Modeling Landslides with a Material Instability Criterion - Multiscale Geomechanics: From Soil to Engineering Projects - page 234

238 Multiscale Geomechanics

- ω k the weight affected at each integration point k (Gauss quadrature);
- j e the determinant of the Jacobian matrix of element e .

For the same reasons as the ones discussed at the material point level, it is worth
normalizingtheglobalsecond-orderworkinordertofollowitsevolution.Khoa[KHO 05]
suggested the following expression:

D 2 W

D 2 W n =

k ω k j e

[8.33]

t dσ k dε k

e e

k

The variation of this quantity D 2 W n , called normalized global second-order work
is displayed in Figure 8.13. On this curve, D 2 W n highly increases at the outset
of loading, then passes through a peak and decreases to tend in principle towards
zero at the effective failure state. From a theoretical point of view, if we overcome
numerical imperfections, global second-order work should vanish before (in case of
diffuse failure) or at the same time as the convergence loss which corresponds to the
vanishing of the determinant of the global tangent stiffness matrix. It is for this reason
that, when computational convergence is lost, we can assume that the body loses its
ultimate global stability.

60

step 25

40

step 75

step 85

20

step 96

0

0 100 200 300 400

Upper water table level (m)
Figure 8.13. D 2 W n change during the wetting

8.4. Conclusion

The subject of this chapter has been the local second-order work criterion, a
condition which allows the material instability of elastoplastic materials to be detected.
For associated materials (which validate normality rule), we have found that this
conditionisattainedatthesametimeastheplasticitylimit.Fornon-associatedmaterials,
however, we see that it can be reached not only before the plasticity limit, but also before
a Rice strain localization criterion.