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for Blow up, periodicity, and parametric resonances in viscoelastic shells

===Blow up, periodicity, and parametric resonances in viscoelastic shells===
Walter Lacarbonara <br>
Department of Structural Engineering, University of Rome La Sapienza

This work addresses an extensive global treatment of radial motions of compressible nonlinearly viscoelastic cylindrical and spherical shells under time-dependent pressures. The shells are described by a geometrically exact 2-dimensional theory in which the shells suffer thickness strains as well as stretching of their base surfaces.  The global treatment furnishes a variety of conditions on a general class of material properties and on the pressure terms ensuring that there are solutions existing for all times, there are unbounded globally defined solutions, there are solutions that blow up in finite time, and there are periodic solutions.  Within the class of radial periodic motions, we then employ local methods (higher-order asymptotics) to study stability and bifurcation near the well-known parametric resonances caused by the time-dependent pressures.  The formidable effect of the nonlinear viscoelastic constitutive laws on the parametric instability diagrams and the post-critical parametric-resonance motions is discussed.
This work shows that choosing a limited family of constitutive functions, 
inspired by the desire to get a simple nonlinear model extending that for linear elasticity, can cause one to overlook important effects and dangerous instabilities.

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