G. Geymonat

Determination of the symmetries of an experimentally determined stiffness tensor : Application to acoustic measurements

Abstract For most materials, the symmetry group is known a priori and deduced from the realization process. This allows many simplifications for the measurements of the stiffness tensor. We deal here with the case where the symmetry is a priori unknown, as for biological or geological materials, or when the process makes the material symmetry axis uncertain (some composites, monocrystals). The measurements are then more complicated and the raw stiffness tensor obtained does not exhibit any symmetry in the Voigts matrical form, as it is expressed in the arbitrarily chosen specimens base. A complete ultrasonic measurement of this stiffness tensor from redundant measurements is proposed. In a second time, we show how to make a plane symmetry pole figure able to give visual information about the quasi-symmetries of a raw stiffness tensor determined by any measurement method. Finally we introduce the concept of distance from a raw stiffness tensor to one of the eight symmetry classes available for a stiffness tensor. The method provides the nearest tensor (to the raw stiffness tensor) possessing a chosen symmetry class, with its associated natural symmetry base.

Bifurcation and Localization in Rate-Independent Materials. Some General Considerations

This work deals with some aspects of bifurcation and localization phenomena for solids made of rate-independent materials. Only the theoretical developments are presented. Physical non-linearities (plasticity, damage, ...) and geometrical non-linearities are taken into account. The analysis is limited to quasi-static loadings. A full and complete analysis of the rate problem for incrementally linear solids is carried out. The first order rate problem is formulated and analysed in the framework of modern theory of linear elliptic boundary value problems. Three conditions are necessary and in the same time sufficient for this problem to be well-posed. These conditions are local in nature and are used to describe localization phenomena.

On the rigidity of certain surfaces with folds and applications to shell theory

In the asymptotic theory of thin elastic shells the rigidity of the mid-surface with kinematic boundary conditions plays an important role. Rigidity is understood in the sense of infinitesimal (linearized) rigidity, i.e., the displacements vanish provided the variation of the first fundamental form vanishes. In this case the surface is also called “stiff”, as it cannot undergo pure bendings. A stiff surface is imperfectly stiff or perfectly stiff when the origin respectively does or does not belong to the essential spectrum of the boundary-value problem. These questions are investigated in the framework of Douglis-Nirenberg elliptic systems, with boundary conditions and transmission conditions at the folds. The index properties ensures quasi-stiffness, i.e. stiffness up to a finite number of degrees of freedom. The concept of perfect stiffness is linked with estimates for the rigidity system at an appropriate level of regularity for the data and the solution. It is proved that surfaces with folds are never perfectly stiff. It is also shown that the transmission conditions at the folds contain more conditions than those satisfying the Shapiro-Lopatinskii property. This leads to certain rigidity properties of the folds. Some examples are given.

A two-field finite element formulation for elasticity coupled to damage

A two-field variational formulation is proposed for continuum damage mechanics problems. This formulation is applied to the numerical solution, via the finite element method, of initial boundary value problems arising in elasticity coupled to damage; the nodal variables are the displacement and damage fields. Simple examples are presented to demonstrate the advantages and limitations of this formulation.

Numerical analysis for eversion in elastic spherical caps equilibrium

Abstract The present work exhibits some qualitative and numerical results concerning the everted equilibrium shape of a thin elastic spherical cap.

The maximal thickness for everted equilibrium shapes of elastic spherical caps

This work is devoted to a qualitative and numerical study of the solution path of a nonlinear functional equation describing the everted equilibrium shapes of an elastic spherical cap.

Constructive Methods for Abstract Differential Equations and Applications

The Dirichlet problem in a conical domain for some elliptic equations can be reduced by separation of variables to a linear abstract differential equation of first or second order in a Hubert space H. We review some results giving the solution u as a superposition of exponential H-valued polynomials \( {e^{{ - {\lambda_k}t}}}{p_k}(t) \).