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On the elastic deformation of non-saturated swelling soils

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Summary

This paper aims at completing and extending the theories as they have been applied to swelling media for the last 50 years, to swelling non-saturated soils. However, having regard to the complicated behaviour of swelling soils, it was thought necessary to keep the state of stress as simple as possible when discussing swelling in cylindrical specimens in which drainage is “completely” prevented. Definitions of the parameters are attemptedbased on equilibrium thermodynamics. Contributions to swelling stress calculation, when expansion is considered in relation to vapor pressure and moisture content, are also given.

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Abbreviations

u w :

hydrostatic pressure of soil water

σ:

total stress

σ′:

effective stress

m c :

soil mass

m w :

water mass

H=(m w/m c):

moisture content (gravimetric)

A=(m w/m c)** (i=x,y,z orr, θ,z):

moisture content at constant swelling pressure (gravimetric)

n=(m w/m c)h :

moisture content at constant vapor pressure (gravimetric)

n a :

amount of a constituant phase

h v :

vapor pressure of soil water

h 0 :

vapor pressure of pure water

h=(h v/h 0):

relative vapor pressures

M :

molecular weight of water

\(\bar v\) :

specific volume of the soil water vapor

R :

gas constant (8.3144 Joules/mole·oK)

T :

absolute temperature

P :

swelling pressure of an isotropic soil swelling without constrain

S :

entropy of a sample of volumeV

O :

oncotic energy

O r :

residual oncotic energy

U :

internal energy

F :

Helmholtz free energy

G :

Gibb's free energy

\(\bar G_\alpha = \left( {\frac{{\partial G}}{{\partial n_\alpha }}} \right)_{T,P,P_i ,n_b } \) :

partial molar Gibb's free energy of constituenta

\(\mu _\alpha = \bar G_a \) :

chemical potential which equals the partial molar Gibb's free energy of constituenta

μ v :

chemical potential of water vapor

μ v :

chemical potential of soil water

V :

volume

V H :

volume at constant moisture content

W a :

specific energy for the adsorbed water

W a0 :

specific osmotic energy of adsorbed solutes

W 0 :

specific osmotic energy of free solutes

x, y, z :

Cartesian coordinate system

r, θ,z :

cylindrical coordinate system

u, v, z w :

displacement components along the strain variablesx, y, andz

z w :

height of the water column

P i (i=x, y, z):

loads along the strain variablesx, y, z

P i (i=x, y, z) orr, θ,z :

mechanical pressure along the strain variables

X i (i=x, y, z orr, θ,z):

swelling pressures along the strain variablesx, y, z

P k :

hydrostatic swelling pressure defined byp k=(X x+X y+X z)/3

P k, H :

hydrostatic swelling pressure under constant moisture content

l i (i=x, y, z):

extential displacements along the strain variablesx, y, z

s i * (i=x, y, z orr, θ,z):

differential swelling

s i (i=x, y, z orx, θ,z):

differential swelling per unit volume

s k :

hydrostatic differential swelling defined bys k=(s x+s y+s z)/3

s h :

differential swelling under constant vapor pressure

Q′ :

effective swelling pressure defined as the energy encompassing all the unknown effects contributing to swelling against the mechanical pressure and the vapor pressure of soil water per unit volume

Q :

coefficient of swelling

E i (i=r, θ,z orx, y, z):

Young's moduli in the directioni

E i, h (i=x, y, z):

Young's moduli at constant vapor pressure

B h :

Bulk modulus under constant vapor pressure

G ij. h (i, j=x, y, z):

Shear modulus under constant vapor pressure

v ij (i, j=r, θ,z orx, y, z):

Poisson's coefficient which characterizes the compression in the directioni for tension in the directionj, etc.

e i (i=r, θ,z):

final radial, tangential and axial strains

e i, s (i=r, θ,z):

radial, tangential and axial strains due to swelling

\(\bar X_i ,\bar x_i ,(i = r,\theta ,z)\) :

loads along the strain variablesr, θ,z

G ij (i, j=x, y, z):

rigidity modulus

g :

acceleration due to gravity

a :

radius of the cylinder

r 1,r 2 :

roots of the characteristic equation

W :

work done by the surroundings on the system

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Author notes

  1. T. K. Karalis

    Present address: Department of Civil Engineering, Democritos University of Thrace, 67100, Xanthi, Greece

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    Karalis, T.K. On the elastic deformation of non-saturated swelling soils. Acta Mechanica 84, 19–45 (1990). https://doi.org/10.1007/BF01176086

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