The double Burgers model of fractured rock masses considering creep fracture damage

2.1. Crack initiation

The normal and tangential stresses applied on the main crack surface in compressive-shear states yield (Fig. 1):

where ${\sigma }_{ne}$ and ${{\tau }_n}_e$ are normal and tangential stresses applied on the main crack surface, respectively, ${\sigma }_1$ and ${\sigma }_3$ are the maximum and minimum principal stresses, respectively, $\psi$ is the angle between the crack and the maximum principal stress.

According to the maximum circumferential stress theory, the initial crack propagates at the tip of main crack along the direction of the maximum circumferential stress, with a crack initiation angle ${\theta }_c$ of 70.5°. The stress intensity factor (SIF) ${{\rm K}}_{{\rm I}}$ at the main crack tip is [13]:

where $f$ is the friction coefficient of the crack surface, and $a$ is the half of crack length.

When the SIF at main crack tip is higher than the fracture toughness ${{\rm K}}_{{\rm I}C}$ of rock,the wing crack initiates.

Fig. 1Schematic diagram of wing crack initiation at the main crack tip

2.2. Transient propagation of the wing crack

The previous experimental studies [1, 6-8] indicated that the wing crack propagates approximately along the direction of the maximum principle stress, after it initiates, as shown in Fig. 2.

Fig. 2Wing crack propagation

The influence of the main crack on the SIF of wing crack tip can be reflected by the effective shear stress ${\tau }_e$ applied on the main crack [13]:

According to Horri and Nemat-Nasser wing crack model [14], the SIF of the wing crack tip can be expressed:

where $l$ is the propagation length of wing crack, $\psi$ is the angle between the wing crack and the main crack.

The SIF at the wing crack tip continuously decreases with the wing crack propagation, according to Eq. (5). The crack transient propagation length can be obtained, by assuming that ${{\rm K}}_{{\rm I}}$ is ${{\rm K}}_{IC}$ in Eq. (5).

2.3. Creep propagation of wing crack

Previous experimental results [3, 5, 7, 15] showed that a subcritical crack growth at very low velocity (10^-2-10^-9 m/s) can be observed when the SIF is between the threshold stress intensity for subcritical crack growth ${{\rm K}}_0$ and the fracture toughness ${{\rm K}}_{IC}$ [3, 5, 15-17], which is crack creep propagation. When ${{\rm K}}_{{\rm I}}$ is lower than ${{\rm K}}_0$, crack propagation ceases. The ultimate crack propagation length $l_n$ can be obtained, by assuming that ${{\rm K}}_{{\rm I}}$ equals ${{\rm K}}_0$ in Eq. (5). The subcritical crack propagation length $l_2$ is a difference between the ultimate crack propagation length $l_n$ and crack transient propagation length $l_1\left(l_2=l_n-l_1\right)$ (Fig. 3).

Experimental studies on the subcritical crack propagation showed that the rate of the creep fracture is [3, 5, 15-17]:

where, $l_s$ is an arbitrary value in range of subcritical propagation length, $A$ and $n$ are parameters related to subcritical crack propagation.

Eq. (6) shows that the crack propagation rate decreases with the increase in crack propagation length. When the SIF decreases to the threshold stress intensity for subcritical crack growth ${{\rm K}}_0$, the crack propagation ceases [15-17].

Fig. 3The evolution curve of the SIF at the wing crack tip

4.1. Damage deformation caused by crack creep fracture

The strain increment of the fractured rock masses, $d{\epsilon }_{ij}$, composes of the strain increment of rock matrix, $d{\epsilon }_{ij}^0$ and the damage strain increment of the cracks, $d{\epsilon }_{ij}^m$:

The increment of the linear elastic strain of the rock matrix is:

where $S_{ijkl}^0$ is elastic flexibility tensor for rock matrix.

The damage strain increment, caused by crack slippage and propagation, can be obtained, using the Rice thermodynamic theory [19, 20]:

where $f_{\alpha }(\sigma ,H)$ is a set of thermodynamic forces conjugated to the internal variables ${\xi }_{\alpha }$, the ${\sigma }_{ij}$ is the stress tensor, $H$ is the present condition of the internal variable, and $V_0$ denotes the volume of a representative volume element (RVE).

The average slip $b_1$ of the points on $P{P}^{\text{'}}$ is equal to the average Mode II crack opening displacement induced by the effective shear stress ${\tau }_e$ (see Fig. 7(a)):

where, $v_0$ is the Passion ratio, $E_0$ is the elastic modulus.

The increment of inelastic strain induced by crack slippage can be expressed [19, 20]:

where, ${\omega }_0=N{a}^2/A_0$ is the initial crack density parameter, $N$ is the number of (non-interacting) cracks having the same orientation $\phi$, $A_0$ is the area of the representative unit, $\overline{b_1}=b/a$, a normalized slip, $\theta$ is the angle between the two coordinate systems $\theta =\pi /$2$-\psi$.

Then, Fig. 7 shows the increment of the nonlinear damage strain, caused by wing crack propagation. According to the equilibrium conditions of the plane and the Mohr-Coulomb criterion, the balanced force ${\tau }_{e1}$ on the main crack plane is:

The average opening of the mode II crack, $\overline{b_2}$ induced by ${\tau }_{e1}$, due to wing crack propagation is:

Fig. 7a) Model of wing crack in compressive-shear state, b) increment of the inelastic damage strain induced by wing crack propagation

a)

b)

The inelastic residual energy, generated by wing crack propagation is [20]:

where:

The increment of the inelastic residual energy can be expressed as [20]:

So, a set of thermodynamic forces $f_{\alpha }(\sigma ,H)$ conjugated to the internal variables $\overline{b_2}$ and $L$ can be expressed as:

The increment of the nonlinear damage strain is:

According to Eq. (24), the increment of the nonlinear damage strain, caused by wing crack propagation can be obtained. Thus, the damage deformation is time-dependent due to creep propagation of wing crack[21, 22].

4.2. Double Burgers model

The Burgers creep model of rock matrix includes the Kelvin model and the Maxwell model, thus, the corresponding constitutive model, in the form of stress and strain deviation, yields [10-12]:

where, $E_{r1}$, $E_{r2}$, ${\eta }_{r1}$and ${\eta }_{r2}$ are the elastic modulus of controlling delay, the elastic shear modulus, the rate of determining the delayed elasticity and the viscous flow rate, respectively.

The stress tensor ${\sigma }_{ij}$ at a certain point can be decomposed into the deviatoric stress tensor ($S_{ij}$) and the spherical stress tensor (${\sigma }_m$). Similarly, the strain tensor ${\epsilon }_{ij}$ at a certain point can be decomposed into the deviatoric strain tensor ($e_{ij}$) and the spherical strain tensor (${\epsilon }_m$). These parameters have a specific relationship under an elastic state, which can be expressed as follows:

The following assumptions are made (Ottosen 1986): (1) creep is sensitive to deviatoric stress tensor ($S_{ij}$), but the spherical stress tensor (${\sigma }_m$) does not cause the creep; (2) the Poisson’s ratio of the rock does not change over time in the process of creep. Based on the above assumptions, the Burgers model yields:

Double Burgers model of fractured rock masses is proposed in order to study the creep behaviors of fractured rock masses [23]. The implementation steps are as follows: (1) Determine creep parameters of the rock matrix based on Burgers creep model; (2) Determine creep propagation length of rock cracks based on equivalent Burgers model for crack creep fracture; (3) Determine the inelastic damage strain, caused by creep fracture of rock cracks; (4) Determine whole creep strain of fractured rock masses according to Eq. (12). To implement the numerical simulation of double Burgers model of fractured rock masses, the inelastic damage deformation caused by crack creep fracture is embedded into the Burgers model in FLAC^3D.

5.1. Numerical results of the specimen containing ordered cracks

Fractured rock specimen with a height of 2.0 m and a width of 1.0 m has two sets of ordered cracks [24, 25]. Table 3 shows the distribution parameters of the original cracks. According to the previous study [16], the creep parameters of rock matrix are shown in Table 4. The parameters for subcritical crack propagation and equivalent burgers model of marble crack are listed in Tables 1 and 2, respectively. The maximum and minimum principle stresses applied on specimens are set to 35 MPa and 1.65 MPa, respectively.

The inelastic damage strain, caused by creep fracture of rock cracks is calculated according to Eq. (24). The axial and lateral inelastic damage strains are plotted in Fig. 8, where the lateral damage strain is much higher than the axial damage strain. For example, the lateral damage strain is 2.68 times of the axial damage strain at creep time of 176 h, because the contribution of wing propagation to lateral damage strain is larger than axial damage strain [26].

Figs. 9(a) and (b) show the comparisons of creep curves of fractured rock specimen with and without consideration of creep damage, respectively. It is noted that both the lateral and axial creep strains with consideration of creep damage are higher than those without consideration of creep damage. Especially, the lateral creep strain with consideration of creep damage is much higher than that without consideration of creep damage. For example, the lateral creep strain when creep damage is considered is 1.48 times of this when creep damage is ignored at creep time of 176 h. The creep ratio at steady creep stage with and without consideration of creep damage is 7.18×10^-9 h^-1 and 4.33 h^-1, the former is 1.66 times of the latter.

Fig. 8Inelastic damage strain curves of ordered rock cracks group

Fig. 9Comparisons of creep curves of fractured rock specimen with and without consideration of creep damage for: a) lateral creep strain, b) axial creep strain

a)

b)

5.2. Numerical results of the specimens containing random cracks

Fractured rock specimens containing random cracks is a square with a height of 20 m and a width of 20 m as shown in Fig. 10. The distribution parameters of the random cracks are listed in Table 5.

Fig. 10Fractured rock specimens containing random cracks

The creep damage strain-time curve of the fractured rock masses is plotted in Fig. 11. The axial and lateral creep damage strains nonlinearly increase with the increase in creep time. The lateral damage strain is much higher than the axial damage strain. The lateral damage strain is 3.46 times of the axial damage strain at creep time of 200 h.

Fig. 11The creep damage strain-time curve of the fractured rock masses

Fig. 12 shows the creep curves of fractures rock masses, when creep damage is considered and ignored. It is noted the crack creep fracture significantly contributes to the lateral creep damage, so the lateral creep strain with consideration of creep damage is much higher than that without consideration of creep damage, however, the contribution of crack creep fracture to the lateral creep strain relatively is smaller.

Fig. 12Comparisons of creep curves with and without consideration of creep damage

For example, the lateral and axial creep strains when creep damage is considered are 1.40 and 1.03 times of those when creep damage is ignored at creep time of 200 h, respectively. The above rules for fractured rock specimens containing random cracks are the same to fractured rock specimens containing random cracks.