We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions $$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right),u^{\text{'}\text{'}\text{'}}\left(t\right)\right),\text{a.e.}t\in [0,1],\hfill \\ u\left(0\right)=a,u^{\text{'}}\left(0\right)=b,u\left(1\right)=c,u^{\text{'}\text{'}}\left(1\right)=d,\hfill \end{array}\right.$$ where the nonlinear term $f(t,u_0,u_1,u_2,u_3)$ is a strong Carathéodory function. By constructing suitable height functions of the nonlinear term $f(t,u_0,u_1,u_2,u_3)$ on bounded sets and applying the Leray-Schauder fixed point theorem, we prove that the equation has a solution provided that the integration of some height function has an appropriate value.

Solvability of the rational contact with limited interpenetration of different kind of viscolastic plates is proved. The biharmonic plates, von Kármán plates, Reissner-Mindlin plates, and full von Kármán systems are treated. The viscoelasticity can have the classical (``short memory'') form or the form of a certain singular memory. For all models some convergence of the solutions to the solutions of the Signorini contact is proved provided the thickness of the interpenetration tends to zero.

We present a general numerical method for calculating effective elastic properties of periodic structures based on the homogenization method. Some concrete numerical examples are presented.

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...

On se propose d’étudier la stabilité d’une poutre flexible homogène, encastrée à une extrémité. À l’autre extrémité est attachée une masse ponctuelle où on applique un moment proportionnel à la vitesse de déplacement angulaire. On montre par une analyse spectrale que le taux optimal de décroissance de l’énergie est déterminé par l’abscisse spectrale du générateur infinitésimal du semi-groupe associé au problème.

We study the stability of a flexible beam clamped at one end. A mass is attached at the other end, where a control moment is applied. The boundary control is proportional to the angular velocity at the end. By spectral analysis, we prove that the optimal decay rate of the energy is given by the spectrum of the generator of the semigroup associated to the system.

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...

Dans ce travail, nous étudions une équation des poutres d’Euler-Bernoulli, on contrôle par combinaison linéaire de vitesse et vitesse de rotation appliquées à l’une des extrémités du système. Tout d’abord nous montrons que le problème est bien posé et qu’il y a stabilité uniforme sous certaines conditions portant sur les coefficients de feedback. Puis nous estimons le taux optimal de décroissance de l’énergie du système par la méthode de Shkalikov.

The paper deals with the problem of equilibrium stability of prismatic, homogeneous, intrinsically isotropic, viscoelastic beams subjected to the action of constant compressive axial force in the light of Lyapounov's stability theory. For a class of functional expressions of creeping kernels characteristic of no-aging viscoelastic materials of the hereditary type, solution of the governing integro-differential equations is given. Referring to polymeric materials of the PMMA type, numerical results...

We introduce a model of a vibrating multidimensional structure made of a n-dimensional body and a one-dimensional rod. We actually consider the anisotropic elastodynamic system in the n-dimensional body and the Euler-Bernouilli beam in the one-dimensional rod. These equations are coupled via their boundaries. Using appropriate feedbacks on a part of the boundary we show the exponential decay of the energy of the system.