Collaborative Research: Strain-limiting Cosserat Rods with Applications to Modeling Biological Fibers

A vast array of objects encountered throughout science, engineering, and everyday life have two physical dimensions that are much smaller than a third. Examples include collagen fibers, DNA, RNA, beams, pipelines, pencils, and cellphone charger cords. This research project concerns the theory of Cosserat rods: bodies that are parameterized by a curve with two perpendicular directions, the two smaller physical dimensions, specified at each point. This theory models the bodies response to applied forces using partial differential equations, expressing balance of linear momentum, angular momentum, energy, and entropy. The research will investigate and validate a new class of Cosserat rod models, describing the stretch-limiting behavior of biological fibers, including DNA and RNA. Apart from determining the models’ fundamental properties from the mathematical physics point of view, the results of the research will contribute to the understanding and prediction of how fibers bend, twist, and stretch during biological processes. The project will provide research training opportunities for graduate students. The research project will focus on new nonlinear, thermodynamically consistent, strain-limiting constitutive relations between the bending, twisting, and stretching strains and the contact couples and forces in a Cosserat rod. In the static Green elastic setting, the research will include establishing energetic stability and bifurcation results for helical equilibrium states, and the quantitative comparison of predictions to the associated, available, experimental data. In the dynamic viscoelastic setting, the balance equations are a system of parabolic partial differential equations with different field equations, resulting from specifying different dissipation rates. The investigators will focus on obtaining global well-posedness, Lyapunov stability and long-time asymptotic results for solutions to such nonlinear field equations with experimentally natural boundary conditions. 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.