Mathematical Sciences: Nonlinear Continuum Mechanics

9404580科尔曼 本文将研究弹性杆动力学的非线性理论的控制方程，该理论在杆的厚度、曲率、扭转和拉伸方面的误差在2阶以内。由于该理论并不预先假设位移和旋转很小，因此其控制方程不是线性的。最近的数值实验表明，这些方程是近似的，但不是完全可积的孤子理论的意义。作为第一步的发展理论的振动的结构，其元素服从非线性理论，信息将寻求有关的定性性质的解决方案的边界数据适当的悬臂梁等单一元素。杆理论中正在研究的其他问题与DNA分子三级结构理论中出现的问题有关。要有一个理论的动态杆，占内部耗散，理论的弹性杆将推广到一个适当的非线性粘弹性材料。在一个单独的研究领域，将研究非线性效应在固体稳定性理论中对表面扩散引起的形状变化的作用。最近的研究表明，在圆柱体抗表面扩散稳定性的线性化理论中，一个常见的结果是：对于波长小于未扰动圆柱体周长的所有正弦扰动，圆柱体都是稳定的，当控制方程中的非线性项被考虑时，必须进行修改。杆的研究对于控制结构振动的方法的设计、理解影响直升机旋翼桨叶运动稳定性的因素、理解DNA中的超螺旋现象以及发现称为拓扑异构酶的酶的作用机制具有重要意义。物体抗表面扩散稳定性的研究与微电子器件中欧姆导体的耐久性和微机电器件的设计有关。

9404580 Coleman A study will be made of the governing equations of a nonlinear theory of the dynamics of elastic rods that is complete to within an error of order 2 in a rod's thickness, curvature, twist, and stretch. As the theory does not presuppose that displacements and rotations are small, its governing equations are not linear. Recent numerical experiments have indicated that these equations are approximately but not completely integrable in the sense of the theory of solitons. As a first step toward the development of a theory of the vibrations of structures whose elements obey the nonlinear theory, information will be sought about the qualitative properties of solutions for boundary data appropriate to such single elements as cantilever beams. Other problems under investigation in rod theory bear on issues that arise in theories of the tertiary structure of DNA molecules. To have a theory of the dynamics of rods that accounts for internal dissipation, the theory of elastic rods will be generalized to one appropriate to nonlinear viscoelastic materials. In a separate area of research, a study will be made of the role of nonlinear effects in the theory of the stability of solid bodies against changes of shape caused by surface diffusion. Recent research has shown that a familiar result in the linearized thery of the stability of circular cylinders against surface diffusion, to the effect that a cylinder is stable against all sinusoidal perturbations of wavelength less than the circumference of the unperturbed cylinder, must be modified when nonlinear terms in the governing equations are taken into account.The research to be performed under the project on the mechanics of rods is expected to have implications for the design of methods of controlling the vibration of structures, for understanding factors that affect the stability of motions of helicopter rotor blades, for obtaining an understanding of the phenomenon of supercoiling in DNA, and for finding the mechanisms of the action of the enzymes called topoisomerases. The work on the stability of bodies against surface diffusion bears on several important problems concerning the durability of ohmic conductors in micro-electronic devices and the design of micro-electromechanical devices.