Numerical Modeling: An Efficient Tool for Analyzing the Behavior of Constructions - Multiscale Geomechanics: From Soil to Engineering Projects - page 258

One part of tensor σ at point x can be obtained by a double projection:

v t · σ ∆∆ · w = v t ∆ · σ · w ∆ [9.42]

which gives

σ ∆∆ = O t

∆ · σ · O ∆

[9.43]

For a given plane of normal n passing through point x , considering the tangent
projection below:

O ∆ = O t = I − n ⊗ n [9.44]

where t is the unit vector of the same orientation as the tangent stress in the plane
perpendicular to n , the stress vector and tensor can be written using the following
decomposition:
- for the constraint vector:

σ · n =(σ·n) t +σ nn n [9.45]

with σ nn being the normal stress and

(σ · n) t = O t · (σ · n) [9.46]

- for the stress tensor:

σ = σ tt +(σ·n) t ⊗n+n⊗(n·σ) t +σ nn n ⊗ n [9.47]

with

σ tt = σ − σ · n ⊗ n − n ⊗ n · σ + σ nn n ⊗ n [9.48]

We can consider situations where plasticity mechanisms are written in terms of σ tt
or in a general manner σ ∆∆ . In practice, we can imagine three typical situations:
1) Stratified rocky materials, where prior to the loads applied weakness planes were
already created. This could be of interest to the engineer. Further plastic deformations
occur preferably along these planes distributed in space. Behavior is governed by
normal and tangential stresses applied on planes of normal n : σ nn and σ nt (or τ ).
The strains ε nn and ε nt (or γ ) will also appear. The necessary number of weak planes
are chosen. However, a question should be answered: are these chosen mechanisms,
based on their weakness planes alone, sufficient to model the behavior of the solid or
should additional isotropic mechanisms (Mohr-Coulomb or Hoek and Brown, which
are often used in rock mechanics) be added, according to the microstructure of the rock
mass. Therefore the mechanism is implemented, so that the user has the possibility of