ELASTIC MEMBRANE EQUATION WITH MEMORY TERM AND NONLINEAR BOUNDARY DAMPING:GLOBAL EXISTENCE,DECAY AND BLOWUP OF THE SOLUTION

In this paper we consider the Elastic membrane equation：with memory term and nonlinear boundary damping： Under some appropriate assumptions on the relaxation function h and with certain initial data, the global existence of solutions ：and a general decay for the energy are established using the multiplier technique. Also, ＇we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.

CAVITATION IN HOOKEAN ELASTIC MEMBRANES

An exact solution to cavitation is found tension of ac lass ofCauchy elastic membranes. The constitutive relationship of materialsis based on Hookean elastic law and finite logarithmic strain mea-Sure. A variable transformation is used in solving the two-pointboundary-value problem of nonlinear ordinary Different equation. Asimple formula to calculate the critical stretch for cavitation isderived. As the nu- Merical results, the bifurcation curvesdescribing void nucleation and suddenly rapidly growth of the cavityAre obtained. The boundary layers of displacements and stresses nearthe cavity wall are observed.

VIBRATION OF ARBITRARILY SHAPED MEMBRANES WITH ELASTICAL SUPPORTS AT POINTS

This paper presents a new method for solving the vibration of arbitrarily shaped membranes with ela.stical supports at points. The reaction forces of elastical supports at points are regarded as unknown external forces acting on the membranes. The exact solution of the equation of motion is given which includes terms representing the unknown reaction forces. The frequency equation is derived by the use of the linear relationship of the displacements with the reaction forces of elastical supports at points. Finally the calculating formulae of the frequency equation of circular membranes are analytically performed as examples and the inherent frequencies of circular membranes with symmetric elastical supports at two points are numerically calculated.

EXISTENCE OF MINIMIZING SOLUTIONSAROUND“EXTENDED STATES”FOR A NONLINEARLY ELASTICCLAMPED PLANE MEMBRANE

The formal asymptotic analysis of D. Fox, A. Raoult & J.C. Simo[10] has justified the twodimensional nonlinear "membrane" equations for a plate made of a Saint Venant-Kirchhoff material.This model, which retains the material-frame indifference of the original three dimensional problem in the sense that its energy density is invariant under the rotations of R3, is equivalent to finding the critical points of a functional whose nonlinear part depends on the first fundamental form of the unknown deformed surface.The author establishes here, by the inverse function theorem, the existence of an injective solution to the clamped membrane problem around particular forces corresponding physically to an "extension" of the membrane. Furthermore, it is proved that the solution found in this fashion is also the unique minimizer to the nonlinear membrane functional, which is not sequentially weakly lower semi-continuous.

Torsional wave in a circular micro-tube with clogging attached to the inner surface

In the present paper, we study the torsional wave propagation along a micro-tube with clog- ging attached to its inner surface. The clogging accumulated on the inner surface of the tube is modeled as an ＂elastic membrane＂ which is described by the so-called surface elasticity. A power-series solution is particularly developed for the lowest order of wave propagation. The dispersion diagram of the lowest-order wave is numerically presented with the surface （clogging） effect.