Showing papers by "Luca Placidi published in 2016"

A review on 2D models for the description of pantographic fabrics

Abstract: A review on models for pantographic fabrics, a new promising kind of metamaterials, is presented. We treat those models that are able to capture the peculiar effects conferred by their specific microstructure and that can be generalized for the description of more complex metamaterials. For each approach, model formulation and modeling assumptions are discussed along with the presentation of numerical solutions in exemplary cases and no attempt is made to model damage and failure phenomena.

A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model

Abstract: A one-dimensional displacement second-gradient damage continuum theory has been already presented within the framework of a variational approach. Damage is associated with strain concentration. Therefore, not only non-local effects of displacement second-gradient modelling should be considered in a comprehensive model, but also any plastic effects. The aim of this paper is therefore to extend such a model to plasticity. The action is intended to depend not only with respect to first and second gradient of displacement field and to a scalar damage field, but also to further two internal variables, i.e. the accumulated plastic tension and the accumulated plastic compression. A constitutive prescription on the stiffness is given in terms of the scalar damage parameter in a usual way, i.e. as in many other works, it is prescribed to decrease as far as the damage increases. On the other hand, the microstructural material length (i.e. the square of the constitutive function in front of the squared displacement second-gradient term in the action functional) is prescribed to increase as far as the damage increases, being this last assumption connected to the interpretation that a damage state induces a microstructure in the continuum and that such a microstructure is more important as far as the damage increases. Initial damage threshold and yield stresses are naturally introduced in the action in front of linear terms, respectively, of damage and plastic internal variables. The hardening matrix is also introduced in a natural way as the coefficient matrix in front of the quadratic terms of the two plastic internal variables. At a given value of damage and plastic parameters, the behaviour is referred to second-gradient linear elastic material. However, the damage and plastic evolutions make the model not only nonlinear, but also inelastic. The second principle of thermodynamics is considered by assuming that the scalar damage and plastic parameters do not decrease their values in the process of deformation, and this implies a dissipation for the elastic strain energy. A novel result of this investigation, where displacement second-gradient and plastic effects are combined, is that the distributed and concentrated external double forces do not make work on the displacement gradient but only to its elastic part and this means that the displacement gradient cannot be prescribed, at the border, independently of the plastic internal variables.

A second gradient formulation for a 2D fabric sheet with inextensible fibres

Abstract: We present numerical simulations of rectangular woven fabrics made of two, initially orthogonal, families of inextensible fibres. We consider an energy functional which includes both first and second gradients of the displacement. The energy density is expressed in terms of the angles between the fibres directions, using trigonometric functions and their gradients. In particular, we focus on an energy density depending on the squared tangent of the shear angle, which automatically satisfies some natural properties of the energy. The numerical results show that final configurations obtained by the second gradient energies are smoother than the first gradient ones. Moreover, we show that if a second gradient energy is considered, the shear energy is better uniformly distributed.

Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation

Abstract: In this paper, the dynamics of a cantilever beam subjected to harmonic excitations and to the contact of an obstacle is studied with the help of experimental and numerical investigations. The steel flexible structure is excited close to the free end with a shaker and may come into contact with a deformable and dissipative obstacle. A technique for modeling contact phenomena using piece-wise linear dynamics is applied. A finite-dimensional modal model is developed through a Galerkin projection. Concentrated masses, dampers and forces are considered in the equations of motion in such a way that the boundary conditions are those of a cantilever beam. Numerical studies are conducted by assuming finite-time contact duration to investigate the frequency response of the impacted beam for different driving frequencies. Experimental results have been extrapolated through a displacement laser sensor and a load cell. The comparison between numerical and experimental results show many qualitative and quantitative similarities. The novelty of this paper can be synthetized in (a) the development of experimental results that are in good agreement with the numerical implementation of the introduced model; (b) the development of a comprehensive contact model of the beam with an unilateral, deformable and dissipative obstacle located close to the tip; (c) the possibility of accounting for higher modes for the cantilever beam problem, and hence of analyzing how the response varies when moving the excitation (and/or the obstacle) along the beam, and of investigating the effect of the linearly elastic deformability of the built‐in end of the beam; (d) an easy and intuitive solution to the problem of accounting for spatially singular masses, dampers, springs and forces in the motion equations; (e) the possibility of accounting for finite gap and duration of the contact between beam and obstacle.

First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model

Abstract: 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 our model by superimposing the experimentally-based profile of the reflection coefficient (plotted as function of the frequency of the traveling waves) with the analogous profile obtained via simulations based upon the relaxed micromorphic model. We end up with the determination of 5 out of 6 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 modulus $\mu_{c}$, still remains undetermined, since experimental data concerning the transmission properties of the considered interface for transverse incident waves are not yet available. A fundamental result of the present paper is the estimate of the non-locality intrinsically associated to the underlying microstructure of the metamaterial. As a matter of fact, we appraise that the characteristic length $L_{c}$ measuring the non-locality of the considered phononic crystal is of the order of $1/3$ of the diameter of the considered fluid-filled holes.

First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model

Abstract: 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 modulus μ c , 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 length L c measuring the non-locality of the phononic crystal is of the order of 1 3 of the diameter of its fluid-filled holes.

Deformation of an elastic magnetizable square rod due to a uniform electric current inside the rod and an external transverse magnetic field

Abstract: We find the deformation and stresses in an infinite rod of an electric conducting material with square normal cross-section, carrying uniform electric current and subjected to an external, initially uniform magnetic field. The complete solution of the uncoupled problem is obtained using a boundary integral method. The results are discussed in detail.

The influence of hydrostatic stress on the frequency equation of flexural waves in a magnetoelastic transversly isotropic circular cylinder

Abstract: In this paper, we investigated the influence of initial stress on the frequency equation of flexural waves in a transversely isotropic circular cylinder permeated by a magnetic field. The problem is represented by the equations of elasticity taking into account the effect of the magnetic field as given by Maxwell's equations in the quasi-static approximation. The free stress conditions on the inner and outer surfaces of the hollow circular cylinder were used to form a frequency equation in terms of the wavelength, the cylinder radii, the initial stress and the material constants. The frequency equations have been derived in the form of a determinant involving Bessel functions and its roots given the values of the characteristic circular frequency parameters of the first three modes for various geometries. These roots, which correspond to various modes, have been verified numerically and represented graphically in different values for the initial stress. It is recognized that the flexural elastic waves in a solid body propagated under the influence of initial stress can be differentiated in a clear manner from those propagated in the absence of an initial stress. We also observed the initial stress has a great effect on the propagation of magnetoelastic flexural waves. Therefore this research is theoretically useful to convey information on electromagnetic properties of the material: for example through a precise measurement of the surface current induced by the presence of the magnetic field.