Non-Standard Plate and Shell Models in Solid Mechanics

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Plate and shell theories are fundamental to the field of structural mechanics. They are essentially two dimensional representations of thin structures and afford a convenient and effective means for engineers to analyze the strength and safety of diverse types of components, ranging from the domes of buildings to aircraft fuselages, ship hulls and automobile bodies. Thin shells also constitute the basic components of an enormous range of biological structures, including cells, bones and parts of the eye, to name just a few. Thin structures are ubiquitous in nature and technology because they afford by far the most efficient arrangements for sustaining loads while minimizing weight. Current engineering theories for plate and shell structures are based on plausible assumptions about how the material of the plate deforms in three dimensions, to aid in the formulation of a tractable two-dimensional model. However, it is known that these assumptions are sometimes not realized in practice. This award supports fundamental advances in these theories that do not rely on such assumptions, while taking into account the effects of plasticity and fiber reinforcement in the constituent materials. Results from this research will enable the analysis of more efficient aerospace, automotive and civil structures, and will therefore benefit the U.S. economy and society in general. The techniques established here will provide engineers with improved analysis tools and will also benefit emerging fields such as biomechanics. These methods and results will be disseminated in the form of a monograph currently in progress, and through course lectures and specialized short courses.Modern work on theories of plates and shells emphasizes rigorous dimension reduction procedures such as asymptotic expansions or gamma convergence. These are invariably based on the underlying assumption that the plate or shell behaves elastically. In contrast, in the older literature one finds models that account for more general material behavior, such as plasticity and the anisotropy induced by fiber reinforcement, albeit based on ad hoc assumptions about the underlying kinematics which may not be realized in general applications. The present research is designed to fill the gap between existing models that address such effects in ad-hoc approaches, and modern models that are confined to purely elastic behavior but are overall more rigorous. The research therefore has the potential to significantly expand the reach of structural and solid mechanics. The research team will define new models for the inelastic behavior in shells, based on dimension reduction approaches, to include plasticity and diffusion of species within the shell material. The team will also establish models that account for the effects of intrinsic flexural and torsional stiffness of the fibers in fiber-reinforced plates and shells. This is expected to yield a substantial improvement over existing theories that account only for the anisotropy conferred by the fibers. These theoretical advances will be supported by extensive numerical simulations.