Remarks about Signorini's problem in linear elasticity
Remarks on Carleman estimates and exact controllability of the Lamé system
In this paper we established the Carleman estimate for the two dimensional Lamé system with the zero Dirichlet boundary conditions. Using this estimate we proved the exact controllability result for the Lamé system with with a control locally distributed over a subdomain which satisfies to a certain type of nontrapping conditions.
Remarks on materials with memory
The mistaken notion that consistency with fading memory should require uniqueness is refuted by citation of the sources. Indeed, insistence on uniqueness would exclude many examples from non-linear elasticity and would exclude materials capable of exhibiting hysteresis.
Remarks on the theory of elasticity
In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the norm of the deformation gradient and a nonlinear function of the determinant of the gradient. Non–interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector in 1995. It applies, however, only if some -norm of the gradient with is controlled...
Remarks on -quasiconvexity, interpenetration of matter, and function spaces for elasticity
Remarks to weakly continuous inverse operators and an application in hyperelasticity
Representation of solutions of some boundary value problems of elasticity by a sum of the solutions of other boundary value problems.
Residual a posteriori error estimators for contact problems in elasticity
This paper is concerned with the unilateral contact problem in linear elasticity. We define two a posteriori error estimators of residual type to evaluate the accuracy of the mixed finite element approximation of the contact problem. Upper and lower bounds of the discretization error are proved for both estimators and several computations are performed to illustrate the theoretical results.
Resolution of a fixed point problem by an incremental method and application in nonlinear elasticity
Résumé d'une Conférence sur la théorie mathématique de l'élasticité, faite aux élèves de l'École Polytechnique (promotion de 1877-1879).
Scaling laws for non-euclidean plates and the isometric immersions of riemannian metrics
Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the proper...
Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics
Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the proper scaling....
Sensitivity analysis of a nonlinear obstacle plate problem
We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results of [9, 10] for the linear case, is done by application of an abstract variational result [6], where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that...
Sensitivity Analysis of a Nonlinear Obstacle Plate Problem
We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results of [9,10] for the linear case, is done by application of an abstract variational result [6], where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that...
Shape and topological sensitivity analysis in domains with cracks
The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural...
Shape optimization of elastic axisymmetric plate on an elastic foundation
An elastic simply supported axisymmetric plate of given volume, fixed on an elastic foundation, is considered. The design variable is taken to be the thickness of the plate. The thickness and its partial derivatives of the first order are bounded. The load consists of a concentrated force acting in the centre of the plate, forces concentrated on the circle, an axisymmetric load and the weight of the plate. The cost functional is the norm in the weighted Sobolev space of the deflection curve of radius....
Shape optimization of elasto-plastic axisymmetric bodies
A minimization of a cost functional with respect to a part of a boundary is considered for an elasto-plastic axisymmetric body obeying Hencky's law. The principle of Haar-Kármán and piecewise linear stress approximations are used to solve the state problem. A convergence result and the existence of an optimal boundary is proved.
Singular Perturbations For Heart Image Segmentation Tracking
In this note we give a result of convergence when time goes to infinity for a quasi static linear elastic model, the elastic tensor of which vanishes at infinity. This method is applied to segmentation of medical images, and improves the 'elastic deformable template' model introduced previously.
Singularity problems in linear elastodynamics
Small amplitude homogenization applied to models of non-periodic fibrous materials
In this paper, we compare a biomechanics empirical model of the heart fibrous structure to two models obtained by a non-periodic homogenization process. To this end, the two homogenized models are simplified using the small amplitude homogenization procedure of Tartar, both in conduction and in elasticity. A new small amplitude homogenization expansion formula for a mixture of anisotropic elastic materials is also derived and allows us to obtain a third simplified model.