#### 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.

#### 1. Introduction

Let and ; a real tensor is called an elasticity tensor, if

Consider the following optimization problem with an elasticity tensor [1, 2]:

Qi et al. introduced the following definition of M-eigenvalues of an elasticity tensor [3, 4].

Definition 1. (see [3, 4]). Let be an elasticity tensor, if there exist nonzero vectors, and , and a real number , such thatwhereThen, is called an M-eigenvalue of , and the nonzero vectors and are called the corresponding M-eigenvectors.

Qi in [3, 5, 6] presented some basic studies for tensor computations and approximations. Li et al. [710], Bu et al. [11], Che et al. [12], and Zhao et al. [13, 14] worked on analyzing the M-eigenvalues for various elasticity tensors. The authors in [15] proposed a tensor-based FTV model for the three-dimensional image deblurring problem, and some properties for Z-eigenvalues of tensor are given in [1618]. Letwhere and are symmetric matrices with entries

And, assume that is the minimal eigenvalue of a matrix , is the maximal eigenvalue of a matrix , and is the spectral radius of a matrix . In 2021, Li et al. established the following bounds for M-eigenvalues of an elasticity tensor.

Theorem 1. (see [19]). Letbe an elasticity tensor andbe an M-eigenvalue of. Then,whereand

Theorem 2. (see [19]). Letbe an elasticity tensor andbe the M-spectral radius of. Then,where

The following necessary and sufficient condition for strong ellipticity for general anisotropic elastic materials is presented by Han et al. [20].

Theorem 3. (see [20]). Letbe an elasticity tensor. The strong ellipticity condition holds, i.e.,for all nonzero vectors if and only if the smallest M-eigenvalue of is positive.

One application of the lower bound in Theorem 1 is to identify the strong ellipticity condition of an elasticity tensor, and the upper bound in Theorem 2 is given to accelerate convergence of the WQZ-algorithm [19]. In this paper, by breaking into disjoint subsets and its complement, new S-type upper bounds for the M-spectral radius of an elasticity tensor are given in Section 2. In Section 3, S-type sufficient conditions are also given to identify the strong ellipticity condition of an elasticity tensor.

#### 2. S-Type Upper Bounds

In this section, we give S-type upper bounds for the largest M-eigenvalues of an elasticity tensor, and the relationship between the S-type upper bounds and existed upper bounds is also established. The sets , , , and are defined by and , , and .

Theorem 4. Letbe an elasticity tensor andbe the M-spectral radius of. Then,where

Proof. Let be an M-eigenvalue of with the M-eigenvectors , ,Obviously, at least one of and is nonzero.
Case I. If , from the -th equation of we haveThen, we can getTaking modulus in the above equation, we haveThen,If , similarly we can getMultiplying (20) with (21), we haveTherefore,If , thenwhich means that (23) also holds.
Case II. . If , by inequality (5), then ; it yields that (7) also holds. If , by inequality (6), then ; it yields that (7) also holds.
Let and , from the -th equation of we haveand similarly, we can get

We compare the S-type upper bounds in Theorem 4 with the results in [19], which shows that our new S-type upper bounds are always tighter than the results in [19].

Theorem 5. Letbe an elasticity tensor. Then,

Proof. If , thenorWe only proof the following case, and the other case can be proved similarly. Iffrom the proof of Theorem 4,Let , , thenFrom inequalities (20) or (21), there is an with ; for this index , we haveand therefore,

In 2009, the following WQZ-algorithm was presented to compute the largest M-eigenvalue of an elasticity tensor [4].

 Step 0: given a tensor , vectors and . Set and , where with the entries as follows: Step 1: compute Output , . Step 2: find the largest M-eigenvalue of the tensor : where

The following example in [4] is taken to show that the tighter upper bound can accelerate convergence of the WQZ-algorithm.

Example 1. Consider the tensor of Example 4.1 in [4, 21], whereIn [4], is taken as follows:Let ; by Corollary 2 in [22], we haveBy Theorem 2, we haveLet ; by Theorem 4, we have

Example 2. Consider the elasticity tensor of CaMg(CO3)2-dolomite [21], whose nonzero entries areIn [4], is taken as follows:Let ; by Corollary 2 in [22], we haveBy Theorem 2, we haveLet ; by Theorem 4, we haveIn Figure 1, we can find that, when taking , the sequence generated in the WQZ-algorithm converges to the largest M-eigenvalue more rapidly than taking and .

#### 3. S-Type M-Eigenvalue Inclusion Sets and Strong Ellipticity Conditions

In this section, based on the S-type M-eigenvalue inclusion sets of an elasticity tensor, S-type sufficient conditions for strong ellipticity conditions are given. Let and , we need the following lemma.

Lemma 1. (see [23]). Letbe an elasticity tensor. Then, the strong ellipticity condition holds if and only if the matrix(or) is positive definite for each nonzero(or).

Theorem 6. Letbe an elasticity tensor andbe an M-eigenvalue ofwith the M-eigenvectors,. Then,where

Proof. Let be an M-eigenvalue of with the M-eigenvectors and ,Obviously, at least one of and is nonzero.
Case I. If , from the -th equation of we haveThen, we can getTaking modulus in the above equation, we haveThen,If , similarly we can getMultiplying (53) with (54), we haveso that . If ; then,which means that .
Case II. . Without loss of generality, let , by inequality (8), then ; it yields that .
Let and , from the -th equation of similarly we can get .

Theorem 7. Letbe an elasticity tensor. If there existsorsuch thatorthen the strong ellipticity condition holds.

Proof. Let be an M-eigenvalue of and . From Theorem 6, we obtain . If , there are and such thatorThen,andwhich contradicts . Therefore, . Then, by Theorem 3, the strong ellipticity condition holds for the elasticity tensor .
If , the second conclusion can be obtained similarly.

The following sufficient conditions for strong ellipticity are given by Li et al. [19].

Theorem 8. Letbe an elasticity tensor. Iforthen the strong ellipticity condition holds.

Based on the above theorems, we introduce the definitions strictly diagonally dominated (M-SDD) and S-type strictly diagonally dominated (M-SSDD) elasticity tensors, which are based on the eigenvalues of matrices of and .

Definition 2. Let be an elasticity tensor. Iforthen the elasticity tensor is called strictly diagonally dominated(M-SDD).

Definition 3. Let be an elasticity tensor. If there exists or such thatorthen the elasticity tensor is called S-type strictly diagonally dominated(M-SSDD).

Next, we give the relationships between the M-SDD elasticity tensor and the M-SSDD elasticity tensor.

Theorem 9. Letbe an elasticity tensor. Ifis an M-SDD elasticity tensor, thenis an M-SSDD elasticity tensor.

Proof. If is an M-SDD elasticity tensor, we only prove the following case; the other case can be proved similarly. For all ,Then, for all and ,which imply thatand then is an M-SSDD elasticity tensor.

Now, the following example is explored to show the efficiency of the results in Theorems 8 and 9.

Example 3. Let be an elasticity tensor, whereand other .
Obviously, we haveLet , by direct computation, we haveandThen, satisfies the sufficient conditions of Theorem 7, and the conditions of Theorem 8 do not hold by and . Therefore, the strong ellipticity condition holds for the elasticity tensor by Theorem 7. In fact, the smallest M-eigenvalue of is 3.5.
Let ; by Theorem 11 in [22], we havewhere are defined in [22], which shows that the conditions of Theorem 11 in [22] do not hold.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the New Academic Talents and Innovative Exploration Fostering Project (Qian Ke He Pingtai Rencai [2017]5727-21), Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2020]094 and [2022]017), Science and Technology Foundation of Guizhou Province (Qian Ke He Ji Chu ZK[2021] Yi Ban 014), joint science and technology fund project of Zunyi Science and Technology Bureau and Zunyi Normal University (Zun Shi Ke He HZ[2020]30 and [2020]27), Zunshi 2020 Academic New Talent Cultivation and Innovation Exploration Project (Zunshi XM [2020] no. 1-12), and Zunshi He Difangchanye (Zi[2020]07).