Analytical Solutions for One-Dimensional Large-Strain Nonlinear Consolidation of Soft Soils by Considering a Time-Dependent Drainage Boundary

Acknowledgments

Notation

A₁

$-{\displaystyle \frac{2\beta [1-I_c(\alpha -1)]}{M(b_m-\beta )}\frac{q_u}{{\sigma }_0^{\mathrm{\prime }}}{N}_q^{-I_c(\alpha -1)}}$;

A₂

$-{\displaystyle \frac{2\beta [1-I_c(\alpha -1)]}{M[b_m-(k+1)\beta ]}\frac{q_u}{{\sigma }_0^{\mathrm{\prime }}}{N}_q^{-I_c(\alpha -1)}}$;

A₃

$-{\displaystyle \frac{2\beta [1-I_c(\alpha -1)]}{M(b_n-\beta )}\frac{q_u}{{\sigma }_0^{\mathrm{\prime }}}{N}_q^{-I_c(\alpha -1)}}$;

A₄

$-{\displaystyle \frac{2\beta [1-I_c(\alpha -1)]}{M[b_n-(k+1)\beta ]}\frac{q_u}{{\sigma }_0^{\mathrm{\prime }}}{N}_q^{-I_{\mathrm{c}}(\alpha -1)}}$;

$A_{\alpha }$

[−I_c(α − 1)][−I_c(α − 1) − 1]…[−I_c(α − 1) − k + 1];

a

Lagrangian coordinate;

a_j−1,a_j

starting point and ending coordinate of the jth integral interval;

B

βH²/c_v0;

b_m

c_v0(M/H)²;

b_n

c_v0v₀(M/H)²;

c_F

large-strain consolidation coefficient, and $c_F=(k_{v0}/{\gamma }_w)({\sigma }_0^{\mathrm{\prime }}/I_c)({\sigma }_0^{\mathrm{\prime }}/{\sigma }^{\mathrm{\prime }}{)}^{I_c(\alpha -2)-1}$;

c_v0

initial consolidation coefficient, and $c_{v0}=k_{v0}{\sigma }_0^{\mathrm{\prime }}/(I_c{\gamma }_w)$;

e

void ratio;

e₀

initial void ratio;

f(t)

$-\beta [1-I_c(\alpha -1)]{\displaystyle \frac{q_u}{{\sigma }_0^{\mathrm{\prime }}}\exp (-\beta t){\left[N_q-\frac{q_u}{{\sigma }_0^{\mathrm{\prime }}}\exp (-\beta t)\right]}^{-I_c(\alpha -1)}}$;

f_m(t)

$-{\displaystyle \frac{2\beta [1-I_c(\alpha -1)]}{M}\frac{q_u}{{\sigma }_0^{\mathrm{\prime }}}\exp (-\beta t)N_q^{-I_{\mathrm{c}}(\alpha -1)}}$ $\left[1+\sum _{k=1}^{\mathrm{\infty }}{\displaystyle A_{\alpha }\frac{{(-1)}^k}{k!}{\left(\frac{q_u}{N_q{\sigma }_0^{\mathrm{\prime }}}\right)}^k\exp (-k\beta t)}\right]$;

H

thickness of soft soil layer;

I_c

compression index in the bilogarithmic coordinate system;

I_k

permeability index in the bilogarithmic coordinate system;

k_v

coefficient of permeability;

k_v0

initial coefficient of permeability;

M

[(2m − 1)π]/2;

N

number of equal integral intervals;

N_q

$({\sigma }_0^{\mathrm{\prime }}+q_u)/{\sigma }_0^{\mathrm{\prime }}$;

q_u

uniform instantaneous load;

S(a, t)

settlement at depth a in Lagrangian coordinate;

S_t

settlement at the top surface at time t;

S_∞

final settlement at the surface of the soil layer;

T_v

c_v0t/H²;

t

time;

U_st

average degree of consolidation defined by deformation;

U_pt

average degree of consolidation defined by stress;

u

excess pore-water pressure;

v₀

$[1+N_q^{I_c(2-\alpha )+1}]/2$;

$w_{\beta }(t)$

${\left[N_q-{\displaystyle \frac{q_u}{{\sigma }_0^{\mathrm{\prime }}}\exp (-\beta t)}\right]}^{1-I_c(\alpha -1)}-N_q^{1-I_c(\alpha -1)}$;

α

parameter of bilogarithmic permeability mode;

β

interface parameter;

γ_w

specific weight of water;

σ′

effective stress; and

${\sigma }_0^{\mathrm{\prime }}$

initial effective stress.

References