Cosserat Rotations Research Articles

Cracks of higher modes in Cosserat continua

Rotational degrees of freedom in Cosserat continua give rise to higher fracture modes. Three new fracture modes correspond to the cracks that are surfaces of discontinuities in the corresponding components of independent Cosserat rotations. We develop a generalisation of J- integral that includes these additional degrees of freedom. The obtained path-independent integrals are used to develop a criterion of crack propagation for a special type of failure in layered materials with sliding layers. This fracture propagates as a progressive bending failure of layers – a “bending crack that is, a crack that can be represented as a distribution of discontinuities in the layer bending. This situation is analysed using a 2D Cosserat continuum model. Semi-infinite bending crack normal to layering is considered. The moment stress concentrates along the line that is a continuation of the crack and has a singularity of the power − 1/4. A model of process zone is proposed for the case when the breakage of layers in the process of bending crack propagation is caused by a crack (microcrack in our description) growing across the layer adjacent to the crack tip. This growth is unstable (in the moment-controlled loading), which results in a typical descending branch of moment stress – rotation discontinuity relationship and hence in emergence of a Barenblatt-type process zone at the tip of the bending crack.

A finite-strain elastic–plastic Cosserat theory for polycrystals with grain rotations

We investigate geometrically exact generalized continua of Cosserat micropolar type. A variational form of these models is recalled and extended to finite-strain elasto-plasticity based on the multiplicative decomposition of the deformation gradient. The stress driving the plastic evolution is the Eshelby energy momentum tensor. No plastic Cosserat rotation is introduced and the plastic spin is set to zero. It is argued that the traditional Cosserat couple modulus μ c should be set to zero for polycrystal specimens liable to fracture in shear, still leading to a complete Cosserat theory with independent rotations in the geometrically exact case in contrast to the infinitesimal, linearized model. A geometrical linearization of the presented finite-strain plasticity model is already shown to be well posed.

Modelling strain localization in granular materials using micropolar theory: mathematical formulations

It has been known that classical continuum mechanics laws fail to describe strain localization in granular materials due to the mathematical ill-posedness and mesh dependency. Therefore, a non-local theory with internal length scales is needed to overcome such problems. The micropolar and high-order gradient theories can be considered as good examples to characterize the strain localization in granular materials. The fact that internal length scales are needed requires micromechanical models or laws; however, the classical constitutive models can be enhanced through the stress invariants to incorporate the Micropolar effects. In this paper, Lade's single hardening model is enhanced to account for the couple stress and Cosserat rotation and the internal length scales are incorporated accordingly. The enhanced Lade's model and its material properties are discussed in detail; then the finite element formulations in the Updated Lagrangian Frame (UL) are used. The finite element formulations were implemented into a user element subroutine for ABAQUS (UEL) and the solution method is discussed in the companion paper. The model was found to predict the strain localization in granular materials with low dependency on the finite element mesh size. The shear band was found to reflect on a certain angle when it hit a rigid boundary. Applications for the model on plane strain specimens tested in the laboratory are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd.

A novel approach to examining double-shearing type models for granular materials

A novel method, designated as the Rotation of Principal Axes Method (RPAM), capable of examining the double-shearing type kinematic models for granular materials is presented herein. A planar velocity field, which is proposed to represent a continuous rotation of principal strain rate axes, is applied to each model to analyse the rotation of principal stress axes. The proposed approach was proven to show main features of the double-shearing model, the double-sliding free-rotating model, and the revised double-shearing model, in a simple way interesting to geo-researchers. Furthermore, the RPAM was efficient in investigating the choice of a Cosserat rotation rate in kinematic theories and determining a key model parameter in the revised double-shearing model.

FE-studies on rapid flow of bulk solids in silos

This paper contains results of numerical modelling of the onset of silo flow for granular material in a model silo with convergent walls. The calculations were performed with a finite element method based on a polar elasto-plastic constitutive relation by Muhlhaus. It differs from the conventional theory of plasticity by the presence of Cosserat rotations and couple stresses using a mean grain diameter as a characteristic length. The characteristic length causes that numerical results do not depend upon the mesh discretisation. The model tests on rapid silo flow of glass beads performed by Renner in a glass hopper with a large wall inclination from the bottom were numerically simulated. The FE-calculations were performed for plane strain by taking into account inertial forces and linear viscous damping. A satisfactory agreement between numerical and experimental results was obtained. In addition, the FE-calculations were performed for very rough walls. Advantages and limitations of a continuum approach for simulations of rapid silo flow were outlined.

Numerical simulation of filling in silos with a polar hypoplastic constitutive model

The behaviour of granular materials during silo filling is investigated with a polar hypoplastic model. The model is an alternative to elastoplastic constitutive models and can capture the salient features of granular bodies with consideration of shear localization. It is of the rate type, incrementally non-linear, and is established within the framework of a Cosserat continuum. It takes into account Cosserat rotations and couple stresses using the mean grain diameter as a characteristic length. Numerical simulations of silo filling with dry dense sand are performed with a Cosserat type approach implemented in a finite element method for model silos with parallel and slightly convergent walls. The silo walls are smooth, rough, and very rough. Because of the presence of a characteristic length in the constitutive model, the numerical results are independent of the spatial discretization. The results show that polar boundary conditions with consideration of Cosserat rotations and wall roughness realistically describe the interface behaviour between the silo wall and the silo fill. The Cosserat effect, which is manifested by the appearance of grain rotations, is only significant in the wall shear zone. A quantitative comparison between the numerical calculations and the model tests carried out in a plane strain bin and hopper shows satisfactory agreement. The advantages of the approach used are outlined.

Numerical simulation of shear band formation with a polar hypoplastic constitutive model

Shear band formation in dry sand during biaxial compression tests is investigated with a polar hypoplastic model. The model is of the rate type and incrementally non-linear. It is established within the framework of a Cosserat continuum. It takes into account Cosserat rotations and couple stresses using the mean grain diameter as a characteristic length. The calculations of biaxial tests with a finite element method are performed for two different meshes and two different mean grain diameters. Initiation of shear bands is triggered by a small inhomogeneity in the initial material properties. From the presence of a characteristic length the calculated thickness of shear zones is almost independent of the spatial discretisation if the size of finite elements is not greater than five characteristic lengths. The numerical results show that the polar effect manifested by the appearance of grain rotations is significant in a shear zone. A quantitative comparison between the numerical calculations and the experimental results shows satisfying agreement.

Experimental and numerical study of sand–steel interfaces

AbstractExperimental and numerical studies on and sand–steel interfaces are presented. Emphasis is laid on the effect of boundary conditions of the whole system and of localized deformation. The experiments with different roughness of steel surface, sand density, normal stress and grain size are carried out in a plane strain apparatus, a parallely guided direct shear apparatus and in a planar silo model with a movable bottom and parallel steel walls. During the test in the plane strain apparatus the localized zone is observed with the help of X‐rays. The results indicate a significant effect of wall roughness and boundary conditions of the whole system on the wall friction angle and the thickness of the localized zone along the steel surface.An elastoplastic constitutive model established within the framework of a Cosserat continuum, capable of describing isotropic hardening, softening and dilatancy, is implemented in a finite element code. The model differs from the conventional theory of plasticity due to the presence of Cosserat rotation and couple stress using the mean grain diameter as the characteristic length. Finite element simulations of simple shear tests are presented. The additional boundary condition along the steel plate, characteristic of the Cosserat continuum, allows for modelling the different roughness of the steel plate with consideration of grain rotations. A comparison between the numerical calculations and the experimental results shows acceptable agreement.

Numerical Study on sand and steel interfaces

Simple shear tests of sand along steel plates have been simulated numerically using a finite element method and an elasto-plastic Cosserat model. The Cosserat model differs from the conventional theory of plasticity due to the fact that the Cosserat rotations, the couple stresses and the mean grain diameter as a characteristic length are taken into account. The additiona Cosserat boundary condition along the steel plate allows a more detailed description of the wall roughness. Comparisons between the numerical calculations and the experimental results show acceptable agreement.