Lorenzo Freddi

Thin notched beams

The three-dimensional elasticity energy for a notched rod under axial loads is shown to converge in a variational sense to a class of one-dimensional models with a localized compliance when the beam becomes very slender and when the ratio between the depth and the width of each notch vanishes in a suitable way. Also the case of notched thermal conductors is considered, in a parallel treatment, because it falls in a similar but simpler framework.

The energy density of martensitic thin films via dimension reduction

Martensitic thin films have recently attracted much interest because of their applications in the construction of actuators, sensors and micromachines. These technological applications have generated a new interest in nonlinear membranes theories. The purpose of the present paper is to obtain, by means of a rigorous dimension reduction, the energy density of a thin film, and hence to deduce a membrane model capable of describing the microstructure observed in martensitic materials. Phase transformations have been successfully modelled within the theory of non-linear elasticity by Ball and James [5]. Within this approach, equilibrium configurations are found by minimizing the total free energy of the body. In the spirit of the works on dimension reduction (see for instance [1, 3]) we start by considering the total free energy

Dimension reduction of a crack evolution problem in a linearly elastic plate

A two-dimensional model which describes the evolution of a crack in a plate is deduced from a three-dimensional lin- early elastic Griffiths type model. The result is achieved by adopting the framework of energetic solutions for rate-independent processes, to model three-dimensional fracture evolution, in conjunction with a variational dimension reduction procedure.

ANISOTROPIC INHOMOGENEOUS RECTANGULAR THIN-WALLED BEAMS ∗

This paper is devoted to the asymptotic analysis of the problem of linear elasticity for an anisotropic and inhomogeneous body occupying, in its reference configuration, a cylindrical domain with a rectangular cross section with sides proportional to

Nonlinear thin-walled beams with a rectangular cross-section - Part II

\varepsilon

A variational model for anisotropic and naturally twisted ribbons

and

Homogenization of optimal control problems for functional differential equations

\varepsilon^2

Relaxed optimal control problems and applications to shape optimization

and clamped on one of its bases. The sequence of solutions

Saint-Venant's theory for beams with multi-connected cross-section: Justification and error estimate

u^\varepsilon

Residually stressed beams

of the equilibrium problem is shown to converge in an appropriate topology, as