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In this review, we discuss some conditions for achieving non-interpenetration and self-contact of solids, in particular for regular, inextensible, and closed elastic rods. We establish some equivalences between conditions that were stated sometimes independently, underlying their local or global character. We then examine three conditions related to virtual displacements and to topological characters of knots, that can be generalized to filaments, considering the midline of the loop as an inextensible regular knot.

We perform a variational analysis of an elastic membrane spanning a closed curve which may sustain bending and torsion. First, we deal with parametrized curves and linear elastic membranes proving the existence of equilibria and finding first-order necessary conditions for minimizers computing the first variation. Second, we study a more general case, both for the boundary curve and for the membrane, using the framed curve approach. The infinite dimensional version of the Lagrange multipliers' method is applied to get the first-order necessary conditions. Finally, a numerical approach is presented and employed in several numerical test cases.

Dopo una breve introduzione alla Meccanica dei Continui solidi, vengono presi in considerazione alcuni modelli che possono descrivere il comportamento elastico di un muscolo, un materiale non isotropo ma trasversalmente isotropo. Una delle sue peculiarità è quella di produrre sia uno sforzo passivo, analogo ai materiali non biologici, sia uno sforzo attivo responsabile della potenza necessaria al movimento: verranno quindi presentati gli strumenti modellistico-matematici che servono per descrivere questo fenomeno. Questi modelli possono essere inoltre utili per indagare il comportamento di alcune patologie legate all'invecchiamento, tra cui la sarcopenia, caratterizzata dalla perdita di massa e di funzionalità del tessuto muscolare., After a brief introduction of the Continuum Mechanics of Solids, some models of a muscle, which is a transversely isotropic material, are considered. A feature of this material is to produce not only a passive stress, analogous to ordinary materials, but also an active stress, needed to produce power: the main mathematical tool used to describe this phenomenon will be introduced. These kind of models could help understanding some pathologies due to ageing, like the sarcopenia, which consists in a loss of muscle mass and muscle performance.