The rigorous derivation of linear elasticity from finite elasticity by means of $\Gamma$-convergence is a well-known result, which has been extended to different models also beyond the elastic regime. However, in these results the applied forces are usually assumed to be dead loads, that is, their density in the reference configuration is independent of the actual deformation. In this paper we begin a study of the variational derivation of linear elasticity in the presence of live loads. We consider a pure traction problem for a nonlinearly elastic body subject to a pressure live load and we compute its linearization for small pressure by $\Gamma$-convergence. We allow for a weakly coercive elastic energy density and we prove strong convergence of minimizers.

We first investigate spectral properties of the Neumann–Poincaré (NP) operator for the Lamé system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for the Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, it is polynomially compact and its spectrum on two-dimensional smooth domains consists of eigenvalues that accumulate to two different points determined by the Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on discs and ellipses. Using these resonances occurring at eigenvalues is considered. We also show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.

This paper derives a higher order hybrid stress finite element method on quadrilateral meshes for linear plane elasticity problems. The method employs continuous piecewise bi-quadratic functions in local coordinates to approximate the displacement vector and a piecewise-independent 15-parameter mode to approximate the stress tensor. Error estimation shows that the method is free from Poisson-locking and has second-order accuracy in the energy norm. Numerical experiments confirm the theoretical results.

We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains. The technique is based on a smooth coordinate transformation, which maps an unbounded domain into a unit square. Arbitrary geometries are defined by suitable level-set functions. The equations are discretized by classical nine-point stencil on interior points, while boundary conditions and high order reconstructions are used to define the field variables at ghost-points, which are grid nodes external to the domain with a neighbor inside the domain. The linear system arising from such discretization is solved by a multigrid strategy. The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources. The method is suitable to treat problems in which the geometry of the source often changes (explore the effects of different scenarios, or solve inverse problems in which the geometry itself is part of the unknown), since it does not require complex re-meshing when the geometry is modified. Several numerical tests are successfully performed, which asses the effectiveness of the present approach.

A phase field approach for structural topology optimization which allows for topologychanges and multiple materials is analyzed. First order optimality conditions arerigorously derived and it is shown via formally matched asymptoticexpansions that these conditions converge to classical first order conditions obtained inthe context of shape calculus. We also discuss how to deal with triple junctions wheree.g. two materials and the void meet. Finally, we present severalnumerical results for mean compliance problems and a cost involving the least square errorto a target displacement.

We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linearelasticity. We show that the UWVF of Navier’s equation can be derived as an upwinddiscontinuous Galerkin method. Using this observation, error estimates are investigatedapplying techniques from the theory of discontinuous Galerkin methods. In particular, wederive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and thenan error estimate in the L2(Ω) norm in terms of the bestapproximation error. Our final result is an L2(Ω) norm errorestimate using approximation properties of plane waves to give an estimate for the orderof convergence. Numerical examples are presented.

Dans le cadre d'une formation en mécanique, les travaux pratiques de mécanique des milieux déformables sont nécessaires aux étudiants afin qu'ils puissent mettre en pratique des notions abstraites tels que les tenseurs des contraintes et des déformations. Afin de mieux adapter ces séances de travaux pratiques aux attentes des étudiants actuels, nous avons choisi de décomposer ces séances en parties expérimentales et numériques. L'objectif de cet article est de présenter un exemple de travaux pratiques (compression d'un disque le long d'un diamètre) et de montrer les apports pédagogiques de la démarche adoptée pour l'étudiant.

In this note we give a result of convergence when time goes to infinity for aquasi static linear elastic model, the elastic tensor of which vanishes atinfinity. This method is applied to segmentation of medical images, and improvesthe 'elastic deformable template' model introduced previously.

This work is concerned with the equilibrium configurations of elastic structuresin contact with Coulomb friction. We obtain a variational formulation of thisequilibrium problem. Then we propose sufficient conditions for the existence ofan infinity of equilibrium configurations with arbitrary small frictioncoefficients. We illustrate the result in two space dimensions with asimple example.

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.

We consider the linearized elasticity system in a multidomain of ${\bf R}^3$ . This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius $r^{\varepsilon}$ . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and $r^{\varepsilon}$ tend to zero simultaneously, with $r^{\varepsilon}\gg\varepsilon^2$ , we identify the limit problem. This limit problem involves six junction conditions.

We present a new stabilized mixed finite element method for the linear elasticity problem in $\mathbb{R}^2$ . Theapproach is based on the introduction of Galerkin least-squares terms arising from the constitutive andequilibrium equations, and from the relation defining the rotation in terms of the displacement. We show thatthe resulting augmented variational formulation and the associated Galerkin scheme are well posed, and thatthe latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions,respectively. In particular, the discrete scheme allows the utilization of Raviart–Thomas spaces of lowestorder for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for therotation. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, whichyields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace isthen approximated by piecewise linear elements on an independent partition of the Neumann boundary whose meshsize needs to satisfy a compatibility condition with the mesh size associated to the triangulation of thedomain. Several numerical results illustrating the good performance of the augmented mixed finite elementscheme in the case of Dirichlet boundary conditions are also reported.

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.