New lower bounds for the minimum M-eigenvalue of elasticity M-tensors and applications

M-eigenvalues of elasticity M-tensors play an important role in nonlinear elasticity and materials. In this paper, we present several new lower bounds for the minimum M-eigenvalue of elasticity M-tensors and propose numerical examples to illustrate the efficiency of the obtained results. As applications, we provide several checkable sufficient conditions for the strong ellipticity and positive definiteness of irreducible elasticity M-tensors.


Introduction
A tensor A = (a ijkl ) ∈ E 4,n is called a fourth-order real partially symmetric tensor if a ijkl = a jikl = a ijlk , i, j, l, k ∈ [n], where [n] = {1, 2, . . . , n}. The tensor of elastic moduli for a linearly anisotropic elastic solid is a fourth-order real partially symmetric tensor [1], and the components of such a tensor are considered as the coefficients of the following optimization problem: [n] a ijkl x i x j y k y l , s.t. x T x = 1, y T y = 1, x, y ∈ R n .
(1.1) Problem (1.1) has applications in the ordinary ellipticity and strong ellipticity and nonlinear elastic materials analysis . The strong ellipticity condition is stated as f (x, y) > 0 for all nonzero vectors x, y ∈ R n , which guarantees the existence of solutions of basic boundary-value problems of elastostatics and ensures an elastic material to satisfy some mechanical properties [29]. In fact, the KKT condition of (1.1) can be regarded as the following definition of M-eigenvalues.

Definition 1.1 ([1]) Let
A ∈ E 4,n . If there are λ ∈ R and x, y ∈ R n \{0} such that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Axy 2 = λx, Ax 2 y = λy, x T x = 1, y T y = 1, (1.2) where (Axy 2 ) i = j,k,l∈ [n] a ijkl x j y k y l , and (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 A, and x, y are called the corresponding left and right Meigenvectors of A, respectively. Furthermore, Han et al. revealed that the strong ellipticity condition holds if and only if the smallest M-eigenvalue is positive [1]. Recently, Ding et al. [30] investigated a fourthorder structured partially symmetric tensors named elasticity M-tensors, and some sufficient conditions for the strong ellipticity were provided. Since the strong ellipticity condition and M-positive definiteness can be identified by the smallest M-eigenvalue, He et al. [31] proposed some lower bounds for the minimum M-eigenvalue of elasticity M-tensors.
In this paper, we present several new bounds for the minimum M-eigenvalue of elasticity M-tensors. We prove that the bounds are tighter than those proposed in [31]. Numerical examples illustrate the efficiency of the obtained results. As applications, we give some checkable sufficient conditions for the strong ellipticity and positive definiteness of elasticity tensors.

Main results
For an elasticity tensor A ∈ E 4,n , its M-spectral radius is denoted by The identity tensor I = (e ijkl ) ∈ E 4,n is defined by |a ijll | , To continue, we need the following definitions and technical results.
If A is not reducible, then we say that A is irreducible. {a iill }.
Proof By Theorem 2.1 suppose that x = {x i } n i=1 > 0 ∈ R n and y = {y l } n l=1 > 0 ∈ R n are the corresponding left and right M-eigenvectors, respectively. Let a ppll x p y 2 l , that is, Furthermore, Multiplying (2.1) and (2.2), we have which means that that is, a iitt x 2 i y t .
Multiplying (2.4) and (2.5), we have which means that . Then the conclusion follows.
Next, we compare the bound in Theorem 2.3 with that in Theorem 2.4 and obtain the following conclusion. Thus which means that From (2.9) we have Then Therefore In what follows, we propose another lower bound for τ (A).

Theorem 2.6 Let
Proof a ijkt x i x j y k a iitt x 2 i y t .
Let β t = min i∈ [n] {a iitt }. It follows from Theorem 2.2 that that is, , multiplying (2.6) and (2.13), we have which means that Next, we compare the bound in Theorem 2.3 with that in Theorem 2.6 and obtain the following result. Since which means that Since which means that β l -C l (A) , and the desired result follows.
The following example shows the superiority of the conclusions obtained in Theorems 2.4 and 2.6.    Their comparison is drawn in Fig. 1, which reveals that our bounds are tighter than those of [31].  Hence A is positive definite, and the strong ellipticity condition holds.
The following example reveals that Theorems 3.1 and 3.2 can identify the positive definiteness of elasticity M-tensors.

Availability of data and materials
The data used to support the findings of this study are available from the corresponding author upon request.