Limit Analysis of Collapse States in Cellular Solids

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.

In addition to the academic beneficiaries identified in this case for support, the potential impact of the proposed research may be broken down into the following areas.

KNOWLEDGE AND PEOPLE. The limit analysis of elastic cellular structures is an important issue which has received little attention from the mathematical community to date. The long-term ambition is to make the new results useful for modern practical applications in bio-medical and material engineering, and also to help to identify new applications. From an applied mathematics point of view, the immediate goal is to inform the engineering research community who would then be able to test the mathematical results on a wide range of man-made materials of engineering interest.

LAM is an effective and welcome communicator within both the mathematical and engineering communities, as demonstrated by her track record, and her presentations at conferences and research seminars during the project will enhance the visibility of cellular materials and also encourage young researchers from both these communities into the area.

As a direct outcome, the project will train a young postdoctoral researcher in the solid mechanics underlying the physical model of cellular materials and the mathematical and numerical techniques needed to understand and analyse these materials.

In order to validate the computational results on real physical structures, Prof Sam Evans (Cardiff, Engineering) has agreed to carry out experimental tests on cellular structures made of a nonlinear elastic material. This will offer opportunities for feed-back on the computational models and feed-forward for the design of new model structures of engineering interest for further testing and validation.

Mathematical findings resulting from the project will also be included in the Finite Elasticity course for the fourth year MMath programme at Cardiff University, established by LAM in 2013-14. A new postgraduate-level course in Nonlinear Elasticity will also be introduced by LAM as part of the MAGIC (Mathematics Access Grid Instruction and Collaboration) programme funded by EPSRC. The MAGIC group contains 20 Universities across England and Wales, including Cardiff.

ECONOMY AND SOCIETY. Cellular materials can be found in most things where light-weight, shock-absorbing, multi-functional materials are required, and although it is hard to make a precise estimate of how much they cost or what the value of those protected by them is, it is not difficult to appreciate their overwhelming importance for the economy and society. For example, many foams and sponges designed for cushioning and re-usability can be found in everyday life as well as in several industrial areas, e.g. microelectronics, aerospace, pharmaceutical and food processes.

In the UK, the development of new flexible scaffolds for tissue re-growth in biomedical applications is a rapidly growing multidisciplinary area of Biomaterials and Tissue Engineering, which in turn underpins two of the current EPSRC Challenge Themes, namely Manufacturing the Future and Healthcare Technologies. In this context, the results of the proposed research will contribute to the development of associated research areas in material engineering, and may ultimately be used to optimize the design of cellular materials for highly flexible stents and soft tissue scaffolds, which are under continuing research and development.

For these complex materials to be understood and optimised with respect to their mechanical response, reliable computer models based on rigorous mathematical mechanical analysis are needed and may open the way for new applications.