Analysis of Deformation, Buckling, and Fracture of Materials: From Composite Materials to Thin Domains

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Manufactured devices and structures are often composites containing slender parts. To avoid structural failures and predict their mechanical behavior, one needs to understand material mechanical behavior under loading, namely the rigidity, flexibility, buckling, and fracture of structures, and to derive the effective behavior of composites, which broadly speaking amounts to establishing tight bounds on a composite's effective properties. Existing engineering thin structure theories mostly rely on formal asymptotic expansions and approximations, and while generally they predict the deformation and fracture of thin materials quite accurately, they fail to do so in some cases for reasons not well understood, since often there is not a mathematically rigorous and comprehensive mechanism for verification of the regime of validity, even when the structural geometry is simple. On the other hand, existing mathematical thin structure theories still contain several unsolved or unverified regimes concerning the geometry of the structure and the energy and loading magnitudes. This project aims to tackle these shortcomings in the modeling of deformation, buckling, and fracture of thin structures, by building new tools to obtain mathematically rigorous thin structure theories depending on geometric parameters of the structure, and to study composites and printed materials to derive new bounds on their effective properties. The project will provide research training opportunities for graduate students. The first part of the project deals with continuum and fracture mechanics. While the mathematical theory of deformation and rigidity for developable shells is well understood, it is less so for constant-sign Gaussian curvature shells and thin domains. At the same time the study of buckling of thin structures is known to be complex and underdeveloped. It is known that the linear geometric rigidity of shells with pinned thin faces depends on the Gaussian curvature of the shell, and this project aims at showing that this is indeed the case in the nonlinear setting too and at improving the theories for constant-sign Gauss curvature thin domains. The project plans also to tackle the modeling of buckling of thin structures by means of a new slender structure buckling theory, and, on the side of fracture mechanics, to derive a mathematically rigorous shell fracture Griffith theory for shells with vanishing or constant-sign Gaussian curvature. The second part of the project intends to derive new bounds on the effective properties of composites and printed materials, known as metamaterials, by means of the so-called extremal quasiconvex quadratic forms. The project will use techniques and tools from applied analysis and real algebraic and convex geometry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.