Numerical Modeling: An Efficient Tool for Analyzing the Behavior of Constructions - Multiscale Geomechanics: From Soil to Engineering Projects - page 260

Table 9.1 summarizes the yield functions and the hardening laws for the different
variants of the family of models.

9.3.5. Mechanical behavior of non-saturated soil

To model the mechanical behavior of a saturated soil, we define the Terzaghi's
effective stress tensor [TER 67] that applies only to the skeleton:

σ = σ + p w I

[9.54]

where:
- σ is the Cauchy total stress tensor;
- σ is the Cauchy effective stress tensor;
- p w is the pore pressure in the fluid; and
- I is the second-order unit tensor.

Biot [BIO 41, BIO 57] proposed the following relation generalizing Terzaghi's
postulate to materials with compressible solid phase:

σ = σ + αp w I

[9.55]

where α depends on the contrast between the stiffness of the material constituting the
matrix ( K s ) and that of the skeleton ( K ). This coefficient may vary between zero and
one ( α =1− K s ). It is obvious that the behavior of unsaturated soil is dependent on
both stresses and suction. However, the debate on whether to use one or the other of
the stress tensors is still open.

The approaches used fall into two families. In the first and oldest one, our attempt
is to proceed as in saturated soils. The problem, then, is how to define the effective
stress tensor. Under this approach, we only partially question Terzaghi's postulate.
With σ ∗ Cauchy net stress tensor and p g the gas phase pressure, it is assumed that the
decomposition of stress tensor in the form:

σ ∗ = σ − p g I = σ + σ f [9.56]

is still valid. σ f represents the portion of stresses taken up by the fluid. In the case of an
unsaturated soil, we can distinguish several possible formulations of σ f : the traditional
formulation is to define σ f in a form close to the saturated case of the type:
σ f = pI

[9.57]

where p represents the total pressure of the fluid in the media. According to the mixture
theory: