Abstract

Nonlinear elastic models are widely used to describe the elastic response of crystalline solids, for example, the well-known Cauchy–Born model. While the Cauchy–Born model only depends on the strain, effects of higher-order strain gradients are significant and higher-order continuum models are preferred in various applications such as defect dynamics and modeling of carbon nanotubes. In this paper we rigorously derive a higher-order nonlinear elasticity model for crystals from its atomistic description in one dimension. We show that, compared to the second-order accuracy of the Cauchy–Born model, the higher-order continuum model in this paper is of fourth-order accuracy with respect to the interatomic spacing in the thermal dynamic limit. In addition we discuss the key issues for the derivation of higher-order continuum models in more general cases. The theoretical convergence results are demonstrated by numerical experiments.

中文翻译：

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具有周期边界条件的原子链的高阶非线性弹性模型的先验分析

摘要

非线性弹性模型被广泛用于描述结晶固体的弹性响应，例如，著名的柯西-伯恩模型。尽管柯西—波恩模型仅取决于应变，但高阶应变梯度的影响非常明显，并且在各种应用（例如缺陷动力学和碳纳米管建模）中，高阶连续谱模型更为可取。在本文中，我们从一维原子的原子描述中严格得出了晶体的高阶非线性弹性模型。我们证明，与Cauchy-Born模型的二阶精度相比，本文的高阶连续谱模型在热力学极限中相对于原子间间距具有四阶精度。此外，我们讨论了在更一般的情况下推导高阶连续谱模型的关键问题。数值实验证明了理论收敛的结果。