Study of the Rotation of Blocks in a Two-Dimensional Blocky Rock Mass under the Impact Load and Torque

Data Availability Statement

Acknowledgments

Notation

a₁, a₂, a₃, a₄

coefficients of vertical interlayers determined by contact conditions;

b₁, b₂, b₃, b₄

coefficients of horizontal interlayers determined by contact conditions;

c

viscous damping coefficient of interlayers;

CSE

compressive strain energy (strain energy generated by axial tensile or compressive deformation);

E

input energy of external loads to the blocky rock masses;

E_k(i,j)(t)

translational kinetic energy of the block (i, j);

E_k(t)

translational kinetic energy of the whole blocky rock masses;

E_r(i,j)(t)

rotational kinetic energy of the block (i, j);

E_r(t)

rotational kinetic energy of the whole blocky rock masses;

E_x, E_y

elastic modulus of the vertical interlayer and horizontal interlayer;

F_x, F_y

force components acting on the block;

(i, j)

coordinate positions of the block in the x- and y-directions;

H₀

Heaviside function;

J

moment of inertia of the block;

L₁, L₂, L₃, L₄

interfaces between the block (i, j) and the four surrounding interlayers;

2L_x

size of blocks in the x-direction;

2L_y

size of blocks in the y-direction;

L_z

size of blocks in the z-direction;

2l_b

thickness of boundary interlayers;

2l_x

thickness of vertical interlayers;

2l_y

thickness of horizontal interlayers;

m

mass of the block;

M₀

torque acting on the block;

n_x, n_y

number of horizontal and vertical blocks;

P₀

amplitude of impact load acting on the block;

p_x, p_y

resultant stress components of the block in the x- and y-directions;

U_0x(i,j)

strain energy density of the vertical interlayer (i, j);

U_tx(i,j)

strain energy of the vertical interlayer (i, j);

U_0y(i,j)

strain energy density of the horizontal interlayer (i, j);

U_ty(i,j)

strain energy of the horizontal interlayer (i, j);

u

translational displacement of the block in the x-direction;

u′

displacement considering the rotation at any point of the block in x-direction;

u_c,i,j

normal deformation of the vertical interlayer (i, j);

u_l

normal displacement of interlayers;

u_s,i,j

shear deformation of the horizontal interlayer (i, j);

v

translational displacement of the block in y-direction;

v_c,i,j

normal deformation of the horizontal interlayer (i, j);

v_{e(i, j)}

velocity in the x-direction, y-direction, or the total velocity of the block (i, j);

v_s,i,j

shear deformation of the vertical interlayer (i, j);

v′

displacement considering the rotation at any point of the block in y-direction;

v_l

shear displacement of interlayers;

β_1x, β_2x, β_3x, β_4x; β_1y, β_2y, β_3y, β_4y

defined coefficients;

γ_1x, γ_2x, γ_3x, γ_4x; γ_1y, γ_2y, γ_3y, γ_4y

defined coefficients;

φ

rotation angle of the block;

λ

lame constant of interlayers;

μ

shear modulus of interlayers;

ρ

density of blocks;

υ

Poisson’s ratio;

μ_x, μ_y

shear modulus of the vertical interlayer and horizontal interlayer;

λ_x, λ_y

lame constant of the vertical interlayer and horizontal interlayer;

ω

angular frequency;

Δt

time step of finite difference method;

μ_b

shear modulus of boundary interlayers;

λ_b

lame constant of boundary interlayers;

Δu_A,i,j, Δu_B,i,j, Δu_C,i,j

normal deformation of the vertical interlayer at Points A, B, and C; and

Δv_D,i,j, Δv_E,i,j, Δv_F,i,j

normal deformation of the horizontal interlayer at Points D, E, and F.

References