Non-locality as a regularization mechanism in elastodynamics

IF 1.9 4区 工程技术 Q3 MECHANICS
J.R. Fernández , R. Quintanilla
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引用次数: 0

Abstract

It is well known that the solutions to the elastodynamic problem do not satisfy continuous dependence properties on the initial values, and/or supply terms, when the elastic tensor fails to be positive. In fact, the behavior of the solutions can be very explosive since the elements of the spectrum can go to infinite. Therefore, it is very relevant to identify thermomechanical mechanisms regularizing the behavior of the solutions. So, the main aim of this note is to show, from an analytical point of view, how the non-locality, in the sense of Eringen, is a mechanism satisfying this property of regularization of the solutions. It is worth noting that such system has not been previously studied from an analytical point of view. We firstly obtain the existence of the solutions to this problem, even when we do not assume any positivity on the elastic tensor. This result is proved with the help of the linear semigroups theory; however, even with these regularizing effects, the solutions to this problem are unstable. A particular easy one-dimensional problem is also considered. The extension of the existence and instability results to the thermoelastic case is pointed out later. Finally, we also study the spatial behavior of the solutions to the problem in the case that the region is a semi-infinite cylinder, and we obtain a Phragmen–Lindelöf alternative of the exponential type. This result is also relevant because a similar result, without considering regularizing terms, is unknown if the elastic tensor is not positive definite.

作为弹性动力学正则化机制的非位置性
众所周知,当弹性张量不为正时,弹性力学问题的解不满足初始值和/或供给项的连续依赖特性。事实上,由于频谱元素可以达到无限大,解的行为可能非常具有爆炸性。因此,找出规范解的行为的热力学机制非常重要。因此,本论文的主要目的是从分析的角度来说明,艾林根意义上的非局部性是如何满足解的正则化这一特性的。值得注意的是,以前从未从分析角度研究过这种系统。我们首先得到了这个问题的解的存在性,即使我们不假设弹性张量的任何正向性。这一结果是在线性半群理论的帮助下证明的;然而,即使有这些正则化效应,该问题的解也是不稳定的。我们还考虑了一个特别简单的一维问题。随后,我们指出了存在性和不稳定性结果在热弹性情况下的扩展。最后,我们还研究了在区域为半无限圆柱体的情况下问题解的空间行为,并得到了指数型的 Phragmen-Lindelöf 替代结果。这一结果也很重要,因为如果弹性张量不是正定的,那么在不考虑正则项的情况下,类似的结果也是未知的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
4.10
自引率
4.20%
发文量
114
审稿时长
9 months
期刊介绍: Mechanics Research Communications publishes, as rapidly as possible, peer-reviewed manuscripts of high standards but restricted length. It aims to provide: • a fast means of communication • an exchange of ideas among workers in mechanics • an effective method of bringing new results quickly to the public • an informal vehicle for the discussion • of ideas that may still be in the formative stages The field of Mechanics will be understood to encompass the behavior of continua, fluids, solids, particles and their mixtures. Submissions must contain a strong, novel contribution to the field of mechanics, and ideally should be focused on current issues in the field involving theoretical, experimental and/or applied research, preferably within the broad expertise encompassed by the Board of Associate Editors. Deviations from these areas should be discussed in advance with the Editor-in-Chief.
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