Theoretical Solution for Stress and Strain Distributions Induced by Generalized Elastic Constant Variation in Rock Mass

Acknowledgments

Notation

A

area of domain subject to surcharge traction;

A_P,2 and A_R,2

areas of partial and remaining domains subject to surcharge traction for Case II;

$\overrightarrow{\mathit{C}}$, E, G, and K

general elastic constant matrix, and Young’s, shear, and bulk moduli;

${\overrightarrow{\mathit{C}}}_0$, E₀, G₀, and K₀

general elastic constant matrix, Young’s modulus, shear modulus, and bulk modulus of whole domain prior to elastic constant change;

${\overrightarrow{\mathit{C}}}_1$, E₁, G₁, and K₁

ultimate general elastic constant matrix, Young’s modulus, shear modulus, and bulk modulus of whole domain for Case I;

${\overrightarrow{\mathit{C}}}_{\mathit{P},0}$ and ${\overrightarrow{\mathit{C}}}_{\mathit{R},0}$

general elastic constant matrixes of partial and remaining domains prior to elastic constant change;

${\overrightarrow{\mathit{C}}}_{\mathit{P},2}$ and ${\overrightarrow{\mathit{C}}}_{\mathit{R},2}$

ultimate general elastic constant matrixes of partial and remaining domains for Case II;

d/dt and δ

total and partial time derivatives (known as rate);

${\overrightarrow{\mathit{d}}}_s$ and ${\overrightarrow{\mathit{d}}}_g$

displacement vectors associated with surface traction and body force;

${\overrightarrow{\mathit{d}}}_{\mathit{P},2}$ and ${\overrightarrow{\mathit{d}}}_{\mathit{R},2}$

displacement vectors associated with surface traction within partial and remaining domains, respectively, induced by variation in elastic constant for Case II;

E_h and E_v

Young’s moduli in transverse isotropic plane and in plane normal to transverse isotropy;

E_P,0 and E_R,0

Young’s moduli of partial and remaining domains prior to elastic constant change;

E_P,2 and E_R,2

ultimate Young’s moduli of partial and remaining domains for Case II;

$\overrightarrow{\mathit{g}}$ and g

body force vector and gravity;

G_P,0 and G_R,0

shear moduli of partial and remaining domains prior to elastic constant change;

G_P,2 and G_R,2

ultimate shear moduli of partial and remaining domains for Case II;

h₀

displacement of centroid of whole domain prior to elastic constant change;

h₁

displacement of centroid of whole domain induced by variation in elastic constant for Case I;

h_T,1

ultimate displacement of centroid of rock mass for Case I;

h_P,2 and h_R,2

displacements of centroid of partial and remaining domains induced by variation in elastic constant for Case II;

I_1,0 and s_ij,0

first stress invariant and deviator stresses within whole domain prior to elastic constant change;

I_1,1 and s_ij,1

ultimate first stress invariant and deviator stresses within whole domain for Case I;

I_1,P,0 and I_1,R,0

first stress invariants within partial and remaining domains prior to elastic constant change;

I_1,P,2 and I_1,R,2

ultimate first stress invariants within partial and remaining domains for Case II;

K_P,0 and K_R,0

bulk moduli of partial and remaining domains prior to elastic constant change;

K_P,2 and K_R,2

ultimate bulk moduli of partial and remaining domains for Case II;

m and M

mass of rock and self-weight of rock mass;

$\overrightarrow{\mathit{P}}$

surcharge traction (line load, surcharge pressure, or both);

${\overrightarrow{\mathit{P}}}_{\mathit{P},2}$ and ${\overrightarrow{\mathit{P}}}_{\mathit{R},2}$

surface tractions applied on area of partial and remaining domains for Case II;

R

ratio of elastic constant after change to elastic constant before change, for example, $R={\overrightarrow{\mathit{C}}}_1/{\overrightarrow{\mathit{C}}}_0$ for Case I, $R={\overrightarrow{\mathit{C}}}_{\mathit{P},2}/{\overrightarrow{\mathit{C}}}_{\mathit{P},0}$ for partial domain for Case II, and $R={\overrightarrow{\mathit{C}}}_{\mathit{R},2}/{\overrightarrow{\mathit{C}}}_{\mathit{R},0}=1$ for remaining domain for Case II;

s_ij,P,0 and s_ij,R,0

deviator stresses within partial and remaining domains prior to elastic constant change;

s_ij,P,2 and s_ij,R,2

ultimate deviator stresses within partial and remaining domains for Case II;

t and T

time and temperature;

W

work input to system by external forces;

W₀

work done to whole domain by gravity prior to elastic constant change;

W₁

work done to whole domain by gravity induced by variation in elastic constant for Case I;

W_P,0 and W_R,0

work done to partial and remaining domains by gravity prior to elastic constant change;

W_P,2 and W_R,2

work done to partial and remaining domains by gravity induced by variation in elastic constant for Case II;

W_T,1

ultimate work done to whole domain for Case I;

W_T,P,2 and W_T,R,2

ultimate work done to partial and remaining domains for Case II;

→

superimposed symbol that represents a vector;

Δ, ρ, and v

variable increment, material density, and Poisson’s ratio;

$\overrightarrow{\mathit{\epsilon }}$ and $\overrightarrow{\mathit{\sigma }}$

strain and stress vectors;

${\overrightarrow{\mathit{\epsilon }}}_0$ and ${\overrightarrow{\mathit{\sigma }}}_0$

strain and stress vectors within whole domain prior to elastic constant change;

${\overrightarrow{\mathit{\epsilon }}}_1$ and ${\overrightarrow{\mathit{\sigma }}}_1$

ultimate strain and stress vectors within whole domain for Case I;

${\epsilon }_d$ and ${\epsilon }_v$

deviatoric and volumetric strains;

${\epsilon }_{d,0}$ and ${\epsilon }_{v,0}$

deviatoric and volumetric strains within whole domain prior to elastic constant change;

${\epsilon }_{d,1}$ and ${\epsilon }_{v,1}$

ultimate deviatoric and volumetric strains within whole domain for Case I;

${\epsilon }_{d,P,0}$ and ${\epsilon }_{v,P,0}$

deviatoric and volumetric strains within partial domain prior to elastic constant change;

${\epsilon }_{d,P,2}$ and ${\epsilon }_{v,P,2}$

ultimate deviatoric and volumetric strains within partial domain for Case II;

${\epsilon }_{d,R,0}$ and ${\epsilon }_{v,R,0}$

deviatoric and volumetric strains within remaining domain prior to elastic constant change;

${\epsilon }_{d,R,2}$ and ${\epsilon }_{v,R,2}$

ultimate deviatoric and volumetric strains within remaining domain for Case II;

${\overrightarrow{\mathit{\epsilon }}}_{\mathit{P},0}$ and ${\overrightarrow{\mathit{\sigma }}}_{\mathit{P},0}$

strain and stress vectors within partial domain prior to elastic constant change;

${\overrightarrow{\mathit{\epsilon }}}_{\mathit{P},2}$ and ${\overrightarrow{\mathit{\sigma }}}_{\mathit{P},2}$

ultimate strain and stress vectors within partial domain for Case II;

${\overrightarrow{\mathit{\epsilon }}}_{\mathit{R},0}$ and ${\overrightarrow{\mathit{\sigma }}}_{\mathit{R},0}$

strain and stress vectors within remaining domain prior to elastic constant change;

ψ

strain energy density;

ψ₀

strain energy density within whole domain prior to elastic constant change;

ψ_P,0 and ψ_R,0

strain energy density prior to elastic constant change within partial and remaining domains;

ψ_P,2 and ψ_R,2

ultimate strain energy density for Case II within partial and remaining domains;

ψ_T,1

ultimate strain energy density within whole domain for Case I;

Ψ, Q, S, and U

enthalpy, heat that system gains or losses, entropy, and internal energy;

Ω

volume of domain subject to body force;

Ω₀

volume of whole domain subject to body force prior to elastic constant change;

Ω₁

volume of whole domain subject to body force for Case I;

Ω_P,0 and Ω_R,0

volumes of partial and remaining domains subject to body force prior to elastic constant change; and

Ω_P,2 and Ω_R,2

volumes of partial and remaining domains subject to body force for Case II.

References