Mathematical Sciences: Nonlinear Continuum Mechanics

9404580科尔曼将对弹性杆动力学的非线性理论的控制方程进行研究，该理论在杆的厚度、曲率、扭转和拉伸方面的误差完全在2阶以内。由于该理论不假定位移和转动很小，所以它的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意义上是近似但不完全可积的。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，将寻求关于适用于悬臂梁等单单元的边界数据解的定性性质的信息。杆状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。为了建立考虑内部耗散的杆的动力学理论，将弹性杆理论推广到适用于非线性粘弹性材料的理论。在另一个单独的研究领域，将研究非线性效应在防止表面扩散引起的形状变化的固体稳定性理论中的作用。最近的研究表明，当考虑到控制方程中的非线性项时，圆柱体抗表面扩散稳定性的线性化理论中的一个常见结果，即圆柱体对小于未扰动圆柱体周长的所有正弦扰动是稳定的，必须进行修正。在杆的力学项目下进行的研究有望对控制结构振动的方法的设计、对直升机旋翼桨叶运动稳定性的影响因素的理解、对DNA中的超卷现象的理解以及对称为拓扑异构酶的酶的作用机制的发现有所启示。物体抗表面扩散稳定性的研究涉及到微电子器件欧姆导体的耐久性和微电子机械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.