Solutions for One-Dimensional Rheological Consolidation of a Clay Layer with Threshold Hydraulic Gradient under Multistage Loading

Acknowledgments

Notation

a

E₁/E₀;

b

k_vη₁/(γ_wH²);

C₁

(bx₁ + a + 1)/[b(x₁ − x₂)];

C₂

(bx₂ + a + 1)/[b(x₁ − x₂)];

c_v

coefficient of consolidation, c_v = k_vE₀/γ_w;

E₀

modulus of an independent spring in Merchant rheological model;

E₁

modulus of a spring in the Kelvin body;

e

natural index;

F_m(s)

L[T_m(t)], Laplace transformation of T_m(t);

H

thickness of the clay layer;

h

depth of the moving boundary at time t;

h_∞

final depth of the moving boundary;

i

hydraulic gradient;

i₀

threshold hydraulic gradient existing in the water flow of clays;

i₁

the critical hydraulic gradient between the exponential and linear relationship;

I₁₁

${\displaystyle \frac{RX{T}_{vc}\sin M}{M}{x}_1}$;

I_1j

${\displaystyle \frac{q_u(T_{v2j-1}-T_{v2j-2})}{(q_j-q_{j-1})}\frac{RX\sin M}{M}{x}_1}$;

I₂₁

${\displaystyle \frac{RX{T}_{vc}\sin M}{M}{x}_2}$;

I_2j

${\displaystyle \frac{q_u(T_{v2j-1}-T_{v2j-2})}{(q_j-q_{j-1})}\frac{RX\sin M}{M}{x}_2}$;

j

positive integer, 1, 2, 3, …, n−1, n;

k

positive integer, 1, 2, 3, …;

k_v

coefficient of permeability;

L[T_m(t)]

Laplace transformation of T_m(t);

l

positive integer, 1, 2, 3, …, p−1;

M

(2m − 1)π/2;

m

positive integer, 1, 2, 3, …;

m₁

the exponent of exponential relationship at gradients lower than i_1;

N

total thin layers divided in the numerical model;

n

total stages of the multistage loading;

p

positive integer, 1, 2, 3, …, N;

Q(s)

L[q(t)], Laplace transformation of q(t);

q_j

stable value of the jth-stage loading;

q_u

final value of the multistage loading;

q(t)

a multistage loading;

R

i₀γ_wH/q_u;

r

positive integer, 1, 2, 3, …;

S_∞

the final settlement of soil samples;

S_t

the settlement of soil samples at time t;

s

Laplace transform variable which is a complex number;

t

time;

t_2j−1

time when the jth-stage loading increases to the stable value q_j;

t_2j−2, t_2j

initial and final time of the jth-stage loading;

t_c

time when the multistage loading increases to the final value q_u;

T_m(t)

function of time t;

$T_m^{\mathrm{\prime }}(t)$

derivative of T_m(t);

T_p

time when the moving boundary reaches the pth thin layer;

T_v

time factor, and T_v = k_vE₀t/(γ_wH²);

$T_v^k$

T₂ + (k − 1)ΔT_v, the final time of the kth time interval;

T_vc

c_vt_c/H²;

T_vj

c_vt_j/H²;

U_pt

average degree of consolidation in terms of excess pore-water pressure;

u

excess pore-water pressure;

u(z, ∞)

final residual excess pore-water pressure in the clay due to threshold gradient;

V

u/q_u;

$V_l^k$

value of V at the lth spatial node when $T_v=T_v^k$;

v

average velocity of water flow in the clay;

W

q(t)/q_u;

W⁰

q(0)/q_u;

W^k

value of W at time $T_v^k$;

w

u − i₀γ_wz;

X

h/H;

x₁

${\displaystyle -\frac{1}{2b}\left[(b{M}^2/X^2+a+1)-\sqrt{{(b{M}^2/X^2+a+1)}^2-4ab{M}^2/X^2}\right]}$;

x₂

${\displaystyle -\frac{1}{2b}\left[(b{M}^2/X^2+a+1)+\sqrt{{(b{M}^2/X^2+a+1)}^2-4ab{M}^2/X^2}\right]}$;

Z

z/H;

Z_l

lΔZ;

z

depth;

γ_w

specific weight of water;

ΔT_v

time interval;

ΔZ

1/N;

ɛ_z

vertical strain;

η₁

viscosity coefficient of a dashpot in the Kelvin body;

λ

ΔT_v/(ΔZ)²;

${\sigma }_z^{\mathrm{\prime }}$

vertical effective stress; and

τ

integration variable.

References