The Concept of Effective Stress in Unsaturated Soils - Multiscale Geomechanics: From Soil to Engineering Projects - page 151

pressure changes [BI0 41; COU 95; DEB 90; LOR 00; RIC 76]. At the origin,
effective stress was the macroscopic stress of the solid skeleton. In the case of
saturated soils, the expression which is generally used is [NUR 71; SKE 61]

σ' = σ - α p w δ [6.1]

where σ' is the effective stress tensor, σ the total stress tensor, δ the unit tensor, p w
the pore fluid pressure and α the Biot coefficient.

In the case of an isotropic material:

K
K

[6.2]

T
s

α =− 1

where K T is the compressibility modulus of the solid phase and K s , the
compressibility modulus of the solid grains. If the solid constituent is supposed to be
incompressible ( K s  ∝; α = 1), the previous relationship becomes Terzaghi's
effective stress [TER 36]:

σ' = σ - p w δ [6.3]

In the case of unsaturated soils, effective stress was expressed for the first time
by Bishop [BIS 59; BIS 63] as

σ' = σ - p a δ + χ ( p a - p w ) δ [6.4]

Here p a is the pore-air pressure and χ is an effective stress parameter, called
Bishop's parameter, which is a function of the degree of saturation (equal to 1 for
saturated soils and 0 for dry soils). Effective stress can also be written as

σ' = σ net + χ p c δ [6.5]

with

σ net = σ - p a δ : net total stress

p c = p a - p w : capillary pressure.

[BIS 61] and [BIS 63] showed, first on the basis of experimental results, that the
shear strength and volume change remained the same when the net total stress σ net