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

: 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.

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]. For example, a fourth-order partially symmetric tensor with n = 2 or 3, called the elasticity tensor, can be used in the two/three-dimensional field equations for a homogeneous compressible nonlinearly elastic material for static problems without body forces [27]. 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 A · xy 2 = λx Ax 2 y· = λy x x = 1 y y = 1, 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 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. In Section 2, some preliminary results are recalled. In Section 3, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors. Numerical examples are proposed to verify the efficiency of the obtained results.

Preliminaries
In this section, we firstly introduce some definitions and important properties of elasticity M-tensors [24,26,27].
Lemma 1 (Theorem 1 of [27]). M-eigenvalues always exist. 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]). The M-spectral radius of any nonnegative tensor in E 4,n is exactly its greatest M-eigenvalue. Furthermore, there is a pair of nonnegative M-eigenvectors corresponding to the M-spectral radius.
In the following, we characterize M-eigenvectors of elasticity M-tensors without irreducibility conditions. Proof. Since A is a elasticity M-tensor, there exist a nonnegative tensor B ∈ E 4,n and a real number

Bounds for the Minimum M-Eigenvalue of Elasticity M-Tensors
In this section, we establish sharp bounds for τ M (A). We begin our work by collecting the information of max i,j∈N,i =j

Lemma 6.
For any x ∈ R n , if x 2 1 + x 2 2 + · · · + x 2 n = 1, where λ denotes Lagrange multiplier. For all i = j, deriving the above equation x i and x j respectively, we get 2x 2 Hence, we obtain x 2 i = x 2 j , λ = 1 2 . Particularly, set 2 Remark 1. For the right M-eigenvector y ∈ R n , we can establish similar conclusions that max Without irreducible conditions, we propose a sharp upper bound for the minimum M-eigenvalue of elasticity M-tensors.
Setting a feasible solution of (3) From Lemma 2 and (4), it holds that Next, we propose sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors.
Proof. Let τ M (A) be the minimum M-eigenvalue of A. By Lemma 5, there exist nonnegative left and right M-eigenvectors (x, y) corresponding to τ M (A). On one hand, setting Setting α p = min l∈N {a ppll }, by (5) and Lemma 6, one has It follows from (6) and definition of ω p that On the other hand, setting y t = max Letting β t = min i∈N {a iitt }, by (8) and Lemma 6, we have It follows from (9) and definition of m t that which shows From (7) and (10), the result follows. 2 Now, we are at a position to prove that the bound in Theorem 2 is tighter than that of Theorem 3.1 of [26].
Since α i − a iill ≤ 0 and On the other hand, it follows from Theorem 3.1 of [26] that Following the similar arguments in the proof of (11), we obtain It follows from (11) and (12) that 2 Choosing x q as a component of x with the second largest modulus, we obtain another sharp lower bound for τ M (A).  ((a pp11 y 2 1 + · · · + a ppnn y 2 n ) − τ M (A))x p = − n ∑ j, k, l = 1, k = l a pjkl x j y k y l − n ∑ j, l = 1, j = p a pjll x j y 2 l .
Setting α p = min l∈N {a ppll }, from (13) and Lemma 6, we obtain From the q-th equation of τ M (A)x = Axy 2 and α q = min l∈N {a qqll }, we yield which implies When Then, solving for τ M (A), we have where It follows from (17) and (18) that On the other hand, set y t ≥ y s ≥ max Letting β t = min i∈N {a iitt }, by (19) and Lemma 6, we obtain Using (20), we yield Recalling s-th equation of τ(A)y = Ax 2 y and β s = min which implies Then, solving for τ(A), we obtain where θ t,s (A) = (β t + 1 From (23) and (24), it holds that Thus, the desired result holds. 2 In the following, we use Example 3.1 of [26] to show the superiority of our results.
Example 1. Let A = (a ijkl ) ∈ E 4,2 be an elasticity M-tensor, whose entries are The bounds via different estimations given in the literature are shown in Table 1: By computations, we obtain that the minimum M-eigenvalue and corresponding with left and right M-eigenvectors are(τ M (A),x,ȳ) = (2.4534, (0.8398, 0.5430), (0.7071, 0.7071)). It is easy to see that the results given in Theorems 3.1-3.3 are sharper than some existing results [26]. It is noted that Theorems 2 and 3 have their own advantages. Theorem 3 can estimate the lower bound of the minimum M-eigenvalue more accurately, but the calculation of Theorem 2 is simpler.
Ding et al. [24] pointed out that a tensor is M-positive if and only if its smallest M-eigenvalue is positive. In the following, the results given in Theorems 2 and 3 can exactly check the positiveness of the elasticity M-tensor A.  Table 2. From Theorems 2 and 3, we obtain 0.77 ≤ τ M (A) ≤ 1.55, which shows that A is M-positive definite. However, the existing results of [26] cannot identify the M-positiveness of A.
For the medium-sized tensors, we show the validity of the estimations by our theorems.  Table 3.

Conclusions
In this paper, we exactly characterized the information of eigenvectors without irreducible conditions. Further, we proposed a new upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors by establishing new eigenvalue inequality. Numerical examples were proposed to verify the efficiency of the obtained results.