It has been known since the pioneering works by Piola, Cosserat, Mindlin, Toupin, Eringen, Green, Rivlin and Germain that many micro-structural effects in mechanical systems can be still modeled by means of continuum theories. When needed, the displacement field must be complemented by additional kinematical descriptors, called sometimes microstructural fields. In this paper, a technologically important class of fibrous composite reinforcements is considered and their mechanical behavior is described at finite strains by means of a second-gradient, hyperelastic, orthotropic continuum theory which is obtained as the limit case of a micromorphic theory. Following Mindlin and Eringen, we consider a micromorphic continuum theory based on an enriched kinematics constituted by the displacement field u and a second-order tensor field psi describing microscopic deformations. The governing equations in weak form are used to perform numerical simulations in which a bias extension test is reproduced. We show that second-gradient energy terms allow for an effective prediction of the onset of internal shear boundary layers which are transition zones between two different shear deformation modes. The existence of these boundary layers cannot be described by a simple first-gradient model, and its features are related to second-gradient material coefficients. The obtained numerical results, together with the available experimental evidences, allow us to estimate the order of magnitude of the introduced second-gradient coefficients by inverse approach. This justifies the need of a novel measurement campaign aimed to estimate the value of the introduced second-gradient parameters for a wide class of fibrous materials

In this paper, we propose to use a second gradient, 3D orthotropic model for the characterization of the mechanical behavior of thick woven composite interlocks. Such second-gradient theory is seen to directly account for the out-of-plane bending rigidity of the yarns at the mesoscopic scale which is, in turn, related to the bending stiffness of the fibers composing the yarns themselves. The yarns' bending rigidity evidently affects the macroscopic bending of the material and this fact is revealed by presenting a three-point bending test on 0 • /90 • and ±45 • specimens of composite interlocks. These specimens differ one from the other for the different relative direction of the yarns with respect to the edges of the sample itself. Both types of specimens are independently seen to take advantage of a second-gradient modeling for the correct description of their macroscopic bending modes. The results presented in this paper are essential for the setting up of a correct continuum framework suitable for the mechanical characterization of composite interlocks. The few second-gradient parameters introduced by the present model are all seen to be associated with peculiar deformation modes of the mesostructure (bending of the yarns) and are determined by inverse approach. Although the presented results undoubtedly represent an important step toward the complete characterization of the mechanical behavior of fibrous composite reinforcements, more complex hyperelastic second-gradient constitutive laws must be conceived in order to account for the description of all possible mesostructure-induced deformation patterns.Mathematics Subject Classification. 74-XX · 74Bxx · 74B20.

In this paper, a new consistent dynamic model is proposed, aimed at studying linear vibrations induced in an elastic wire by a bilaterally constrained single mass moving with a constant velocity. Starting from a variational formulation, through the Hadamard’s condition, a corrective term to the local linear stiffness is determined in the continuum model as a function of the moving mass velocity; in this way, the boundary conditions are properly found. The representation of the solutions of the hyperbolic equations governing the motion of the wire presents some difficulties, which are solved by means of a suitable coordinate transformation in a time-invariant domain and a judicious choice of the set of shape functions, to be used in the discrete formulation of the problem. This new description allows an easy estimation of high-order deformations that are neglected by a purely linear approach. When the mass velocity is sufficiently high, displacements near the supports show high gradients: in these cases, it is necessary to use an unknown velocity or introduce an advanced mechanical model in order to correctly describe the motion of the mass. Numerical examples confirm the stability of the proposed solution in all conditions examined

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A critical review of three paradoxical phenomena, occurring in the dynamic stability of finite-dimensional autonomous mechanical systems, is carried out. In particular, the well-known destabilization paradoxes of Ziegler, due to damping, and Nicolai, due to follower torque, and the less well known failure of the so-called ‘principle of similarity’, as a control strategy in piezo-electro-mechanical systems, are discussed. Some examples concerning the uncontrolled and controlled Ziegler column and the Nicolai beam are discussed, both in linear and nonlinear regimes. The paper aims to discuss in depth the reasons of paradoxes in the linear behavior, sometimes by looking at these problems in a new perspective with respect to the existing literature. Moreover, it represents a first attempt to investigate also the post-critical regime.

The Nicolai problem concerning the stability of a quasisymmetric cantilever beam embedded in a three-dimensional space, under a compressive dead load and a follower torque, is addressed. The effect of external and internal damping on stability is investigated. The partial differential equations of motion, accounting for the pretwist contribution, are recast in weak form via the Galerkin method, and a linear algebraic problem, governing the stability of the rectilinear configuration of the beam, is derived. Perturbation methods are used to analytically compute the eigenvalues, starting with an unperturbed, undamped, symmetric, untwisted beam, axially loaded, in both the subcritical and critical regimes. Accordingly, an asymmetry parameter, the torque, the damping, and the load increment are taken as perturbation parameters. Maclaurin series are used for semisimple eigenvalues occurring in subcritical states, and Puiseux series for the quadruple-zero eigenvalue existing at the Euler point. Based on the eigenvalue behavior described by the asymptotic expansions, the stability domains are constructed in the two or three-dimensional space of the bifurcation parameters. It is found that dynamic bifurcations occur in the subcritical regime, and dynamic or static bifurcations in the critical regime. It is shown that stability is governed mostly by the bifurcation of the lowest eigenvalue. In all cases the Nicolai paradox is recovered, and the beneficial effects of asymmetry and damping are highlighted.

Buckling of uniformly and not uniformly compressed tower buildings, resting on Winkler type soil, is investigated. An equivalent beam is introduced, able to capture the essential behavior of the building. It is a 3D Timoshenko beam, modeled in the framework of a direct approach, whose constitutive law is derived via a homogenization procedure, which includes the effect of the column prestress. The continuous model is discretized via finite differences, and a linear bifurcation analysis is carried out by solving an algebraic eigenvalue problem. Numerical results are shown for sample problems, aimed at detecting the structural behavior, and illustrating the role of some mechanical parameters. Results supplied by the equivalent beam model are compared with those derived by finite element analyses, carried out on three-dimensional frames.

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