Some inequalities for the minimum M-eigenvalue of elasticity M-tensors

He, Jingsong; Xu, Guangjun; Liu, Yanmin

doi:10.3934/jimo.2019092

Cited by 10 publications

(14 citation statements)

References 16 publications

(23 reference statements)

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“…In this section, we firstly introduce some definitions and important properties of elasticity M-tensors [24,26,27].…”

Section: Preliminariesmentioning

confidence: 99%

“…If x and y are left and right M-eigenvectors of A, associated with an M-eigenvalue λ, then then λ = Ax 2 y 2 . Lemma 2 (Lemma 2.3 of [26]). Let A = (a ijkl ) ∈ E 4,n be an irreducible elasticity tensor and τ M (A) be the minimal M-eigenvalues of A, then Lemma 4 (Theorem 4.1 of [24]).…”

Section: Preliminariesmentioning

confidence: 99%

“…A fourth-order partially symmetric tensor is useful in nonlinear elastic material analysis [3,[16][17][18][19][20][21][22][23][24][25][26][27]. Ostrosablin [16] first constructed a complete system of eigentensors for the fourth rank tensor of elastic modulus, and Nikabadze [18] generalized these results and constructed a full system of eigentensors for a tensor of any even rank, as well as a complete system of eigentensor-columns for a tensor-block matrix of any even rank [22,23].…”

Section: Introductionmentioning

confidence: 99%

“…To identify the strong ellipticity in elastic mechanics, Han et al [25] introduced M-eigenvalues of a fourth-order partially symmetric tensor. For λ ∈ R, x, y ∈ R n , if Based on structural properties of elasticity M-tensors, He et al [26] proposed some bounds for the minimum M-eigenvalue under irreducible conditions. However, some of information eigenvectors x on elasticity M-tensors is not fully mined, such as max i,j∈N,i =j |x i ||x j | ≤ 1 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Meanwhile, irreducibility is a relatively strict condition for elasticity M-tensor. Inspired by these observations, we want to present sharp bounds for the minimum M-eigenvalue of elasticity M-tensors by describing eigenvectors precisely without irreducible conditions, which improve existing results in [26]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Sharp Bounds on the Minimum M-Eigenvalue of Elasticity M-Tensors

Zhang¹,

Sun

Wang

2020

Mathematics

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The M-eigenvalue of elasticity M-tensors play important roles in nonlinear elastic material analysis. In this paper, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors without irreducible conditions, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.where (A · xy 2 ) i = ∑ j,k,l∈ [n] a ijkl x j y k y l , (Ax 2 y·) l = ∑ i,j,k∈ [n] a ijkl x i x j y k , then the scalar λ is called an M-eigenvalue of the tensor A, and x and y are called left and right M-eigenvectors of A associated with the M-eigenvalue. Then the M-spectral radius of A is denoted byRecently, tensors with special structures, such as nonnegative tensors, M-tensors and H-tensors, are becoming the focus in recent research [2,24,[28][29][30][31]. Some effective algorithms for finding eigenvalue and the corresponding eigenvector have been implemented [1,24,[32][33][34][35]. For example, Bozorgmanesh et al. [32] propose an algorithm that can solve E-eigenvalue problem faster. However, it is very difficult for these algorithms to compute all M-eigenvalues or E-eigenvalues. Thus, some researchers turned to investigating eigenvalue inclusion sets [4,7,[36][37][38][39][40][41]. Particularly, some bounds for the minimum H-eigenvalue of nonsingular M-tensors have been proposed [2,28,30,42,43]. Ding et al. [24] introduced a structured partially symmetric tensor named elasticity M-tensors and established important properties of elasticity M-tensors and nonsingular elasticity M-tensors.

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“…In this section, we firstly introduce some definitions and important properties of elasticity M-tensors [24,26,27].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%