One-dimensional von K\'arm\'an models for elastic ribbons

By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von K\'arm\'an plates, passing to the limit as the width of the plate goes to zero. The one-dimensional model found starting from the "linearized" von K\'arm\'an energy corresponds to that of a linearly elastic beam that can twist but can deform in just one plane; while the model found from the von K\'arm\'an energy is a non-linear model that comprises stretching, bendings, and twisting. The "constrained" von K\'arm\'an energy, instead, leads to a new Sadowsky type of model

A variational model for anisotropic and naturally twisted ribbons

We consider thin plates whose energy density is a quadratic function of the difference between the second fundamental form of the deformed configuration and a "natural" curvature tensor. This tensor either denotes the second fundamental form of the stress-free configuration, if it exists, or a target curvature tensor. In the latter case, residual stress arises from the geometrical frustration involved in the attempt to achieve the target curvature: as a result, the plate is naturally twisted, even in the absence of external forces or prescribed boundary conditions. Here, starting from this kind of plate energies, we derive a new variational one-dimensional model for naturally twisted ribbons by means of Gamma-convergence. Our result generalizes, and corrects, the classical Sadowsky energy to geometrically frustrated anisotropic ribbons with a narrow, possibly curved, reference configuration

No-tension bodies: A reinforcement problem

In this work we show that the framework put forward by Lucchesi et al. (2006) to study the equilibrium configurations of panels made of no-tension material can be easily extended to the case of a no-tension material with a reinforcing tensile resistant unidimensional material. This kind of bodies could be used to describe reinforced concrete structures. By solving the equilibrium equations we find a family of solutions each of which is characterized by a singular curve where the stress in the no-tension material concentrates. We show that among these, the curve that minimizes the maximum stress resembles the line tension found experimentally on reinforced concrete beams.© 2012 Elsevier Masson SAS. All rights reserved

A Corrected Sadowsky Functional for Inextensible Elastic Ribbons

The classical theory of ribbons, developed by Sadowsky and Wunderlich, has recently received renewed attention. Here, by means of Gamma-convergence, we re-examine the derivation of the limit energy of an inextensible, isotropic, elastic strip as the width goes to zero. We find that this rigorously derived functional agrees with the classical Sadowsky functional only for "large" curvature of the centerline of the strip

Energy Based Global-Local Strategies with Adaptive Mesh Refinement for the Phase-Field Approach to Brittle Fracture

This paper analyses different discretization procedures and compares their numerical performances in the solution of phase field approach to fracture problem. A predictor energetic principle is employed to determine the active regions where damage evolves and, by the usage of a global/local strategy, mesh adaptive refinement or a combination of the two techniques, smaller displacement and damage problems are solved. The computational costs of the simulations are therefore drastically reduced without lowering the accuracy of the results. Initially, the effectiveness and accuracy of the different strategies are analysed and compared. After, the effects of the active zones on the performance and precision of the results is investigated via a parametric analysis. Two different numerical examples are presented in order to validate and show the efficiency of the proposed optimization strategies in lowering the computational costs and CPU times required to perform the numerical simulations

Residually stressed beams

In this paper we derive a theory for a linearly elastic residually stressed rod through an asymptotic analysis based on Γ-convergence

Quasistatic delamination of sandwich-like Kirchhoff-Love plates

A quasistatic rate-independent adhesive delamination problem of laminated plates with a finite thickness is considered. By letting the thickness of the plates go to zero, a rate-independent delamination model for a laminated Kirchhoff-Love plate is obtained as limit of these quasistatic processes. The same dimension reduction procedure is eventually applied to processes which are sensitive to delamination modes, namely opening vs. shearing is distinguishe

Optimal control of the transmission rate in compartmental epidemics

We introduce a general system of ordinary differential equations that includes some classical and recent models for the epidemic spread in a closed population without vital dynamic in a finite time horizon. The model is vectorial, in the sense that it accounts for a vector valued state function whose components represent various kinds of exposed/infected subpopulations, with a corresponding vector of control functions possibly different for any subpopulation. In the general setting, we prove well-posedness and positivity of the initial value problem for the system of state equations and the existence of solutions to the optimal control problem of the coefficients of the nonlinear part of the system, under a very general cost functional. We also prove the uniqueness of the optimal solution for a small time horizon when the cost is superlinear in all control variables with possibly different exponents in the interval (1,2]. We consider then a linear cost in the control variables and study the singular arcs. Full details are given in the case n=1 and the results are illustrated by the aid of some numerical simulations.Comment: Accepted by Mathematical Control and Related Field

Optimal Control Problems with Weakly Converging Input Operators in a Nonreflexive Framework

The variational convergence of sequences of optimal control problems with state constraints (namely inclusions or equations) with weakly converging input multi-valued operators is studied in a nonre\ub0exive abstract framework, using \ua1-conver- gence techniques. This allows to treat a lot of situations where a lack of coercivity forces to enlarge the space of states where the limit problem has to be imbedded. Some concrete applications to optimal control problems with measures as controls are given either in a nonlinear multi-valued or nonlocal but single-valued framework