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During the excavation of deep coal and rock mass, the radial stress of the free face changes from three-dimensional compression state to two-dimensional stress, bearing the combined action of dynamic disturbance and static load at the same time. With that, many mines suffer from dynamic disasters, such as coal and gas outburst, rock burst and rock caving during deep mining excavation, which is often accompanied by plate crack, spalling and other disasters, seriously affecting the stability of stope and roadway. Taking thin rectangular coal and rock mass as the research object, the dual equation of the free vibration was derived and the exact solution model of the free vibration was established with the use of Hamilton dual system. Based on the action characteristics of the uniform impact load, the effective mode of the forced vibration was obtained by using the Duhamel integral principle and the orthogonality of the mode function. Based on the third strength theory and the numerical simulation results, the dynamic damage process and development trend of coal and rock mass were analyzed under uniform impact load. It was concluded that the starting position of dynamic damage can be judged by the first order main mode, and the development direction and trend of the damage can be judged by the fifth and sixth order main modes. It was concluded that the vibration mode functions of coal and rock mass with four side fixed (C-C-C-C), the two sides fixed and simply supported on the other (S-C-S-C) are mainly composed of three modes that are the first order (dominant frequency), the fifth order and the sixth order, from which the dynamic damage mechanism is preliminarily studied.

With the increasing demand for mineral resources and the continuous consumption of shallow resources, the mining of domestic and foreign coal mines and metal mines has gradually shifted to the deep [

Therefore, this research mainly focused on the dynamic response characteristic and failure mechanism of thin coal and rock mass based on rectangular thin plate model. The Hamilton dual system was used to study the dynamic response characteristics of thin coal and rock mass under forced vibration. Besides, based on the characteristics of uniform impact load, the effective vibration mode solution of thin coal and rock mass under forced vibration was studied by using Duhamel integral principle, and the dynamic damage failure process and mechanism of thin coal and rock mass were analyzed, which are beneficial to revealing the mechanism of coal rock dynamic disaster.

The size requirements of the thin rectangular plate (the ratio of the thickness to the width of the plate is between 1/5 and 1/80). The thin plate model can resist bending and torsion and can also bear the stress in the plane. And the neutral plane was used to establish the coordinate system as shown in

Supposing the density of the thin plate was ρ, the thickness was h, the size was a × b, the elastic modulus was E, and the Poisson’s ratio was υ. Under the impact load q ( x , y , t ) , according to the transient equilibrium conditions of the internal mechanics of the thin plate, the differential equation of the forced vibration of the thin plate can be derived [

D ∇ 4 w ( x , y , t ) + ρ h ∂ 2 w ( x , y , t ) ∂ t 2 = q ( x , y , t ) (1)

Among them, w ( x , y , t ) was the deflection, D was the bending stiffness.

Firstly, the homogeneous equation of Equation (1) was solved.

When q ( x , y , t ) = 0 , the free vibration differential equation of the thin plate was [

D ∇ 4 w ( x , y , t ) + ρ h ∂ 2 w ( x , y , t ) ∂ t 2 = 0 (2)

Supposing that the vibration of a thin plate had the following change rule of the harmonic oscillator with time:

w ( x , y , t ) = W ( x , y ) e i w t (3)

Among them, ω was the natural frequency and W ( x , y ) was the deflection mode function of the thin plate. Substituting Equation (3) into Equation (2), then:

∂ 4 W ∂ 4 x + ∂ 4 W ∂ 2 x ∂ 2 y + ∂ 4 W ∂ 4 y = k 4 W (4)

Among them, k 4 = ρ h ω 2 / D .

In order to decouple the physical quantities in the control equations, it was necessary to introduce the Hamilton dual equations [

If θ = ∂ W ∂ x , from the force analysis results of the thin plate, the relationship between the physical quantities were:

∂ θ ∂ x = − M x D − υ ∂ 2 W ∂ y 2 (5)

∂ V x ∂ x = D ( 1 − υ 2 ) ∂ 4 W ∂ y 4 − υ ∂ 2 M x ∂ y 2 − ρ ω 2 W (6)

∂ M x ∂ x = V x + 2 D ( 1 − υ ) ∂ 2 θ ∂ y 2 (7)

Among them, M_{x}, M_{y}, M_{xy}, Q_{x}, Q_{y}, V_{x}, V_{y} were the bending moment, torque, transverse shear force, and transverse total shear force of the thin plate, respectively. If vector v = [ W , θ , − V x , M x ] , then the dual equation of Hamilton system was:

v ′ = H v (8)

That was,

[ ∂ W ∂ x ∂ θ ∂ x ∂ ( − V x ) ∂ x ∂ M x ∂ x ] = [ 0 1 0 0 − υ ∂ 2 ∂ y 2 0 0 − 1 D − D ( 1 − υ 2 ) ∂ 4 ∂ y 4 + ρ ω 2 h 0 0 υ ∂ 2 ∂ y 2 0 2 D ( 1 − υ ) ∂ 2 ∂ y 2 − 1 0 ] ∗ [ W θ − V x M x ]

Based on the symplectic geometry method [

W ( x ) = a 1 e i β 1 x + b 1 e − i β 1 x + c 1 e β 2 x + d 1 e − β 2 x = A 1 cos β 1 x + B 1 sin β 1 x + C 1 cosh β 2 x + D 1 sinh β 2 x (9)

W ( y ) = a 2 e i α 1 y + b 2 e − i α 1 y + c 2 e α 2 y + d 2 e − α 2 y = A 2 cos α 1 y + B 2 sin α 1 y + C 2 cosh α 2 y + D 2 sinh α 2 y (10)

Among them:

α 1 = k 2 + μ 2 , α 2 = k 2 − μ 2

β 1 = k 2 + λ 2 , β 2 = k 2 − λ 2

α 1 2 + α 2 2 = β 1 2 + β 2 2 = 2 k 2

α 1 2 + β 1 2 = k 2 , α 2 2 + β 2 2 = 3 k 2 , α 2 2 − β 1 2 = k 2

The density of sheet coal bodies is h, the thickness is h, and the size is a × b, and the elastic modality is E. The four-side solid sheet coal body (C-C-C-C) along the y direction boundary condition is:

W ( x , 0 ) = 0 , ∂ W ( x , 0 ) / ∂ y = 0 ;

W ( x , b ) = 0 , ∂ W ( x , b ) / ∂ y = 0 .

Substitute (10) available:

[ 1 0 1 0 0 α 1 0 α 2 cos α 1 b sin α 1 b cosh α 2 b sinh α 2 b − α 1 sin α 1 b α 1 cos α 1 b α 2 sinh α 2 b α 2 cosh α 2 b ] ∗ [ A 2 B 2 C 2 D 2 ] = [ 0 0 0 0 ]

If the above formula had a solution, the determinant of the coefficient matrix on the left was zero, and the frequency equation along the y direction was:

1 − cos α 1 b cosh α 2 b sin α 1 b sinh α 2 b = α 1 2 − α 2 2 2 α 1 α 2 (11)

To make C_{2} = 1, the combined frequency equation solves:

A 2 = − 1 , B 2 = k 1 α 1 / α 2 , C 2 = 1 , D 2 = − k 1

Among them,

Thus,

W ( y ) = − cos α 1 y + α 2 α 1 k 1 sin α 1 y + cosh α 2 y − k 1 sinh α 2 y

Similarly,

W ( x ) = − cos β 1 x + β 2 β 1 k 2 sin β 1 x + cosh β 2 x − k 2 sinh β 2 x

Among them,

k 2 = cos β 1 a − cosh β 2 a β 2 β 1 sin β 1 a − sinh β 2 a

Therefore, the vibration equation of the free vibration of the Thin Coal and Rock Mass (C-C-C-C) is:

W ( x , y ) = W ( x ) W ( y ) = ( − cos β 1 x + β 2 β 1 k 2 sin β 1 x + cosh β 2 x − k 2 sinh β 2 x ) × ( − cos α 1 y + α 2 α 1 k 1 sin α 1 y + cosh α 2 y − k 1 sinh α 2 y ) (12)

As for a thin plate with two sides fixed and other two sides simply supported (S-C-S-C), the boundary conditions along the x direction (simple supported side) were:

W ( 0 , y ) = 0 , ∂ 2 W ( 0 , y ) / ∂ x 2 = 0 ;

W ( a , y ) = 0 , ∂ 2 W ( a , y ) / ∂ x 2 = 0 .

Substitute (9) available,

[ 1 0 1 0 − β 1 2 0 β 2 2 0 cos β 1 a sin β 1 a cosh β 2 a sinh β 2 a − β 1 2 cos β 1 a − β 1 2 sin β 1 a β 2 2 cosh β 2 a β 2 2 sinh β 2 a ] ∗ [ A 1 B 1 C 1 D 1 ] = [ 0 0 0 0 ]

Available from the upper formula, A 1 = C 1 = 0 ;

If there is an upper solution, the left coefficient matrix is zero, resulting in a frequency equation along the x direction:

sin β 1 a = 0

Make, B_{1} = 1, D_{1} = 0. So:

W ( x ) = sin β 1 x

W ( y ) = − cos α 1 y + α 2 α 1 k 1 sin α 1 y + cosh α 2 y − k 1 sinh α 2 y

Therefore, the vibration function of the free vibration of Thin Coal and Rock Mass (S-C-S-C) is:

W ( x , y ) = W ( x ) W ( y ) = sin β 1 x × ( − cos α 1 y + α 2 α 1 k 1 sin α 1 y + cosh α 2 y − k 1 sinh α 2 y ) (13)

The form of the solution of the non-homogeneous Equation (1) can be written as:

w ( x , y , t ) = ∑ n = 1 ∞ W n ( x , y ) ϕ n ( t ) (14)

Substituting Equation (13) into Equation (1):

∑ n = 1 ∞ [ D ∇ 4 W n ( x , y ) ϕ n ( t ) + ρ h W n ( x , y ) ϕ ″ n ( t ) ] = q ( x , y , t )

Because,

D ∇ 4 W n ( x , y ) = ρ h ω n 2 W n ( x , y )

Thus,

∑ n = 1 ∞ ρ h W n ( x , y ) [ ω n 2 ϕ n ( t ) + ϕ ″ n ( t ) ] = q ( x , y , t )

Both sides multiply by W m ( x , y ) at the same time, according to the orthogonality of the deflection vibration mode function, it can be derived [

∬ Ω ρ h W m ( x , y ) W n ( x , y ) d x d y = 0 , ( m ≠ n )

Integrating over the entire plane area, then:

∬ Ω ρ h W n 2 ( x , y ) [ ω n 2 ϕ n ( t ) + ϕ ″ n ( t ) ] d x d y = ∬ Ω q ( x , y , t ) W n ( x , y ) d x d y

If,

M n = ∬ Ω ρ h W n 2 ( x , y ) d x d y

P n ( t ) = ∬ Ω q ( x , y , t ) W n ( x , y ) d x d y

Then,

ϕ ″ n ( t ) + ω n 2 ϕ n ( t ) = 1 M n P n ( t )

According to Duhamel’ integral,

ϕ n ( t ) = 1 M n ω n ∫ 0 t P n ( τ ) sin ω n ( t − τ ) d τ

Assuming that the pressure of the explosion impact load on the thin plate could be regarded as an instantaneous uniform load, it had the following form:

q ( x , y , t ) = q 0 δ ( t − t 1 )

So:

ϕ n ( t ) = 1 M n ω n ∫ 0 t P n ( τ ) sin w n ( t − τ ) d τ = ∬ Ω q 0 W n ( x , y ) d x d y M n ω n ∫ 0 t δ ( τ − t 1 ) sin ω n ( t − τ ) d τ = ∬ Ω q 0 W n ( x , y ) d x d y M n ω n sin ω n ( t − t 1 )

Therefore, the solution of the vibration control Equation (1) of the thin plate under uniform impact was:

w ( x , y , t ) = ∑ n = 1 ∞ W n ( x , y ) ϕ n ( t ) ∑ n = 1 ∞ ∬ Ω q 0 W n ( x , y ) d x d y M n ω n W n ( x , y ) sin ω n ( t − t 1 ) = ∑ n = 1 ∞ A n W n ( x , y ) sin ω n ( t − t 1 ) ( t ≥ t 1 ) (15)

The parameter values of the thin plate were: h = 0.06 m, ρ = 2800 kg/m^{3}, E = 72 × 10^{9} Pa, υ = 0.3, a × b = 3 m × 3.6 m. Combined with the frequency equation of the Thin Coal and Rock Mass (C-C-C-C), the Newton iteration method can be used to solve the main vibration mode. The calculated values of the first 10 order vibration parameters were shown in _{n}/q_{0} was about 9.4% of the 1st order coefficient; the 6th order coefficient A_{n}/q_{0} was about 4.9% of the 1st order coefficient.

Parameters | 1^{st} order | 2^{nd} order | 3^{rd} order | 4^{th} order | 5^{th} order | 6^{th} order | 7^{th} order | 8^{th} order | 9^{th} order | 10^{th} order |
---|---|---|---|---|---|---|---|---|---|---|

β_{1} | 1.443 | 1.301 | 2.573 | 2.483 | 1.227 | 3.643 | 2.404 | 3.589 | 2.346 | 2.346 |

β_{2} | 2.145 | 3.247 | 2.949 | 3.765 | 4.437 | 3.895 | 4.802 | 4.514 | 5.930 | 5.930 |

α_{1} | 1.122 | 2.104 | 1.019 | 2.001 | 3.015 | 0.975 | 2.940 | 1.935 | 3.904 | 3.851 |

α_{2} | 2.329 | 2.795 | 3.778 | 4.042 | 3.479 | 5.243 | 4.494 | 5.433 | 4.248 | 5.083 |

k | 1.828 | 2.474 | 2.767 | 3.189 | 3.255 | 3.771 | 3.797 | 4.078 | 4.079 | 4.509 |

parameter Order | w_{n} | M_{n} | ∬ Ω W n ( x , y ) d x d y | A_{n}/q_{0} |
---|---|---|---|---|

1^{st} order | 308 | 5.787 × 10^{3} | 13.589 | 7.624 × 10^{−6} |

2^{nd} order | 563 | 7.293 × 10^{3} | 8.1 × 10^{−12} | 1.973 × 10^{−18} |

3^{rd} order | 705 | 1.310 × 10^{4} | 7.0 × 10^{−12} | 7.581 × 10^{−19} |

4^{th} order | 936 | 6.482 × 10^{3} | 0 | 0 |

5^{th} order | 975 | 1.267 × 10^{3} | 8.503 | 7.171 × 10^{−7} |

6^{th} order | 1309 | 2.587 × 10^{4} | 12.565 | 3.71 × 10^{−7} |

7^{th} order | 1328 | 6.584 × 10^{3} | −4.3 × 10^{−11} | −4.918 × 10^{−18} |

8^{th} order | 1531 | 9.105 × 10^{3} | 3.1 × 10^{−9} | 2.798 × 10^{−16} |

9^{th} order | 1532 | 2.071 × 10^{4} | 4.9 × 10^{−10} | 1.544 × 10^{−17} |

10^{th} order | 1872 | 8.215 × 10^{3} | 0 | 0 |

Combined with the frequency equation, can be derived from

Based on the third strength criterion, the dynamic distribution of the maximum shear stress in the thin plate (C-C-C-C) was shown in Figures 2-4. It can be seen

Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

β_{1} | 1.047 | 2.094 | 1.047 | 2.094 | 1.047 | 3.142 | 2.094 | 3.142 | 1.0472 | 3.142 |

α_{1} | 1.183 | 1.052 | 2.127 | 2.031 | 3.025 | 0.992 | 2.960 | 1.959 | 3.908 | 2.896 |

α_{2} | 1.896 | 3.143 | 2.592 | 3.591 | 3.368 | 4.552 | 4.187 | 4.855 | 4.180 | 5.303 |

k | 1.580 | 2.343 | 2.371 | 2.917 | 3.201 | 3.294 | 3.626 | 3.702 | 4.046 | 4.273 |

parameter Order | W_{n} | M_{n} | ∬ Ω W n ( x , y ) d x d y | A_{n}/q_{0} |
---|---|---|---|---|

1^{st} order | 230 | 1.407 × 10^{3} | 7.171 | 2.511 × 10^{−5} |

2^{nd} order | 506 | 3.867 × 10^{3} | 1.427 × 10^{−13} | 7.271 × 10^{−20} |

3^{rd} order | 518 | 1.080 × 10^{3} | −2.848 × 10^{−13} | −5.09 × 10^{−19} |

4^{th} order | 784 | 1.722 × 10^{3} | 2.275 × 10^{−25} | 1.685 × 10^{−31} |

5^{th} order | 943 | 9.980 × 10^{2} | 2.54 | 2.699 × 10^{−6} |

6^{th} order | 999 | 8.898 × 10^{3} | 6.171 | 6.943 × 10^{−7} |

7^{th} order | 1210 | 1.301 × 10^{3} | 3.229 × 10^{−14} | 2.051 × 10^{−20} |

8^{th} order | 1261 | 2.976 × 10^{3} | 6.691 × 10^{−11} | 1.78310^{−17} |

9^{th} order | 1507 | 9.636 × 10^{3} | 0 | 0 |

10^{th} order | 1680 | 1.856 × 10^{3} | 1.045 | 3.352 × 10^{−7} |

from the figures that the maximum shear stress of the first order main mode was mainly distributed in the middle area of the four sides; the maximum shear stress of the fifth order main mode was mainly distributed in the middle area of the two short sides, the center area of the thin plate and the center area along the long axis of the thin plate; the maximum shear stress of the sixth order main mode was mainly distributed in the middle area of two long edges, the center area of the sheet and the center area along the short axis of the sheet.

Therefore, the dynamic failure process was as follows: under the uniform impact load, the location of the rectangular plate (C-C-C-C) prone to damage was located in the middle of the four sides and the central area of the rectangular plate, mainly along the central area of the long axis of the plate, and then along the central area of the short axis (x axis).

Based on the third strength theory, the dynamic distribution of the maximum shear stress in the thin plate (S-C-S-C) under the impact load was shown in Figures 5-8. It can be seen from the figure that the maximum shear stress of the first order main mode were mainly distributed in the middle region of the short side and four concentration regions of the long side; the maximum shear stress of the fifth order mode were mainly distributed in the middle region of the two short sides and the center region along the long axis of the thin plate The maximum shear stress of the sixth order mode were mainly distributed along the center of the short axis of the thin plate; the maximum shear stress of the tenth order mode were distributed in each area.

The dynamic failure process was as follows: under the impact load, the damage of the Thin Coal and Rock Mass (S-C-S-C) was located in the middle of the short side and the four concentration areas of the long side; it was mainly along the long axis of the thin plate, followed by the central area along the short axis.

Based on Ls-DYNA software, the PLASTIC_KINEMATIC material model was used to simulate and analyze the destruction process of thin plates. The dynamic failure process of the thin Coal and Rock Mass (C-C-C-C) was as follows: first, the crack occurred at the center of the short side of the thin plate, and the damage occurred at the center of the long side of the thin plate, then the central area of the thin plate. The damage direction mainly developed along the long axis, and the second along the short axis or width. In the end, the main damage area were around the thin plate, the central area, and the area along the central long axis, as shown in

The dynamic failure process of the Both sides are firmly supported and both sides simply supported Thin Coal and Rock Mass (S-C-S-C) was as follows: first, cracks occurred in the middle of the short side of the thin plate and the four concentration areas on the long side, and then the damage occurred on the long side of the thin plate. The failure direction of the thin plate mainly developed along the long axis. It would develop toward the center of the sheet al.ong the short axis or width, as shown in

Under the impact load, the vibration of thin plate (C-C-C-C) was dominated by the first order (dominant frequency), followed by the fifth order and the sixth order. According to

Similarly, the analysis of the vibration of the thin plate (S-C-S-C) was dominated by the first order (main frequency), followed by the fifth order, the sixth order, and the tenth order. From Figures 5-8, it can be seen that the maximum shear of the first order main mode were mainly distributed in the middle of the short side and the four concentration areas of the long side, which were also the locations where the thin plate first break; the maximum shear of the fifth order mode were mainly distributed in the center of the two short sides and along the long axis of the thin plate, which was the main direction of crack development;

the maximum shear of the 6^{th} order mode were mainly distributed along the short axis, which was also the main direction of crack development. Eventually, the crack would develop towards the central area of the thin plate; the maximum shear of the 10^{th} order mode was distributed in each area of the thin plate, but its amplitude was relatively small even can be ignored. The entire destruction development process was consistent with the destruction process simulated in

Based on the above analysis, it can be concluded that the initial failure position of the thin plate (C-C-C-C) and the thin plate (S-C-S-C) can be inferred from the dynamic distribution characteristics of the first order mode (main frequency), and the development direction and trend of the failure can be judged from the fifth and sixth order main modes. The three main modes constitute the effective main vibration mode under the impact load of the rectangular Thin Coal and Rock Mass mode.

1) Based on the rectangular thin plate model, the dual equation of Hamilton system for free vibration of thin coal and rock mass was established; based on the orthogonality of mode function and Duhame integral, the dynamic deflection distribution equation of thin coal and rock mass under uniform impact load was derived.

2) Based on the third strength theory and numerical simulation results, the dynamic damage process of thin coal and rock mass under impact load was analyzed, the initial failure position of the rectangular thin plate can be inferred from the dynamic distribution characteristics of the first order mode (main frequency), and the development direction and trend of the failure can be judged from the fifth and sixth order main modes.

3) The first order main mode, fifth and sixth order main modes constitute the thin Coal and Rock Mass (C-C-C-C) and (S-C-S-C) effective main vibration mode equation synthetically under uniform impact load.

State Key Research and Development Program Funding (2016 YFC0801800), National Natural Science Foundation Of China Project (51804311), China Mining University (Beijing) Deep Rock and Soil Forces and Underground Engineering National Key Laboratory Open Fund Project (SKLGDUEK1818).

The authors declare no conflicts of interest regarding the publication of this paper.

Li, F., Sha, F.F., Zhang, M.B., He, Z.J., Dong, X.H. and Ren, B.R. (2020) Dynamic Failure Model of Thin Coal and Rock Mass under Uniform Impact Load. Engineering, 12, 699-714. https://doi.org/10.4236/eng.2020.1210049