Limit Analysis of Collapse States in Cellular Solids

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Cellular solids are two or three dimensional bodies divided into cells, the walls of which are made of a solid material capable of undertaking (large) elastic deformations without plastic failure or fracture. Due to their exceptional mechanical efficiency, they are ubiquitous in nature and industry, yet they are less well understood than almost any other class of materials. Among the best known mechanical qualities of these structures are their high strength-to-weight ratio and their energy absorption capacity, which are due the inextricable relation between the geometric architecture and the constitutive properties of the underlying solid mater. The aim of this project is to gain insight into the properties of cellular structures of nonlinear elastic material and their overall response under loads by providing both a new framework to understand the mechanical properties of these structures and rigorous mathematical techniques for the analysis of their large elastic deformations caused by the application of external loads. Taking into account the interplay between the geometry and the mechanical qualities of the elastic cell walls, novel numerical methods will be devised to compute effective lower and upper bounds for the critical load causing densification by cell closure in various cellular structures, and the bounds gap will be used as an indicator for the computational error. In this context, the proposed investigation and non-standard numerical procedures are novel and have many potential applications. For example, the development of new flexible stents and scaffolds for soft tissue re-growth in biomedical applications is a rapidly growing multidisciplinary area of biomaterials and tissue engineering, and many foams and sponges designed for cushioning and re-usability can also be found in everyday life as well as in several industrial areas, e.g. microelectronics, aerospace, pharmaceutical and food processes. For these complex materials to be understood and optimised with respect to their mechanical response, reliable computer models supported by rigorous mathematical mechanical analysis are needed, and may also open the way for new applications.