone and the support zone. The frozen surrounding rock is assumed to conform to the ideal elastoplastic model and Mohr–Coulomb yield criterion in their model. Although many progresses have been made in theoretical studies (Zhang et al., 2003; Lai et al., 2000a,b; Gao et al., 2012; Feng et al., 2014), they only calculate frost heaving pressure under one freezingthawing cycle, and cannot consider the eﬀect of the number of freezingthawing cycles on frost heaving pressure. However, in fact, tunnels will inevitably undergo many freezing-thawing cycles in cold region, accordingly, rock is damaged due to many cycles. So how to consider rock damage caused by the freezing-thawing cycles on frost heaving pressure is a signiﬁcant problem to solve. In next section, the calculation method of tunnel surrounding rock frost heaving pressure based on the elastic theory is ﬁrstly introduced. Then, in Section 3, the deterioration law of surrounding rock elastic modulus is obtained by considering rock damage caused by the freezing-thawing cycles. Finally, in Section 4, we propose the model for the tunnel surrounding rock frost heaving pressure by considering the rock freezing-thawing damage.

Vtotal = πc 2 + απ [c 2 − b2]

(1)

We also assume that the expansion of outer wall of the frozen surrounding rock is Δh , thus, its radius will be c + Δh . Accordingly, the total volume Vtotal of the model after freezing in Fig. 1 will be

Vtotal = π (c + Δh)2

(2)

So Eq. (1) should be equal to Eq. (2), and we obtain

(α + 1) c 2 − αb2 − c

Δh =

(3)

The frost heaving pressure in the tunnel is mainly caused by water in the surrounding rock freezing into ice. The frozen surrounding rock of volume expansion trends to move outside and inside walls, and what we usually care about is the frost heaving pressure σf acting on the lining by the inner wall of frozen zone. It can be seen from Fig. 1 that the tunnel lining is only acted by the frost heaving pressure σf which can be seen as a thick-walled cylinder only subjected to external pressure, and thus it is a typical axisymmetric issue. In order to solve the problem, the polar coordinate system is adopted. It is assumed that σθ , σr , σz , εθ , εr and εz are the tangential, radial and axial stress and strain of the circular section, respectively, and the stress satisﬁes σθ > σz > σr (according to the rule of rock mechanics, the positive compressive stress, and the negative tensile stress). Then, the unsupported tunnel can be assumed to be a thick-walled cylinder which is only subjected to the external pressure, which is the frost heaving pressure σf . The basic equations are as follows. The equilibrium diﬀerential equation is

2. Mathematical model 2.1. Theoretical model The calculation model is shown in Fig. 1. For simplicity, the tunnel lining, frozen zone and non-frozen zone in surrounding rock are assumed to be three axial symmetrical elastic bodies which completely contact with each other. The tunnel is assumed to be circular in an inﬁnite region. In order to conduct an elastoplastic analysis of cold region tunnel surrounding rock, the following assumptions are made: (1) tunnel is subjected to hydrostatic stress and its cross-section is circular; (2) frozen and non-frozen surrounding rock, and support structure are all assumed to be isotropic and elastic; (3) shear stresses, acting on the interface between the lining and the surrounding rock and the interface between the frozen surrounding rock and the non-frozen surrounding rock, are disregarded; (4) analysis is based on a plane-

dσr σ − σθ + r =0 dr r

(4)

The geometric equation is

εr =

du dr

εθ =

u r

(5)

where u is the radial displacement of tunnel surrounding rock. The elastic constitutive equation is

⎧ εr = ⎪ ⎨ ⎪ εθ = ⎩

1 − v12 E1 1 − v12 E1

(σ − (σ − r

v1 σ 1 − v1 θ

θ

v1 σ 1 − v1 r

) )

(6)

where E1 and υ1 are the tunnel lining elastic modulus and Poisson's ratio, respectively. The stress boundary condition is:

σr |r = a = 0,

σr |r = b = σf

(7)

The displacement of tunnel lining under the frost heaving pressure σf can be obtained by solving Eqs. (4)–(7):

u1 (r ) = − Fig. 1. Calculation model of tunnel frost heaving pressure. a, b and c are the inner radius of the lining, the inner radius of the frozen zone (or outer radius of the lining), and the outer radius of the frozen zone, respectively. E1, v1, E2, v2, E3, v3 are the elastic modulus and Poisson’s ratio of the lining, frozen and nonfrozen surrounding rock, respectively (Feng et al., 2014).

b2σf E1

(b2

−

a2)

2 ⎡ (1 − ν1 ) r + (1 + ν1 ) a ⎤ ⎥ ⎢ r ⎦ ⎣

(8)

At the outer diameter of tunnel lining, namely r = b, the displacement δ1 is:

δ1 = − 402

bσf

2 2 k bσ ⎡ a + b − ν1⎤ = − 1 f 2 − a2 ⎥ E1 ⎢ b E1 ⎣ ⎦

(9)

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where k1 = b2 − a2 − ν1. The inner and external walls of frozen surrounding rock are subjected to the frost heaving pressure σf and the non-frozen surrounding rock pressure σh , respectively. Therefore, the frozen surrounding rock can be assumed to an axisymmetric problem, subjected to both internal and external pressures. Its displacement can be obtained with the method stated above:

u2 (r ) =

(1 − ν2 )[b2σf − σh (c + Δh)2] r E2 [(c +

Δh)2

−

b 2]

+

(1 + ν2 )(σf − σh ) b2 (c + Δh)2 E2 [(c + Δh)2 − b2] r (10)

where E2 and υ2 are the elastic modulus and Poisson's ratio of frozen surrounding rock, respectively. At the inner wall of frozen zone, r = b, the displacement δf1 is

b ⎧ ⎡ (c + Δh)2 + b2 2(c + Δh)2 + ν2 ⎤ σf − δf 1 = σh ⎫ 2 2 ⎥ E2 ⎨ (c + Δh)2 − b2 ⎬ ⎣ (c + Δh) − b ⎦ ⎩⎢ ⎭

Fig. 2. Variation of the rock frost heaving ratio α with the rock void ratio n.

ratio, Xia et al. (2013) proposed the calculation method of the saturated rock frost heaving ratio in open system:

(11)

At the outer wall of frozen zone, r = c + Δh , and the displacement δf2 is

δf 2 = −

α = 2.17\% ηn

(16)

where η is the water-heat transfer inﬂuence factor. When the rock is sensitive to frost heaving, η = 1.58, otherwise η = 1. n is the rock void ratio, which is easy to obtain through experiments. The comparison between the calculation results from Xia et al. (2013) and the experimental data by Matsuoka (1990b) is shown in Fig. 2, where we can see that they generally agree with each other.

c + Δh ⎧ 2b2 (c + Δh)2 + b2 σ −⎡ σ − ν2 ⎤ σh ⎫ 2 − b2 f 2 2 f ⎢ ⎥ E2 ⎨ ( c + Δ h ) ⎣ (c + Δh) − b ⎦ ⎬ ⎭ ⎩ (12)

The inner wall of non-frozen surrounding rock is subjected to the surrounding rock pressure σh , and its external wall is inﬁnite without any pressure. Therefore, the non-frozen surrounding rock can be seen as an axisymmetric problem with inner pressure. The displacement δ2 of the inner wall of the non-frozen surrounding rock can be obtained with the method stated above:

2.2. Parametric sensitivity analysis

where E3 and υ3 are the elastic modulus and Poisson's ratio of the nonfrozen rock, respectively. According to the contact relationship among the lining, frozen zone and non-frozen zone, we obtain:

In view of the results obtained by Tan et al. (2011), for a cold region circular tunnel in Fig. 1, the relevant calculation parameters are adopted as shown in Table 1. Therefore, the initial frost heaving ratio α of the surrounding rock can be calculated from Eq. (16) to be 0.0189%. Then, the frost heaving pressure σf = 0.20 MPa can be obtained from Eq. (15). In order to analyze the inﬂuence of the above parameters on frost heaving pressure, the parametric sensitivity analysis is adopted to study the inﬂuence of the above parameters on the calculation results.

⎧ δ1 − δf 1 = 0 δ − δf 2 = Δh ⎨ ⎩ 2

(1) Eﬀect of the frost heaving ratio α of the surrounding rock on frost heaving pressure

c δ2 = (1 + ν3 ) σh E3

(13)

(14)

The frost heaving pressure σf can be obtained by solving Eqs. (9), (11)–(14):

σf =

The eﬀect of the frost heaving ratio α of the surrounding rock on frost heaving pressure is shown in Fig. 3. When the value of α gradually increases from 0 to 0.01%, 0.5%, 1%, 1.5% and 2%, the frost heaving pressure increases from 0 MPa to 2.3, 11.5, 22.97, 34.41 and 45.83 MPa, respectively. It shows that the frost heaving pressure almost increases linearly with increasing α, which can also be observed from Eq. (15). It indicates that the value of α has a signiﬁcant eﬀect on its frost heaving pressure. Meanwhile, it can also be seen from Eq. (16) that α is also related to the rock void ratio and frost heaving sensitivity, of which the rock void ratio is an important factor aﬀecting the rock frost heaving ratio. The main reason leading to the rock frost failure is the volume expansion caused by the water in the microcracks frozen into ice. Therefore, in order to reduce the frost heaving failure of rock, in practical engineering, we can reduce the rock void ratio and the inﬁltration of water by grouting. It is noted that the material here is assumed to be elastic, while in practical engineering, the plastic failure in the surrounding rock and lining occurs when the frost heaving pressure reaches a certain value, therefore, it will not produce so high

Δh ⎡ E1 k3 + v2 E1 k 42− E2 k 4 k1 ⎤ ⎡ c (1 + v3 ) 2E1 k2 ⎣ E3 ⎣

where

⎦

k1 =

a2 + b2

− ν1,

b2 − a2 Δh)2 − b2 , Δh

+

k2 ν 2 E2

−

k2 k3 ⎤ E2 k 4 ⎦

k2 = c + Δh ,

+

2k2 b2 E2 k 4

(15)

k3 = (c + Δh)2 + b2 ,

= (α + 1) c 2 − αb2 − c . k 4 = (c + From Eq. (15), the frost heaving pressure in the surrounding rock is not only related to the geometric dimension of the model (e.g., the inner diameter a of the lining, the inner diameter b of the frozen zone or the outer diameter of the lining, and the outer diameter c of the frozen zone), but is also related to the elastic constant of the model (e.g., the lining elastic modulus and Poisson's ratio (E1, v1), frozen surrounding rock (E2, v2) and non-frozen surrounding rock (E3, v3). Above all, it is also related to the surrounding rock frost heaving ratio α. It is easy to obtain the ﬁrst two kinds of parameters and we do not discuss it here again. Here we mainly discuss the determination of the rock frost heaving ratio α. The freezing experiment on the saturated rocks by Matsuoka (1990a) showed that the rock frost heaving ratio is aﬀected by the volume expansion (when water in the microcracks is frozen into ice), water-heat transfer and the eﬀect of rock on frost heaving. Meanwhile, Matsuoka (1990b) obtained the frost heaving ratio of various types of rock through experiments, which is about 0.1% ∼0.5%. Considering the eﬀect of water-heat transfer on the rock frost heaving

Table 1 The calculation parameters of the tunnel elastic model.

403

a/m

b/m

c/m

E1/GPa

v1

E2/GPa

v2

n0

E3/GPa

v3

η

3.0

3.6

5.0

10

0.3

37.64

0.25

0.0067

37.64

0.25

1.3

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Fig. 5. Variation of the frost heaving pressure with E1 and v1.

Fig. 3. Variation of the frost heaving pressure with the frost heaving ratio α of the surrounding rock.

the frost heaving pressure is shown in Fig. 5. It can be seen that the frost heaving pressure increases approximately linearly with increasing E1, whereas v1 has less eﬀect on the frost heaving pressure. With increasing E1, the ability of the tunnel lining to resist the surrounding rock deformation increases, so that the frost heaving pressure of the surrounding rock cannot be released in time, and accordingly the frost heaving pressure acting on the tunnel lining increases. Secondly, assume that the elastic modulus E2 and Poisson's ratio v2 of the surrounding rock in the frozen area are 20, 25, 30, 35, 40 GPa and 0.15, 0.2, 0.25, 0.3, 0.35, respectively, the frost heaving pressure is shown in Fig. 6. It can be seen that although the frost heaving pressure increases with increasing E2 and v2, its increase extent is not obvious. This is because the frost heaving pressure is mainly controlled by the frost heaving coeﬃcient of the rock in the frozen zone, while the eﬀect of its elastic constants is not much. Finally, we assume that the elastic modulus E3 and Poisson's ratio v3 of the surrounding rock in non-frozen zone are 20, 25, 30, 35, 40 GPa and 0.15, 0.2, 0.25, 0.3, 0.35, respectively, the frost heaving pressure is shown in Fig. 7. It can be seen that the frost heaving pressure decreases with increasing E3 and the decrease extent is rather more, however, v3 has a relative less eﬀect on frost heaving pressure. That is because the non-frozen surrounding rock can be regarded as an inﬁnite elastic body subjected to frost heaving pressure. As its elastic modulus increases, the displacement of its inner wall decreases, and the expansion of the surrounding rock in the frozen zone caused by the frost heaving accordingly moves towards the lining, which reduces the frost heaving pressure acting on the non-frozen surrounding rock.

Fig. 4. Variation of the frost heaving pressure with various a keeping c = 5 m as a constant, and various c keeping a = 3 m as a constant.

frost heaving pressure. (2) Eﬀect of the model size on frost heaving pressure Assume that the inner diameter of the lining a is 1.5 m, 2 m, 2.5 m, 3 m (keeping the outer diameter of the frozen zone c = 5 m), the corresponding frost heaving pressures are 0.69, 0.54, 0.37, 0.20, 0.03 MPa, Fig. 4. If we assume that the outer diameter of the frozen zone c is 3.5 m, 4 m, 4.5 m, 5 m, 5.5 m, 6 m (keeping the value of a = 3 m), the corresponding frost heaving pressures are 0.04, 0.11, 0.20, 0.32, 0.5 MPa. It indicates that frost heaving pressure decreases and increases with increasing a and c, respectively. This is because when the outer diameter b is invariant, the lining thickness decreases with increasing a, accordingly, the frost heaving pressure on the lining decreases. When the lining thickness is little, the deformation constraint acting on the surrounding rock by the lining is low, and therefore the expansion deformation of the surrounding rock caused by frost heaving pressure is large. The release ratio of the frost heaving pressure becomes large, accordingly, the residual frost heaving pressure acting on the lining is little. On the contrary, with increasing c, the volume of the frost heaving rock increases, accordingly, the frost heaving pressure increases. Moreover, the frost heaving pressure of the surrounding rock is factor of 0.04 and 12.5 of its original value when the values of a and c increase by a factor of 2.3 and 1.7 of their original sizes, respectively. It shows that the frost heaving pressure is greatly inﬂuenced by the values of a and c. Therefore, the surrounding rock can be allowed to produce certain frost heaving expansion to release part of the frost heaving pressure which reduces the frost heaving pressure on the tunnel lining. This can reduce the possibility of tunnel failure.

3. Variation of surrounding rock elastic modulus under freezingthawing cycles 3.1. Initiation and propagation of the microcrack under frost heaving pressure The rock properties such as the initial void ratio, pore size distribution and mineral content have dramatic inﬂuences on rock deterioration in cold regions (Mutlutűk et al., 2004; Yavuz et al., 2006). Therefore, the rock mechanical property such as strength and compressibility remarkably deteriorates under freezing–thawing cycles (Altindag et al., 2004; Beier and Sego, 2009; Hale and Shakoor, 2003;

(3) Eﬀect of the material mechanical property on the frost heaving pressure Assume that the tunnel lining elastic modulus E1 and Poisson's ratio v1 are 5, 10, 15, 20, 25 GPa and 0.15, 0.2, 0.25, 0.3, 0.35, respectively,

Fig. 6. Variation of the frost heaving pressure with E2 and v2. 404

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¡ b ÷

y

2b

x

2l

¡ ÷ l

Fig. 9. The frost heaving propagation model of the microcrack. Fig. 7. Variation of the frost heaving pressure with E3 and v3.

microcrack will propagate in both x and y directions, shown in Fig. 9. The following assumptions are made in studying the heaving propagation of the microcrack: ① The microcrack is elliptical before and after frost heaving, namely, the microcrack shape and center position are the same, only its size changes. ② The deformation of the rock particle is neglected. ③ The microcrack is saturated all the time and the linear elastic fracture mechanics is adopted to analyze the rock mechanical behavior. ④ The propagation of the microcrack is stable. When the temperature is low, water in the microcrack will freeze into ice with volume expansion. Because of the constraint of the microcrack walls, the expansion pressure will be produced by ice. According to Griﬃth's energy release ratio theory, the elastic strain energy will generate in the medium around the microcracks. When the crack tip stress intensity factor K1 is larger than the rock fracture toughness KIC, the microcrack begins to propagate and the elastic strain energy is released. So there is:

Karaca et al., 2010; Proskin et al., 2010), which ﬁnally aﬀects the stability of the tunnels. The mechanism for rock freezing–thawing damage is as follows. Water in the microcracks expands about 9% of the original volume when rock is frozen, which induces tensile stress concentration and make the microcracks propagate. When rock is thawed, water ﬂows through the newly formed microcracks which induce further damage (Chen et al., 2004; Hori and Morihiro, 1998). In the macroscopic viewpoint, the rock damage caused by the freezing–thawing cycles is the reduction of the rock elastic modulus. Therefore, the eﬀect of the freezing–thawing cycles on the rock elastic modulus is studied with mesomechanics. The irregular microcracks in rock are regarded as the elliptical cavities (fractures) in plane, shown in Fig. 8. The major axis of the ellipse is 2l, the minor axis is 2b, and b < < l. The inner wall of the microcracks is subjected to uniform normal frost heaving pressure p, after N freezing-thawing cycles, the half-length of microcracks is changed to lN. Assume the microcrack develops along the direction of the minimum strain energy density factor, in other words, the microcrack instability propagation occurs when the minimum strain energy density factor Smin reaches the corresponding critical value Sc of the material. Under the plane strain condition, the strain energy density factor of the type I microcrack tip zone is (Sih, 1973):

S = a11 KI2

W=Z−U

where W is the work done by the frost heaving pressure, Z is the stored elastic strain energy around the microcracks, U is the reduced total potential energy of the whole system. Assume that the elastic strain energy can be released completely during the process of crack propagation, there is:

W = −U

(17)

∂ 2S >0 ∂θ 2

(18)

W = 4pl × Δb

The wing crack initiation angle θ = 0 can be obtained from Eq. (18), namely, the microcrack self-similar propagation happens. The microcrack initiation criterion is

Smin = Sc =

1 − 2vr 2 KIC 4πGr

(21)

When the stress intensity factor is larger than or equal to the fracture toughness, the microcrack begins to propagate, and the crack after propagation is shown in Fig. 9 with dashed curve. The frost heaving pressure does work along the normal direction of the microcrack wall, therefore, the left-hand side of Eq. (21) can be expressed as:

1

where a11 = 16πG [(3 − 4νr − cos θ)(1 + cos θ)], Gr is rock shear modr ulus, νr is rock Poisson's ratio, and θ is the wing crack initiation angle.

∂S = 0, ∂θ

(20)

(22)

The reduction of the system potential energy on the right-hand side of Eq. (21) can be expressed with the work done by the frost heaving pressure:

U = −2Y × Δl

(19)

(23)

where KIC is the rock fracture toughness.

where Y is the microcrack Griﬃth energy release ratio and its calculation method is:

3.2. Propagation length of the microcrack under frost heaving pressure

Y=

(24)

ErT

When microcrack is subjected to frost heave, an even-distributed frost heaving pressure will act on its inner wall, accordingly, the

where is the rock elastic modulus at T temperature, according to Xi et al. (2014), increasing when the temperature decreases. So we describe that ErT = mEr , ErT and Er are rock elastic modulus at the low temperature T and ordinary temperature, respectively, and m is the elastic modulus ampliﬁcation factor due to temperature decreasing. It is related to the temperature T, usually taken as 1–2. Substituting Eqs. (22)–(24) into Eq. (21) yields:

y p x

2b

p2 πl p2 πl = T mEr Er

Δb =

Y Δl 2pl

(25)

According to the volume change of water in the microcrack before and after its phase change, we obtain:

2l Fig. 8. The microcrack model under the frost heaving pressure.

πlb + ΔVi = π (l + Δl)(b + Δb) 405

(26)

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density parameter β can be expressed as:

where Δl is wing-crack propagation length, Δb is normal displacement increment of wing crack. From Eqs. (25) and (26), the equation for the microcrack propagation length can be obtained:

A (Δl)2 + B (Δl) + C = 0

β = ρ (N (Δl))2

Secondly, Rostásy et al. (1980) concluded that the number of microcracks basically remained constant after many freezing-thawing cycles. Only the longer microcracks propagated, while the shorter microcracks remain closed due to the squeezing of other microcracks. In other words, with increasing the freezing-thawing cycles, the number of microcracks which are able to propagate will be less and less. According to Griﬃth microcrack propagation theory, the number of microcracks that are activated and propagating in unit volume obeys the exponential distribution (Margolin and Adams, 1985)

(27)

where A = πY , B = π (2plb + lG ) , C = −2plΔVi . The relationship between the crack propagation length and the frost heaving pressure under the plane strain condition can be obtained by solving Eq. (27):

Δl =

−(2plb + lY ) π +

(2plb + lY )2π 2 + 8πYpl × ΔVi 2Yπ

(28)

ρ = ρ0 e−l / lc

It can be seen that it is necessary to determine the expansion volume ΔVi of ice and the ice expansion pressure p against the microcrack wall when solving Eq. (28). First, the calculation method of ΔVi is discussed. Without considering the constraint of the microcrack wall, ice in the microcrack will expand freely. However, in fact, the microcrack wall will exert a reactive pressure on ice which equals to the value of p, accordingly, ice will produce the elastic strain. According to the elasticity theory, the volumetric strain ε v of ice under plane strain conditions is:

p 3(1 − 2νi ) εv = p= Ki Ei

EN 1 = E0 1 + πN 2ρ0 [ (2.09 − 2qmE0/(πKi )) − 1]2 l 2e−l / lc

b Δb = l Δl

(29)

(30)

(31)

4. Model for surrounding rock frost heaving pressure under freezing-thawing cycles

Combing Eqs. (25) and (31) yields:

p=

2qErT 2qmEr = π π

From Section 3, we see that the freezing-thawing cycles will cause the damage to the rock, leading to a decrease in the rock elastic modulus. While in Section 2, the calculation method of the surrounding rock frost heaving pressure, namely, Eq. (15) is proposed according to the elastic theory, which does not consider the eﬀect of the variation of the rock elastic modulus and void ratio with the number of freezingthawing cycles. Therefore, it is only applicable for solving the initial frost heaving pressure of the surrounding rock. While in fact, the practical engineering rock often undergoes many freezing-thawing cycles during its operation. Therefore, we consider here the damage of the surrounding rock caused by the freezing-thawing cycles and proposed the model for the frost heaving pressure of the surrounding rock under the freezing-thawing cycles. Here the eﬀect of the number of the freezing-thawing cycles on the rock in the frozen zone is only considered, then, the surrounding rock frost heaving pressure σf (N ) at the Nth freezing-thawing cycle can be solved with Eqs. (37) and (15):

(32)

Substituting Eqs. (24), (30) and (32) into Eq. (28) yields:

Δl = [ (2.09 − 2qmEr /(πKi )) − 1] l

(33)

3.3. Variation of rock elastic modulus with freezing-thawing cycles Based on the meso-damage theory, an averaging method is adopted to reﬂect the rock macroscopic mechanical properties. Here, MoriTanaka method (Mori and Tanaka, 1973) is adopted to consider the interaction among the microcracks under two-dimensional conditions and the calculation of the rock eﬀective elastic modulus can be obtained:

EN 1 = E0 1 + πβ

(37)

The rock elastic modulus EN after N freezing-thawing cycles can be obtained with Eq. (37). In order to validate this proposed method, the experimental data obtained by Tan et al. (2011) is adopted here. The test rock is granite taken from the Galongla Mountain in Tibet of China. The rock is cylindrical with 50 mm in diameter and 100 mm high, and its physical and mechanical parameters are shown in Table 2. Assume that the microcrack half-length l is 9.2e−6 m, half width b is 1.6e−7 m, the microcrack distribution parameter lc is 5e−5 m, and the microcrack density ρ is 4.2e8/m2. The ice modulus Ei and Poisson's ratio υi are 600 MPa and 0.35, respectively, and m = 1.5 is adopted. Therefore, the variation of the rock elastic modulus with the number of freezingthawing cycles is shown in Fig. 10. It can be observed that the rock elastic modulus gradually decreases with increasing the number of freezing-thawing cycles which ﬁts well with the experimental results in Tan et al. (2011).

We now discuss the calculation of the frost heaving pressure p for the microcrack. It can be seen from Fig. 9 that the microcrack size will change after propagation but its shape will still remain unchanged, therefore we assume that:

q=

(36)

where ρ is the number of microcracks with a radius larger than length l in unit volume, ρ0 is the total number of microcracks, and lc is microcracks distribution parameter. Substituting Eqs. (33), (35) and (36) into Eq. (34) yields

where Ei, υi and Ki are ice elastic modulus, Poisson's ratio and bulk modulus, respectively. Assume that the volume expansion ratio of ice is 9% when it expands freely, the volume change of the microcrack under the expansion pressure p is:

ΔVi = πlb (1.09 − ε v ) = πlb (1.09 − p / Ki )

(35)

(34) Table 2 The physical and mechanical parameters of the rock sample.

where E0 is the rock initial elastic modulus; EN is the elastic modulus of a rock after N freezing-thawing cycles; β is the microcrack density parameter and β = ρ (Δl)2 , ρ is the number of the microcracks per unit area with its propagation half-length Δl after freezing and thawing. First, assume that the microcrack is always saturated, the frost heaving pressure is only related to the temperature, so we assume that the microcrack propagation length after every freezing-thawing cycle is always the same. After N freezing-thawing cycles, the microcrack

Rock type

md/g

ms/g

ρd/g/cm3

n0

σuc/MPa

E0/GPa

v0

Granite

521.87

523.13

2.77

0.0067

135.73

37.64

0.25

md, ms—the dry and saturated mass; ρd—dry density; n0—initial void ratio; σuc—uniaxial compressive strength; E0, v0—initial elastic modulus and Poisson’s ratio. 406

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Fig. 11. Variation of the frost heaving pressure with the number of freezingthawing cycles. Fig. 10. Variation of the rock elastic modulus with the number of freezingthawing cycles.

shown in Fig. 11 after 150 freezing-thawing cycles. It can be found that the rock frost heaving ratio is only 0.000189 and its corresponding frost heaving pressure is only 0.2 MPa at the beginning of frost heaving. Thereafter, with increasing the number of the freezing-thawing cycles, the rock void ratio increases, accordingly, the rock frost heaving ratio increases. Finally, the frost heaving pressure of the surrounding rock increased to 12.93 MPa, which is by factor of 64.7 of the original. As the number of the freezing-thawing cycles increases, the frost heaving pressure gradually reaches a steady state. This is because the frost heaving pressure is mainly due to the volume expansion of the water in the microcracks when freezing into ice. In contrast, the volume expansion of rock particles caused by frost is almost negligible. At the initial stage, the rock void ratio is little, accordingly, the frost heaving pressure due to frost is small. While increasing the number of the freezing-thawing cycles, the microcrack propagates under the frost heaving pressure, leading to the increase of the rock damage and void ratio. Assume that the rock is always saturated, the water content in the rock increases also with the rock void ratio, accordingly, the frost heaving pressure of the surrounding rock increases. However, as shown in Fig. 10, with increasing the number of freezing-thawing cycles, the elastic modulus of the frozen rock ﬁnally tends to be a constant value. Accordingly, the rock damage, void ratio and water content also tend to be a constant value, and the frost heaving pressure of the surrounding rock tends to be a constant value with increasing the number of the freezing-thawing cycles. Therefore, it is assumed that the rock frost heaving pressure mainly depends on the volume expansion resulting from the water in the microcracks when it is freezing into ice. In order to validate the above viewpoint, the rock frost heaving ratio is assumed to be constant, namely 0.000189. It is assumed that the freezing-thawing cycles only lead to the decrease in the rock elastic modulus, accordingly, the variation of the rock frost heaving pressure with the number of the freezing-thawing cycles is shown in Fig. 12. It can be observed that as the number of the freezing-thawing cycles increases from 0 to 150, the rock elastic modulus gradually decreases from 37.64 GPa to 15.15 GPa. Accordingly, the rock frost heaving pressure gradually decreases from 0.198 MPa to 0.144 MPa, and the maximum decrease ratio is only 27.3%. Comparing to the result in Fig. 11, its variation magnitude is very little, which indicates that the surrounding rock elastic modulus has a very limited eﬀect on its frost heaving pressure. Therefore, it can be considered that the rock frost heaving pressure is mainly caused by the frost expansion of water in microcracks. In practical engineering, in order to mitigate the surrounding rock failure induced by the frost heaving, the drainage measures should be adopted to prevent water inﬁltration into the tunnel

Δh

σf (N ) = ⎡ ⎣

E1 k3 + v 2 E1 k 4 − E2(N ) k 4 k1 2E1 k22

⎤ ⎡ c (1 + v3 ) + ⎦ ⎣ E3

k2 ν 2 E2(N )

−

k2 k3 ⎤ E2(N ) k 4

⎦

+

2k2 b2 E2(N ) k 4

(38) where E2(N ) =

E2 1 + πN2ρ0 [ (2.09 − 2qmE2 / (πKi)) − 1]2 l2e−l / lc

is the elastic modulus

of the rock in the frozen zone subjected to N freezing-thawing cycles. We assume that the eﬀect of the freezing-thawing cycles on the Poisson's ratio of the rock in the frozen zone is negligible. Meanwhile with increasing the number of the freezing-thawing cycles, the rock elastic modulus decreases, accordingly, the rock void ratio increases. It can be seen from Eq. (16) that with increasing the rock void ratio, the rock frost heaving ratio will signiﬁcantly increase, and ﬁnally it will aﬀect the surrounding rock frost heaving pressure. Therefore, the eﬀect of the freezing-thawing cycles on the rock void ratio is discussed below. Liu et al. (2015) assume that the damage variable after the freezing-thawing cycles can be deﬁned as:

DN = 1 −

2 EN 1 − nN VpN =1− E0 1 − n 0 Vp2

(39)

where E0 and EN are the rock initial elastic modulus and the elastic modulus after N freezing-thawing cycles respectively, n0 and nN are the rock initial void ratio and the void ratio after N freezing-thawing cycles respectively, Vp and VpN are the rock initial longitudinal wave velocity and the longitudinal wave velocity after N freezing-thawing cycles. If assume Vp = VpN , the rock void ratio nN after N freezing-thawing cycles can be obtained with Eq. (39): (40)

nN = 1 − (1 − n 0 ) EN / E0

Then substituting Eq. (40) into Eq. (16), the rock frost heaving ratio αN after N freezing-thawing cycles can be obtained:

αN = 2.17\% η [1 − (1 − n 0 ) EN / E0]

(41)

Substituting Eq. (41) into Eq. (38), the frost heaving pressure σf (N ) of the surrounding rock at the Nth freezing-thawing cycles can be solved. Next the variation of the surrounding rock frost heaving pressure with the freezing-thawing cycles is illustrated with the calculation example below. The tunnel shown in Fig. 1 is taken as an example, and its calculation parameters are shown in Table 3. Assume that the rock is saturated during the freezing-thawing cycles. Then, after 150 freezingthawing cycles, the relationship between the surrounding rock frost heaving pressure σf (N ) and the number of the freezing-thawing cycles is Table 3 The calculation parameters of the tunnel damage model. a/m

b/m

c/m

E1/GPa

v1

E2/GPa

v2

n0

E3/GPa

v3

Ei/MPa

vi

l/m

b/m

lc/m

ρ/m2

η

m

3.0

3.6

5.0

10

0.3

37.64

0.25

0.0067

37.64

0.25

600

0.35

9.2e−6

1.6e−7

5e−5

4.2e8

1.3

1.5

407

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Fig. 12. Variation of the frost heaving pressure and rock elastic modulus with the number of freezing-thawing cycles.

surrounding rock. 5. Conclusions In this work, the following conclusions are obtained: (1) Based on the theory of elasticity, the frost heaving pressure calculation model for the circular tunnel surrounding rock is established. It indicates that the frost heaving pressure of surrounding rock is not only related to the model geometrical dimensions and the material mechanical parameters, but also related to the frost heaving ratio α of the surrounding rock. (2) Based on the Griﬃth fracture theory and the mesoscopic damage theory, the variation of rock elastic modulus with the number of freezing-thawing cycles is established. It indicates that the rock elastic modulus gradually decreases with increasing the number of freezing-thawing cycles. (3) Considering the eﬀects of freezing-thawing cycles on the rock elastic modulus, void ratio and frost heaving ratio, a model for the tunnel surrounding rock frost heaving pressure based on damage mechanics is proposed by considering the eﬀect of the freezingthawing cycles on the rock damage. The example shows that with the increase in the number of freezing-thawing cycles, the frost heaving pressure of surrounding rock increases gradually and tends to a certain value. Therefore, it is believed that the frost heaving pressure is mainly caused by the frozen expansion of water in microcracks when it is frozen into ice. Thus, in practical engineering, in order to reduce the frost heaving damage of surrounding rock, water drainage measures should be adopted to control the inﬁltration of water. Acknowledgment This study is supported by the Fundamental Research Funds for the Central Universities (2017) of China, and National Natural Science Funds (41162009) of China. References Altindag, R., Alyildiz, I.S., Onargan, T., 2004. Mechanical property degradation of ignimbrite subjected to recurrent freeze–thaw cycles. Int. J. Rock Mech. Min. Sci. 41, 1023–1028. Asaad, A., Kevin, B., Xavier, B., Ákos, T., Muzahim, A., 2015. Critical degree of saturation: a control factor of freeze-thaw damage of porous limestones at Castle of Chambord. France. Eng. Geol. 185, 71–80.

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