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

在科学、工程和日常生活中遇到的大量物体都有两个比第三个小得多的物理尺寸。例子包括胶原纤维、DNA、RNA、光束、管道、铅笔和手机充电线。该研究项目涉及Cosserat杆理论：由具有两个垂直方向的曲线参数化的身体，两个较小的物理尺寸，在每个点指定。该理论使用偏微分方程模拟物体对作用力的响应，表达线性动量、角动量、能量和熵的平衡。该研究将调查和验证一类新的Cosserat杆模型，描述生物纤维（包括DNA和RNA）的拉伸限制行为。除了从数学物理的角度确定模型的基本属性外，研究结果将有助于理解和预测纤维在生物过程中如何弯曲、扭曲和拉伸。该项目将为研究生提供研究培训机会。该研究项目将集中在新的非线性，一致的，应变限制的本构关系之间的弯曲，扭转和拉伸应变和接触对和Cosserat杆的力。在静态绿色弹性设置，研究将包括建立螺旋平衡状态的能量稳定性和分叉结果，以及预测的定量比较，相关的，可用的，实验数据。在动态粘弹性环境中，平衡方程是一个抛物型偏微分方程组，由于指定不同的耗散率，导致不同的场方程。研究人员将专注于获得全局适定性，李雅普诺夫稳定性和长期渐近结果的解决方案，这种非线性场方程与实验自然边界条件。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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.