Rock Dynamics and Applications – State of the Art – Zhao & Li (eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00056-8

Numerical simulation on rock failure process under combined static and dynamic loading W.C. Zhu, L.L. Niu, J. Wei, Y. Bai & C.H. Wei

Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang, China

ABSTRACT: In this study, a damage-based model for simulating the damage and failure process of rock under combined static and dynamic loading is introduced, and it is implemented into RFPA-Dynamics (Rock Failure Process Analysis for Dynamics) and COMSOL Multiphysics, in order to simulate rock failure under combined static and dynamic loading conditions. The rock failure under combined static and dynamic loading during the SHPB test is numerically simulated and the mechanism associated with the dynamic strength increase under combined static and dynamic loading is clarified. In addition, the rock blasting under in-situ geo-stress is simulated as a combined static, dynamic and static process, i.e. the rock damage process under consecutive contribution of quasi-static geo-stress, blasting stress wave and quasi-static explosion gas pressure. Based on which, the effect of geo-stress condition on the blasting damage of rock is examined. 1 Introduction The failure process of rock subjected to combined static and dynamic loading constitutes the mechanism of many engineering applications such as rockburst prediction, rock fragmentation, as well as rock drilling and blasting. During deep underground mining, the existence of high static geo-stress concentration in rockmass is a prerequisite for the occurrence of rockburst, external disturbances, such as unloading due to excavation and dynamic disturbance excited by blasting may be key factors to trigger rockburst around the underground opening (Li et al. 2008, Zhu et al. 2010). In this regard, it is of great significance to study the damage and failure of rock under a combined static and dynamic loading. During the rock blasting, the rock is originally under a pre-existing geo-stress condition, the detonating of explosive results in two types of loadings applied on the borehole wall, namely a stress wave pulse and an explosion gas pressure with longer duration. The stress wave is responsible for initiation of the crushing zone and the surrounding radial fractures, while the explosion gas pressure further extends the fractures (Kutter and Fairhurst 1971, Ma and An 2008). Therefore, the rock blasting under in-situ geo-stress is simulated as a combined static, dynamic and static process, i.e. the rock damage process under consecutive contribution of quasi-static geo-stress, blasting stress wave and quasi-static explosion gas pressure. In this work, a general damage model for rock failure under static or dynamic loading is introduced and it is implemented into the Rock Failure Process Analysis for Dynamics (RFPA-Dynamics) to simulate the deformation and failure process of rock under combined static and dynamic loading during the SHPB tests. Also, this rock damage model is implemented into commercial FEM software, COMSOL Multiphysics (COMSOL 2008), to simulate the rock blasting, when rock damage is considered to be the consecutive contribution of quasi-static geo-stress, blasting stress wave and quasi-static explosion gas pressure. 395

2 Elastic damage model of rock Initially the rock is assumed elastic, with constitutive relationship defined by a generalized Hooke’s law. As illustrated in Figure 1, the damage of rock in tension or shear is initiated when its state of stress satisfies the maximum tensile stress criterion or the Mohr-Coulomb criterion, respectively, as expressed by:

F1 ≡ σ 1 − ft 0 = 0 or

F2 ≡ −σ 3 + σ 1 (1 + sinθ ) θ (1 − sin φ ) − fc 0 = 0

(1)

where ft0 and fc0 are uniaxial tensile and compressive strength (Pa), respectively, θ is internal frictional angle, and F1 and F2 are two damage threshold functions. Under any stress conditions, the tensile strain criterion is applied preferentially. According to the elastic damage theory, the elastic modulus of an element degrades monotonically as damage evolves, and the elastic modulus of damaged rock is expressed as follows: E = (1 − D )E0

(2)

where D represents the damage variable, and E and E0 are the elastic moduli of the damaged and the undamaged material (Pa), respectively. In this kind of numerical simulation, the element as well as its damage is assumed isotropic. According to Figure 1, the damage variable can be calculated as:

D = 1− 1−

F1 < 0 and F2 < 0

0

εt 0 ε1

n

εc0 ε3

n

F1 = 0 and dF1 > 0

(3)

F2 = 0 and dF2 > 0

where εt0 and εc0 are maximum tensile principal strain and maximum compressive principal strain when damage occurs, and n is a constitutive coefficient and it is 2.0. In this respect, the damage variable calculated with Eq. (3) is always from 0 to 1.0 regardless of what kind of damage it may suffer. It should be noted that, in the numerical implementation of Eq. (3), the tensile damage is always preferable to shear one, that is to say, the maximum tensile stress criterion is firstly used to judge whether the elements damage in tension or not, only the elements that do not damage in tensile mode will be checked its shear damage with the Mohr-Coulomb criterion.

Figure 1. The elastic damage-based constitutive law under uniaxial stress condition.

396

3 rock failure under combined static and dynmaic loading The numerical model is established according to the SHPB tests of rock done with the apparatus designed by Li et al. (2008). The rock specimen is sandwiched between two steel bars (incident bar and transmitted bar). The rock specimen, as well as two steel bars, is included in the numerical model. The applied static boundary stresses (denoted as ps), are applied incrementally until pre-specified static stresses are attained. Then the incident triangular stress pulse pd(t) is input, because this incident waveform is similar to the half-sine shape and is of great help in eliminating oscillations during wave propagation in the bars. During the numerical simulations, the stress in rock specimen is retrieved with different methods, i.e. σs ( = (σi + σr + σt)/2) is calculated according the data of incident wave σi, reflected wave σr and transmitted wave σt. σt may also represent the stress in rock specimen if the stress equilibrium at two ends of the specimen is achieved, and σave is stress averaged over 5 typical points in the specimen. Figure 2 shows the numerical results of a rock specimen under a combined static and dynamic loading, where the stress-strain curve, damage distribution, strain rate and stress equilibrium factor are presented. Because the rock specimen is highly stressed, its damage initiates at t = 4 µs (just after the stress wave travels into the rock specimen). Also, the value

Figure 2. Failure process of rock specimen under combined static-dynamic loading (ps = 140 MPa).

397

Figure 3. Variation of combined static-dynamic strength with static stress (DIF is dynamic increase factor of rock strength, ps is the static pre-compression stress, and σc is the uniaxial static compressive strength).

of σs is close to that of σave although it under-estimated the post-peak brittleness of the stress-strain curve. The strain rates changes from about 87.1 s-1 at ti = 4.0 µs for damage initiation to 103.0 s-1 at tf = 17.0 µs for peak stress. The stress equilibrium factor Fs is 0.98 when damage initiates at ti = 4.0 µs, and it increases to 1.03 at t1 = 6.2 µs. In general, in the pre-peak region, constant strain rate and stress equilibrium can be maintained. However, in the post-peak region of the stress-strain curve, stress equilibrium is lost, and the strain rate also increases dramatically. As shown in Figure 3, the strength of the rock specimen (denoted with σs) increases continuously with the elevated static stress σs. When different homogeneity indices of 1.5, 3.0 and 6000.0, as defined in Zhu et al. (2010), are specified to the rock specimens, the Dynamic Increase Factor (DIF) of the rock strength calculated using and σave varies considerably (Fig. 3). The DIF (characterized by σs) increases gradually with static stress. However, for the DIF characterized by σave, it increases until a peak and then decreases with increasing static stress. In this regard, the rock strength denoted by σave shows a similar tendency to experimental response determined by Li et al. (2010). In addition, this tendency seems more distinct with increasing heterogeneity of the rock specimen. Compared to the heterogeneous rock specimens (m = 1.5 or m = 5.0), the DIF of a homogeneous rock specimen (m = 6000.0) characterized by σave is nearly constant, which indicates that rock heterogeneity is one factor that may lead to the increased rock strength under combined static and dynamic loading. 4 blasting damage of rock under in-situ stress It is the blasting stress wave and explosion gas pressure that contribute to the rock fragmentation during rock blasting (Kutter and Fairhurst 1971). The blasting stress wave initiates the primarily radial cracks, and the quasi-static explosion gas pressure may result in the increase of the crushed zone radius, the extension of existing cracks and possible creation of new radial cracks. Figure 4 shows the model setup for the blasting damage when two boreholes are detonated simultaneously. The boundary stresses of σbx and σby for quasi-static stress analysis are applied in X and Y directions, respectively; the p and pg are blasting stress wave and explosion gas pressure that are applied consecutively to the boundary of the borehole. The stress-time history is a general form of a pulse function, which can be used to represent a large range of the borehole pressure (Cho et al. 2003, Ma and An 2008), expressed as, 398

Figure 4. The numerical model for blasting damage in rock under in-situ stress (σbx and σby are oundary stresses for quasi-static analysis applied in X and Y directions, respectively; p and pg are b blasting stress wave and explosion gas pressure applied for blasting damage analysis).

Figure 5. The development of damage during rock blasting under different in-situ geo-stress conditions.

p = p0ξ e− α t − e− β t

(4)

where p is the stress at time t (Pa), p0 is the peak stress (Pa), and α and β are constants. For convenient representation of the rising and decaying phase, two constants, i.e., ξ = 1 /(e −α t0 − e − β t0 ) and t0 = (1 /( β − α ))ln( β / α ), are defined. The quasi-static explosion gas pressure, is assumed to be expressed with a Weibull function expressed as, pg ( s ) = pg 0

m s u0 s0

m− 1

399

s m exp − s0

(5)

where pg is the explosion gas pressure (Pa), pg0 is the reference explosion gas pressure related to the peak value of pg. pg0 is roughly specified based on previous experiences according to the equation of state of explosion products as outlined in (Persson et al. 1994), s is the loading step for explosion gas pressure, and m is a shape parameter. As shown in Figure 5, when the blasting stress wave is applied, the crushed zone is produced around the borehole, around which, the tensile radial cracks may also initiate. As time elapses, the crushed zone may extend and the radial cracks propagate further. After the quasi-static explosion gas pressure is applied, it may contribute a lot to the formation and propagation of the existing radial cracks. In this regard, the numerical model proposed above can at least qualitatively capture the damage zone development around the borehole during blasting. The incorporation of in-situ geo-stress into the numerical simulation leads to the shrinkage of the damage zone, due to the confinement of geo-stress on the rock. Furthermore, the extent of the damage zone is closely related to the in-situ stress because the lateral pressure coefficient controls the stress distribution around the boreholes before the blasting loading is applied. As shown in Figure 5, the crack propagation direction generally coincides with the maximum compressive principal stress. Acknowledgements The present work is funded by National Science Foundation of China (Grant Nos. 51222401, 51128401, 50934006), the Faculty and Staff Exchange Grants of Sino-Swiss Science and Technology Cooperation (SSSTC) Scheme (Grant No. EG092011), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110042110035), the China-South Africa Joint Research Programme (Grant No. 2012DFG71060), and the Fundamental Research Funds for the Central Universities of China (Grant Nos. N110201001 and N100601004). This support is gratefully acknowledged. References Cho, S.H., Ogata, Y., Kaneko, K. 2003. Strain rate dependency of the dynamic tensile strength of rock. International Journal of Rock Mechanics and Mining Sciences 40: 763–777. COMSOL AB. 2008. COMSOL Multiphysics Version 3.5, User’s Guide and Reference Guide. (ww. comsol.com). Kutter, H.K., Fairhurst, C. 1971. On the fracture process in blasting. International Journal of Rock Mechanics and Mining Sciences 8: 181–202. Li, X.B., Zhou, Z.L., Lok, T.S., Hong, L., Yin, T.B. 2008. Innovative testing technique of rock subjected to coupled static and dynamic loads. International Journal of Rock Mechanics and Mining Sciences 45: 739–748. Li, X.B., Gong, X.Q., Zhao, J., Gao, K., Yin, T.B. 2010. Test study of impact failure of rock subjected to one-dimensional coupled static and dynamic loads (in Chinese). Chinese Journal of Rock Mechanics and Engineering 29 (2): 251–260. Ma, G.W., An, X.M. 2008. Numerical simulation of blasting-induced rock fractures. International Journal of Rock Mechanics and Mining Sciences 45: 966–975. Persson, P.A., Holmberg, R., Lee, J. 1994. Rock Blasting and Explosives Engineering. CRC Press, pp. 100–142. Zhu, W.C., Li, Z.H., Zhu, L., Tang, C.A. 2010. Numerical simulation on rockburst of underground opening triggered by dynamic disturbance. Tunnelling and Underground Space Technology 25(5): 587–599.

400

Numerical simulation on rock failure process under combined static and dynamic loading W.C. Zhu, L.L. Niu, J. Wei, Y. Bai & C.H. Wei

Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang, China

ABSTRACT: In this study, a damage-based model for simulating the damage and failure process of rock under combined static and dynamic loading is introduced, and it is implemented into RFPA-Dynamics (Rock Failure Process Analysis for Dynamics) and COMSOL Multiphysics, in order to simulate rock failure under combined static and dynamic loading conditions. The rock failure under combined static and dynamic loading during the SHPB test is numerically simulated and the mechanism associated with the dynamic strength increase under combined static and dynamic loading is clarified. In addition, the rock blasting under in-situ geo-stress is simulated as a combined static, dynamic and static process, i.e. the rock damage process under consecutive contribution of quasi-static geo-stress, blasting stress wave and quasi-static explosion gas pressure. Based on which, the effect of geo-stress condition on the blasting damage of rock is examined. 1 Introduction The failure process of rock subjected to combined static and dynamic loading constitutes the mechanism of many engineering applications such as rockburst prediction, rock fragmentation, as well as rock drilling and blasting. During deep underground mining, the existence of high static geo-stress concentration in rockmass is a prerequisite for the occurrence of rockburst, external disturbances, such as unloading due to excavation and dynamic disturbance excited by blasting may be key factors to trigger rockburst around the underground opening (Li et al. 2008, Zhu et al. 2010). In this regard, it is of great significance to study the damage and failure of rock under a combined static and dynamic loading. During the rock blasting, the rock is originally under a pre-existing geo-stress condition, the detonating of explosive results in two types of loadings applied on the borehole wall, namely a stress wave pulse and an explosion gas pressure with longer duration. The stress wave is responsible for initiation of the crushing zone and the surrounding radial fractures, while the explosion gas pressure further extends the fractures (Kutter and Fairhurst 1971, Ma and An 2008). Therefore, the rock blasting under in-situ geo-stress is simulated as a combined static, dynamic and static process, i.e. the rock damage process under consecutive contribution of quasi-static geo-stress, blasting stress wave and quasi-static explosion gas pressure. In this work, a general damage model for rock failure under static or dynamic loading is introduced and it is implemented into the Rock Failure Process Analysis for Dynamics (RFPA-Dynamics) to simulate the deformation and failure process of rock under combined static and dynamic loading during the SHPB tests. Also, this rock damage model is implemented into commercial FEM software, COMSOL Multiphysics (COMSOL 2008), to simulate the rock blasting, when rock damage is considered to be the consecutive contribution of quasi-static geo-stress, blasting stress wave and quasi-static explosion gas pressure. 395

2 Elastic damage model of rock Initially the rock is assumed elastic, with constitutive relationship defined by a generalized Hooke’s law. As illustrated in Figure 1, the damage of rock in tension or shear is initiated when its state of stress satisfies the maximum tensile stress criterion or the Mohr-Coulomb criterion, respectively, as expressed by:

F1 ≡ σ 1 − ft 0 = 0 or

F2 ≡ −σ 3 + σ 1 (1 + sinθ ) θ (1 − sin φ ) − fc 0 = 0

(1)

where ft0 and fc0 are uniaxial tensile and compressive strength (Pa), respectively, θ is internal frictional angle, and F1 and F2 are two damage threshold functions. Under any stress conditions, the tensile strain criterion is applied preferentially. According to the elastic damage theory, the elastic modulus of an element degrades monotonically as damage evolves, and the elastic modulus of damaged rock is expressed as follows: E = (1 − D )E0

(2)

where D represents the damage variable, and E and E0 are the elastic moduli of the damaged and the undamaged material (Pa), respectively. In this kind of numerical simulation, the element as well as its damage is assumed isotropic. According to Figure 1, the damage variable can be calculated as:

D = 1− 1−

F1 < 0 and F2 < 0

0

εt 0 ε1

n

εc0 ε3

n

F1 = 0 and dF1 > 0

(3)

F2 = 0 and dF2 > 0

where εt0 and εc0 are maximum tensile principal strain and maximum compressive principal strain when damage occurs, and n is a constitutive coefficient and it is 2.0. In this respect, the damage variable calculated with Eq. (3) is always from 0 to 1.0 regardless of what kind of damage it may suffer. It should be noted that, in the numerical implementation of Eq. (3), the tensile damage is always preferable to shear one, that is to say, the maximum tensile stress criterion is firstly used to judge whether the elements damage in tension or not, only the elements that do not damage in tensile mode will be checked its shear damage with the Mohr-Coulomb criterion.

Figure 1. The elastic damage-based constitutive law under uniaxial stress condition.

396

3 rock failure under combined static and dynmaic loading The numerical model is established according to the SHPB tests of rock done with the apparatus designed by Li et al. (2008). The rock specimen is sandwiched between two steel bars (incident bar and transmitted bar). The rock specimen, as well as two steel bars, is included in the numerical model. The applied static boundary stresses (denoted as ps), are applied incrementally until pre-specified static stresses are attained. Then the incident triangular stress pulse pd(t) is input, because this incident waveform is similar to the half-sine shape and is of great help in eliminating oscillations during wave propagation in the bars. During the numerical simulations, the stress in rock specimen is retrieved with different methods, i.e. σs ( = (σi + σr + σt)/2) is calculated according the data of incident wave σi, reflected wave σr and transmitted wave σt. σt may also represent the stress in rock specimen if the stress equilibrium at two ends of the specimen is achieved, and σave is stress averaged over 5 typical points in the specimen. Figure 2 shows the numerical results of a rock specimen under a combined static and dynamic loading, where the stress-strain curve, damage distribution, strain rate and stress equilibrium factor are presented. Because the rock specimen is highly stressed, its damage initiates at t = 4 µs (just after the stress wave travels into the rock specimen). Also, the value

Figure 2. Failure process of rock specimen under combined static-dynamic loading (ps = 140 MPa).

397

Figure 3. Variation of combined static-dynamic strength with static stress (DIF is dynamic increase factor of rock strength, ps is the static pre-compression stress, and σc is the uniaxial static compressive strength).

of σs is close to that of σave although it under-estimated the post-peak brittleness of the stress-strain curve. The strain rates changes from about 87.1 s-1 at ti = 4.0 µs for damage initiation to 103.0 s-1 at tf = 17.0 µs for peak stress. The stress equilibrium factor Fs is 0.98 when damage initiates at ti = 4.0 µs, and it increases to 1.03 at t1 = 6.2 µs. In general, in the pre-peak region, constant strain rate and stress equilibrium can be maintained. However, in the post-peak region of the stress-strain curve, stress equilibrium is lost, and the strain rate also increases dramatically. As shown in Figure 3, the strength of the rock specimen (denoted with σs) increases continuously with the elevated static stress σs. When different homogeneity indices of 1.5, 3.0 and 6000.0, as defined in Zhu et al. (2010), are specified to the rock specimens, the Dynamic Increase Factor (DIF) of the rock strength calculated using and σave varies considerably (Fig. 3). The DIF (characterized by σs) increases gradually with static stress. However, for the DIF characterized by σave, it increases until a peak and then decreases with increasing static stress. In this regard, the rock strength denoted by σave shows a similar tendency to experimental response determined by Li et al. (2010). In addition, this tendency seems more distinct with increasing heterogeneity of the rock specimen. Compared to the heterogeneous rock specimens (m = 1.5 or m = 5.0), the DIF of a homogeneous rock specimen (m = 6000.0) characterized by σave is nearly constant, which indicates that rock heterogeneity is one factor that may lead to the increased rock strength under combined static and dynamic loading. 4 blasting damage of rock under in-situ stress It is the blasting stress wave and explosion gas pressure that contribute to the rock fragmentation during rock blasting (Kutter and Fairhurst 1971). The blasting stress wave initiates the primarily radial cracks, and the quasi-static explosion gas pressure may result in the increase of the crushed zone radius, the extension of existing cracks and possible creation of new radial cracks. Figure 4 shows the model setup for the blasting damage when two boreholes are detonated simultaneously. The boundary stresses of σbx and σby for quasi-static stress analysis are applied in X and Y directions, respectively; the p and pg are blasting stress wave and explosion gas pressure that are applied consecutively to the boundary of the borehole. The stress-time history is a general form of a pulse function, which can be used to represent a large range of the borehole pressure (Cho et al. 2003, Ma and An 2008), expressed as, 398

Figure 4. The numerical model for blasting damage in rock under in-situ stress (σbx and σby are oundary stresses for quasi-static analysis applied in X and Y directions, respectively; p and pg are b blasting stress wave and explosion gas pressure applied for blasting damage analysis).

Figure 5. The development of damage during rock blasting under different in-situ geo-stress conditions.

p = p0ξ e− α t − e− β t

(4)

where p is the stress at time t (Pa), p0 is the peak stress (Pa), and α and β are constants. For convenient representation of the rising and decaying phase, two constants, i.e., ξ = 1 /(e −α t0 − e − β t0 ) and t0 = (1 /( β − α ))ln( β / α ), are defined. The quasi-static explosion gas pressure, is assumed to be expressed with a Weibull function expressed as, pg ( s ) = pg 0

m s u0 s0

m− 1

399

s m exp − s0

(5)

where pg is the explosion gas pressure (Pa), pg0 is the reference explosion gas pressure related to the peak value of pg. pg0 is roughly specified based on previous experiences according to the equation of state of explosion products as outlined in (Persson et al. 1994), s is the loading step for explosion gas pressure, and m is a shape parameter. As shown in Figure 5, when the blasting stress wave is applied, the crushed zone is produced around the borehole, around which, the tensile radial cracks may also initiate. As time elapses, the crushed zone may extend and the radial cracks propagate further. After the quasi-static explosion gas pressure is applied, it may contribute a lot to the formation and propagation of the existing radial cracks. In this regard, the numerical model proposed above can at least qualitatively capture the damage zone development around the borehole during blasting. The incorporation of in-situ geo-stress into the numerical simulation leads to the shrinkage of the damage zone, due to the confinement of geo-stress on the rock. Furthermore, the extent of the damage zone is closely related to the in-situ stress because the lateral pressure coefficient controls the stress distribution around the boreholes before the blasting loading is applied. As shown in Figure 5, the crack propagation direction generally coincides with the maximum compressive principal stress. Acknowledgements The present work is funded by National Science Foundation of China (Grant Nos. 51222401, 51128401, 50934006), the Faculty and Staff Exchange Grants of Sino-Swiss Science and Technology Cooperation (SSSTC) Scheme (Grant No. EG092011), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110042110035), the China-South Africa Joint Research Programme (Grant No. 2012DFG71060), and the Fundamental Research Funds for the Central Universities of China (Grant Nos. N110201001 and N100601004). This support is gratefully acknowledged. References Cho, S.H., Ogata, Y., Kaneko, K. 2003. Strain rate dependency of the dynamic tensile strength of rock. International Journal of Rock Mechanics and Mining Sciences 40: 763–777. COMSOL AB. 2008. COMSOL Multiphysics Version 3.5, User’s Guide and Reference Guide. (ww. comsol.com). Kutter, H.K., Fairhurst, C. 1971. On the fracture process in blasting. International Journal of Rock Mechanics and Mining Sciences 8: 181–202. Li, X.B., Zhou, Z.L., Lok, T.S., Hong, L., Yin, T.B. 2008. Innovative testing technique of rock subjected to coupled static and dynamic loads. International Journal of Rock Mechanics and Mining Sciences 45: 739–748. Li, X.B., Gong, X.Q., Zhao, J., Gao, K., Yin, T.B. 2010. Test study of impact failure of rock subjected to one-dimensional coupled static and dynamic loads (in Chinese). Chinese Journal of Rock Mechanics and Engineering 29 (2): 251–260. Ma, G.W., An, X.M. 2008. Numerical simulation of blasting-induced rock fractures. International Journal of Rock Mechanics and Mining Sciences 45: 966–975. Persson, P.A., Holmberg, R., Lee, J. 1994. Rock Blasting and Explosives Engineering. CRC Press, pp. 100–142. Zhu, W.C., Li, Z.H., Zhu, L., Tang, C.A. 2010. Numerical simulation on rockburst of underground opening triggered by dynamic disturbance. Tunnelling and Underground Space Technology 25(5): 587–599.

400