On anisotropic invariants of a symmetric tensor: Crystal classes, quasi-crystal classes and others

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 454, Issue 1972, 1998, Pages 1217-1240

  Xiao, H ^a,b  

^a RUHR UNIVERSITY BOCHUM   (Germany)

^b PEKING UNIVERSITY   (China)

Author keywords

Anisotropy; Crystal classes; Invariants; Irreducible functional bases; Quasi crystal classes; Symmetric tensor

Indexed keywords

EID: 0038447567     PISSN: 13645021     EISSN: None     Source Type: Journal    
DOI: 10.1098/rspa.1998.0203     Document Type: Article

References (40)

1. Adkins, J. E. 1958 Dynamic properties of resilient materials: constitutive equations. Phil. Trans. R. Soc. Land. A 250, 519-534.
2. Adkins, J. E. 1960a Symmetry relations for orthotropic and transversely isotropic materials. Arch. Ration. Mech. Analysis 4, 193-213.
3. Adkins, J. E. 19G06 Further symmetry relations for transversely isotropic materials. Arch. Ration. Mech. Analysis 5, 263-274.
4. Basista, M. 1985 Tensor functions representations as applied to deriving constitutive relations for skewed anisotropy. Z. Angew. Math. Mech. 65, 151-158.
5. Bass, J. 1968 Cours de mathématiques, vol. I. Paris: Masson.
6. Boehler, J. P. 1979 A simple derivation of representations for non-polynomial constitutive equa-tions in some cases of anisotropy. Z. Angew. Math. Mech. 59, 157-167.
7. Boehler, J. P. (ed.) 1987 Applications of tensor functions in solid mechanics. CISM Courses and Lectures no. 292. Berlin: Springer.
8. Bringen, A. C. & Maugin, G. A. 1989 Electrodynamics of continua. I. Foundations and solid media. Berlin: Springer.
9. Green, A. E. & Adkins, J. E. 1960 Large elastic deformations and nolinear continuum mechanics. Oxford: Clarendon.
10. Hahn, T. 1987 Space-group symmetry. International tables for crystallography, vol. A, 2nd edn. Dordrecht: Reidel.
11. Kiral, E. & Bringen, A. C. 1990 Constitutive equations of nonlinear electromagnetic-elastic crystals. Berlin: Springer.
12. Lew, J. S. 1967 Material symmetry restrictions for certain locally compact symmetry groups. //J. J. Math. 11, 202-216.
13. Liu, I. S. 1982 On representations of anisotropic invariants. Int. J. Engng Sci. 20, 1099-1109.
14. Pennisi, S. & Trovato, M. 1987 On irreducibility of Professor G. F. Smith's representations for isotropic functions. Int. J. Engng Sci. 25, 1059-10G5.
15. Pipkin, A. C. & Rivlin, R. S. 1959 The formulation of constitutive equations in continuum physics I. Arch. Ration. Mech. Analysis 4, 129-144.
16. Pipkin, A. C. & Wineman, A. S. 1963 Material symmetry restrictions on non-polynomial constitutive equations. Arch. Ration. Mech. Analysis 12, 420-426.
17. Sénéchal, M. 1995 Quasicrystals and geometry. Cambridge University Press.
18. Smith, G. F. 1962a Further results on the strain-energy function for anisotropic elastic materials. Arch. Ration. Mech. Analysis 10, 108-118.
19. Smith, G. F. 19626 On the yield condition of anisotropic materials. Q. Appl. Math. 20, 241-247.
20. Smith, G. F. 1971 On isotropic functions of symmetric tensors, skewsymmetric tensors and vectors. Int. J. Engng Sci. 9, 899-916.
21. Smith, G. F. 1982 On transversely isotropic functions of vectors, symmetric second-order tensors and skewsymmetric second-order tensors. Q. Appl. Math. 15, 308-314.
22. Smith, G. F. 1994 Constitutive equations for anisotropic and isotropic materials. New York: Elsevicr.
23. Smith, G. F. & Rivlin, R. S. 1957 Stress-deformation relations for anisotropic solids. Arch. Ration. Mech. Analysis 1, 107-112.
24. Smith, G. F. & Rivlin, R. S. 1958 The strain-energy functions for anisotropic elastic materials. Trans. Am. Math. Soc. 88, 175-193.
25. Spencer, A. J. M. 1971 Theorey of invariants. In Continuum physics (ed. A. C. Bringen), vol. I. New York: Academic.
26. Truesdell, C. & Noll, W. 1965 The nonlinear field theories of mechanics. In Handbuch der Physik, Dd., vol. III/3. Berlin: Springer.
27. Vainshtein, B. K. 1994 Modern crystallography. 1. Fundamental of crystals. Berlin: Springer.
28. Wang, C. C. 1970 A new representation theorem for isotropic functions. Part I. Arch. Ration. Mech. Analysis 36, 166-197
29. Wineman, A. S. & Pipkin, A. C. 1964 Material symmetry restrictions on constitutive equations. Arch. Ration. Mech. Analysis 17, 184-214.
30. Xiao, H. 1995 General irreducible representations for constitutive equations of elastic crystals and transversely isotropic elastic solids. J. Elasticity 39, 47-73.
31. Xiao, H. 1996a On anisotropic scalar functions of a single symmetric tensor. Proc. R. Soc. Land. A 452, 1545-1561.
32. Xiao, H. 19966 On minimal representations for constitutive equations of anisotropic elastic materials. J. Elasticity 45, 13-32
33. Xiao, H. 1996c On isotropic extension of anisotropic tensor functions. Z. Angew. Math. Mech. 76, 205-214.
34. Xiao, H. 1997a A unified theory of representations for scalar-, vector- and second-order tensorvalued anisotropic functions of vectors and second-order tensors. Arch. Mech. 49, 995-1039.
35. Xiao, H. 19976 On constitutive equations of Cauchy elastic solids: all kinds of crystals and quasicrystals. J. Elasticity. (In the press.)
36. Zheng, Q. S. 1993a Two-dimensional tensor function representations for all kinds of material symmetry. Proc. R. Soc. Land. A 443, 127-138.
37. Zheng, Q. S. 19936 On transversely isotropic, orthotropic and relatively isotropic functions of symmetric tensors, skewsymmetric tensors and vectors. Parts I-V. Int. J. Engng Sci. 31, 1399-1453.
38. Zheng, Q. S. fe Boehler, J. P. 1994a The description, classification and reality of material and physical symmetries. Acta Mcchanica 102, 73-89.
39. Zheng, Q. S. &: Boehler, J. P. 19946 Tensor function representations as applied to formulating constitutive laws for clinotropic materials. Acta Mechanica Sinica 10, 336-348.
40. Zwillinger, D. (ed.) 1996 Standard mathematical tables and formulae. Boca Raton, FL: CRC.