Search

This paper describes phenomena associated to the microstructure, such as the local bending stiffness of the yarns, with a second gradient parameter associated to terms in the energy with higher ord...

In the present contribution we show that the relaxed micromorphic model is the only non-local continuum model which is able to account for the description of band-gaps in metamaterials for which the kinetic energy accounts separately for micro and macro-motions without considering a micro-macro coupling. Moreover, we show that when adding a gradient inertia term which indeed allows for the description of the coupling of the vibrations of the microstructure to the macroscopic motion of the unit cell, other enriched continuum models of the micromorphic type may allow the description of the onset of band-gaps. Nevertheless, the relaxed micromorphic model proves to be yet the most effective enriched continuum model which is able to describe multiple band-gaps in non-local metamaterials., Comment: arXiv admin note: text overlap with arXiv:1607.07385

The classical models used for describing the mechanical behavior of woven fabrics do not fully account for the whole set of phenomena that occur during the testing of such materials. This lack of precision is mainly due to the absence of energy terms related to the microstructural properties of the fabric and, in particular, to the bending stiffness of the yarns. The importance of the bending stiffness on the overall mechanical behavior of woven reinforcements, if already essential for the complete description of balanced fabrics, becomes even more important in the case of unbalanced ones. In this paper it is shown that the unbalance in the bending stiffnesses of the warp and weft yarns produces macroscopic effects that are extremely visible: we mention, for example, the asymmetric S-shape of a woven interlock subjected to a Bias Extension Test. We propose to introduce a constrained micromorphic model and, simultaneously, a discrete model that are both able to account for (i) the angle variation between warp and weft tows, (ii) the unbalance in the bending stiffness of the yarns and (iii) the relative slipping of the tows. The introduced constrained micromorphic model is rigorously framed in the spirit of the Principle of Virtual Work for the study of the equilibrium of continuum bodies. A suitable constraint is introduced in such micromorphic model by means of Lagrange multipliers in the strain energy density and the resulting constrained model is seen to be a particular second gradient one. The main advantage of using such constrained micromorphic model is that the kinematical and traction boundary conditions that can be imposed on some sub-portions of the boundary of the considered body take a natural and unique meaning. The discrete model is set up by opportunely interconnecting Euler–Bernoulli beams with different bending stiffnesses in the two directions by means of rotational and translational elastic springs. The main strength of such discrete model is that the slipping of the tows is described in a rather realistic way. Suitable numerical simulations are presented for both the continuum and the discrete models and a comparison between the simulations and the experimental results is performed showing a definitely good agreement.

International audience; The classical continuum models used for the woven fabrics do not fully describe the whole set of phenomena that occur during the testing of those materials. This incompleteness is partially due to the absence of energy terms related to some micro-structural properties of the fabric and, in particular, to the bending stiffness of the yarns. To account for the most fundamental microstructure-related deformation mechanisms occurring in unbalanced interlocks, a second-gradient, hyperelastic, initially orthotropic continuum model is proposed. A constitutive expression for the strain energy density is introduced to account for i) in-plane shear deformations, ii) highly different bending stiffnesses in the warp and weft directions and iii) fictive elonga-tions in the warp and weft directions which eventually describe the relative sliding of the yarns. Numerical simulations which are able to reproduce the experimental behavior of unbalanced carbon interlocks subjected to a Bias Extension Test are presented. In particular, the proposed model captures the macroscopic asymmetric S-shaped deformation of the specimen, as well as the main features of the associated deformation patterns of the yarns at the mesoscopic scale.

In this paper, we propose the first estimate of some elastic parameters of the relaxed micromorphic model on the basis of real experiments of transmission of longitudinal plane waves across an interface separating a classical Cauchy material (steel plate) and a phononic crystal (steel plate with fluid-filled holes). A procedure is set up in order to identify the parameters of the relaxed micromorphic model by superimposing the experimentally based profile of the reflection coefficient (plotted as function of the wave-frequency) with the analogous profile obtained via numerical simulations. We determine five out of six constitutive parameters which are featured by the relaxed micromorphic model in the isotropic case, plus the determination of the micro-inertia parameter. The sixth elastic parameter, namely the Cosserat couple modulusμc, still remains undetermined, since experiments on transverse incident waves are not yet available. A fundamental result of this paper is the estimate of the non-locality intrinsically associated with the underlying microstructure of the metamaterial. We show that the characteristic lengthLcmeasuring the non-locality of the phononic crystal is of the order of13of the diameter of its fluid-filled holes.