Novak I, Truskinovsky L
Department of Cell Biology, University of Connecticut, Farmington, CT 06030, USA.
Physique et Mécanique des Milieux Hétérogènes, CNRS-UMR 7636, PSL, ESPCI, 10 Rue Vauquelin, 75005 Paris, France
Philos Trans A Math Phys Eng Sci. 2017 May 13;375(2093). doi: 10.1098/rsta.2016.0160.
The complexity of fracture-induced segmentation in elastically constrained cohesive (fragile) systems originates from the presence of competing interactions. The role of discreteness in such phenomena is of interest in a variety of fields, from hierarchical self-assembly to developmental morphogenesis. In this paper, we study the analytically solvable example of segmentation in a breakable mass-spring chain elastically linked to a deformable lattice structure. We explicitly construct the complete set of local minima of the energy in this prototypical problem and identify among them the states corresponding to the global energy minima. We show that, even in the continuum limit, the dependence of the segmentation topology on the stretching/pre-stress parameter in this problem takes the form of a devil's type staircase. The peculiar nature of this staircase, characterized by locking in rational microstructures, is of particular importance for biological applications, where its structure may serve as an explanation of the robustness of stress-driven segmentation.This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications.'
在弹性约束的内聚（易碎）系统中，骨折诱导分割的复杂性源于竞争相互作用的存在。离散性在这类现象中的作用在从分层自组装到发育形态发生的各种领域都备受关注。在本文中，我们研究了一个与可变形晶格结构弹性相连的易碎质量 - 弹簧链中的分割问题，该问题具有解析解。我们明确构建了这个典型问题中能量局部极小值的完整集合，并在其中识别出对应全局能量极小值的状态。我们表明，即使在连续极限情况下，该问题中分割拓扑对拉伸/预应力参数的依赖呈现出魔鬼阶梯的形式。这种阶梯的特殊性质，其特点是锁定在有理微结构中，对于生物学应用尤为重要，在生物学中其结构可用于解释应力驱动分割的稳健性。本文是主题为“通过复杂介质中的不稳定性进行图案化：理论与应用”的特刊的一部分。
