## Variational Models and Methods in Solid and Fluid Mechanics

July 12, 2010 — July 16, 2010

Coordinators:

• Francesco Dell'Isola (Università "La Sapienza", Roma, Italy)
• Sergey Gavrilyuk (Polytech Marseille, France)

“For this would be agreed by all: that Nature does nothing in vain nor labours in vain”. Olympiodorus, Commentary on Aristotle’s Meteora translated by Ivor Thomas in the Greek Mathematica Works Loeb Classical Library

“La nature, dans la production de ses effets, agit toujours par les voies les plus simples”. Pierre de Fermat

Variational formulation of the governing equations of solid and fluid mechanics is a classical but a very challenging topic. Variational methods give an efficient and elegant way to formulate and solve mathematical problems that are of interest for engineers. This formulation allows us an easier justification of the well-posedness of mathematical problems, the study of stability of particular solutions, a simpler implementation of numerical methods. Often, mechanical problems are posed in a variational context by their nature. Hamilton’s principle of stationary (or least) action is the conceptual basis of practically all models in physics. The variational formulation is also useful for obtaining simpler approximate asymptotical models.

In this course, three fundamental aspects of the variational formulation of mechanics will be presented: physical, mathematical and applicative ones.
The first aspect concerns the investigation of the nature of real physical problems with the aim of finding the best variational formulation suitable to those problems. A deep knowledge of the physical problems is needed to determine the Lagrangian function of the system and the nature of the admissible variations. In many problems of mechanics and physics, the functionals being minimized depend on parameters which can be considered as random variables. Variational structure of such problems brings considerable simplifications in their study.
The second aspect is the study of the well-posedeness of those mathematical problems which need to be solved in order to draw previsions from the formulated models. Often, the emphasis is put on integrable non-linear equations. It is worth to notice that most non-linear systems of continuum mechanics represent non-integrable systems. However, their variational formulation implies a special structure of the governing equations which should be used for their mathematical study.
The third aspect is related to the direct application of variational analysis to solve real engineering problems. A Rayleigh-Hamilton principle is used to establish boundary conditions at discontinuity surfaces in porous media. New variational models of fracture mechanics are presented and solved. Continuous structures to which are attached special sets of resonators (discrete as well as continuous) that generate amplitude decays in its impulse response are also treated.

The course is addressed to doctoral students, young and senior researchers and engineers interested in this field.

Keywords: Continuum Mechanics, Fracture and Damage Mechanics, Wave Motions in Solids