703
Numerical investigations of shear strain localization in an elasto-plastic Cosserat
material
Investigations numériques sur les déformations en cisaillement dans un matériau élastoplastique
de type Cosserat
Ebrahimian B.
School of Civil Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran
Noorzad A.
Faculty of Water and Environmental Engineering, Power and Water University of Technology, Tehran, Iran
ABSTRACT: The phenomenon of strain localization in narrow zones, called shear bands, is mainly related to the micro-structure of
granular materials. This phenomenon cannot be modeled properly within the framework of classical continuum especially in the post-
bifurcation regime due to the lack of characteristic length of the micro-structure. For finite element calculations, Cosserat (micro-
polar) continuum is an effective regularization technique to remove the numerical difficulties when shear localization occurs. The
paper presents numerical investigations of shear strain localization in plane shearing of an infinite granular layer as well as biaxial
compression of a specimen using micro-polar (Cosserat) continuum approach. It is shown that the micro-polar effects i.e., Cosserat
rotations, micro-curvatures and couple stresses are significant in the emerged shear bands. Shear banding pattern is significantly
affected by the prescribed micro-polar boundary conditions of entire system as well as geometry of specimen. It is confirmed that the
proposed elasto-plastic Cosserat model is capable to predict the evolution of micro-polar effects within the shear band.
RÉSUMÉ : Le phénomène de localisation des contraintes dans des zones étroites, appelées bandes de contraintes, est principalement
présent dans des microstructures de matériaux granulaires. Ce phénomène ne peut pas être modélisé correctement au moyen de
continuum classique, en particulier pour un régime de postbifurcation, à cause du manque de critères de longueur des microstructures.
Pour des calculs par éléments finis, un continuum Cosserat (micropolaire) est un moyen technique qui permet de supprimer les
difficultés numériques lorsque des cisaillements apparaissent. Cet article présente les investigations numériques de déformations en
cisaillement en plan pour une couche granulaire infinie ainsi que la compression biaxiale d’un spécimen en ayant recours à un
continuum micropolaire (Cosserat). Il est montré que les effets micropolaires, comme les rotations Cosserat, les micros-courbures et
les couples de contraintes sont significatifs dans les bandes de cisaillement apparentes. La structure de bandes de cisaillement est
affectée de manière significative par les conditions micropolaires aux limites du système complet ainsi que par la géométrie du
spécimen. Il est confirmé que le modèle élastoplastique Cosserat proposé est en mesure de prévoir l’évolution d’effets micropolaires
dans la bande de cisaillement.
KEYWORDS: strain localization; micro-polar (Cosserat) continuum; characteristic length; granular materials; micro-polar effects.
1 INTRODUCTION
The evolution of shear bands in granluar bodies is strongly
related to the micro-properties of material (Hall et al. 2010).
Shear band thickness is influenced by the soil grain size which
cannot be modeled properly with classical continuum models
due to the lack of a material characteristic length. As a
consequence, the shear band thickness is characterized by the
element size in finite element simulations, and the predicted
load-displacement curves are unreliable in the post-bifurcation
regime (de Borst 1991). In order to overcome this shortcoming
of classical continuum models and deal with such a complex
phenomenon within the framework of continuum mechanics,
micro-polar or the so-called Cosserat continuum models may be
used, which offer the possibility to include the mean grain
diameter as characteristic length (Mühlhaus 1986). The
presence of characteristic length allows taking into account the
microscopic inhomogeneities triggering shear localization (e.g.
grain size, size and spacing of micro-defects) observed
experimentally in granular materials. In this paper, an elasto-
plastic Cosserat continuum model is proposed which takes into
account micro-rotations (Cosserat rotations), micro-curvatures,
non-symmetric shear stresses, and couple stresses. The mean
grain diameter as characteristic length is also incorporated into
the model formulations. For plane strain condition,
implementation of the model in a finite element program is
outlined. Due to the presence of a characteristic length of the
micro-structure, the considered boundary value problems are
mathematically well-posed and the shear band thickness
predicted form finite element calculations is mesh independent,
provided the element size is small enough. The performance of
the present model is demonstrated by the numerical simulations
of large monotonic plane shearing and biaxial comression
leading to fully developed plastic flow. The focus of the
investigations is on studying the evolution of micro-polar
effects within the granular body. The influence of additional
non-standard Cosserat boundary conditions on the pattern of
shear banding is also considered. In particular, it is investigated
how the rotation resistance of soil grains in contact with
boundaries influences the location and evolution of shear
localization.
2 THE ELASTO-PLASTIC COSSERAT MODEL
According to Vardoulakis and Sulem (1995), the objective or
Cosserat strain rate tensor can be defined as
1
1
n
n
n
n
n ij
n ij
ij
ij
E
c
(1)
where,
ij
= classical strain rate tensor;
ij
= classical spin
tensor;
E
c
ij
= Cosserat spin tensor which is given by
c
ij
ijk k
e
c
(2)
where,
e
ijk
= Ricci permutation tensor; and
c
k
= Cosserat
rotation. The micro-curvature vector of deformation or the
gradient of soil grain rotation can be given by
