Brittle fracture on plates and shells

Predicting how thin structures undergo applied stresses is of crucial importance in various industrial sectors, such as automotive, aerospace, and marine industries. The corresponding mathematical challenge is to elaborate reliable models for membranes, plates, and shells, able to capture the physical features of such lower dimensional structures. In the last decades, the mathematical community has focused on the rigorous derivation of 2-dimensional models in the framework of elasticity. However, to asses the safety of many engineering structures, the description of the sole elastic behavior of a thin structure is insu_cient, and failure phenomena, such as brittle fracture, have to be considered. This leads us to the study of dimension reduction problems in the context of variational fracture mechanics and free discontinuity functionals. By means of variational convergence, we aim at a rigorous justification of models of brittle fracture on membranes, plates, and shells, and at the development of new numerical methods for fracture simulation on thin structures. We will consider crack processes in linear and nonlinear elasticity, in membrane and in bending regimes, their phase-field approximations, and their numerical implementation with Finite Element Method. The most challenging tasks of our analysis will be (i) to handle the various energy landscapes of brittle fracture, ranging from membrane to bending theory and from linear to nonlinear and finite elasticity, (ii) to develop the correct functional setting for describing brittle fracture on thin structures, focusing in particular on the non- Euclidean nature of membranes and shells, and (iii) to elaborate a solid background for numerical simulations of fracture evolutions on membranes, shells, and plates. We will incorporate fracture phenomena in dimension reduction problems, justifying the most common models of fracture on lower dimensional objects, and establishing a sound foundation to the applied research performed in recent years. At the same time, we will contribute to the investigation of GSBV and GSBD spaces by extending their de_nition to non-Euclidean settings and by studying their influence on new dimension reduction problems. Our theoretical investigation will further result in the development of computational methods for crack evolution on thin structures. The program will be developed by the applicant Stefano Almi in collaboration with the co-applicant Joachim Schöberl (TU Wien) and the international partners Manuel Friedrich, Francesco Solombrino, and Emanuele Tasso.

During the funding period we have focused on the analysis of the variational model of fracture on plates and shells. We have been able to find the appropriate functional setting for such models in the realm of Free Discontinuity problems, providing structural properties and initiating the investigation of necessary and sufficient conditions for the existence of equilibrium configurations. We have further considered the modelling and rigorous justification of non-interpenetration of matter in fracture mechanics and related problems in elasticity. In the framework of linearisation of nonlinear models of elasticity, we have analysed the dependence on variable exponents and non-standard growth conditions, typically used to model non-homogeneous materials. Finally, we have studied the concentration effects due to the interplay between multi-well potential in elasticity and microscopic defects, which are known to lead to non-elastic phenomena at the macroscopic level.