Scale model test study of influence of joints on blasting vibration attenuation

Abstract

Ground vibrations are an integral part of the process of rock blasting. The analysis of blasting vibration attenuation is the basis for a blasting risk assessment. To study the influence of rock joints on the blasting vibration attenuation, an autoclaved aerated concrete block was used as a similar material of rock in model tests of blasting vibration propagation. The attenuation process of the blasting vibration was physically simulated. The attenuation behaviors of the blasting vibrations in different directions to joint strike were fitted. Additionally, a positive correlation between the attenuation parameters and the directions was obtained. Furthermore, the Hilbert–Huang transform (HHT) based on the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) was used to analyze the vibration velocity signals. The characteristics and variation of the energy distribution were studied in the time–frequency domain. The results showed that the energy amplitude decreased as the propagation distance increased. The energy distribution gradually evolved from concentrated to discrete in the frequency domain. The low-frequency components were always present in the velocity signals at each measurement point, but the high-frequency components constantly changed due to the reflection and transmission of the waves after encountering the joint surfaces. The results of this study provide a reference for the prediction and control of the blasting vibration effect in jointed rock masses.

Appendix

Similarity means that different systems undergo the same physical variation process with similar geometric shapes. For a certain physical quantity, the ratio between two systems is called the similarity ratio, which is expressed by C_x, where the subscript x represents the specific physical quantity. There are mutual constraint relationships between the similarity ratios of different physical quantities (Song 2016). The knowledge of certain ratios in two similar systems enables the analyst or experimenter to infer the value of a certain physical quantity if the values of the other physical quantities are known.

Physical quantities are classified as basic or derived physical quantities. The basic physical quantities exist independently of the other physical quantities, and the derived physical quantities are derived from the basic physical quantities. Dimensions are used to distinguish the types of physical quantities. Dimensions are classified as basic or derived dimensions. Basic dimensions are the dimensions of the basic physical quantities, such as the dimensions of force [F], time [T], and length [L]. Derived dimensions are derived from the basic dimensions, and their distinguishing feature is that they can be expressed in the form of power functions of the basic dimensions. Moreover, some physical quantities have no dimensions, which are called zero dimensions, such as angles, strains, and Poisson’s ratios. The similarity ratio of a physical quantity with a zero dimension is 1. In addition, physical quantities with the same dimensions have the same ratios.

In view of the research focus of this model test, the density, volumetric weight, gravitational acceleration, angle, Poisson’s ratio, strain, length, Brazilian tensile strength, saturated uniaxial compressive strength, elastic modulus, P-wave velocity, time, and frequency were selected as similarity variables. For convenience of expression, a physical quantity symbol subscripted by p denotes a prototype physical quantity, and a symbol subscripted by m denotes a model physical quantity.

• (1) The similarity ratios of physical quantities with zero dimension were as follows:

$$ \mathrm{similarity}\ \mathrm{ratio}\ \mathrm{of}\ \mathrm{strain},\kern0.5em {C}_{\varepsilon }=\frac{\epsilon_{\mathrm{p}}}{\epsilon_{\mathrm{m}}}=1, $$

(26)

$$ \mathrm{similarity}\ \mathrm{ratio}\ \mathrm{of}\ \mathrm{Poisson}^{\prime}\mathrm{s}\ \mathrm{ratio},\kern0.5em {C}_v=\frac{v_{\mathrm{p}}}{v_{\mathrm{m}}}=1, $$

(27)

$$ \mathrm{similarity}\ \mathrm{ratio}\ \mathrm{of}\ \mathrm{angle},\kern0.5em {C}_{\theta }=\frac{\theta_{\mathrm{p}}}{\theta_{\mathrm{m}}}=1. $$

(28)

• (2) The elastic modulus and strength (including the Brazilian tensile strength and saturated uniaxial compressive strength) have the same dimensions, so their corresponding similarity ratios were the same:

$$ {C}_{\sigma }=\frac{\sigma_{\mathrm{p}}}{\sigma_{\mathrm{m}}}={C}_E=\frac{E_{\mathrm{p}}}{E_{\mathrm{m}}}. $$

(29)

• (3) Since the prototype and model were in the same gravitational field, the similarity ratio of the gravitational acceleration was C_g = 1.

• (4) The similarity ratios of the density and geometric length were set as follows:

$$ \mathrm{similarity}\ \mathrm{ratio}\ \mathrm{of}\ \mathrm{density},\kern0.5em {C}_{\rho }=\frac{\rho_{\mathrm{p}}}{\rho_{\mathrm{m}}}=3.5, $$

(30)

$$ \mathrm{similarity}\ \mathrm{ratio}\ \mathrm{of}\ \mathrm{length},{C}_L=\frac{L_{\mathrm{p}}}{L_m}=13. $$

(31)

• (5) Based on the relationship between the volumetric weight and density,

$$ {C}_{\gamma }={C}_g{C}_{\rho }=3.5. $$

(32)

• (6) Dimensional analysis was used to determine the similarity ratio of the strength.

γ and L were taken as known physical quantities, and σ was an unknown physical quantity. The dimensions of these three physical quantities are expressed in terms of the basic dimensions, [γ] = [FL⁻³], [L] = [L], [σ] = [FL⁻²].

A power function for σ in terms of γ and L was defined as follows:

$$ \sigma ={\gamma}^{x_1}{L}^{x_2}. $$

(33)

where x₁ and x₂ are unknown exponentials of the power function. Equation (33) is expressed in dimensional form as follows:

$$ {FL}^{-2}={\left({FL}^{-3}\right)}^{x_1}\bullet {(L)}^{x_2}. $$

(34)

The solution is $\left\{\begin{array}{c}{x}_1=1\\ {}{x}_2=1\end{array}\right.$, and thus,

$$ \sigma =\gamma L. $$

(35)

Correspondingly, the similarity ratio of the strength is

$$ {C}_{\sigma }=\frac{\sigma_{\mathrm{p}}}{\sigma_{\mathrm{m}}}={C}_{\gamma }{C}_L=45.5. $$

(36)

• (7) Dimensional analysis is used to determine the similarity ratio of the P-wave velocity, σ and ρ were taken as known physical quantities, and vp was an unknown physical quantity. The forms expressed by the basic dimensions are as follows: [σ] = [FL⁻²], [ρ] = [FT²L⁻⁴], and [vp] = [LT⁻¹].

A power function for vp in terms of σ and ρ was defined as follows:

$$ vp={\sigma}^{x_3}{\rho}^{x_4}, $$

(37)

where x₃ and x₄ are unknown exponentials. Equation (37) can be transformed into a dimensional expression as follows:

$$ {LT}^{-1}={\left({FL}^{-2}\right)}^{x_3}\bullet {\left({FT}^2{L}^{-4}\right)}^{x_4}. $$

(38)

The solution is $\left\{\begin{array}{c}{x}_3=\frac{1}{2}\\ {}{x}_4=-\frac{1}{2}\end{array}\right.$, and thus,

$$ vp={\sigma}^{\frac{1}{2}}{\rho}^{-\frac{1}{2}}. $$

(39)

The similarity ratio of the P-wave velocity is

$$ {C}_{vp}=\frac{vp_{\mathrm{p}}}{vp_{\mathrm{m}}}={C_{\sigma}}^{\frac{1}{2}}{C_{\rho}}^{-\frac{1}{2}}=3.6. $$

(40)

• (8) The similarity ratios of the time and frequency are respectively:

$$ {C}_t=\frac{t_{\mathrm{P}}}{t_{\mathrm{m}}}=\frac{C_L}{C_{vp}}=3.6, $$

(41)

$$ {C}_f=\frac{f_{\mathrm{P}}}{f_{\mathrm{m}}}=\frac{1}{C_t}=\frac{1}{3.6}=0.28. $$

(42)

The similarity ratios of the above physical quantities are summarized in Table 2.