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General semi-analytical solutions to one-dimensional consolidation for unsaturated soils

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Abstract

This paper presents general semi-analytical solutions to Fredlund and Hasan’s one-dimensional (1D) consolidation equations for unsaturated soils subject to different initial conditions, homogeneous boundaries and time-dependent loadings. Two variables are introduced to transform the two-coupled governing equations of pore-water and pore-air pressures into an equivalent set of partial differential equations (PDEs), which are solved with the Laplace transform method. The pore-water and pore-air pressures and settlement are obtained in the Laplace transform domain. The Crump’s method is used to perform inverse Laplace transform to obtain the solutions in the time domain. The present solutions are more general in practical applications and show good agreement with the previous solutions in the literature.

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Abbreviations

A :

parameter influencing loading magnitude

a :

linear loading rate

a a :

changing rate of initial pore-air pressure along depth

a w :

changing rate of initial pore-water pressure along depth

b :

loading parameter controlling rate of exponential loading

b a :

initial pore-air pressure at top surface

b w :

initial pore-water pressure at top surface

C a :

interactive constant with respect to air phase

C w :

interactive constant with respect to water phase

C av :

coefficient of volume change with respect to air phase

C wv :

coefficient of volume change with respect to water phase

C aσ :

consolidation coefficient for air phase

C wσ :

consolidation coefficient for water phase

c :

parameter influencing loading amplitude

g :

gravitational acceleration

h :

depth of soil layer

k a :

coefficient of air permeability

k w :

coefficient of water permeability

M :

molecular mass of air

m a1k :

coefficient of air volume change with respect to change in γ-ua

m a2 :

coefficient of air volume change with respect to change in u a-u w

m w1k :

coefficient of water volume change with respect to change in γ-ua

m w2 :

coefficient of water volume change with respect to change in u a-uw

n 0 :

initial porosity

Q(s):

result of Laplace transform of \(\frac{{\partial q\left( t \right)}}{{\partial t}}\) upon time t

q 0 :

initial surcharge

R :

universal gas constant

S r0 :

initial degree of saturation

T :

absolute temperature

t :

time

u a :

pore-air pressure

u atm :

atmospheric pressure

\(\overline u _a^0\) :

absolute pore-air pressure

u 0a :

initial pore-air pressure

u w :

pore-water pressure

u 0w :

initial pore-water pressure

w :

settlement

w*:

normalized settlement

z :

investigated depth

γ w :

unit weight of water

ε v :

total volumetric strain

θ :

angular frequency for sinusoidal loading function

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Authors and Affiliations

  1. Department of Civil Engineering, Shanghai University, Shanghai, 200444, China

    Lei Wang, De’an Sun & Aifang Qin

Authors
  1. Lei Wang
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  2. De’an Sun
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  3. Aifang Qin
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Corresponding author

Correspondence to De’an Sun.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 41372279 and 41630633)

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Wang, L., Sun, D. & Qin, A. General semi-analytical solutions to one-dimensional consolidation for unsaturated soils. Appl. Math. Mech.-Engl. Ed. 38, 831–850 (2017). https://doi.org/10.1007/s10483-017-2209-8

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  • DOI: https://doi.org/10.1007/s10483-017-2209-8

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