Margherita Solci

A derivation of linear elastic energies from pair-interaction atomistic systems

Pair-interaction atomistic energies may give rise, in the framework of the passage from discrete systems to continuous variational problems, to nonlinear energies with genuinely quasiconvex integrands. This phenomenon takes place even for simple harmonic interactions as shown by an example by Friesecke and Theil [19]. On the other hand, a rigorous derivation of linearly elastic energies from energies with quasiconvex integrands can be obtained by

Interaction of a Bulk and a Surface Energy with a Geometrical Constraint

\Gamma

The Sardinia Radio Telescope: A comparison between close-range photogrammetry and finite element models:

-convergence following the method by Dal Maso, Negri and Percivale [14]. We show that the derivation of linear theories by

Asymptotic analysis of Lennard-Jones systems beyond the nearest-neighbour setting: A one-dimensional prototypical case

\Gamma

Motion of Discrete Interfaces Through Mushy Layers

-convergence can be obtained directly from lattice interactions in the regime of small deformations. Our proof relies on a lower bound by comparison with the continuous result, and on a direct Taylor expansion for the upper bound. The computation is carried over for a family of lattice energies comprising interactions on the triangular lattice in dimension two.

A homogenization result for interacting elastic and brittle media

This study is an attempt to generalize in dimension higher than two the mathematical results in [E. Bonnetier and A. Chambolle, SIAM J. Appl. Math., 62 (2002), pp. 1093–1121]. It is the study of a physical system whose equilibrium is the result of a competition between an elastic energy inside a domain and a surface tension, proportional to the perimeter of the domain. The domain is constrained to remain a subgraph. It is shown by Bonnetier and Chambolle that several phenomena appear at various scales as a result of this competition. In this paper, we focus on establishing a sound mathematical framework for this problem in a higher dimension. We also provide an approximation, based on a phase‐field representation of the domain.

A RELAXATION RESULT FOR ENERGIES DEFINED ON PAIRS SET-FUNCTION AND APPLICATIONS

The Sardinia Radio Telescope (SRT), located near Cagliari (Italy), is the world’s second largest fully steerable radio telescope endowed with an active-surface system. Its primary mirror has a quasi-parabolic shape with a diameter of 64 m. The configuration of the primary mirror surface can be modified by means of electro-mechanical actuators. This capability ensures, within a fixed range, the balancing of the deformation caused, for example, by loads such as self-weight, thermal effects and wind pressure. In this way, the difference between the ideal shape of the mirror (which maximizes its performances) and the actual surface can be reduced. In this paper the authors describe the characteristics of the SRT, the close-range photogrammetry (CRP) survey developed in order to set up the actuator displacements, and a finite element model capable of accurately estimating the structural deformations. Numerical results are compared with CRP measurements in order to test the accuracy of the model.

INTERFACIAL ENERGIES ON PENROSE LATTICES

We consider a one-dimensional system of Lennard-Jones nearest- and next-to-nearest-neighbour interactions. It is known that if a monotone parameterization is assumed then the limit of such a system can be interpreted as a Griffith fracture energy with an increasing condition on the jumps. In view of possible applications to a higher-dimensional setting, where an analogous parameterization does not always seem reasonable, we remove the monotonicity assumption and describe the limit as a Griffith fracture energy where the increasing condition on the jumps is removed and is substituted by an energy that accounts for changes in orientation (‘creases’). In addition, fracture may be generated by ‘macroscopic’ or ‘microscopic’ cracks.

Double-porosity homogenization for perimeter functionals

We study the geometric motion of sets in the plane derived from the homogenization of discrete ferromagnetic energies with weak inclusions. We show that the discrete sets are composed by a ‘bulky’ part and an external ‘mushy region’ composed only of weak inclusions. The relevant motion is that of the bulky part, which asymptotically obeys to a motion by crystalline mean curvature with a forcing term, due to the energetic contribution of the mushy layers, and pinning effects, due to discreteness. From an analytical standpoint, it is interesting to note that the presence of the mushy layers implies only a weak and not strong convergence of the discrete motions, so that the convergence of the energies does not commute with the evolution. From a mechanical standpoint it is interesting to note the geometrical similarity of some phenomena in the cooling of binary melts.

Interfacial energies on quasicrystals

We consider energies modelling the interaction of two media parameterized by the same reference set, such as those used to study interactions of a thin film with a stiff substrate, hybrid laminates or skeletal muscles. Analytically, these energies consist of a (possibly non-convex) functional of hyperelastic type and a second functional of the same type such as those used in variational theories of brittle fracture, paired by an interaction term governing the strength of the interaction depending on a small parameter. The overall behaviour is described by letting this parameter tend to zero and exhibiting a limit effective energy using the terminology of Gamma-convergence. Such energy depends on a single state variable and is of hyperelastic type. The form of its energy function highlights an optimization between microfracture and microscopic oscillations of the strain, mixing homogenization and high-contrast effects.