Geoscience Reference
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254
Multiscale Geomechanics
elastoplastic tensor connecting the elastoplastic strain rate (increment) to the stress rate
(increment) is dependent on the direction of loading. In the viscoplastic models, this
condition, based on Perzyna's theory [PER 66], has been eliminated. This constraint
is also absent in incremental and hypoplastic models [DAR 82, DAF 86]. In these
incremental models, the different terms of the elastoplastic tangent tensor are defined
from the test results. Their generalization to load paths, different from those for which
the terms of the tensor were evaluated, is indirect. Although the behavior when loading
and unloading can be modeled, they are not yet adapted to cyclic loading [KOL 00].
A vast literature on different types of models for soil behavior and their formulations can
be found in the following documents: [CAM 00, HIC 02, MAN 87, MES 93, ZIE 99].
The models presented in this chapter belong to the first category and are situated
withintheframeworkofincrementalplasticity.Thischoiceisfurtherjustifiedbythefact
that these models were destined to be integrated into computing finite element software
entirely suited to this type of modeling. Moreover, since the displacements are part of
the unknown variables of the system to be solved, we must choose a formulation based
on the imposed strains. We proceed now to the mathematical formulation of a generic,
simplified law that contains all the important features of the behavior described above.
9.3.3.
Simplified model
As previously mentioned, the mean stress and shear (or deviatoric stress) are factors
that govern the behavior of soils. Very often axisymmetrical triaxial tests, easy to
perform, are used to characterize the behavior of geomaterials. Constitutive laws of
isotropic materials are formulated in terms of invariants such as the mean stress
p
, the
deviator stress
q
, the compressibility
v
and deviatoric strain
or deviatoric strains
given by
p
=
tr
(σ
)/3; s
= σ
− p
I; q =(
2
tr
(s
⊗ s
))
1/2
ε
v
=
tr
(ε); ε = ε −
ε
3
I; =(
3
tr
(ε ⊗ ε))
1/2
The behavior of soils is also dependent on the third invariant, but as we shall see in
the formalism of the multimechanism model we have adopted, even though the stress
orientation has been taken into account, the third invariant does not come into play
explicitly. Given the fact that during the triaxial test, the two quantities
p
0
and q evolve,
it is difficult to specify the role played by each factor. From a pedagogical point of
view, it would be more convenient to reason in terms of normal stress and shear stress.
Unfortunately, the laboratory tests used to study these quantities are often difficult to
perform, or they are tests in which the stress and strain fields are not homogeneous,
hence unsuitable for establishing constitutive laws.
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