Mathematical Sciences: Nonlinear Continuum Mechanics

[404580] Coleman将研究弹性杆的非线性动力学理论的控制方程，该方程在杆的厚度、曲率、扭转和拉伸的误差范围内为2阶。由于该理论没有预设位移和旋转很小，因此其控制方程不是线性的。最近的数值实验表明，这些方程在孤子理论的意义上是近似可积的，但不是完全可积的。作为发展其单元服从非线性理论的结构振动理论的第一步，将寻求有关适用于悬臂梁等单一单元的边界数据解的定性性质的信息。杆理论研究的其他问题与DNA分子三级结构理论中出现的问题有关。为了有一个考虑内耗散的杆的动力学理论，弹性杆的理论将被推广到一个适用于非线性粘弹性材料的理论。在一个单独的研究领域，将研究非线性效应在固体的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，当考虑到控制方程中的非线性项时，必须修正圆柱对表面扩散稳定性的线性化理论中一个熟悉的结果，即圆柱对波长小于未受扰动圆柱周长的所有正弦扰动都是稳定的。在该项目下进行的杆力学研究预计将对控制结构振动的方法的设计、对影响直升机旋翼叶片运动稳定性的因素的理解、对DNA超卷曲现象的理解以及对被称为拓扑异构酶的酶的作用机制的发现产生影响。物体抗表面扩散稳定性的研究涉及到微电子器件中欧姆导体的耐久性和微电子器件的设计等几个重要问题。

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.