Cesare Davini

A complete list of invariants for defective crystals

The classical theory of continuous distributions of dislocations has traditionally focused on the Burgers’ vectors and the dislocation density tensor as descriptions of defectiveness. We prove that, generally, there is an infinite number of tensor densities with similarly descriptive properties, and that there is a functional basis for this list which strictly includes the Burgers’ vectors and dislocation density. Moreover the changes of state which preserve these densities turn out to represent slip in certain surfaces associated with crystal geometry, so that the basic mechanism of plasticity emerges naturally from abstract ideas which neither anticipate nor involve the kinematics of particular types of crystal defects.

On defect-preserving deformations in crystals

Abstract In the context of a continuum theory of crystals with defects, one can define a particular list of invariants so that elements of this list do not change when the crystal is deformed elastically. Here we characterize (most of) the deformations which leave these elements unchanged and find that these “defect-preserving” deformations.strictly include the elastic deformations. This characterization allows a judgement of the completeness of the list of invariants. Furthermore it turns out that the defect-preserving deformations which are not elastic generally involve some kind of rearrangement, or slip, of the crystal lattice, and that if one admits such deformations in variational principles determining equilibria of the lattice, then the crystal is necessarily weak, in some sense; for example, the crystal may be able to equilibrate only simple pressure over its boundary.

An Unconstrained Mixed Method for the Biharmonic Problem

In this work we present a finite element method for the biharmonic problem based on the primal mixed formulation of Ciarlet and Raviart [A mixed finite element method for the biharmonic equation, in Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed., Academic Press, New York, 1974, pp. 125--143]. We introduce a dual mesh and a suitable approximation of the constraint that enables us to eliminate the auxiliary variable with no computational effort. Thus, the discrete problem turns to be governed by a system of linear equations with symmetric and positive definite coefficients and can be solved by classical algorithms. The construction of the stiffness matrix is obtained by using Courant triangles and can be done with great efficiency.

Relaxed Notions of Curvature and a Lumped Strain Method for Elastic Plates

The paper proposes an approximation method, lumped strain method (LSM), for elastic plates based on generalized notions of the Gaussian and mean curvatures and on a relaxation of the energy functional to the space of continuous piecewise linear functions. The method adapts ideas from the theory of

Some remarks on the continuum theory of defects in solids

\Gamma

Generalized Hessian and External Approximations in Variational Problems of Second Order

-convergence. We restrict our attention to the case of the simply supported plate and state the proof of convergence of the method. We give an application to the rhombic plate and compare the results with those obtained by standard finite element approximations.

The Gaussian stiffness of graphene deduced from a continuum model based on Molecular Dynamics potentials

A succinct account of the classical continuum theory of defects is given and some critical remarks on its relevance for the mechanics of the model are discussed.

A geometric classification of inhomogeneities in continua with dislocations

We introduce a suitable notion of generalized Hessian and show that it can be used to construct approximations by means of piecewise linear functions to the solutions of variational problems of second order. An important guideline of our argument is taken from the theory of the Γ-convergence. The convergence of the method is proved for integral functionals whose integrand is convex in the Hessian and satisfies standard growth conditions.

A new material property of graphene: The bending Poisson coefficient

We consider a discrete model of a graphene sheet with atomic interactions governed by a harmonic approximation of the 2nd-generation Brenner potential that depends on bond lengths, bond angles, and two types of dihedral angles. A continuum limit is then deduced that fully describes the bending behavior. In particular, we deduce for the first time an analytical expression of the Gaussian stiffness, a scarcely investigated parameter ruling the rippling of graphene, for which contradictory values have been proposed in the literature. We disclose the atomic-scale sources of both bending and Gaussian stiffnesses and provide for them quantitative evaluations.

Homogenization of linearly elastic honeycombs

SommarioIl lavoro propone una classificazione geometrica delle disomogeneità, nel senso di Noll, in mezzi con distribuzioni continue di dislocazioni, partendo da una nozione di equivalenza associata con cambiamenti omogenei del riferimento uniforme.SummaryThe paper proposes a geometric classification of inhomogeneities, in the sense of Noll, in bodies with continuous distribution of dislocations, based on the equivalence notion associated with homogeneous changes of the uniform reference.