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Anisotropic laminates with a negative Poisson's ratio for at least some directions are called auxetic. In this paper, we consider the conditions for optimizing the auxeticity of an orthotropic laminate, namely: for a laminate composed by a given material, (i) how to obtain the lowest, i.e. the highest negative, Poisson's ratio and (ii) how to maximize the auxetic zone, i.e. the set of directions where the Poisson's ratio is negative. It is shown that in both the cases the optimal solution is found on the boundary of the feasible domain and in particular that it can be obtained using angle-ply sequences of identical layers. The polar method with dimensionless moduli is employed for representing the anisotropic behavior of the laminate, which allows, on the one hand, to reduce the dimensionality of the problem and, on the other hand, to have an effective mathematical representation of anisotropy by dimensionless invariants., Comment: 16 pages, 9 figure, 3 tables. arXiv admin note: text overlap with arXiv:2403.16057

The problem of obtaining anisotropic auxetic composite laminates, i.e. having a negative Poisson's ratio for at least some directions, is examined in this paper. In particular, the possibility of obtaining auxeticity stacking uni-directional identical plies is considered. It is shown that if the ply is composed by isotropic matrix and fibers, then it is impossible to obtain totally auxetic orthotropic laminates, i.e. auxeticity for each direction, unless at least one among matrix and fibers is auxetic itself. Moreover, it is shown what are the conditions, in terms of the mechanical properties of the constituents and of the volume fraction of the fibers, to fabricate uni-directional plies with which to realize laminates having a negative Poisson's ratio for some directions. Several existing materials are also examined. All the analysis is done using the polar formalism, very effective for the study of plane anisotropic problems., Comment: 21 figures

The paper addresses the problem of finding the necessary and sufficient conditions to be satisfied by the engineering moduli of an anisotropic material for the elastic energy to be positive for each state of strain or stress. The problem is solved first in the most general case of a triclinic material and then each possible case of elastic syngony is treated as a special case. The method of analysis is based upon a rather forgotten theorem of linear algebra and, in the most general case, the calculations, too much involved, are carried out using a formal computation code. New, specific bounds, concerning some of the technical constants, are also found., Comment: Preprint

The complete set of bounds for the technical constants of an elastic layer, plate or laminate is given. The bounds are valid in general, also for completely anisotropic bodies. They are obtained transforming the polar bounds previously found. These bounds complete the knowledge of classical elasticity at least in the two-dimensional case and are useful in several situations, e.g., for determining the correct feasibility domain in design problems or as necessary conditions for accepting the results of laboratory tests on anisotropic layers., Comment: 11 pages, no figures, no tables

We consider in this paper the general properties of laminates designed to be isotropic in extension and in bending and with a coupling between the in- and out-of plane responses. In particular, we analyze the mathematical properties of the tensors describing the elastic and thermal behavior and the mechanical consequences of these properties. The differences, from the mathematical and mechanical point of view, between the hybrid laminates, i.e. composed by layers of different materials, and those made of identical plies, are pointed out and analyzed. The polar formalism for planar tensors is used in this study., Comment: 31 pages, 9 figures, 6 tables

The analysis of the mathematical and mechanical properties of thermoelastic coupling tensors in anisotropic laminates is the topic of this paper. Some theoretical results concerning the compliance tensors are shown and their mechanical consequences analyzed. Moreover, the case of thermally stable laminates, important for practical applications, is also considered. The study is carried out in the framework of the polar method, a mathematical formalism particularly well suited for the analysis of planar anisotropic problems, introduced by Prof. G. Verchery in 1979., Comment: 35 pages, 18 figures

This paper contains a set of theoretical results concerning the coupling tensor B of an anisotropic laminate and of its compliance corresponding B. The theoretical analysis and the mechanical results are obtained through an extensive use of the so-called polar formalism, introduced as early as 1979 by Prof. G. Verchery.

The paper concerns an experimental study on the wind pressures over the surface of a worldwide known Gothic Cathedral: Notre Dame of Paris. The experimental tests have been conducted in the CRIACIV wind tunnel, Prato (Italy), on a model of the Cathedral at the scale 1:200 reproducing the atmospheric boundary layer. Two types of tests have been conducted: with or without the surrounding modeling the part of the city of Paris near the Cathedral. This has been done, on the one hand, for evaluating the effect of the surrounding buildings onto the wind pressure distribution on the Cathedral, and, on the other hand, to have a wind pressure distribution plausible for any other Cathedral with a similar shape. The tests have been done for all the wind directions and the mean and peak pressures have been recorded. The results emphasize that the complex geometry of this type of structures is responsible for a peculiar aerodynamic behavior that does not allow estimating correctly the wind loads on the various parts of the Cathedral based on codes and standards, which are tailored for ordinary regular buildings.

This text is the support for the course of Modeling of Solids, of the Master of Mechanics of the University Paris-Saclay - Curriculum MMM: Mathematical Methods for Mechanics, held at Versailles. The course is the continuation of the course Continuum Mechanics - Solids, and as such it is an introduction, for graduate students, to some typical topics of the theory of solid bodies. The different arguments are dealt with in a simple, succinct way, the objective being to give to students the fundamentals of each argument. Only static problems are considered, being the dynamic of structures dealt with in other courses., Comment: 249 pages

This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text focuses on some subjects, with the intention of providing the reader with the main algebraic tools necessary for a modern course in continuum mechanics. The presentation of tensor algebra and analysis is intentionally done in Cartesian coordinates, that are normally used for classical problems. Then, an entire chapter is devoted to the passage to curvilinear coordinates and to the formalism of co- and contra-variant components. The tensor theory and results are specially applied to introduce some subjects concerning differential geometry of curves and surfaces. Also in this case, the presentation is mainly intended for applications to continuum mechanics., Comment: 168 pages, 7 chapters, more than 100 exercices, 49 figures

The timber roofing structure (charpente or combles, in French) of the cathedral Notre-Dame, destroyed by the fire of April 15th, 2019, is studied. The aim is twofold: on the one hand, it is interesting to evaluate the structural behavior of the original wooden structure in view of the reconstruction of the cathedral's roof. On the other hand, its structural analysis, never done before, can help to shed a light on the design process used by the masterbuilders of the XIIIth century, and to reconstruct, at least in part, the structural thought and knowledge of the ancient builders., Comment: Preprint; 34 pages, 18 figures, 13 tables

Machine Learning methods and, in particular, Artificial Neural Networks (ANNs) have demonstrated promising capabilities in material constitutive modeling. One of the main drawbacks of such approaches is the lack of a rigorous frame based on the laws of physics. This may render physically inconsistent the predictions of a trained network, which can be even dangerous for real applications. Here we propose a new class of data-driven, physics-based, neural networks for constitutive modeling of strain rate independent processes at the material point level, which we define as Thermodynamics-based Artificial Neural Networks (TANNs). The two basic principles of thermodynamics are encoded in the network's architecture by taking advantage of automatic differentiation to compute the numerical derivatives of a network with respect to its inputs. In this way, derivatives of the free-energy, the dissipation rate and their relation with the stress and internal state variables are hardwired in the network. Consequently, our network does not have to identify the underlying pattern of thermodynamic laws during training, reducing the need of large data-sets. Moreover the training is more efficient and robust, and the predictions more accurate. Finally and more important, the predictions remain thermodynamically consistent, even for unseen data. Based on these features, TANNs are a starting point for data-driven, physics-based constitutive modeling with neural networks. We demonstrate the wide applicability of TANNs for modeling elasto-plastic materials, with strain hardening and strain softening. Detailed comparisons show that the predictions of TANNs outperform those of standard ANNs. TANNs ' architecture is general, enabling applications to materials with different or more complex behavior, without any modification.