Theoretical Solution for Cavity Expansion in Crushable Soil

Acknowledgments

Notation

A_p, A_q

intermediate variable defined in Eqs. (39) and (40);

a₀, a

initial and created cavity radius;

B

breakage;

B_B, B_q

intermediate variable defined in Eqs. (43) and (44);

B_p

breakage at the EP boundary;

C_p, C_q, C_B

intermediate variable defined in Eqs. (45)–(47);

C₁

integration constant;

D_r

relative density of the soil;

dλ

nonnegative multiplier;

E_B

breakage energy;

E_c

critical breakage energy;

e_max, e_min

maximum and minimum void ratio of the soil;

e₀

initial void ratio of the soil;

F(x, B)

current cumulative GSD by mass;

F_u(x)

final cumulative GSD by mass;

F₀(x)

initial cumulative GSD by mass;

f

yield function;

G

shear modulus;

K

bulk modulus;

k

k 1 for cylindrical cavity and k 2 for spherical cavity;

k₁, k₂, k₃

three intermediate variables defined in Eq. (22);

M

friction coefficient at the critical state;

N_LP

normalized limit expansion pressure;

n₀

initial porosity of the soil;

p′

mean effective stress;

$p_c^{\mathrm{\prime }}$

critical comminution pressure;

$p_p^{\mathrm{\prime }}$

mean effective stress at the EP boundary;

$p_0^{\mathrm{\prime }}$

initial mean effective stress;

q

deviatoric stress;

q_p

deviatoric stress at the EP boundary;

r

radial position of soil particle;

r_p

radius of the plastic zone;

t

time;

u_w

pore pressure of the soil;

u_w0

initial pore pressure of the soil;

u_wp

pore pressure of soil at the EP boundary;

u_r

radial displacement of the soil;

v

radial expansion velocity of the soil;

v_p

radial velocity at the EP boundary;

x

grain size;

δ

ratio between bulk modulus and shear modulus;

ɛ_p, ɛ_q

total volumetric and shear strains;

${\epsilon }_p^e,{\epsilon }_p^p$

elastic and plastic volumetric strains;

${\epsilon }_q^e,{\epsilon }_q^p$

elastic and plastic shear strains;

${\epsilon }_r,{\epsilon }_{\theta }$

radial and circumferential strain;

η

similarity variable;

μ

Poisson’s ratio of soil;

${\sigma }_{a,l}^{\mathrm{\prime }}$

effective radial stress at the cavity wall;

${\sigma }_r^{\mathrm{\prime }},{\sigma }_{\theta }^{\mathrm{\prime }}$

effective radial and circumferential stress;

${\sigma }_{rp}^{\mathrm{\prime }},{\sigma }_{\theta p}^{\mathrm{\prime }}$

effective radial and circumferential stress at the EP boundary;

υ

specific ratio of the soil;

υ_p

specific ratio of the soil at the EP boundary;

υ₀

initial specific ratio of the soil;

φ

internal frictional angle;

ω

coupling angle; and

ϑ

grading index.

References