×

zbMATH — the first resource for mathematics

On the averaging of symmetric positive-definite tensors. (English) Zbl 1094.74010
Summary: We present properly invariant averaging procedures for symmetric positive-definite tensors which are based on different measures of nearness of symmetric positive-definite tensors. These procedures intrinsically account for the positive-definite property of the tensors to be averaged. They are independent of the coordinate system, preserve material symmetries, and more importantly, they are invariant under inversion. The results of these averaging methods are compared with the results of other methods including that proposed by S. Cowin and G. Yang [J. Elasticity 46, No. 2, 151–180 (1997; Zbl 0902.73006)] for the case of the elasticity tensor of generalized Hooke’s law.

MSC:
74B05 Classical linear elasticity
15B48 Positive matrices and their generalizations; cones of matrices
15A72 Vector and tensor algebra, theory of invariants
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] K. Aleksandrov and L. Aisenberg, A method of calculating the physical constants of polycrystalline materials. Sov. Phys. Dokl. 11 (1966) 323–325.
[2] T. Ando, C.-K. Li and R. Mathias, Geometric means. Linear Algebra Appl. 385 (2004) 305–334. · Zbl 1063.47013
[3] P.G. Batchelor, M. Moakher, D. Atkinson, F. Calamante and A. Connelly, A rigorous framework for diffusion tensor calculus. Magn. Reson. Med. 53 (2005) 221–225.
[4] R. Bhatia, Matrix Analysis. Springer, Berlin Heidelberg New York (1997). · Zbl 0863.15001
[5] S.C. Cowin and M.M. Mehrabadi, On the structure of the linear anisotropic symmetries. J. Mech. Phys. Solids 40 (1992) 1459–1472. · Zbl 0837.73013
[6] S.C. Cowin and G. Yang, Averaging anisotropic elastic constant data. J. Elast. 46 (1999) 151–180. · Zbl 0902.73006
[7] R.F.S. Hearmon, The elastic constants of piesoelectric crystals. J. Appl. Phys. 3 (1952) 120–123.
[8] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978). · Zbl 0451.53038
[9] R. Hill, The elastic properties of crystalline aggregate. Proc. Phys. Soc. A 65 (1952) 349–354.
[10] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Transl. from Russian by G.A. Yosifian. Springer, Berlin Heidelberg New York (1994).
[11] C.S. Kenney and A.J. Laub, Condition estimates for matrix functions. SIAM J. Matrix Anal. Appl. 10 (1989) 191–209. · Zbl 0684.65039
[12] S. Kullback and R.A. Leibler, On information and sufficiency. Ann. Math. Stat. 22 (1951) 79–86. · Zbl 0042.38403
[13] S. Lang, Fundamentals of Differential Geometry. Springer, Berlin Heidelberg New York (1999). · Zbl 0932.53001
[14] S. Matthies and M. Humbert, On the principle of a geometric mean of even-rank symmetric tensors for textured polycrystals. J. Appl. Crystallogr. 28 (1995) 254–266.
[15] M.M. Mehrabadi, S.C. Cowin and J. Jaric, Six-dimensional orthogonal tensor representation of the rotation about an axis in three dimensions. Int. J. Solids Struct. 32 (1995) 439–449. · Zbl 0865.73006
[16] M.M. Mehrabadi and S.C. Cowin, Eigentensors of linear anisotropic elastic materials. Quart. J. Mech. Appl. Math. 43(1) (1990) 15–41. · Zbl 0698.73002
[17] M. Moakher, Means and averaging in the group of rotations. SIAM J. Matrix Anal. Appl. 24 (2002) 1–16. · Zbl 1028.47014
[18] M. Moakher, A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26 (2005) 735–747. · Zbl 1079.47021
[19] M. Moakher and P.G. Batchelor, Symmetric positive definite matrices: From geometry to applications and visualization. Chapter 17 In: J. Weickert and H. Hagen (eds.), Visualization and Image Processing of Tensor Fields. Springer, Berlin Heidelberg New York (2005).
[20] A. Morawiec, Calculation of polycrystal elastic constants from single crystal data. Phys. Status Solidi, B 154 (1989) 535–541.
[21] P.R. Morris, Averaging fourth-rank tensors with weight functions. J. Appl. Phys. 40 (1969) 447–448.
[22] W. Pusz and S.L. Woronowicz, Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8 (1975) 159–170. · Zbl 0327.46032
[23] A. Reuss, Berechnung der Fliebgrenze von Mischkristallen auf grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech. 9 (1929) 49–58. · JFM 55.1110.02
[24] S. Smith, Optimization techniques on Riemannian manifolds. In: Fields Institute Communications, Vol. 3, AMS (1994) pp. 113–146. · Zbl 0816.49032
[25] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications II. Springer, Berlin Heidelberg New York (1988). · Zbl 0668.10033
[26] G.E. Trapp, Hermitian semidefinite matrix means and related matrix inequalities–an introduction. Linear and Multilinear Algebra 16 (1984) 113–123. · Zbl 0548.15013
[27] W. Voigt, Ueber die Beziehung zwischen den beiden Elasticitätconstanten isotroper Körper. Ann. Phys. (Leipzig). 38 (1889) 573–587. · JFM 21.1039.01
[28] W. Voigt, Lehrbuch der Kristallphysik. Teubner, Leipzig (1928).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.