course: Mechanics over spaces of maps between manifolds: a general model-building tool for the mechanics of complex bodies
speaker: Paolo Maria Mariano (Università di Firenze e Scuola Normale Superiore)abstract: Mechanics constructed over jets of appropriate fiber bundles involving abstract paracompact manifolds constitutes a theoretic model-building framework for complex bodies, bodies in which substructural changes from nano-to-mesoscopic level prominently influence the gross behavior in a way in which consequent substructural interactions cannot be neglected and have to be described directly. For these bodies each material element is a system and can be represented as microcanonical, canonical or grand-canonical ensemble. In the last case one allows migration of substructures driven by the competition of substructural interactions from place to place (a universal mechanism). Such interactions satisfy own pointwise balances which are peculiar because in general their integral counterparts do not make geometrical sense (unless the manifold of substructural shapes coincides with a linear space) and are not related with any Killing field of the metric in the ambient space. An essential question is whether and in what sense balances of substructural interactions are covariant in conservative and dissipative cases. The answer is non-trivial: Balances of substructural interactions in the bulk of the base manifold, across "surface" and "line" defects are covariant with respect to the action of generic Lie groups on the manifold of substructural shapes. Surface and line balances of standard interactions are covariant with respect to the action of the group of automorphism of the ambient space. The evolution equations of "surface", "line" and "point" defects follow from invariance with respect to the action of the special group of isocoric diffeomorphims over the base manifold and are generalizations of the motion-by-curvature. Take note that substructural interactions affect even strongly the evolution of defects, in addition, in the case of complex fluids they are responsible of the nucleation of topological transitions in the flow. Remarks above constitute the main subject of my lectures. In the last part of them I will discuss some aspects of the existence of ground states for general complex bodies with decomposed free energies, underlying the link between the topological features of the manifold of substructural shapes and some prominent aspects of the existence results.
timetable:
Wed 3 Oct, 17:00 - 18:00, Aula Dini
Thu 4 Oct, 10:00 - 11:00, Aula Dini
Fri 5 Oct, 18:00 - 19:00, Aula Dini
Sat 6 Oct, 11:00 - 12:00, Aula Dini
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