New S-Type Bounds of M-Eigenvalues for Elasticity Tensors with Applications

Zhuanzhou Zhang,Jun He,Zerong Ren,Yanmin Liu,Xian-Ming Gu

doi:10.1155/2021/9742682

Abstract

In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, S-type upper bounds for the M-eigenvalues of elasticity tensors are established. Finally, S-type sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the S-type M-eigenvalue inclusion sets.

Highlights

IntroductionLet A (aijkl) ∈ Rm×n×m×n be an elasticity tensor, if there exist nonzero vectors, x ∈ Rm and y ∈ Rn, and a real number λ ∈ R, such that
En, λ is called an M-eigenvalue of A, and the nonzero vectors x and y are called the corresponding M-eigenvectors
E following necessary and sufficient condition for strong ellipticity for general anisotropic elastic materials is presented by Han et al [20]

Summary

Introduction

Let A (aijkl) ∈ Rm×n×m×n be an elasticity tensor, if there exist nonzero vectors, x ∈ Rm and y ∈ Rn, and a real number λ ∈ R, such that. Let A (aijkl) ∈ Rm×n×m×n be an elasticity tensor and λ be an M-eigenvalue of A. en, max􏼈δ1, δ2􏼉 ≤ λ ≤ min􏼈θ1, θ2􏼉,. Let A (aijkl) ∈ Rm×n×m×n be an elasticity tensor and ρM(A) be the M-spectral radius of A. E following necessary and sufficient condition for strong ellipticity for general anisotropic elastic materials is presented by Han et al [20].

S-Type Upper Bounds

S-Type M-Eigenvalue Inclusion Sets and Strong Ellipticity Conditions