Modeling, Analysis, and Computation in Nonlinear Elasticity

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This project is focused on the mathematical prediction of mechanical behavior in highly deformable elastic structures and solids, including both soft elastic systems and brittle materials. Soft structures occur naturally in biological systems such as skin and bio-membranes and are also manufactured as thin films and elastomers. The prediction of their initial instability, such as wrinkling, and post-critical pattern formation under loading and/or growth play a prominent role in this part of the project. For brittle solids (such as ceramics or steels at low temperature), we will use a new approach to fracture, predicting the initiation and formation of cracks under incremental loading. Overall, the project aims to provide new classes of continuum-mechanical models and novel approaches to their analyses, leading to a quantitative understanding of the mechanical behavior of these systems. The work has a range of potential applications – from bio-molecular structures to engineering machines and structures. This project includes opportunities for the research training of graduate students. Classes of nonlinear models of elastic-surface structures, soft elastic solids, and brittle solids will be analyzed. The main goals of the work are: (i) To provide properly formulated mechanics-based models. (ii) To obtain rigorous mathematical results, viz., establish existence theorems – the only true way to “ensure that the mathematical description of a physical phenomenon is meaningful” (R. Courant). (iii) To detect new phenomena. Goals (i) and (ii) inform and enrich each other; goal (iii) is enabled by goals (i) and (ii). This research is highly interdisciplinary, requiring tools and perspectives from several fields, e.g., nonlinear continuum mechanics, elliptic PDE systems, bifurcation theory, calculus of variations, numerical methods, symmetry ideas and biophysics, while providing new links between them. The project will provide new results and insights pertaining to surface-creasing formation in soft elastic solids, fracture of brittle materials, post-critical growth in elastic solids and pattern formation in lipid-bilayer structures or bio-membranes.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.