Nonuniqueness of equilibrium states for axisymmetric elastic shells in tension

The problem of nonuniqueness of static axisymmetric solutions for a non-linearly elastic cylindrical shell in which the ends are pulled apart with a constant traction while retaining the radii of its ends fixed is studied. In the elastic case, we prove the existence of buckled states and the possibility of necking. In the hyperelastic case a global existence and nonuniqueness theorem is proved, via the energy criterion.     

Interfacial debonding of two-phase composite with periodic structure

A mixed analytical-numerical (boundary element method) procedure is presented for estimating the effective elastic moduli of a two-phase periodic composite by application of a unit cell. The two-phase composite consists of a metal/polymer matrix and one/three circular ceramic inclusions with adhesive and partial debonding of the interface. The results are displayed numerically with special attention given to development of plastic zones as debonding occurs. Dependence of load-time history is exhibited.     

Non-linear bell-shaped and kink-shaped strain waves in microstructured solids

The evolution of finite-amplitude strain waves is studied in a medium with microstructure when dissipation and energy input are taken into account. The governing non-linear equation for longitudinal strain waves is obtained in the one-dimensional case. The propagation and attenuation or amplification of bell-shaped and kink-shaped waves, whose parameters are defined in an explicit form through the parameters of the microstructured medium, are studied.     

Buckling of a cylindrical shell subject to a wrench

We examine the axisymmetric buckling of a generalized Kirchhoff cylindrical shell under an axial wrench. If the shell is non-linear elastic, we provide a boundary value problem amenable to the bifurcation theory of Poincaré. Using this theory we prove a non-uniqueness theorem and obtain the critical loads and critical states. In the hyperelastic case, by exploiting the variational character of the problem, a global existence and non-uniqueness theorem is also proved.

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Sommario Viene esaminato un problema di buckling per un guscio cilindrico, del tipo di Kirchhoff generalizzato, sotto un carico ai bordi di trazione-torsione. Nel caso elastico non lineare, si perviene ad un problema ai valori al contorno, cui si può applicare la teoria delle biforcazioni di Poincaré. In tal modo si dimostra un teorema di non unicità della soluzione e si ottengono i punti di biforcazione e i carichi critici. Nel caso iperclastico, facendo ricorso all'aspetto variazionale del problema, viene dimostrato un teorema globale di esistenza e non unicità della soluzione.

     

E. Frola (1906–1962): An Attempt Towards an Axiomatic Theory of Elasticity

In a few papers published in the 1940's, the Italian mathematician Eugenio Frola (1906–1962) proposed an axiomatic formulation of the theory of linear elasticity as an autonomous branch of mathematics, and he sketched a program for the logical foundations of that mathematical theory. He gave only some general principles and a few hints, without obtaining any definite axiomatic structure in the sense of the fundamental work of W. Noll and C. Truesdell a few years later. Nevertheless, Frola's attempt of finding rigorous axiomatic foundations of classical elasticity must be acknowledged as a very first step in the right direction.