Method of Numerical Modeling for Constitutive Relations of Clay

INTRODUCTION

In jointed and fractured rock masses, it is often difficult to achieve a smooth fracture plane between blasted holes.McKown (1984) indicated that when the angle between rock structural planes and the blast-induced plane is less than 60°, it is difficult to achieve a smooth-blasting surface.From the viewpoint of rock strength theory, Zhang (1988) pointed out that, when the angle between rock structural plane and blast-induced plane is within 25° to 40°, the blast-fractured surface is jagged in the form of "Z" pattern.

Generally, the factors, which affect blasting results, may be divided into those that can be controlled and those that cannot be controlled.The controllable factors include explosive properties, initiation timing and blast geometry.The uncontrollable factors comprise the rock natural structures, such as joints and fractures, and properties, such as elastic constants, density and strength. Among these, the influence of rock structural planes often contributes to a high degree of variability of blasting results.

This paper presents a theoretical analysis of rock structural plane influence on blasting results based on elasticity and stress wave propagation theory with an emphasis on smooth blasting techniques.Simulated experimental results, using strain and acoustic emission measurements for verifying the theoretical analysis, are also given.

THEORETICAL ANALYSIS

In blasting operations, the rock structure conditions shown in Fig. 1 are often encountered, the rock structural plane and blastinduced fracture plane intersect at angle β, where β is the rock structural plane direction with respect to the plane intersecting the boreholes.A line normal to the rock structural plane is given by 'n', and α is the incident angle of the blast-induced stress wave. In other words, α is the angle between the incident wave (blastinduced stress wave) and 'n' (the normal line of structural plane).By geometry, it can be seen that α+β＝90°.

Figure  1.  Relationship between rock structural plane and blast-induced fracture planes.

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To form a smooth fractured plane without wall damage, the tangential explosive stress component, σ_θ, must be greater than the rock mass dynamic tensile strength of rock and the shear stress component, τ_α, induced on the rock structural plane must be less than rock joint shear strength as determined by Coulomb's criterion

where σ_td is the dynamic tensile strength of the rock joint, while c is the cohesive force on rock structural plane, and ϕ is the friction angle of joint.

According to elasticity theory (Yin, 1990), normal and shear stress components on the rock structural plane in Fig. 1are determined by the following equations

in which the relationship between the tangential stress component, σ_θ, and radial stress component, σ_r, is given by

and η＝C_s/C_p.C_s and C_p are the propagation speed of shear wave and pressure wave in rock masses, respectively.Suppose that K＝1－2η², then

According to the wave theory (Guo, 1982), the following equations can be established.

In the above equations, E is Young's modulus; ρ is the density, and μ is the Poisson ratio of the rock.Using the relations in equation (5), K can be defined as

By substituting equations (2) into the expression for shear strength (equation(1)) and using equations (4) and (6), replacing α with 90°－β

From equation(7), it can be seen that the blast-induced stress components in the rock mass are functions of Poisson's ratio, μ, cohesion, c, friction angle, φ, and the orientation of the structural plane, β.

Previous research by Wu (1997) shows that the rock structural plane orientation has an influence on the stress wave propagation only when incident angle, α, is within a certain range. Shown in Fig. 1, when the angle β ranges from (90°－α₁) to(90°－ α₂), the rock structural plane affects the propagation of blast-induced stress wave.

For incident pressure (P) wave with amplitude A₁, shown in Fig. 2.

Figure  2.  Incident pressure wave on rock structure plane.

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The transmission coefficient (A₅) and the reflection coefficient (A₂) of the P-wave can be calculated by the following (Kolsky, 1983)

Given Poisson's ratio μ＝021 and friction angle ϕ＝10°, 20°, 25°, and 30° respectively, the computed reflection and transmission coefficients are shown in Fig. 3.From Fig. 3, it can be shown that, regardless of the friction angle value, the transmission coefficient (A₅) is minimum while the reflection coefficient (A₂) is maximum when the incident angle α is about 60°(β≈30°).T his indicates that the transmission wave energy is a minimum while the reflection wave energy is a maximum when β is about 30°.T he utilization ratio of explosive energy is the lowest under such blasting conditions.In other words, such blasting conditions cannot achieve a smooth fracture surface.

Figure  3.  Reflection and transmission coefficient with different friction angle ϕ.A₂.reflection coefficient; A_s.transmission coefficient.

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A prerequisite for equation (9) is that the shear stress component on the rock structural plane induced by the incident pressure wave must be larger than or at least equal to the frictional resistance, which can be written as the following form

In Fig. 2, η²＝sin²β/sin²α, so equation (10) can be defined as another form.

Plotting values for α and b using equation (11) when Poisson's ratio, μ, takes on the value of 0.25, it can be shown in Fig. 4 that the rock structural plane affects the propagation of blast-induced wave as the incident angle α ranges from 30°to 80° (the friction angle b is about 18°and β is calculated to range from 10°to 60°under this circumstance).

Figure  4.  Ranges of incident angle α determined by friction angle ϕ.

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Given the theoretical analysis above, it can be stated when the orientation of the rock structural plane and the blast-induced fracture plane are identical, the stresses on the rock structural plane induced by blasting should satisfy the stress-state control equation (7).Rock blasting under this condition will achieve a good smooth surface along the central line of boreholes.If the rock structure angle, β, does not satisfy equation (7) and the joints exhibit a lowly shear strength, the blast-induced surface is usually jagged or in the form of a'Z'.In this case, the rock will be fractured on the structural plane by shear stress, and the tensile stress applied on the structural plane has no effect on the rock breakage.As such, smooth wall fracturing cannot be obtained.

SIMULATED EXPERIMENTS

Blasting experiments in marble and cement mortar were carried out in laboratory conditions using simulated rock structural planes.Marble cores, 50 mm in diameter and 300 mm in length, were prepared with a hole 10 mm in diameter and 60 mm in depth.Shown in Fig. 5, the marble core was cut using a diamond saw to an angle β and glued with a binder to simulate the rock structural plane.An electronic detonator, loaded in the borehole, was used as the explosive source.FoiL-l type strain gages, 3 mm× 15 mm in size, were mounted on both sides of the joint.T he data acquisition system is shown in Fig. 5 and consists of a KD-54super-dynamic amplifier and a CS2092waveform recorder/analyzer.

Figure  5.  Marble core in strain gage experiment and measurement system.

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Table 1 gives the peak strain results of these experiments as a function of β.Figure 6 shows the relationship between ε₂/ε₁ and β, showing that the strain ratio, ε₂/ε₁, is a minimum when β≈30°.T his indicates that a joint orientation of 30°promotes the highest attenuation of explosive energy.In such a case, it will be difficult to achieve a smooth blast-fractured surface.

Table  1.  RESULTS OF STRAIN MEASUREMENT

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Figure  6.  Relationship between ε₂/ε₁ and β.

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A second series of experiments were conducted using a cement mortar (cement∶sand∶water＝1∶3∶1), to simulate the rock mass, cutting joints similar to the previous experiment.Acoustic emission measurements were made for varying values of β using the set-up shown in Fig. 7.

Figure  7.  Cement mortar in acoustic emission experiment.

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The results of these tests are given in Table 2.

Table  2.  EXPERIMENT RESULTS OF ACOUSTIC EMISSION

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Figure 8 shows the relationship between energy per unit sample length, E', and β.It can be seen from Fig. 8 that E' has minimum value when β≈30°, providing similar results to the previous experiments.Typical acoustic waveforms are shown in Fig. 9.

Figure  8.  Relationship between E' and β.

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Figure  9.  Typical acoustic waveform.

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CONCLUSIONS

Both theoretical analysis and simulated experiments in the laboratory show that the influence of rock structural planes contributes to a high degree of variability of blasting results.T he influence of rock structural planes on blasting results depends on the angles between the rock structural planes and the blast-induced fracture planes.It is difficult to achieve better blasting results when the angles between the rock structural planes and the blastinduced fracture planes range from 10° to 60°.Among these angles, 30° is the least favorable angle for achieving a smooth wall. For the angles less than10°and greater than60°, the influence of rock structural planes on blasting results can be ignored.