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. [7–10], 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 [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 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].
| ||||||||||
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).