Si considerano due spazi $S_{\mu }$ e $S_{\nu }^{\ast }$, Riemanniani e a metrica eventualmente indefinita, riferiti a sistemi di co-ordinate $\mathrm{\varnothing }$ e ${\mathrm{\varnothing }}_{\nu }^{\ast }$; e inoltre un doppio tensore $T_{\mathrm{\cdots }}^{\mathrm{\cdots }}$ associato ai punti ${\mathrm{\varnothing }}^{-1}(x)\in S_{\mu }$ e ${\mathrm{\varnothing }}^{\ast -1}(y)\in S^{\ast }$. Si pensa $T_{\mathrm{\cdots }}^{\mathrm{\cdots }}$ dato da una funzione ${\tilde{T}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}$ di $m$ altri tali doppi tensori e di variabili puntuali $x(\in {\mathrm{\Re }}^{\mu })$, $t\in \mathrm{\Re }$ e $y(\in {\mathrm{\Re }}^{\nu })$; poi si considera la funzione composta $${\widehat{T}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}(x,t,y)={\tilde{T}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}[\underset{1,\mathrm{\dots },m}{\underbrace{{\breve{H}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}(x,t,y),\mathrm{\dots },{\breve{H}}_{\mathrm{\cdots }}^{\mathrm{\cdots }}(x,t,y)}},x,t,y].$$ Nella Parte I si scrivono due regole per eseguire la derivazione totale di questa, connessa con una mappa $\widehat{ℰ}$

As a measure of deformation we can take the difference $D\overrightarrow{\phi }-R$, where $D\overrightarrow{\phi }$ is the deformation gradient of the mapping $\overrightarrow{\phi }$ and $R$ is the deformation gradient of the mapping $\overrightarrow{\gamma }$, which represents some proper rigid motion. In this article, the norm $\parallel D\overrightarrow{\phi }{-R\parallel }_{L^p\left(\Omega \right)}$ is estimated by means of the scalar measure $e\left(\overrightarrow{\phi }\right)$ of nonlinear strain. First, the estimates are given for a deformation $\overrightarrow{\phi }\in W^{1,p}\left(\Omega \right)$ satisfying the condition $\overrightarrow{\phi }{|}_{\partial \Omega }=\overrightarrow{\mathrm{id}}$. Then we deduce the estimate in the case that $\overrightarrow{\phi }\left(x\right)$ is a bi-Lipschitzian deformation and $\overrightarrow{\phi }{|}_{\partial \Omega }\ne \overrightarrow{\mathrm{id}}$.

This paper deals with free-energy lower-potentials for some rate-independent one-dimensional models of isothermal finite elastoplasticity proposed in [1]. Extending the thermodynamic arguments of Coleman and Owen [3] to large deformations, the existence, non-uniqueness and regularity of free-energy as function of state are deduced rather than assumed. This approach, along with some optimal control techniques, enables us to construct maximum and minimum free-energy functions and a wide class of differentiable...

The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic...

A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem...

A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem...

Dans ce travail, nous étudions la propriété de base de Riesz et la stabilisation exponentielle pour une équation des poutres d’Euler-Bernoulli à coefficients variables sous un contrôle frontière linéaire dépendant de la position (resp. l’angle de rotation), de la vitesse et de la vitesse de rotation dans le contrôle force (resp. moment). Nous montrons qu’il existe une suite de fonctions propres généralisées qui forme une base de Riesz de l’espace d’énergie considéré, et qu’il y a stabilité exponentielle...

We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter $\epsilon \>0$ and study its asymptotic behavior for $t$ large, as $\epsilon \to 0$. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter $\epsilon$. In order for this to be true the damping mechanism has to have the appropriate scale with respect to $\epsilon$. In the limit as $\epsilon \to 0$ we obtain damped Berger–Timoshenko beam models...

We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter ε > 0 and study its asymptotic behavior for t large, as ε → 0. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter ε. In order for this to be true the damping mechanism has to have the appropriate scale with respect to ε. In the limit as ε → 0 we obtain damped Berger–Timoshenko...

This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients λ, µin three dimensions, $\mathrm{div}\left(\mu (\nabla u+\nabla u^t)\right)+\nabla \left(\lambda \mathrm{div}u\right)+Vu=0$ whereλ and μ are Lipschitz continuous and V∈L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.