The advances in modern mathematics, including the theoretical developments and the computations aspects, make it now possible to study models to describe the deformations of continuum bodies in various regimes, such as the elastic one, the plastic one, and those featuring damage and rupture. Sound mathematical models are important for applications, for the validation of experimental models, and for providing methods to simulate numerically the deformations that a body can undergo.
Mathematical Analysis and Continuum Mechanics provide the right tools to build and study these models. Variational techniques are successfully used to minimise the energy associated with a deformation in order to find the equilibrium configuration of a body subject to external forces. The breadth of application of mechanical theories to everyday-life problems is very wide.
Complex materials, such as polycrystals and multiphase composites, can also be modelled within this general framework. The applications include micromagnetics, nanoelectronics, and the design of new materials, where the mathematical support of the deep understanding of the models can suggest new ways of devising a composite material with ad hoc properties. In these cases, resourceful help comes from the theory of (periodic and stochastic) homogenisation, from the modelling of defects (such as dislocations and disclinations), from averaging processes, and from theories of upscaling, all of which allow to deduce meso- and macroscopic properties of a composite from the information available on its microstructure.
Alongside classical theories, we single out two important generalizations: the theory of structured deformations, providing a multiscale geometry that captures the contributions at the macroscopic level of both smooth and non-smooth geometrical changes at submacroscopic levels, and the theory of peridynamics, offering a non-local approach to continuum mechanics.
The PhD candidate is expected to work on current research topic of interest of the group members. The mathematical preparation will be sharpened to include the necessary techniques (including: functions spaces, functional and convex analysis, calculus of variations, mechanics, relaxation, integral representation of functionals, homogenisation, peridynamics) needed to deal with the problems at hand. The focus of the research will be in problems concerning relaxation and asymptotics of energies for structured deformations, optimal design of materials, evolution problems, possibly including nonlocal models, pattern formation, and microstructures.
Research exchanges with the group members in Torino and Rome will be encouraged, to strengthen the scientific collaboration and disseminate the results. Participation to workshop and conferences is encouraged as well.