## Numerical simulation on rock failure process under

The failure process of rock subjected to combined static and dynamic ... According to the elastic damage theory, the elastic modulus of an element degrades.

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

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.

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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).

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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)