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This paper deals with a nonlinear problem modelling the contact between an elastic body and a rigid foundation. The elastic constitutive law is assumed to be nonlinear and the contact is modelled by the well-known Signorini conditions. Two weak formulations of the model are presented and existence and uniqueness results are established using classical arguments of elliptic variational inequalities. Some equivalence results are presented and a strong convergence result involving a penalized problem...
We present a new stabilized mixed finite element method for the linear elasticity problem in . The
approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and
equilibrium equations, and from the relation defining the rotation in terms of the displacement. We show that
the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that
the latter becomes locking-free and asymptotically locking-free for Dirichlet...
The equilibrium configurations of a one-dimensional variational model that
combines terms expressing the bulk energy of a deformable crystal and its
surface energy are studied. After elimination of the displacement, the
problem reduces to the minimization of a nonconvex and nonlocal functional of
a single function, the thickness. Depending on a parameter which strengthens
one of the terms comprising the energy at the expense of the other, it is
shown that this functional may have a stable absolute...
In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case....
In order to describe a solid which deforms smoothly in some region, but
non smoothly in some other region, many multiscale methods have recently
been proposed. They aim at coupling an atomistic model (discrete
mechanics) with a macroscopic model
(continuum mechanics).
We provide here a theoretical ground for such a coupling in a
one-dimensional setting. We briefly study the general case of a convex
energy, and next concentrate on
a specific example of a nonconvex energy, the Lennard-Jones case....
The quasicontinuum method is a coarse-graining technique for
reducing the complexity of atomistic simulations in a static and
quasistatic setting. In this paper we aim to give a detailed a
priori and a posteriori error analysis for a quasicontinuum
method in one dimension. We consider atomistic models with
Lennard–Jones type long-range interactions and a QC formulation
which incorporates several important aspects of practical QC
methods. First, we prove the existence, the local uniqueness...
We study a mathematical problem modelling the antiplane shear deformation of a viscoelastic body in frictional contact with a rigid foundation. The contact is bilateral and is modelled with a slip-dependent friction law. We present the classical formulation for the antiplane problem and write the corresponding variational formulation. Then we establish the existence of a unique weak solution to the model, by using the Banach fixed-point theorem and classical results for elliptic variational inequalities....
The authors study problems of existence and uniqueness of solutions of various variational formulations of the coupled problem of dynamical thermoelasticity and of the convergence of approximate solutions of these problems.
First, the semidiscrete approximate solutions is defined, which is obtained by time discretization of the original variational problem by Euler’s backward formula. Under certain smoothness assumptions on the date authors prove existence and uniqueness of the solution and establish...
We consider a quasistatic system involving a Volterra kernel modelling
an hereditarily-elastic aging body. We are concerned with the behavior of
displacement and stress fields in the neighborhood of cracks. In this paper, we
investigate the case of a straight crack in a two-dimensional domain with a possibly
anisotropic material law.
We study the asymptotics of the time dependent solution near the crack tips.
We prove that, depending on the regularity of the material
law and the Volterra kernel,...
The subject of topology optimization has undergone an enormous practical development since the appearance of the paper by Bendso e and Kikuchi (1988), where some ideas from homogenization theory were put into practice. Since then, several engineering applications as well as different approaches have been developed successfully. However, it is difficult to find in the literature some analytical examples that might be used as a test in order to assess the validity of the solutions obtained with different...
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