On the derivation of linear elasticity from atomistic models
On the derivation of linear elasticity from atomistic models
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从原子模型推导线弹性
DOI：
10.3934/nhm.2009.4.789
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
B. Schmidt
中科院分区：
文献类型：
--
作者：
B. Schmidt
我们分别从原子模型中导出了当原子数趋于无穷大和原子间距离趋于零时的线弹性能量泛函。我们的方法推广了Braides，Solci和Vitali（2）的最近结果。特别是，我们研究了质量弹簧模型与完整的最近和次最近对相互作用。我们也考虑边界的一部分是自由的边值问题。1.导论.从离散原子模型到连续介质理论的过渡是当前连续介质力学研究的一个活跃领域。对于弹性系统，人们通常参考柯西-玻恩规则从原子相互作用泛函获得宏观能量密度。柯西-玻恩规则指出，粗略地说，每个原子都遵循宏观形变梯度，特别是不考虑微观尺度上的细尺度振荡。对于二维质量弹簧模型，Friesecke和Theil在（8）中证明了柯西-玻恩规则对于接近刚性运动的变形的有效性。Conti、Dolzmann、Kirchheim和Muller在（3）中将他们的结果推广到了任意维度。如果变形梯度非常接近SO（d），即保持刚性运动的取向集，那么我们期望应用线性弹性理论。Dal Maso、Negri和Percivale在（5）中导出了线性弹性的能量泛函作为小位移非线性弹性的极限，从而使这个关系变得严格。(See也是作者的文章（10）关于相关极小值问题的强收敛结果。最近人们注意到，人们可以直接从某些原子对势导出线性弹性泛函：对于一类特殊的对相互作用模型，Braides，Solci和Vitali证明了离散能量泛函收敛于相关连续线性弹性能量泛函的能量泛函（见（2））。在这种设置中，必须处理两个小参数e和δ，分别测量典型的原子间距离和变形与刚性运动集合的局部距离。本文的目的是在三个方向上扩展这些结果。首先，我们将放弃这样的假设，即原子只允许沿着相互作用
We derive linear elastic energy functionals from atomistic models as a -limit when the number of atoms tends to infinity, respec tively, when the interatomic distances tend to zero. Our approach generalizes a recent result of Braides, Solci and Vitali (2). In particular, we study mass spring models with full nearest and next-to-nearest pair interactions. We also consider boundary value problems where a part of the boundary is free. 1. Introduction. The passage from discrete atomic models to continuum theories is an active area of current research in continuum mechanics. For elastic systems one usually refers to the Cauchy-Born rule to obtain macroscopic energy densities from atomistic interaction functionals. The Cauchy-Born rule states that - roughly speaking - each individual atom follows the macroscopic deformation gradient and in particular does not take into account fine scale oscillations on the microscopic scale. For a two-dimensional mass spring model, the validity of the Cauchy-Born rule for deformations close to a rigid motion has been proved by Friesecke and Theil in (8). Their result has been generalized to arbitrary dimensions by Conti, Dolzmann, Kirchheim and Muller in (3). If the deformation gradients are very close to SO(d), the set of orientation pre- serving rigid motions, then we expect linear elasticity theory to apply. This relation has been made rigorous by Dal Maso, Negri and Percivale who derive the energy functional of linear elasticity as a -limit of nonlinear el asticity for small displace- ments in (5). (See also the author's article (10) for a strong convergence result for the associated minimum problems.) Recently it has been noted that one can derive linear elasticity functionals directly from certain atomistic pair potentials: For a special class of pair interaction models Braides, Solci and Vitali prove -convergence of the discre te energy functionals to the energy functional of an associated continuum linear elasticity energy functional (see (2)). In this set-up one has to deal with two small parameters e and δ measuring the typical interatomic distance and the local distance of the deformations from the set of rigid motions, respectively. The aim of the present article is to extend these results in three directions. Firstly, we will drop the assumption that atoms are allowed to interact only along