Modeling Landslides with a Material Instability Criterion - Multiscale Geomechanics: From Soil to Engineering Projects - page 221

min
i=1...,n

det N s =0 with u i ⊂ Z i [8.10]

i

with n the number of tensorial zones of the constitutive model, u i the eigenvector
associated with the null eigenvalue and Z i the tensorial zone examined.

It is of interest to note that a similar analysis can be made in the strain space
and that the condition det (M s )=0 is equivalent to the condition det (N s )=0
for non-associated materials. In fact, it is possible to prove the following relation
[PRU 09a, PRU 09b]:

det N s

det M s =

[8.11]

det N 2

Forassociatedmaterials,thebifurcationdomaincontourcoincideswiththeplasticity

limit surface and relation [8.11] remains valid because det M =1/det N , but the
plasticity limit cannot be expressed in a similar way with the matrix N .

To illustrate the results presented above, Figures 8.2 and 8.3 display limits of the
bifurcation domain as well as instability cones obtained along a drained triaxial path

octolinear model non-linear model

1000 1000

500 500

0 0

0 0

500 500

1000 1000

500 500

1000 1000

0 0

σ 2 (kPa) σ 2 (kPa)

σ 3 (kPa) σ 3 (kPa)

(a) (b)

bifurcation domain
for the non-linear model

σ 1 /p 0

non-linear model
octolinear model
Mohr-Coulomb

σ 2 /p 0 σ 3 /p 0

(c)