Oana Iosifescu

Variational limit of a one-dimensional discrete and statistically homogeneous system of material points

Abstract The energy of a discrete system of material points situated at random on a line and subjected to nearest-neighbor interactions, almost surely converges in a variational sense to a deterministic energy defined on spaces of functions with bounded variation.

LAGRANGE MULTIPLIERS IN INTRINSIC ELASTICITY

In an intrinsic approach to three-dimensional linearized elasticity, the unknown is the linearized strain tensor field (or equivalently the stress tensor field by means of the constitutive equation), instead of the displacement vector field in the classical approach. We consider here the pure traction problem and the pure displacement problem and we show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations (the form of which depends on the type of boundary conditions considered). Using the Babuska-Brezzi inf-sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints. Such results have potential applications to the numerical analysis and simulation of the intrinsic approach to three-dimensional linearized elasticity.

Nonlinear Donati compatibility conditions on a surface — Application to the intrinsic approach for Koiter’s model of a nonlinearly elastic shallow shell

An intrinsic approach to a mathematical model of a linearly or nonlinearly elastic body consists in considering the strain measures found in the energy of this model as the sole unknowns, instead of the displacement field in the classical approach. Such an approach thus provides a direct computation of the stresses by means of the constitutive equation. The main problem therefore consists in identifying specific compatibility conditions that these new unknowns, which are now matrix fields with components in L2, should satisfy in order that they correspond to an actual displacement field. Such compatibility conditions are either of Saint-Venant type, in which case they take the form of partial differential equations, or of Donati type, in which case they take the form of ortho- gonality relations against matrix fields that are divergence-free. The main objective of this paper consists in showing how an intrinsic approach can be successfully applied to the well-known Koiter’s model of a nonlinearly elastic shallow shell, thus providing the first instance (at least to the authors’ best knowledge) of a mathematical justification of this approach applied to a nonlinear shell model (“shallow” means that the absolute value of the Gaussian curvature of the middle surface of the shell is “uniformly small enough”). More specifically, we first identify and justify compatibility conditions of Donati type guaranteeing that the nonlinear strain measures found in Koiter’s model correspond to an actual displacement field. Second, we show that the associated intrinsic energy attains its minimum over a set of matrix fields that satisfy these Donati compatibility conditions, thus providing an existence theorem for the intrinsic approach; the proof relies in particular on an interesting per se nonlinear Korn inequality on a surface. Incidentally, this existence result (once converted into an equivalent existence theorem for the classical displacement approach) constitutes a significant improvement over previously known existence theorems for Koiter’s model of a nonlinearly elastic shallow shell.

Mathematical analysis of a spatially distributed soil carbon dynamics model

The aim of this paper is to study the mathematical properties of a new model of soil carbon dynamics which is a reaction–diffusion–advection system with a quadratic reaction term. This is a spatial version of Modeling Organic changes by Micro-Organisms of Soil model, recently introduced by M. Pansu and his group. We show here that for any nonnegative initial condition, there exists a unique nonnegative weak solution. Moreover, if we assume time periodicity of model entries, taking into account seasonal effects, we prove existence of a minimal and a maximal periodic weak solution. In a particular case, these two solutions coincide and they become a global attractor of any bounded solution of the periodic system.

A MATHEMATICAL MODEL FOR A PSEUDO-PLASTIC WELDING JOINT

An elementary situation in welding involves the perfect assembly of two adherents and a strong adhesive occupying a thin layer. The bulk energy density of the hyperelastic adherents grows superlinearly while that of the pseudo-plastic adhesive grows linearly with a stiffness of the order of the inverse of its thickness e. We propose a simplified but accurate model by studying the asymptotic behavior, when e goes to zero, through variational convergence methods: at the limit, the intermediate layer is replaced by a pseudo-plastic interface which allows cracks to appear.