Sharp Bounds for the Smallest M-eigenvalue of an Elasticity Z-tensor and Its Application

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-05-30 DOI:10.1007/s40840-024-01698-0

Xifu Liu, Jianxing Zhao

引用次数: 0

Abstract

The smallest M-eigenvalue \(\tau _M ({\mathcal {A}})\) of a fourth-order partial symmetric tensor \({\mathcal {A}}\) plays an important role in judging the strong ellipticity condition (abbr. SE-condition) in elastic mechanics. Specifically, if \(\tau _M ({\mathcal {A}})>0\), then the SE-condition of \({\mathcal {A}}\) holds. In this paper, we establish lower and upper bounds of \(\tau _M ({\mathcal {A}})\) via extreme eigenvalues of symmetric matrices and tensors constructed by the entries of \({\mathcal {A}}\). In addition, when \({\mathcal {A}}\) is an elasticity Z-tensor, we establish lower bounds for \(\tau _M ({\mathcal {A}})\) via the extreme C-eigenvalues of piezoelectric-type tensors. Finally, numerical examples show the efficiency of our proposed bounds in judging the SE-condition of \({\mathcal {A}}\).

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弹性 Z 张量最小 M 特征值的锐界及其应用

四阶偏对称张量\({\mathcal {A}}\)的最小M特征值（\(\tau _M ({\mathcal {A}})>0\）在判断弹性力学中的强椭圆性条件（SE-condition）时起着重要作用。具体来说，如果\(\tau _M ({\mathcal {A}})>0\)，那么\({\mathcal {A}}\)的SE条件成立。在本文中，我们通过对称矩阵的极值特征值和由\({\mathcal {A}})条目构造的张量，建立了\(\tau _M ({\mathcal {A}})的下界和上界。）此外，当 \({\mathcal {A}}\) 是弹性 Z 张量时，我们通过压电型张量的极 C 特征值建立了 \(\tau _M ({\mathcal {A}})\) 的下限。最后，数值示例显示了我们提出的边界在判断 \({\mathcal {A}}\) 的 SE 条件时的效率。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.