论文标题
类似Landau的屈曲现象的理论及其在弹性性弱心理中的应用
Landau-like theory for buckling phenomena and its application to the elastica hypoarealis
论文作者
论文摘要
分叉现象在弹性上无处不在，但是它们的研究通常仅限于线性扰动或数值分析，因为第二或更高的变化通常超出了分析治疗之外。在这里，我们回顾了两个关键的数学思想，即分裂的引理和功能的决定性，并展示如何将它们用于得出降低的功能，称为论文中的Landau扩展，从而使我们可以简单但严格地描述Bifurcation场景，包括Equilibribim solutions的稳定性。我们将这些想法应用于一个范式的示例，并在各种软约束的物理系统和生物组织中使用潜在的应用：在压力下可伸缩的弹性环。当拉伸模量足够小时，我们证明了三位一点点的存在，并发现了双重性效应和磁滞。这些结果似乎与心脏细胞最近的一些实验定性一致。
Bifurcation phenomena are ubiquitous in elasticity, but their study is often limited to linear perturbation or numerical analysis since second or higher variations are often beyond an analytic treatment. Here, we review two key mathematical ideas, namely, the splitting lemma and the determinacy of a function, and show how they can be fruitfully used to derive a reduced function, named Landau expansion in the paper, that allows us to give a simple but rigorous description of the bifurcation scenario, including the stability of the equilibrium solutions. We apply these ideas to a paradigmatic example with potential applications to various softly constrained physical systems and biological tissues: a stretchable elastic ring under pressure. We prove the existence of a tricritical point and find bistability effects and hysteresis when the stretching modulus is sufficiently small. These results seem to be in qualitative agreement with some recent experiments on heart cells.
