Biological and bio-inspired locomotion is usually described by recognizing periodic patterns, or gaits, in the movement of limbs or other body parts. But is the evolution of the system actually periodic? Or more properly, relative-periodic, since, presumably, each cycle will propel the animal (or robot) a little bit forward? The answer is often no, due, for instance, to inertia or elasticity. However, we might expect the behaviour to converge asymptotically to a relative-periodic one. In this talk we will introduce this issue considering, as a case study, a dynamic model of rectilinear crawling locomotion.
We study the existence of a global periodic attractor for the reduced dynamics of the model, corresponding to an asymptotically relative-periodic motion of the crawler. The main result is of Massera-type, namely we show that the existence of a bounded solution implies the existence of the global periodic attractor for the reduced dynamics. Additional conditions and a counterexample for the existence of a bounded solution (and therefore of the attractor) will be briefly discussed. We conclude surveying the issue for some related models.
This talk presents an overview of the Irreversible Phase Field Model for Fracture (Irreversible F-PFM) proposed by the speaker and collaborators, which is based on the concept of irreversible gradient flows. An irreversible gradient flow is a constrained gradient flow in which monotonicity in time is enforced, leading to a natural energy-dissipation identity [Akagi–Kimura, 2019; Kimura–Negri, 2021].
The Irreversible F-PFM applies this framework to the Ambrosio Tortorelli regularization of the variational fracture model by Francfort and Marigo, ensuring both the irreversibility of crack evolution and monotonic energy decay (i.e., a consistent dissipation structure). This model is compactly formulated using partial differential equations and is readily amenable to standard finite element solvers, enabling crack propagation simulations without a priori knowledge of the crack path.
After reviewing the mathematical structure of the Irreversible F-PFM, the talk will introduce a range of recent extensions and their finite element simulations, each highlighting how the dissipation structure is preserved or adapted. These include: Irreversible F-PFM under unilateral constraints; Thermo-mechanical fracture models; Phase field modeling of fracking in oil and gas reservoirs; Desiccation-induced fracture modeling; Dynamic phase-field fracture models and seismic fault rupture; Through these examples, we aim to highlight the versatility and robustness of the irreversible phase-field approach in simulating complex fracture phenomena across disciplines.