柔软部分弯曲、伸直和外翻时的褶皱和折痕
Wrinkles and creases in the bending, unbending and eversion of soft sectors
摘要：
我们研究的是什么显然是在自然界中发现的最常见的变形模式之一，科学和工程，即弯曲结构的大弹性弯曲，以及其逆，unbending，这可以超越完全拉直变成外翻。我们发现，这些问题的数学解决方案总是存在的，是唯一的，当固体被建模为一个均匀的，各向同性的，不可压缩的超弹性材料的应变能满足强椭圆条件。我们还提供了显式的渐近解薄部门。当变形足够严重时，弹性材料的压缩侧可能会弯曲，然后可能产生褶皱。我们详细分析了这种不稳定性的Mooney-Rivlin应变能，它涵盖了新胡克模型的情况下，在精确的非线性弹性和弱非线性弹性的三阶弹性材料的发病。特别是相关的理论和数值处理，使我们能够预测皱纹的数量和波长。在实验观察的指导下，我们最后看看折痕的发展，我们通过先进的有限元计算模拟。在某些情况下，线性化分析使我们能够正确地预测折痕的数量和波长，这些折痕的出现只比皱纹早几个百分点。
英文摘要：
We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asympto-tic solutions for thin sectors. When the deformations are severe enough, the compressed side of the elastic material may buckle and wrinkles could then develop. We analyse in detail the onset of this instability for the Mooney-Rivlin strain energy, which covers the cases of the neo-Hookean model in exact non-linear elasticity and of third-order elastic materials in weakly non-linear elasticity. In particular the associated theoretical and numerical treatment allows us to predict the number and wavelength of the wrinkles. Guided by experimental observations we finally look at the development of creases, which we simulate through advanced finite element computations. In some cases the linearised analysis allows us to predict correctly the number and the wavelength of the creases, which turn out to occur only a few percent of strain earlier than the wrinkles.
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