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 $\overrightarrow{\phi }{|}_{\partial \Omega }\ne \overrightarrow{\mathrm{id}}$.

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...

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...

The Cauchy–Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy–Born rule for boundary deformations which are close to rigid motions. This generalizes results of Friesecke...

Quando si pone in un ambiente idrostatico un corpo elastico contenente una certa quantità di gas in una sua cavità, il problema di equilibrio che ne risulta presenta condizioni al bordo di tipo non locale. In questa Nota, la prima di una serie, consideriamo il caso di un contenitore riempito di un gas perfetto e soggetto ad una pressione esterna fissata, ed otteniamo certe identità di media degli sforzi che valgono per tutte le soluzioni del problema di equilibrio (in particolare, per tutte le soluzioni...

In questa Nota, che è il seguito della Nota precedente, proseguiamo lo studio del problema di equilibrio di un involucro elastico pieno di gas, considerando piccole perturbazioni di una configurazione di equilibrio, prodotte da insufflazioni o desufflazioni di piccole quantità di gas. Limitando la nostra attenzione al caso che l'involucro sia una corona sferica che subisce solo deformazioni radiali, mostriamo che, per un valore critico dello spessore della corona, il carattere non locale delle condizioni...

In the present work an extension of a classical Menabrea’s theorem on a variational principle of the second potential energy is considered. Such extension deals with hyperelastic micropolar media without constraints.