双稳态晶格的静态、相容性和相变
Still states of bistable lattices, compatibility, and phase transition
摘要：
二维双稳态晶格是由非线性双稳态杆组成的周期性三角形网络。每个杆的能量是分段二次的，并且有两个极小值。因此，当杆的伸长达到临界值时，杆经历可逆的相变。我们研究了这种格的能量最小化问题。目标是描述当网络中节点数趋于无穷大时系统的有效能量。有效能量最重要的特征是它的“平底”。这意味着对于应变空间中的某个三维集合内的所有应变，有效能量密度为零。平底的出现是因为微观离散模型有大量的不带力的变形状态。我们称这种变形为静态。在本文中，我们提出了一个完整的“平底”集的网络参数的特征。这是通过构建一个家庭的静止状态，其平均应变密集填充集的问题。二维的情况比以前研究的一维链的情况更困难，因为二维网络中的伸长必须满足某些相容性条件，而这些条件在一维情况下不会出现。我们推导出这些条件的小和任意的变形。对于小的变形，提供了完整的分析。
英文摘要：
A two-dimensional bistable lattice is a periodic triangular network of non-linear bi-stable rods. The energy of each rod is piecewise quadratic and has two minima. Consequently, a rod undergoes a reversible phase transition when its elongation reaches a critical value. We study an energy minimization problem for such lattices. The objective is to characterize the effective energy of the system when the number of nodes in the network approaches infinity. The most important feature of the effective energy is its "flat bottom". This means that the effective energy density is zero for all strains inside a certain three-dimensional set in the strain space. The flat bottom occurs because the microscopic discrete model has a large number of deformed states that carry no forces. We call such deformations still states. In the paper, we present a complete characterization of the "flat bottom" set in terms of the parameters of the network. This is done by constructing a family of still states whose average strains densely fill the set in question. The two-dimensional case is more difficult than the previously studied case of one-dimensional chains, because the elongations in two-dimensional networks must satisfy certain compatibility conditions that do not arise in the one-dimensional case. We derive these conditions for small and arbitrary deformations. For small deformations a complete analysis is provided.
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