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of the band having a width equal to 12 times the mean diameter of the grains (see
Figure 3.7a). This is in agreement with the experimental results presented by Oka
and Kazama [OKA 98].
Finally, let us notice that the form of equation [3.19] is compatible with a
homogeneous far-field movement, which also indicates that, if a mean shear strain
rate γ is imposed at the boundary of a domain of width L , the maximum shear rate
possible within a shear band concentrating all the movements in the domain is
limited by the following expression:
γ
kL
[3.20]
Max
Max
()
t
γ
2
3.1.3.8. Localization criterion - shear band evolution
We now consider the localization process as a particular mode of heterogeneity
growth in the motion, induced by the minimum dissipation rule. A localization
criterion can then be established, based on the non-convexity of the specific energy
dissipation rate depending on the correlated fluctuations in strain and density
[FRO 04]. Under usual regular monotonic boundary conditions, the specific energy
dissipation is almost proportional to the strength of the medium expressed in terms
of stress, whose evolution during deformation usually takes one of the forms
presented in the diagram in Figure 3.8a.
As a result, the localization criterion is met early during the movement. This
criterion therefore predicts that the initialization of the strain localization starts well
before the peak of the stress−strain curve, which is experimentally observed
[DES 96].
Once created, their evolution during movement is driven by a criterion that can
also be obtained by the minimum energy dissipation rule. For steady structures in a
permanent regime, the analysis of the heterogeneity by the minimum energy
dissipation rule shows that the shear band evolution tends to maximize the variance
of the deformation amplitudes, i.e. the variance of the shear strain rates [FRO 04].
The comparison of the variances obtained for various admissible deformation
schemes shows that:
− a family of parallel bands tends to converge into a single band that
concentrates the whole movement;
− this single band will narrow until the maximum amplification ratio is reached
(see section 3.1.3.7), with the disappearance of all the movements outside the band.
 
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