Homogenization of interlocking masonry walls

International audienceIn a previous publication the authors proposed a continuous model for describing the mechanical behaviour of ‘Running Bond' masonry walls. Here a different masonry structure is considered. This structure is ‘bi-atomic' in the sense that its pattern consists in two kind of blocks with different. The homogenization by differential expansions technique is used for obtaining an equivalent Cosserat continuum. It is shown that the enriched kinematics of the Cosserat continuum is not sufficient to capture the full dynamic behaviour of this bi-atomic system. However, the Cosserat continuum model behaves well for low frequency waves and for wave lengths 5 to 10 times bigger the elementary cell

Computational multiscale methods for granular materials

AbstractThe fine-scale heterogeneity of granular material is characterized by its polydisperse microstructure with randomness and no periodicity. To predict the mechanical response of the material as the microstructure evolves, it is demonstrated to develop computational multiscale methods using discrete particle assembly-Cosserat continuum modeling in micro- and macro- scales, respectively. The computational homogenization method and the bridge scale method along the concurrent scale linking approach are briefly introduced. Based on the weak form of the Hu-Washizu variational principle, the mixed finite element procedure of gradient Cosserat continuum in the frame of the second-order homogenization scheme is developed. The meso-mechanically informed anisotropic damage of effective Cosserat continuum is characterized and identified and the microscopic mechanisms of macroscopic damage phenomenon are revealed

Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations

In 1922, Cartan introduced in differential geometry, besides the Riemannian curvature, the new concept of torsion. He visualized a homogeneous and isotropic distribution of torsion in three dimensions (3d) by the "helical staircase", which he constructed by starting from a 3d Euclidean space and by defining a new connection via helical motions. We describe this geometric procedure in detail and define the corresponding connection and the torsion. The interdisciplinary nature of this subject is already evident from Cartan's discussion, since he argued - but never proved - that the helical staircase should correspond to a continuum with constant pressure and constant internal torque. We discuss where in physics the helical staircase is realized: (i) In the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d theories of gravity, namely a) in 3d Einstein-Cartan gravity - this is Cartan's case of constant pressure and constant intrinsic torque - and b) in 3d Poincare gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the gauge field theory of dislocations of Lazar et al., as we prove for the first time by arranging a suitable distribution of screw dislocations. Our main emphasis is on the discussion of dislocation field theory.Comment: 31 pages, 8 figure

Determining Cosserat constants of 2D cellular solids from beam models

We present results of a two-scale model of disordered cellular materials where we describe the microstructure in an idealized manner using a beam network model and then make a transition to a Cosserat-type continuum model describing the same material on the macroscopic scale. In such scale transitions, normally either bottom-up homogenization approaches or top-down reverse modelling strategies are used in order to match the macro-scale Cosserat continuum to the micro-scale beam network. Here we use a different approach that is based on an energetically consistent continuization scheme that uses data from the beam network model in order to determine continuous stress and strain variables in a set of control volumes defined on the scale of the individual microstructure elements (cells) in such a manner that they form a continuous tessellation of the material domain. Stresses and strains are determined independently in all control volumes, and constitutive parameters are obtained from the ensemble of control volume data using a least-square error criterion. We show that this approach yields material parameters that are for regular honeycomb structures in close agreement with analytical results. For strongly disordered cellular structures, the thus parametrized Cosserat continuum produces results that reproduce the behavior of the micro-scale beam models both in view of the observed strain patterns and in view of the macroscopic response, including its size dependence

A frictional Cosserat model for the flow of granular materials through a vertical channel

A rigid-plastic Cosserat model has been used to study dense, fully developed flow of granular materials through a vertical channel. Frictional models based on the classical continuum do not predict the occurrence of shear layers, at variance with experimental observations. This feature has been attributed to the absence of a material length scale in their constitutive equations. The present model incorporates such a material length scale by treating the granular material as a Cosserat continuum. Thus localised couple stresses exist and the stress tensor is asymmetric. The velocity profiles predicted by the model are in close agreement with available experimental data. The predicted dependence of the shear layer thickness on the width of the channel is in reasonable agreement with data. In the limit of the ratio of the particle diameter to the half-width of the channel being small, the model predicts that the shear layer thickness scaled by the particle diameter grows.Comment: 17 pages, 12 PostScript figures, uses AmsLaTeX, psfrag and natbib. Accepted for publication in Acta Mechanic

Application of a Cosserat Continuum Model to Non-associated Plasticity

A severe, spurious dependence of numerical simulations on the mesh size and orientation can be observed in elasto-plastic models with a non-associated flow rule. Such mesh dependency effects are quite well-known and vastly investigated in problems with the presence of strain softening in the constitutive relation. However, other material instabilities, like non-associated plastic flow, can also cause mesh sensitivity. Indeed, loss of well-posedness of the problem in quasi-static analyses is the fundamental cause of the observed mesh dependence. It has been known since long that non-associated plastic flow can cause loss of stability, but the consequence for mesh sensitivity, and subsequently, for the difficulty of the equilibrium-finding iterative procedure to converge, have remained largely unnoticed. The present thesis deals with exploring the possibility of using non-standard continua, namely the Cosserat continuum models, in non-associated plasticity problems to tackle the pathological mesh dependencies not only on the size of the elements but also on the orientation of them. The motivation in this thesis for using the Cosserat continuum models for this purpose, lies in the additional parameters that are specified in such a model, i.e. the micro-rotations and characteristic length scale. These features make the Cosserat continuum model a suitable choice to simulate and capture the behavior of the granular and geomaterials. The characteristic length scale helps regularise the strain localisation problems and prevent loss of uniqueness of solution. The mesh size dependency of the results for classical non-associated plasticity models is analysed in depth using an infinitely long shear layer. It is shown that the mesh effect disappears when the standard continuum model is replaced by a Cosserat continuum. Next, the dependence of the shear-band inclination in a biaxial test on the mesh size as well as on the mesh orientation is investigated. The orientation of the developed shear band is found to be dependent on the orientation of the mesh for classical models. Using a Cosserat continuum model, numerical solutions result for shear-band formation which are independent of the size and the orientation of the discretisation

Reduced cosserat continuum as a possible model for granular medium

We considered a nonlinear reduced Cosserat continuum: an elastic medium, whose translations and rotations are independent, the force stress tensor is asymmetric and the couple stress tensor is zero. We suggested the reduced Cosserat continuum as a possible model for granular medium. Granular materials are ubiquitous in our daily lives. They play an important role in many industries, such as mining, agriculture, and construction. We considered a nonlinear reduced Cosserat continuum for reference and current configurations and obtained the set of equations for each configuration