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.
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