Piero Villaggio

Buckling under unilateral constraints

Abstract The classic variational theory for eigenvalue problems is extended so as to define the Euler critical load of a unilaterally constrained beam. The Euler load is obtained by minimising a Rayleigh quotient on a convex subset of a Hilbert space. The variational formulation also provides a method of bounding the buckling load by comparison.

Condition of stability and wave speeds for fluid mixtures

SommarioSi studiano le condizioni algebriche sugli esponenti caratteristici che assicurano la stabilità infinitesima dello equilibrio di una miscela bifase nella classe delle perturbazioni regolari nulle sul contorno della regione contenente la miscela.Dalle condizioni di stabilità si deducono le velocità reali di propagazione di onde della miscela.SummaryWe study the algebraic conditions on the characteristic exponents ensuring the infinitesimal stability of the equilibrium of a two constituent-mixture in the class of regular perturbations vanishing on the boundary of the region containing the mixture.From the stability conditions we deduce the real speeds of wave propagation in the mixture.

Energetic bounds in finite elasticity

SummaryWe study the conditions under which the internal work of deformation in an elastic isotropic body in finite deformations may be bounded by results obtained from a suitably defined linear infinitesimal problem. The values of the constants appearing in the principal inequalities are calculated and discussed for a certain class of extensional deformations.

The Main Elastic Capacities of a Spheroid

In 1869 Kirchhoff [1869] gave a mathematically precise definition of elastic capacity, that is of the total electrical charge to be applied to a conductor in order to generate the constant potential one on it, while the potential at infinity is kept at level zero. According to Kirchhoff’s definition, the capacity of a given solid body D and surface S can be determined by a function u(x, y, z), harmonic in the infinite space outside D, and such that u = 1 on S, and u = 0 at infinite distance. Once u(x, y, z) is known, the capacity is defined by

Zanaboni's treatment of Saint-Venant's principle

C{\text{ = }}\frac{1}{{4\pi }}{\text{ }}\int {\int {\int {\left( {u_{x}^{2} + u_{y}^{2}{\text{ + }}u_{z}^{2}} \right)dx\;dy\;dz{\text{,}}} } }

The Slope of Roofs

(1.1) where the triple integral is extended over the whole space outside D.

Stability conditions for elastic-plastic prandtl-reuss solids

In this paper we study the equilibrium of an elastic body which is simply supported, without friction, by a soft elastic plane. We prove that, if the exterior loads satisfy certain conditions of compatibility, the solution exists and is unique. Moreover, we find which regularity properties of the solution hold, provided the data are sufficiently smooth.

The Dynamics of a Membrane Shock-Absorber

Zanabonis version of Saint-Venants principle, which concerns an elastic body subjected to self-equilibrated loads distributed over a part of its surface, states that the stored energy tends to zero in regions increasingly remote from the load surface. Unlike other formulations, Zanabonis version applies to bodies of arbitrary shape. Here, unnecessarily complicated aspects of the proof are simplified and rendered mathematically precise by appeal to a variant minimum principle. For cylindrically shaped bodies, a new decay estimate is derived that supplements Zanabonis contributions and confirms that his approach provides a viable alternative to other studies of Saint-Venants principle. In conclusion, various generalizations of the original results are briefly discussed, including an extension to nonlinear elasticity.