Multiscale asymptotics for partial wrinkling of thin films in tension and related problems.

在固体力学中，严格地说，膜被定义为没有弯曲刚度的二维弹性表面。由于缺乏抗弯刚度，薄膜不能承受压缩应力，因此，如果外部载荷和边界约束使得薄膜的某些部分产生压缩，则薄膜容易发生屈曲。已经开发了一些简化的（张力场）理论来处理这种情况，并且它们对于预测大规模问题相当有效，例如，褶皱区域的范围和褶皱本身的方向。不幸的是，该编织图案的精细结构（例如，褶皱的数量和它们的幅度）仍然是未知的，因为这种行为基本上取决于存在于大多数膜状结构中的小但非零的弯曲刚度。这个项目将集中在正是这些小尺度的不稳定性过程的特点，通过使用奇异摄动方法。将探讨的问题涉及拉伸环形和矩形薄膜面内载荷下，以及加压圆形薄膜与弹性约束的边界。特别重要的是褶皱的数量对预应力量和膜厚度的依赖性。这些问题在许多实际应用中起着重要作用，从新开发的用于细胞运动的实验技术到高压容器的减压膜的规格。各种薄板理论将被用来探测相应问题的弱非线性区域。在二阶有限弹性理论的帮助下，将对上述几何形状中的一些线性化情况进行补充。该项目的一个显著特点是系统地应用渐近技术来理解应力集中和扭曲现象，并在应用数学、固体力学和流体力学的适当领域之间建立联系。虽然这种技术在流体力学的许多领域都很常见，但它们的力量和多功能性似乎在固体稳定性问题中得到了相当少的认可。实现该项目的目标不仅仅是应用现有方法的问题，而是需要回答关于具有转折点的特征值问题的多渐近结构和适当的力学问题的数学建模的关键问题。

In solid mechanics a membrane is defined, strictly speaking, as a two-dimensional elastic surface with no bending stiffness. Because of this lack of bending rigidity membranes cannot sustain compressive stresses and are thus susceptible to buckling (wrinkling) if the external loads and the boundary constraints are such that compression is developed in some parts of the membrane. A number of simplified (tension field) theories have been developed to deal with such situations and they are fairly efficient for predicting large-scale issues like, for instance, the extent of the wrinkled region and the direction of the wrinkles themselves. Unfortunately, the fine structure of the wrinkling pattern (e.g., the number of wrinkles and their amplitude) remains unknown because this behaviour depends essentially on the small but non-zero bending rigidity present in most membrane-like structures. This project will focus on precisely these small-scale features of the instability process by using singular perturbation methods. The problems that will be explored involve stretched annular and rectangular thin films under in-plane loading, as well as pressurised circular thin films with elastically restrained boundaries. Of special importance will be the dependence of the number of wrinkles on the amount of pre-stress and the thickness of the films. Such issues play an important role in a number of practical applications ranging from the newly developed experimental techniques used in cell locomotion to the specification of pressure-relief membranes for high-pressure vessels. Various thin-plate theories will be used to probe the weakly nonlinear regimes of the corresponding problems. A complementary route for some linearised situations in the geometries mentioned above will be pursued with the help of second-order finite elasticity theory.A distinctive feature of the project will be the systematic application of asymptotic techniques to understanding stress concentration and wrinkling phenomena using, and establishing links between, appropriate areas of applied mathematics, solid mechanics, and hydrodynamics. While such techniques are common-place in many fields of fluid mechanics, their power and versatility seem to have been rather less recognised for stability problems in solids. Achieving the aims of the project is not simply a matter of applying existing methods, but requires key questions to be answered regarding the multi-asymptotic structure of eigenvalue problems with turning points and the mathematical modelling of the appropriate mechanical problems.