Euler elastica as a Γ-limit of discrete bending energies of one-dimensional chains of atoms
Euler elastica as a Γ-limit of discrete bending energies of one-dimensional chains of atoms
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欧拉弹性作为一维原子链离散弯曲能的 Γ 极限
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
J. Wilber
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文献类型：
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作者：
Malena I. Español;D. Golovaty;J. Wilber
在 20 年代，Hencky 提出了一种离散弹性模型，描述由扭转弹簧连接的一系列相同的刚性杆。 Hencky 观察到，随着条形数量的增加而长度的减少，这种离散弹性模型会收敛到欧拉弹性模型，并且亨基的条链模型主要用作欧拉弹性模型的近似。 Hencky 型棒链模型也可以合并到 Frenkel-Kontorova 型离散原子模型中，其中接头和棒分别代表原子和原子间键，而整个原子链与底物或其他链相互作用。然后，可以通过采用适当的离散到连续极限来恢复与该 Frenkel-Kontorova 型模型相对应的连续系统的能量。为离散弹性体开发正确的限制程序可以建立该连续能量的弯曲分量。在本文中，我们使用 γ 收敛来严格证明，当我们考虑的离散弹性模型中的杆长度趋于 0 时，链的弯曲能 γ 收敛到与欧拉弹性相关的连续弯曲能。
In the 1920s, Hencky proposed a discrete elastica model describing a chain of identical rigid bars connected by torsional springs. Hencky observed that this discrete elastica model converges to Euler’s elastica as the number of bars increases while their lengths decrease, and Hencky’s bar-chain model has been used primarily as an approximation of Euler’s elastica. A Hencky-type bar-chain model can also be incorporated into a Frenkel–Kontorova-type discrete atomistic model, where the joints and bars represent the atoms and interatomic bonds, respectively, while the entire chain of atoms interacts with either a substrate or other chains. The energy of a continuum system corresponding to this Frenkel–Kontorova-type model can then be recovered by taking an appropriate discrete-to-continuum limit. Developing a correct limiting procedure for the discrete elastica establishes the bending component of this continuum energy. In this paper we use Γ-convergence to rigorously show that as the bar length in the discrete elastica model we consider goes to 0, the bending energies of the chain Γ-converge to the continuum bending energy associated with Euler’s elastica.