Bernadette Miara

Bio: Bernadette Miara is an academic researcher from University of Paris. The author has contributed to research in topics: Homogenization (chemistry) & Shell (structure). The author has an hindex of 17, co-authored 61 publications receiving 1027 citations. Previous affiliations of Bernadette Miara include École Normale Supérieure & ESIEE.

Existence theorems for two-dimensional linear shell theories

Abstract: We consider linearly elastic shells whose middle surfaces have the most general geometries, and we provide complete proofs of the ellipticity of the strain energies found in two commonly used two-dimensional models: Koiter's model and Naghdi's model.

Multiscale Modeling of Elastic Waves: Theoretical Justification and Numerical Simulation of Band Gaps

Abstract: We consider a three-dimensional composite material made of small inclusions periodically embedded into an elastic matrix; the whole structure presents strong heterogeneities between its different components. In the general framework of linearized elasticity we show that, when the size of the microstructures tends to zero, the limit homogeneous structure presents, for some wavelengths, a negative “mass density” tensor. Hence we are able to rigorously justify the existence of forbidden bands, i.e., intervals of frequencies in which there is no propagation of elastic waves. In particular, we show how to compute these band gaps and illustrate the theoretical results with some numerical simulations.

Piezomaterials for bone regeneration design—homogenization approach ☆

Abstract: Relying on the piezoelectric properties of natural bone we propose a new biomaterial made of an inert perforated piezoelectric matrix filled with living osteoblast cells. We expect that this device will help the process of bone regeneration. In this paper we give some conceptual and numerical tools based on homogenization theory as a starting point in the design of such a “smart system”.

On modifications of Newton's second law and linear continuum elastodynamics

Abstract: In this paper, we suggest a new perspective, where Newton’s second law of motion is replaced by a more general law which is a better approximation for describing the motion of seemingly rigid macroscopic bodies. We confirm a finding of Willis that the density of a body at a given frequency of oscillation can be anisotropic. The relation between the force and the acceleration is non-local (but causal) in time. Conversely, for every response function satisfying these properties, and having the appropriate high-frequency limit, there is a model which realizes that response function. In many circumstances, the differences between Newton’s second law and the new law are small, but there are circumstances, such as in specially designed composite materials, where the difference is enormous. For bodies which are not seemingly rigid, the continuum equations of elastodynamics govern behaviour and also need to be modified. The modified versions of these equations presented here are a generalization of the equations proposed by Willis to describe elastodynamics in composite materials. It is argued that these new sets of equations may apply to all physical materials, not just composites. The Willis equations govern the behaviour of the average displacement field whereas one set of new equations governs the behaviour of the averageweighted displacement field, where the weighted displacement field may attach zero weight to ‘hidden’ areas in the material where the displacement may be unobservable or not defined. From knowledge of the average-weighted displacement field, one obtains an approximate formula for the ensemble averaged energy density. Two other sets of new equations govern the behaviour when the microstructure has microinertia, i.e. where there are internal spinning masses below the chosen scale of continuum modelling. In the first set, the average displacement field is assumed to be observable, while in the second set an average-weighted displacement field is assumed to be observable.

The Periodic Unfolding Method in Homogenization

Abstract: The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104] (with the basic proofs in [Proceedings of the Narvi...

An Introduction to Differential Geometry with Applications to Elasticity

Abstract: Preface Chapter 1. Three-dimensional differential geometry: 1.1. Curvilinear coordinates, 1.2. Metric tensor, 1.3. Volume, areas, and lengths in curvilinear coordinates, 1.4. Covariant derivatives of a vector field, 1.5. Necessary conditions satisfied by the metric tensor the Riemann curvature tensor, 1.6. Existence of an immersion defined on an open set in R3 with a prescribed metric tensor, 1.7. Uniqueness up to isometries of immersions with the same metric tensor, 1.8. Continuity of an immersion as a function of its metric tensor Chapter 2. Differential geometry of surfaces: 2.1. Curvilinear coordinates on a surface, 2.2. First fundamental form, 2.3. Areas and lengths on a surface, 2.4. Second fundamental form curvature on a surface, 2.5. Principal curvatures Gaussian curvature, 2.6. Covariant derivatives of a vector field defined on a surface the Gauss and Weingarten formulas, 2.7. Necessary conditions satisfied by the first and second fundamental forms: the Gauss and Codazzi-Mainardi equations Gauss' theorema egregium, 2.8. Existence of a surface with prescribed first and second fundamental forms, 2.9. Uniqueness up to proper isometries of surfaces with the same fundamental forms, 2.10.Continuity of a surface as a function of its fundamental forms Chapter 3. Applications to three-dimensional elasticity in curvilinear coordinates: 3.1. The equations of nonlinear elasticity in Cartesian coordinates, 3.2. Principle of virtual work in curvilinear coordinates, 3.3. Equations of equilibrium in curvilinear coordinates covariant derivatives of a tensor field, 3.4. Constitutive equation in curvilinear coordinates, 3.5. The equations of nonlinear elasticity in curvilinear coordinates, 3.6. The equations of linearized elasticity in curvilinear coordinates, 3.7. A fundamental lemma of J.L. Lions, 3.8. Korn's inequalities in curvilinear coordinates, 3.9. Existence and uniqueness theorems in linearizedelasticity in curvilinear coordinates Chapter 4. Applications to shell theory: 4.1. The nonlinear Koiter shell equations, 4.2. The linear Koiter shell equations, 4.3. Korn's inequality on a surface, 4.4. Existence and uniqueness theorems for the linear Koiter shell equations covariant derivatives of a tensor field defined on a surface, 4.5. A brief review of linear shell theories References Index.