On the variational limit of a class of nonlocal functionals related to peridynamics
On the variational limit of a class of nonlocal functionals related to peridynamics
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DOI：
10.1088/0951-7715/28/11/3999
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发表时间：
2015-10
期刊：
影响因子：
1.7
通讯作者：
T. Mengesha;Q. Du
中科院分区：
文献类型：
--
作者：
T. Mengesha;Q. Du
本文研究了有界区域上具有周向型非局部弹性能量的变分问题，其结果是非线性非局部方程组。变分问题的适定性是通过仔细研究相关的能量空间建立的。在非局部性消失的情况下，我们用Γ？>-收敛在现有技术的基础上，我们证明了一个Lp-紧性结果（有界域）的基础上，用于研究各种体积约束的最小化问题的变分极限的近边界估计。对于适当形式的能量泛函，我们明确地找到了相应的极限能量。作为一种特殊情况，经典的Navier-Lamé势能被实现为线性化周波能量的极限，为小的均匀应变提供了非局部周波模型与经典力学之间的严格联系。
In this paper we study variational problems on a bounded domain for a nonlocal elastic energy of peridynamic-type which result in nonlinear systems of nonlocal equations. The well-posedness of variational problems is established via a careful study of the associated energy spaces. In the event of vanishing nonlocality we establish the convergence of the nonlocal energy to a corresponding local energy using the method of Γ ?>-convergence. Building upon existing techniques, we prove an Lp-compactness result (on bounded domains) based on near-boundary estimates that is used to study the variational limit of minimization problems subject to various volumetric constraints. For energy functionals in suitable forms, we find the corresponding limiting energy explicitly. As a special case, the classical Navier-Lamé potential energy is realized as a limit of linearized peridynamic energy offering a rigorous connection between the nonlocal peridynamic model to classical mechanics for small uniform strain.