Abstract
In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, Stype upper bounds for the Meigenvalues of elasticity tensors are established. Finally, Stype sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the Stype Meigenvalue 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 Meigenvalues 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 Meigenvalue of , and the nonzero vectors and are called the corresponding Meigenvectors.
Qi in [3, 5, 6] presented some basic studies for tensor computations and approximations. Li et al. [7–10], Bu et al. [11], Che et al. [12], and Zhao et al. [13, 14] worked on analyzing the Meigenvalues for various elasticity tensors. The authors in [15] proposed a tensorbased FTV model for the threedimensional image deblurring problem, and some properties for Zeigenvalues of tensor are given in [16–18]. 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 Meigenvalues of an elasticity tensor.
Theorem 1. (see [19]). Letbe an elasticity tensor andbe an Meigenvalue of. Then,whereand
Theorem 2. (see [19]). Letbe an elasticity tensor andbe the Mspectral 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 Meigenvalue 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 WQZalgorithm [19]. In this paper, by breaking into disjoint subsets and its complement, new Stype upper bounds for the Mspectral radius of an elasticity tensor are given in Section 2. In Section 3, Stype sufficient conditions are also given to identify the strong ellipticity condition of an elasticity tensor.
2. SType Upper Bounds
In this section, we give Stype upper bounds for the largest Meigenvalues of an elasticity tensor, and the relationship between the Stype 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 Mspectral radius of. Then,where
Proof. Let be an Meigenvalue of with the Meigenvectors , ,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 Stype upper bounds in Theorem 4 with the results in [19], which shows that our new Stype 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 WQZalgorithm was presented to compute the largest Meigenvalue of an elasticity tensor [4].

The following example in [4] is taken to show that the tighter upper bound can accelerate convergence of the WQZalgorithm.
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)2dolomite [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 WQZalgorithm converges to the largest Meigenvalue more rapidly than taking and .
3. SType MEigenvalue Inclusion Sets and Strong Ellipticity Conditions
In this section, based on the Stype Meigenvalue inclusion sets of an elasticity tensor, Stype 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 Meigenvalue ofwith the Meigenvectors,. Then,where
Proof. Let be an Meigenvalue of with the Meigenvectors 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 Meigenvalue 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 (MSDD) and Stype strictly diagonally dominated (MSSDD) 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(MSDD).
Definition 3. Let be an elasticity tensor. If there exists or such thatorthen the elasticity tensor is called Stype strictly diagonally dominated(MSSDD).
Next, we give the relationships between the MSDD elasticity tensor and the MSSDD elasticity tensor.
Theorem 9. Letbe an elasticity tensor. Ifis an MSDD elasticity tensor, thenis an MSSDD elasticity tensor.
Proof. If is an MSDD 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 MSSDD 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 Meigenvalue 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]572721), 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. 112), and Zunshi He Difangchanye (Zi[2020]07).