A Freeze-Thaw Fatigue Damage Model for Granite Tunnels

Frozen earth covers a vast area in north China, wherein seasonal frozen earth accounts for 54% and it will undergo freezing and thawing when the surface layer contains water, which brings more challenges to the maintenance and operation works of tunnels located in these areas [1]. Moreover, in some seasonal frozen earth areas, the severe environmental changes can exacerbate the mechanical functions of rock mass, thereby damaging the engineering materials and the structures. To cope with this problem, this paper aims to study the evolution law and mechanism of freeze-thaw damages of granite tunnels and build a freeze-thaw fatigue damage model for the said problem, thereby providing references for the design, construction, and maintenance of tunnel projects.

Rock damage refers to the process in which the properties of the rock gradually degrade under the impact of loads or external environment, the freeze-thaw damages of the rocks are caused by the repeated action of the internal stress of the rocks, and the representation of such damages is the expansion and development of micro cracks inside the rocks. Some fatigue damages occurred in rock samples can be explained by theories of damage concepts summarized in fatigue damage mechanics [2]. In theories about fracture and fatigue [3], materials are prone to brittle fracture when subjected to repeated cyclic stress, because there’re microcracks in the brittle materials themselves. Some scholars regard rocks as brittle materials and pointed out that the damages of rocks in seasonal frozen areas are essentially the fatigue damage, the freeze-thaw damage is the evolution process of internal fatigue damages caused by the development of internal micro cracks in rocks under different mechanisms [4]. Yang et al. [5] analyzed and defined the damages of rocks from macroscopic, microscopic, and mesoscopic perspectives, and clarified the mechanisms of damage mechanics from different angles. Zhang and Yang [6] employed the mesoscopic theories of damage mechanics to construct a model for rocks under repeated loads, defined a damage variable based on mechanics from a macroscopic point of view, and derived the damage equations; the results simulated by the proposed model were compared with the actual results, and the data were in good agreement. Zhou et al. [7] applied and removed loads repeatedly in the test and studied peak strength and other items of the curves attained in the test, and employed fracture mechanics to conclude that rocks are a kind of non-ductile brittle material, its macroscopic mechanical properties exhibited in the mesoscale test are the compaction and expansion process of cracks in the rocks. Field scholars in the world also discovered in their studies that [8-13], the temperature, temperature difference, and change speed of water-ice phase transition, the number of experienced freeze-thaw cycles, and the water content of rock surface all have significant impact on the material properties of the rock. The focus of current research on freeze-thaw damages of rocks is the deterioration of rock damages after subjected to multiple freeze-thaw cycles, however, the evolution process of the fatigue inside the rocks after subjected to freeze-thaw cycles is still unclear to us.

To fill in the research gap mentioned above, this study performed freeze-thaw fatigue damage test on granite tunnels, measured the physical properties of granite rock samples, observed the appearance of rock samples under different numbers of freeze-thaw cycles, and analyzed the expansion state of pores and cracks. Based on the dynamic modulus of elasticity, this paper derived the fatigue evolution equation that describes the freeze-thaw damages of rocks and built a fatigue damage model for granite tunnels, in the hopes of offering useful evidences for the monitoring and maintenance of tunnel projects.

3.1 Rock freeze-thaw fatigue damage model

According to the relationship between physical parameters and the number of freeze-thaw cycles attained in the first chapter, the physical and mechanical parameters of rock samples all decreased with the increase of the number of freeze-thaw cycles. After subjected to 80 freeze-thaw cycles, it’s observed that the physical and mechanical parameters of the rock sample had decreased by more than 30% compared with those before the freeze-thaw cycles. Based on the relationship between the number of cyclic loadings experienced by the material before failure and the fatigue load stress, fatigue can be divided into low-cycle fatigue and high-cycle fatigue. In case of low-cycle fatigue, plastic deformation occurs to the material under each cyclic load, and in this process, the material’s yield stress is generally less than the cyclic load stress, and every freeze-thaw repetition will cause plastic deformation in the material. Thus, it’s concluded that in the freeze-thaw test, the repeated freeze-thaw cycles acted as a low-cycle fatigue effect on the granite.

3.1.1 Determination of damage variable

In order to make the results of the constructed fatigue damage model close to the on-site data, the damage variable was represented by the dynamic modulus of elasticity E_d (MPa), and the calculation equation of E_d is [16]:

$E_{d}=\frac{V_{\rho}^{2} \rho(1+\vartheta)(1-\vartheta)}{1-\vartheta}$        (8)

where, $\rho, V_{\rho}$, and $\vartheta$ respectively represent the density (kg/m³), the longitudinal wave velocity (m/s), and the Poisson's ratio of the rock samples.

Based on the dynamic modulus of elasticity of the rock, the damage variable could be expressed as:

$\mathrm{D}=1-\frac{E_{d}^{\prime}}{E_{d}}$       (9)

where, $E_{d}^{\prime}$ is the dynamic modulus of elasticity (MPa) after damage. Ignoring the changes of Poisson's ratio during the damage process, then by combining above equations, there is:

$\mathrm{D}=1-\frac{\vartheta}{v^{\prime}} \frac{V_{\rho}^{\prime 2}}{V_{\rho}^{2}}$        (10)

where, υ is the volume occupied by per unit mass of the rock under the initial state (m³). “ ’ ” represents the damage.

The increase of crack expansion inside the rock will increase the volume of the rock, then there are:

$v=v_{s}+v_{0}$      (11)

$v^{\prime}=v_{s}+v_{0}+\Delta v_{0}$      (12)

where, $v_{s}, v_{0}$, and $\Delta v_{0}$  are the mineral volume, crack volume, and crack volume change corresponding to per unit mass of the material, their units were all m³, then, the fracture degree $n_{0}$ before the damage process can be expressed as:

$n_{0}=\frac{v_{0}}{v}$      (13)

By combining Eqns. (10), (12), and (13), we can get:

$\mathrm{D}=1-\frac{1-n^{\prime}}{1-n_{0}} \frac{V_{\rho}^{\prime 2}}{V_{\rho}^{2}}$          (14)

According to Eq. (14), we can attain two parameters that are required for the calculation of the damage rate of the dynamic modulus of elasticity, namely the longitudinal wave velocity and the porosity.

3.1.2 The freeze-thaw fatigue evolution equation

To facilitate statistics, for the rock freeze-thaw fatigue damage model studied in this paper, the following assumptions were made:

(1) Under the action of freeze-thaw cycles, the freeze-thaw damages of granite are only caused by the frost-heave force;

(2) Under the action of freeze-thaw cycles, the freeze-thaw damages of granite distribute evenly inside the rock.

In the laboratory test, the expansion rate of damages in rock samples is:

$D^{\prime}=\left(\frac{Y}{X}\right)^{y} p^{\prime}$        (15)

where, $D^{\prime}, Y$, and $p^{\prime}$  respectively represent the rock fatigue damage expansion rate, the energy change rate, and the cumulative plastic strain rate; x and y are relevant parameters.

Under the condition that the force acts on one side, the free energy density φ of the material can be expressed as:

$\varphi=\frac{\sigma^{2}}{2 \rho E(1-D)}$        (16)

$\sigma$, ρ, and E respectively represent the axial stress (MPa), the density of the material (kg/m³), and the elastic modulus of the material (MPa).

Then the energy release rate of rock damages can be expressed as:

$\mathrm{Y}=\rho \frac{\partial \varphi}{\partial D}=\frac{1}{2 E}\left(\frac{\sigma}{1-\mathrm{D}}\right)$          (17)

When one freeze-thaw cycle is finished, the stress generated during the water-ice phase transition will promote the pores and cracks to develop, and meanwhile the water will enter the pores and cracks in the rock, the frost heave force generated in the water-ice phase transition process is greater, which will further expand the pores and cracks inside the rock, therefore, it can be concluded that granite has the feature of strain hardening.

The relationship between cumulative plastic strain and equivalent stress applied for one time is:

$\mathrm{p}=\left[\frac{\sigma}{\mathrm{J}(1-\mathrm{D})}\right]^{\gamma}$         (18)

where, e and E are parameters of the material.

The equation of cumulative plastic strain rate can be expressed as:

$p^{\prime}=e \frac{\sigma^{\prime} \sigma^{\gamma-1}}{J^{\gamma}(1-D)^{\gamma}}$         (19)

where, $\sigma^{\prime}$ represents the change rate of strain.

Then there is: $\Delta \sigma=\sigma_{\max }-\sigma_{\min }$.

As mentioned above, granite has the strain hardening property, so a hardening parameter h was introduced; N represents the number of freeze-thaw cycles; m and n are parameters of the material, then there is:

$\mathrm{h}=(\Delta \sigma)^{m} N^{n}$        (20)

After transformation, m, n, and λ are all material parameters, let J=hF and G=Fλ, then there is:

$p^{\prime}=\frac{e \sigma^{\prime} \sigma^{\gamma-1}}{G^{\gamma}(1-D)^{\gamma}(\Delta \sigma)^{m \gamma} N^{n \gamma}}$            (21)

Combining Eqns. (15), (17), and (21), there is:

$D^{\prime}=\frac{\sigma^{\prime} \sigma^{k-1}}{L(1-D)^{b}(\Delta \sigma)^{m \gamma_{N^{n \gamma}}}}$         (22)

where, k=e+2y, L$=\frac{G^{e}(2 E x)^{y}}{\theta}$.

$\int_{\mathrm{D}}^{\mathrm{D}+\frac{\delta \mathrm{D}}{\delta \mathrm{N}}} \mathrm{dD}=\int_{\sigma_{\min }}^{\sigma_{\mathrm{max}}} \frac{\sigma \sigma^{\mathrm{k}-1}}{\mathrm{LN}^{\mathrm{n} \gamma}(1-\mathrm{D})(\Delta \sigma)^{\mathrm{m} \gamma}} \mathrm{d} \sigma$, by integrating D, we can get:

$\int_{\mathrm{D}}^{\mathrm{D}+\frac{\delta \mathrm{D}}{\delta \mathrm{N}}} \mathrm{dD}=\int_{\sigma_{\min }}^{\sigma_{\max }} \frac{\sigma \sigma^{\mathrm{k}-1}}{\mathrm{LN}^{\mathrm{n} \gamma}(1-\mathrm{D})(\Delta \sigma)^{\mathrm{m} \gamma}} \mathrm{d \sigma}$            (23)

During the freeze-thaw cycles, the minimum value of frost-heave force is 0, namely $\sigma_{\min }$=0, through calculation and derivation, we can get:

$\frac{\mathrm{dD}}{\mathrm{dN}}=\frac{2 \sigma_{\mathrm{max}}^{\mathrm{k}}}{\operatorname{Lm} \alpha(\Delta \sigma)^{\mathrm{m} \gamma_{N^{\mathrm{n}} \gamma}(1-\mathrm{D})^{k}}}$            (24)

During the alternation of freezing and thawing, the frost-heave force didn’t change much, that is, $\sigma_{\max }$ didn’t change; meanwhile, the range of damage variable D could be expressed as:

$\mathrm{D}=\left\{\begin{array}{c}0(N=0) \\ 1\left(N=N_{d}\right)\end{array}\right.$         (25)

where, N_d represents the number of freeze-thaw cycles experienced by granite until its structure fails, then the equation of freeze-thaw fatigue evolution can be attained by introducing N_d and integrating:

$\mathrm{D}=1-\left[1-\left(\frac{N}{N_{d}}\right)^{1-n \gamma}\right]^{\frac{1}{k+1}}$          (26)

Then, the equation of the low-cycle evolution of the rock could be attained, if n$\gamma$=0, then it represents the equation of the high-cycle evolution of the rock.

3.2 The fatigue damage model of granite tunnel

This paper adopted a theoretical equation method for the value assignment of N_d [16, 17]. To judge the criterion of rock structure failure, this paper quoted the criterion that the rock structure is considered failed when the loss of dynamic modulus of elasticity of the concrete material reaches 40%. The linear interpolation method could attain the number of freeze-thaw cycles experienced by the rock when the rock structure fails. According to the physical parameters of tunnel granite shown in Table 5, when the rock had experienced 40 and 80 freeze-thaw cycles, $N_{d}$=130.

$D=1-\left[1-\left(\frac{N}{N_{d}}\right)^{1-n \gamma}\right]^{\frac{1}{k+1}}$

$=1-\frac{1-V_{\rho}^{12} n^{1}}{1-V_{\rho}^{2} n_{0}}$            (27)

Table 5. Physical parameters of granite tunnel

Parameters to be determined are k and $n \gamma$. Let $\mu=1-\gamma$, fitting was performed, the data in Table 5 were substituted into Eq. (27), then there are:

${\left[1-\left(\frac{0}{130}\right)^{x}\right]^{y}=1.000 }$

${\left[1-\left(\frac{5}{130}\right)^{x}\right]^{y}=0.921 }$

${\left[1-\left(\frac{10}{130}\right)^{x}\right]^{y}=0.885 }$

${\left[1-\left(\frac{20}{130}\right)^{x}\right]^{y}=0.821 }$

$\left[1-\left(\frac{40}{130}\right)^{x}\right]^{y}=0.750$         (28)

$\left[1-\left(\frac{80}{130}\right)^{x}\right]^{y}=0.681$        (29)

Fitting was performed in MATLAB to attain the optimal solution, for the values of k and μ, there is:

$\mathrm{D}=1-\left[1-\left(\frac{N}{130}\right)^{0.4746}\right]^{1.0086}$        (30)

Through equation derivation, we can get:

The parameters k and nγ in Eq. (31) are related to the material; also, it is observed from Eq. (30) that the values of k and nγ are not related to $N_{d}$. By substituting different $N_{d}$ values, the fatigue damage evolution equation could be attained as:

$\mathrm{D}=1-\left[1-\left(\frac{N}{130}\right)^{0.4746}\right]^{1.0086}$

$\left(N_{d}=130\right)$        (31)

$\mathrm{D}=1-\left[1-\left(\frac{N}{130}\right)^{0.4746}\right]^{1.0086}$

$\left(N_{d}=145\right)$           (32)

$\mathrm{D}=1-\left[1-\left(\frac{N}{130}\right)^{0.4746}\right]^{1.0086}$

$\left(N_{d}=160\right)$         (33)

This is shown in Figure 8 comparison of model simulation results and actual results showed that they were in good agreement, which had verified the effectiveness of the proposed granite tunnel freeze-thaw fatigue damage model.

8.png

Figure 8. Comparison of model simulation results and test results