The book collects a selection of papers most of which are revised and enriched versions of the contributions presented at the 12th Symposium on Trends of Applications of Mathematics to Mechanics (STAMM) which was sponsored by the International Society for the Interaction between Mathematics and Mechanics (ISIMM) and held in Maiori (Salerno), Italy from September 29th to October 4th, 2002. The Symposium attracted many leading researchers from around the world who are working at the interface between Mathematics and Mechanics. The importance of a close link between these two disciplines have long been recognized and each of them get benefits and stimuli by open problems, methods and results emerging from the other one. The book collects 22 papers which contribute special investigations and more wide presentations of linear and nonlinear problems. It is with the deepest gratitude to the authors that have contributed to the volume and to the publisher, for his highly professional assistance, that the editors submit this book to the international mathematics and mechanics communities.

"The book provides a collection of recent theoretical and methodological advances which can provide support and stimulus to scientists and scholars involved in research activity in the fields of interest."

On the instability of double diffusive convection in porous media under boundary data periodic in space.- Modelling of a free piston problem.- Reflections on frequently used viscoplastic constitutive models.- On hereditary models of polymers.- Edge contact forces in continuous media.- Tangent stiffness of a Timoshenko beam undergoing large displacements.- Qualitative estimates for cross-sectional measures in elasticity.- On nonlinear global stability of Jeffery-Hamel flows.- Energy penalty, energy barrier and hysteresis in martensitic transformations.- On the applicability of generalized strain measures in large strain plasticity.- A nonlocal formulation of plasticity.- Consistent order extended thermodynamics and its application to light scattering.- On instability sources in dynamical systems.- Tangent stiffness of elastic continua on manifolds.- Basic issues in convex homogenization.- Tangent stiffness of polar shells undergoing large displacements.- Global existence of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics.- Central schemes for conservation laws with application to shallow water equations.- Regularized 13 moment equations for rarefied gas flows.- Hydrodynamic calculation for extended differential mobility in semiconductors.- Small planar oscillations of an incompressible, heavy, almost homogeneous liquid filling a container.- Thermodynamics of simple two-component thermo-poroelastic media.