Modeling Landslides with a Material Instability Criterion - Multiscale Geomechanics: From Soil to Engineering Projects - page 219

8.2. Study of the second-order work criterion

8.2.1. Analytical study

Inthissectionwepresentadetailedanalysisofcondition[8.3]givenintheintroduction.
In the following we denote by N the matrix linking strain rate to stress rate, such as:

dε = N dσ [8.4]

and by M the inverse matrix as M = N −1 . In principal axes and for a given tensorial
zone 1 , N takes the following form:

⎡ ⎤

1
E 1

− ν 21
E 2

− ν 31
E 3

⎣ ⎦

− ν 12
E 1

1
E 2

− ν 32
E 3

N = [8.5]

− ν 13
E 1

− ν 23
E 2

1
E 3

With these notations, the shallow link between the second-order work criterion and
the condition given by:

det N s =0 [8.6]

is also described. N s is the symmetric part of N . Using the constitutive relation, we
obtain:

d 2 W =0 ⇔ t dσ N s dσ =0

[8.7]

For all incrementally piece-wise linear models, equation [8.7] can be developed in
the following manner in the principal stress rate space:

ν 12

dσ 1
E 1

+ dσ 2
E 2

+ dσ 3
E 3 E 1 + ν 21

− dσ 1 dσ 2

E 2

[8.8]

ν 32 ν 13

E 3 + ν 23 E 1 + ν 31

− dσ 3 dσ 2 − dσ 1 dσ 3 =0

E 3 E 2

This expression describes an elliptical cone, but real solutions depend on the sign
of the eigenvalues of the quadratic form: the cone can be degenerated. After quadric
reduction, equation [8.8] takes the following form in the diagonal base:

λ 1 X 2 + λ 2 Y 2 + λ 3 Z 2 =0

[8.9]