Multi-field Modelling of Cosserat solids

Abstract

We constructed multi-field generalisations of the Cosserat continuum model on the basis of the square lattice model that takes into account rotational degree of freedom of microstructural elements. This approach allows us to model not only long but also short wavelength fields of displacements and rotations. The accuracy of the wave solutions is examined. The dispersion relations of plane harmonic waves for multi-field Cosserat continuum models are obtained and compared with those of the basic model. Key words:Cosserat solid, Cosserat continuum model, short wavelength solutions INTRODUCTIONContinuum models are effectively used for modelling of solids composed of a large number of microstructural elements and have a wide range of applications from nano-materials to space constructions.They are useful for prediction of physical-mechanical characteristics, obtaining of analytical solutions, constructing of efficient algorithms for numerical simulations, etc. But such models should be corrected in order to describe some essential microstructural effects. The development of more precise models is often based on analysis and rejection of some assumptions of the classical theories. Different classifications of the generalised models are given, for example, in /1, 2/.Several types of generalised models are constructed by taking into account more degrees of freedom of structural elements than in classical models, e.g. /3-8/. In Cosserat and micropolar models not only space displacements but also independent rotations of the elements are taken into consideration and field of 379 Brought to you by | Western University Authenticated Download Date | 6/8/15 11:04 AM Vol. 16, No. 6, 2005 Multi-field Modelling of Cosserat Solidsrotations in addition to the field of displacements is introduced. Structural models are useful for development and applications of field theories. The discrete models of materials with beam-microstructure and grid frameworks, corresponding micropolar models and methods for their obtaining are presented, for example, in /9-12/. The complex-valued Cosserat continuum model was constructed for vibrating grid frameworks in /5/.The variants of lattice models and corresponding Cosserat continuum models were obtained in /13, 14/. Via analysis of the dispersion curves, it has been demonstrated that the Cosserat continuum model gives accurate results for deformation fields with wavelengths larger than approximately six times micro-structural cell distance but produces unrealistic results for short wavelengths. In the present article, we construct generalisations of the Cosserat continuum model applicable not only for long but also for short wavelengths oscillating fields. As a rule, single cell of periodicity and one corresponding vector field of generalised displacements are used for obtaining of continuum models. Rejection of this not self-evident assumption leads to a new class of models. We construct TV-field models on the basis of macrocell /15/, which is consist of Ν simple cells. By using of Ν vector fields...

A discrete model and analysis of one-dimensional deformations in a structural interface with micro-rotations

“…Single-field models were constructed in Section 2 using single vector function with the same degrees of freedom that were chosen to describe kinematics of primitive cell. Considering a larger cell that includes several primitive lattice cells and using, correspondingly, a larger number of fields to describe the deformations of the body leads to the multi-field continuum models (Vasiliev and Miroshnichenko, 2005;Vasiliev et al, 2008;Gonella and Ruzzene, 2010;Vasiliev et al, 2010b). In order to derive equations of 2FMm, we consider a macrocell that includes two neighboring primitive cells.…”