Emilio Barchiesi

Energy approach to brittle fracture in strain-gradient modelling

In this paper, we exploit some results in the theory of irreversible phenomena to address the study of quasi-static brittle fracture propagation in a two-dimensional isotropic continuum. The elastic strain energy density of the body has been assumed to be geometrically nonlinear and to depend on the strain gradient. Such generalized continua often arise in the description of microstructured media. These materials possess an intrinsic length scale, which determines the size of internal boundary layers. In particular, the non-locality conferred by this internal length scale avoids the concentration of deformations, which is usually observed when dealing with local models and which leads to mesh dependency. A scalar Lagrangian damage field, ranging from zero to one, is introduced to describe the internal state of structural degradation of the material. Standard Lamé and second-gradient elastic coefficients are all assumed to decrease as damage increases and to be locally zero if the value attained by damage is one. This last situation is associated with crack formation and/or propagation. Numerical solutions of the model are provided in the case of an obliquely notched rectangular specimen subjected to monotonous tensile and shear loading tests, and brittle fracture propagation is discussed.

An Inverse Method to Get Further Analytical Solutions for a Class of Metamaterials Aimed to Validate Numerical Integrations

We consider an isotropic second gradient elastic two-dimensional solid. Besides, we relax the isotropic hypothesis and consider a D4 orthotropic material. The reason for this last choice is that such anisotropy is the most general for pantographic structures, which exhibit attracting mechanical properties. In this paper we analyze the role of the external body double force \(m^{ext}\) on the partial differential equations and we subsequently revisit some analytical solutions that have been considered in the literature for identification purposes. The revisited analytical solutions will be employed as well for identification purposes in a further contribution.

A Review on Models for the 3D Statics and 2D Dynamics of Pantographic Fabrics

A review on models for the statics of out-of-plane deformable pantographic fabrics is presented, along with a model describing the dynamics of in-plane-only deformable pantographic fabrics. We discuss those models able to describe the mechanical exotic properties conferred by the peculiar microstructure possessed by pantographic metamaterials, when three-dimensional deformations and in-plane dynamics are separately involved. For each approach, model formulation and modelling assumptions are discussed along with the presentation of numerical solutions in exemplary cases, and no attempt is made to model damage and failure phenomena.

Mechanical metamaterials: a state of the art

In this paper, we give a review of the state of the art in the study of mechanical metamaterials. The very attractive property of having a microstructure capable of determining exotic and specifically tailored macroscopic behaviour makes the study of metamaterials a field that is actually in expansion, from both a theoretical and a technological point of view. This work is divided into two sections, describing the phenomenological and theoretical aspects of metamaterials. We first give an overview of some existing metamaterials, such as pentamode materials, auxetic materials, materials with negative mechanical constitutive coefficients and materials with enhanced mechanical properties. We also focus on some emerging areas, such as origami. Then, we present some theoretical studies in the field of mechanical metamaterials, such as those related to first- and second-gradient theories.

Identification of Two-Dimensional Pantographic Structures with a Linear D4 Orthotropic Second Gradient Elastic Model Accounting for External Bulk Double Forces

The present paper deals with the identification of the nine constitutive parameters appearing in the strain energy density of a linear elastic second gradient D4 orthotropic two-dimensional continuum model accounting for an external bulk double force \(m^{ext}\). The aim is to specialize the model for the description of pantographic fabrics, which show such a kind of anisotropy. Analytical solutions for model problems, which are here referred to as the heavy sheet, the non-conventional bending and the trapezoidal cases are recalled from a previous paper and further elaborated in order to perform gedanken experiments. We completely characterize the set of nine constitutive parameters in terms of the materials the fibers are made of (i.e. of the Young’s modulus of the fiber materials), of their cross section (i.e. of the area and of the moment of inertia of the fiber cross sections), of the internal rotational spring positioned at each intersection point between the two families of fibers and of the pitch, i.e. the distance between adjacent pivots. Finally, the remarkable form of the strain energy, derived in terms of the displacement field, is shortly discussed.

A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identification and Numerical Results

In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed for the pantographic beam model. Phase transition and elastic softening are exhibited as well. Using the resulting planar 1D continuum limit homogenized macro-model, by means of FEM analyses, we show some equilibrium shapes exhibiting highly non-standard features. Finally, we conceive that pantographic beams may be used as basic elements in double scale metamaterials to be designed in future.

Introductory remarks about the Volume II of the Complete Works of Gabrio Piola

In this Volume II of the translations into English of the works by Gabrio Piola, we begin from the true first work written by the young Gabrio before 1824, when he was less than 30 years old. The content of the work Sull’applicazione de’ principj della meccanica analitica del Lagrange ai principali problemi. Memoria di Gabrio Piola presentata al concorso del premio e coronata dall’I.R. Istituto di Scienze, ecc. nella solennita del giorno 4 ottobre 1824, Milano, Imp. Regia stamperia, 1825, submitted to respond to a research program proposed by the I.R. Istituto on 4th October 1822, is in some aspects very modern and astonishingly topical.