Giuseppe Geymonat

Mathematical Analysis of a Bonded Joint with a Soft Thin Adhesive

This paper considers the problem of two adherents joined by a soft thin adhesive along their common surface. Using the asymptotic expansion method, the authors obtain a simplified model in which the adhesive is treated as a material surface and is replaced by returning springs. The authors show weak and strong convergence of the exact solution toward the solution of the limit problem. The singularities of the limit problem are analyzed, and it is shown that typically they are logarithmic. Furthermore, the authors investigate the phenomenon of boundary layer by studying the correctors, the extra terms, which must be added to the classical asymptotic expansion to verify the boundary conditions. The correctors show that, contrary to the adherents, in the adhesive there are power-type singularities, which are at the base of the failure of the assemblage.

Identification of mechanical properties by displacement field measurement: a variational approach

We study a parameter identification problem associated with a two-dimensional mechanical problem. In the first part, the experimental technique of determining the displacement field is briefly presented. The variational method proposed herein is based on the minimization of either a separately convex functional or a convex functional that leads to the reconstruction of the elastic tensor and the stress field. These two reconstructed fields are continuous and piecewise linear on a triangulation of the two-dimensional domain. Some numerical and experimental examples are presented to test the performance of the algorithms.

A NEW DUALITY APPROACH TO ELASTICITY

The displacement-traction problem of three-dimensional linearized elasticity can be posed as three different minimization problems, depending on whether the displacement vector field, or the stress tensor field, or the strain tensor field, is the unknown. The objective of this paper is to put these three different formulations of the same problem in a new perspective, by means of Legendre-Fenchel duality theory. More specifically, we show that both the displacement and strain formulations can be viewed as Legendre-Fenchel dual problems to the stress formulation. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new duality approach to elasticity.

A DOMAIN DECOMPOSITION METHOD FOR A BONDED STRUCTURE

We show that the transmission conditions through two elastic bodies bonded by a thin adhesive layer can be written as Robin-type conditions, well-suited for using a domain decomposition algorithm for which we prove convergence. Numerical approximation by means of finite element methods is also presented and analyzed. Convergence of the discrete algorithm is proven as well as optimal error estimates.

Modeling of smart materials with thermal effects: Dynamic and quasi-static evolution

We present a mathematical model for linear magneto-electro-thermo-elastic continua, as sensors and actuators can be thought of, and prove the well-posedness of the dynamic and quasi-static problems. The two proofs are accomplished, respectively, by means of the Hille–Yosida theory and of the Faedo–Galerkin method. A validation of the quasi-static hypothesis is provided by a nondimensionalization of the dynamic problem equations. We also hint at the study of the convergence of the solution to the dynamic problem to that to the quasi-static problem as a small parameter — the ratio of the largest propagation speed for an elastic wave in the body to the speed of light — tends to zero.

On traces of functions in W2,p(Ω) for Lipschitz domains in R3

Abstract We consider the problem of the characterization of the range of the trace operator (γ,γ 1 ): W 2,p (Ω)→R , p∈ ]1,∞[ , defined by the mapping u↦(u |Γ ,∂ n u) , when Ω is a Lipschitz bounded subset of R 3 . R turns out to be a subspace of W 1,p (Γ)× L p (Γ) . To this aim we need to prove a suitable Hodge decomposition for vector fields belonging to L p ( curl ,Ω) , and also to study some properties of the tangential gradient ∇ Γ on a Lipschitz orientable manifold.

Matched asymptotic expansion method for an homogenized interface model

Our aim is to demonstrate the effectiveness of the matched asymptotic expansion method in obtaining a simpli ed model for the influence of small identical heterogeneities periodically distributed on an internal surface on the overall response of a linearly elastic body. The results of some numerical experiments corroborate the precise identi cation of the di fferent steps, in particular of the outer/inner regions with their normalized coordinate systems and the scale separation, leading to the model.

An asymptotic plate model for magneto-electro-thermo-elastic sensors and actuators

We present an asymptotic two-dimensional plate model for linear magneto-electro-thermo-elastic sensors and actuators, under the hypotheses of anisotropy and homogeneity. Four different boundary conditions pertaining to electromagnetic quantities are considered, leading to four different models: the sensor–actuator model, the actuator–sensor model, the actuator model and the sensor model. We validate the obtained two-dimensional models by proving weak convergence results. Each of the four plate problems turns out to be decoupled into a flexural problem, involving the transversal displacement of the plate, and a certain partially or totally coupled membrane problem.

Piezomagnetic tensors symmetries: an unifying tentative approach

Traditionally, one inquires as to what is the form of a physical property tensor invariant under the point group of a crystal. For example, one finds listings showing 16 distinct forms of the piezomagnetic tensor that can arise in a crystal. Alongside each of these 16 distinct forms is listed the point groups which give rise to that specific form (see Birss (1966) for example). This is a classification of the form of piezomagnetic tensors in crystals. We claim however that this classification is quite ambiguous in the sense that it hides the very difference between microscopic and macroscopic symmetries. The topic of the paper is the both a clarification of the difference between material and behaviour symmetries and a classification of piezomagnetic tensors in behaviour equivalences classes, found to be 15.

Mathematical and numerical modeling of plate dynamics with rotational inertia

Abstract We give a presentation of the mathematical and numerical treatment of plate dynamics problems including rotational inertia. The presence of rotational inertia in the equation of motion makes the study of such problems interesting. We employ HCT finite elements for space discretization and the Newmark method for time discretization in FreeFEM++, and test such methods in some significant cases: a circular plate clamped all over its lateral surface, a rectangular plate simply supported all over its lateral surface, and an L-shaped clamped plate.