On the M-Eigenvalues of Elasticity Tensor and the Strong Ellipticity
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On the M-eigenvalues of elasticity tensor and the strong ellipticity condition Hua Xianga∗ Liqun Qib† Yimin Weic‡ a School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China b Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong c School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, P. R. of China January 23, 2018 Abstract Strong ellipticity is an important property in the elasticity theory. In 2009, M-eigenvalues were introduced for the elasticity tensor. It was shown that M-eigenvalues are invariant under coordinate system choices, and the strong ellipticity condition holds if and only if all the M-eigenvalues of the elasticity tensor are positive. Thus, M-eigenvalues are some in- trinsic parameters of the elasticity tensor. In this paper, we show that the M-eigenvalues of the elasticity tensor are closely related with some elastic moduli, such as the bulk modulus, the shear modulus, Lam´e’s first parameter, the P-wave modulus, etc, and the positiveness of the M- eigenvalues are corresponding to some existing conditions for strong ellip- ticity in some special cases, such as the isotropic case, the cubic case, the polar anisotropic case and the tetragonal case. We also present new suf- ficient conditions for the strong ellipticity of the orthotropic case. These, in a certain sense, further reveal the physical meanings of M-eigenvalues. Keywords. M-eigenvalue, Z-eigenvalue, Elasticity tensor, Strong ellipticity, Positive definiteness, Elastic modulus arXiv:1708.04876v2 [math.RA] 22 Jan 2018 1 Introduction The elasticity tensor is a fourth-rank tensor, namely a four-linear map : 3 3 3 3 C C R R R R R. With respect to the basis ∂i in a local coordinate × × × → { } system, the components of is expressed by cijkl = (∂i, ∂j , ∂k, ∂l), where C C ∗E-mail: [email protected]. H. Xiang is supported by the National Natural Science Foundation of China under grants 11571265, 11471253 and NSFC-RGC No.11661161017. †E-mail: [email protected]. L. Qi is supported by the Hong Kong Research Grant Council (Grant No. PolyU 15302114, 15300715, 15301716 and 15300717). ‡E-mail: [email protected]. Y. Wei is supported by the National Natural Science Foun- dation of China under grant 11771099 and International Cooperation Project of Shanghai Municipal Science and Technology Commission under grant 16510711200. 1 i,j,k,l 1, 2, 3 . With respect to the dual basis dxi , the elasticity tensor ∈ { } i j k l { } can be written as = cijkldx dx dx dx . Using these components, we can reformulate the tensorC as C i j k l (w,x,y,z)= wxyz = cijklw x y z , C C i i i i where w = w ∂i, x = x ∂i, y = y ∂i, z = z ∂i, and we adopt the Einstein convention of summing over the repeated indices. When working on the or- thonormal Cartesian coordinate system with a Euclidean metric given by the Kronecker delta, the covariant and contravariant types do not differ, and the indices can be upper or lower, and hence we have the option to work with only subscripts in the remainder of this article. For the fourth-order elasticity or stiffness tensor , the strong ellipticity is defined by the following function to be positive. C 3 f(x, y)= xyxy = Cijklxiyjxkyl > 0, x, y R . C ∀ ∈ Such strong ellipticity condition ensures that the governing differential equations for elastostatics problems be completely elliptic. It is an important property in the elasticity theory associated with uniqueness, instability, wave propagation, etc, and has been studied extensively [2, 3, 8, 9, 10, 11, 16, 17, 21, 22, 23, 25, 26, 27, 28, 31, 32, 33]. In 2009, Qi, Dai and Han [24] introduced the M-eigenvalues for the elas- ticity tensor. Here, the letter “M” means mechanics. It was shown that the strong ellipticity condition holds if and only if the optimal value of a global polynomial optimization problem is positive, which further naturally leads to the M-eigenvalue problem from the optimality condition. The M-eigenvalues θ of the elasticity tensor are defined as follows. C yxy = θx, (1) C· xyx = θy, (2) C · T T under the constraints x x = 1 and y y = 1. Here, ( yxy)i = Cijklyjxkyl and C· ( xyx )l = Cijklxiyjxk in the Einstein summation notation. It was shown that M-eigenvaluesC · are invariant under coordinate system choices, and the strong ellipticity condition holds if and only if all the M-eigenvalue of the elasticity tensor are positive [24]. Thus, M-eigenvalues are some intrinsic parameters of the elasticityC tensor. The M-eigenvalue problem can be reduced to the Z-eigenvalue problem. x3 = ηx, (3) C where xT x = 1, and η is the Z-eigenvalue. As the M-eigenvalue is associated with the strong ellipticity, the Z-eigenvalue is related to the rank-one positive definiteness of . All the Z-eigenvalues are M-eigenvalues of , but not vice versa. Obviously, theC positiveness of Z-eigenvalues provides the necessaC ry conditions for the strong ellipticity of material. We note that there exists the major symmetry Cijkl = Cklij , and the mi- nor symmetry Cijkl = Cjikl and Cijkl = Cijlk . The requirement of the sym- metry reduces the number of different elements to 21. Checking the invari- ance of under a specific transformation Q can further reduces this num- ber for someC special cases. If the material response is to be the same, i.e., 2 Cijkl = QimQjnQkpQlqCmnpq, then it yields the material symmetry relation, which will provide a system of equations that allows reduction in the number of independent elastic moduli [20, 30]. For example, as to an orthotropic material, it has three mutually perpendicular symmetry planes, and this type of material has nine independent material constants. Common examples of such materials include wood and fiber-reinforced composites. Omitting the zero elements, the M-eigenvalue problem (1)-(2) for this case reads C1111y1x1y1 + (C1122 + C1221)y1x2y2 + (C1133 + C1331)y1 x3y3 + C1313 y3x1y3 + C1212 y2x1y2 = θx1, (C2211 + C2112)y2x1y1 + C2222 y2x2y2 + (C2233 + C2332)y2 x3y3 + C2323 y3x2y3 + C2121 y1x2y1 = θx2, (C3311 + C3113)y3x1y1 + (C3322 + C3223)y3 x2y2 + C3333 y3x3y3 + C3232 y2x3y2 + C3131 y1x3y1 = θx3, C1111 x1y1x1 + (C2211 + C1221 )x2y2x1 + (C3311 + C1331 )x3y3x1 + C3131 x3y1x3 + C2121x2y1x2 = θy1 , (C1122 + C2112)x1y1x2 + C2222x2y2x2 + (C3322 + C2332 )x3y3x2 + C3232 x3y2x3 + C1212x1y2x1 = θy2 , (C1133 + C3113)x1y1x3 + (C2233 + C3223)x2y2x3 + C3333x3y3x3 + C2323 x2y3x2 + C1313x1y3x1 = θy3 . To keep accordance with the standard shorthand notation in the theory of elasticity, we need the Voigt notation or the Kelvin notation. Since the minor symmetries are valid, the fourth-order stiffness tensor Cijkl may be written as a6 6 stiffness matrix (cαβ) (a tensor of second order). Due to the major sym- metry,× this matrix is also symmetric. In the Voigt notation, each pair of indices is compressed into one index, and the standard mapping for tensor indices reads as follows. ij = 11 22 33 23, 32 13, 31 12, 21 α⇓ = ⇓⇓⇓⇓1234 ⇓5 ⇓6 For example, in a properly chosen coordinate system, the Voigt matrix of the orthotropic material has the following form. c11 c12 c13 0 0 0 c12 c22 c23 0 0 0 c13 c23 c33 0 0 0 (cαβ )= . 0 0 0 c44 0 0 0 0 0 0 c 0 55 0 0 0 0 0 c66 The elasticity tensor is described by nine parameters: c11 = C1111, c22 = C2222, c33 = C3333, c23 = C2233, c13 = C1133, c12 = C1122, c44 = C2323 = C3232, c55 = C1313 = C3131, and c66 = C1212 = C2121. Considering the symmetries in the elasticity tensor , we naturally introduce the following eigenproblem. C Cijklxkl = ζ xij , (4) where the eigentensor xij is symmetric. Such eigenproblem is closely related to the positive definiteness of , a less general hypothesis than the strong ellipticity, which also guarantees theC uniqueness of solutions in problems of elasticity [13, P.159]. The elasticity stiffness tensor must be positive definite [30, P.48][29, P.34]. That is, The strain energy densityC or elastic potential satisfies 1 C ǫ ǫ > 0, 2 ijkl ij kl 3 where ǫij is any symmetric strain tensor. It physically means that energy is needed to deform an elastic body from its unloaded equilibrium position [14]. The condition for the positiveness of strain energy can be replaced by the less restrictive condition Cijklγij γkl > 0, where the γij need not be symmetric [29, P.34]. Considering the symmetry of C, we rewrite (4) in its equivalent matrix form. C1111 C1122 C1133 √2C1123 √2C1113 √2C1112 x11 x11 C2211 C2222 C2233 √2C2223 √2C2213 √2C2212 x22 x22 C C C √2C √2C √2C x33 x33 3311 3322 3333 3323 3313 3312 = ζ . √2C √2C √2C 2C 2C 2C √2x √2x 2311 2322 2333 2323 2313 2312 23 23 √2C √2C √2C 2C 2C 2C √2x √2x 1311 1322 1333 1323 1313 1312 13 13 √ √ √2C1211 √2C1222 √2C1233 2C1223 2C1213 2C1212 2x12 2x12 The positiveness condition corresponds to impose that all the eigenvalues ζ are positive [30]. What’s more, the positive definiteness of implies strong ellipticity, while the converse statement is not true [13, P.159].C In other words, the positive definiteness conditions are sufficient for the strong ellipticity. In the following, we consider five further simplified cases of the orthotropic material: the isotropic case, the cubic case, the polar anisotropic case, the tetragonal case and the orthotropic case, which have two, three, five, six and nine independent elastic parameters respectively. In this paper, we will show that the M-eigenvalues are closely related with some elastic moduli, such as the bulk modulus, the shear modulus, Lam´e’s first parameter, the P-wave modulus, etc. In fact, all the elastic constants in the diagonal of stiffness matrix (cαβ) are M-eigenvalues.