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

在固体力学中，膜被严格地定义为没有弯曲刚度的二维弹性表面。由于这种弯曲刚度的缺乏，膜不能承受压应力，因此，如果外部载荷和边界约束使得膜的某些部分发生压缩，则容易发生屈曲（起皱）。许多简化的（张力场）理论已经发展出来处理这种情况，它们对于预测大规模的问题相当有效，例如，褶皱区域的范围和褶皱本身的方向。不幸的是，起皱模式的精细结构（例如，起皱的数量及其振幅）仍然未知，因为这种行为本质上取决于大多数膜状结构中存在的小但非零的弯曲刚度。该项目将通过使用奇异摄动方法精确地关注这些不稳定过程的小尺度特征。将探讨的问题包括平面内载荷下的拉伸环形和矩形薄膜，以及具有弹性约束边界的加压圆形薄膜。特别重要的是，褶皱的数量取决于预应力的大小和薄膜的厚度。这些问题在许多实际应用中发挥着重要作用，从用于细胞运动的新开发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讨相应问题的弱非线性机制。对于上述几何中的一些线性化情况，我们将借助二阶有限弹性理论来寻求补充途径。该项目的一个显著特征将是系统地应用渐近技术来理解应力集中和起皱现象，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。虽然这些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上很少得到认可。实现项目目标不是简单地应用现有方法的问题，而是需要回答关于具有拐点的特征值问题的多渐近结构和适当机械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.