不可伸长悬臂梁的解的理论
Theory of Solutions for An Inextensible Cantilever
摘要：
分析了悬臂弹性梁大挠度运动的最新方程。在传统的梁（板）大挠度理论中，非线性恢复力是由于拉伸对弯曲的影响;对于不可伸长的悬臂，弧长保持的强制执行导致准线性刚度效应和惯性效应，这两者都是非线性和非局部的。对于这个模型，通过谱Galerkin方法构造光滑解。需要额外的紧性来达到极限，这是通过一个复杂的过程来获得更高的能量估计。uniform是通过非平凡的非线性分解。通过在运动方程中加入结构（Kelvin-Voigt）阻尼来克服非线性惯性的混杂效应。光滑解的局部适定性首先在不存在非线性惯性效应的情况下显示，然后显示这些惯性效应存在，考虑到结构阻尼。在阻尼的情况下，通过对小数据实现指数衰减，得到了时间上全局的强适定性结果。
英文摘要：
Recent equations of motion for the large deflections of a cantilevered elastic beam are analyzed. In the traditional theory of beam (and plate) large deflections, nonlinear restoring forces are due to the effect of stretching on bending; for an inextensible cantilever, the enforcement of arc-length preservation leads to quasilinear stiffness effects and inertial effects that are both nonlinear and nonlocal. For this model, smooth solutions are constructed via a spectral Galerkin approach. Additional compactness is needed to pass to the limit, and this is obtained through a complex procession of higher energy estimates. Uniqueness is obtained through a non-trivial decomposition of the nonlinearity. The confounding effects of nonlinear inertia are overcome via the addition of structural (Kelvin-Voigt) damping to the equations of motion. Local well-posedness of smooth solutions is shown first in the absence of nonlinear inertial effects, and then shown with these inertial effects present, taking into account structural damping. With damping in force, global-in-time, strong well-posedness result is obtained by achieving exponential decay for small data.
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