On the theory of fourth-order tensors and their applications in computational mechanics

Computer Methods in Applied Mechanics and Engineering

Volumn 189, Issue 2, 2000, Pages 419-438

  Itskov, Mikhail ^a,b  

^a RUHR UNIVERSITY BOCHUM   (Germany)

^b UNIVERSITY OF BAYREUTH   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; ELASTIC MODULI; FUNCTIONS; TENSORS;

COMPUTATIONAL MECHANICS;

MECHANICS;

EID: 0034273134     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(99)00472-7     Document Type: Article

References (42)

1. Y. Başar M. Itskov Finite element formulation of the Ogden material model with application to rubber-like shells Int. J. Numer. Methods Engrg. 42 1998 1279 1305
2. J. Betten, Interpolations methods for tensor functions, in: X.J.R. Avula et al. (Eds.), Mathematical Modelling in Science and Technology, Pergamon Press, New York, 1984
3. J. Betten, Invariants of fourth-order tensors, in: J.P. Boehler (Ed.), Applications of Tensor Functions in Solid Mechanics, CISM courses and lectures No. 292, Wien, 1987, pp. 171–186
4. J. Betten Tensorrechnung für Ingenieure 1987 Teubner Stuttgart
5. J. Betten W. Helisch Irreduzible Invarianten eines Tensors vierter Stufe Z. Angew. Math. Mech. 72 1992 45 57
6. J. Betten W. Helisch Simultaninvarianten bei Systemen zwei- und vierstufiger Tensoren Z. Angew. Math. Mech. 75 1995 753 759
7. J. Betten W. Helisch Tensorgeneratoren bei Systemen von Tensoren zweiter und vierter Stufe Z. Angew. Math. Mech. 76 1996 87 92
8. P. Chadwick R.W. Ogden On the definition of the elastic moduli Arch. Rat. Mech. Anal. 44 1971 41 53
9. P. Chadwick R.W. Ogden A theorem of tensor calculus and its application to isotropic elasticity Arch. Rat. Mech. Anal. 44 1971 54 68
10. P.G. Ciarlet, Mathematical elasticity, Volume I: Three-dimensional elasticity, North-Holland, Amsterdam, 1988
11. S.C. Cowin M.M. Mehrabadi On the structure of the linear anisotropic elastic symmetries J. Mech. Phys. Solids. 40 1992 1459 1472
12. P. Dluzewski P. Rodzik Elastic eigenstates in finite element modelling of large anisotropic elasticity Comput. Methods Appl. Mech. Engrg. 160 1988 325 335
13. T.C. Doyle, J.L. Ericksen, Nonlinear elasticity, Advances in Applied Mechanics IV, Academic Press, New York, 1956
14. M.E. Gurtin, The linear theory of elasticity. in: S. Flugge (Ed.), Handbuch der Physik A, Vol. VI a/2, Springer, Berlin, 1972
15. M.E. Gurtin A short proof of the representation theorem for isotropic, linear stress-strain relations J. Elasticity 4 1974 243 245
16. M. Itskov, The derivative with respect to a tensor: some theoretical aspects and applications, Z. Angew. Math. Mech., in press
17. J.P. Jaric On the representation of symmetric isotropic 4-tensors J. Elasticity 51 1998 73 79
18. M. Kaliske H. Rothert On the finite element implementation of rubber-like materials at finite strains Engrg. Comput. 14 1997 216 232
19. J.K. Knowles On the representation of the elasticity tensor for isotropic materials J. Elasticity 39 1995 175 180
20. L.C. Martins P. Podio-Guidugli A new proof of the representation theorem for isotropic linear constitutive relations J. Elasticity 8 1978 319 322
21. M.M. Mehrabadi S.C. Cowin Eigentensors of linear anisotropic elastic materials Quart. J. Mech. Appl. Math. 43 1990 15 41
22. C. Miehe Computation of isotropic tensor functions Commun. Numer. Methods Engrg. 9 1993 889 896
23. C. Miehe Aspects of the formulation and finite element implementation of large strain isotropic elasticity Int. J. Num. Methods Engrg. 37 1994 1981 2004
24. C. Miehe Comparison of two algorithms for the computation of fourth-order isotropic tensor functions Comput. Struct. 66 1998 37 43
25. M. Mooney A theory of elastic deformations J. Appl. Phys. 11 1940 582 592
26. K.N. Morman The generalised strain measure with application to non homogeneous deformations in rubber-like solids J. Appl. Mech. 53 1986 726 728
27. R.W. Ogden, Non-linear Elastic Deformations, Ellis Horwood Series in Mathematics and its Applications, Chichester, 1984
28. J. Rychlewski On the Hooke's law Prikl. Matem. Mekhan. 48 1984 303 314
29. J. Rychlewski, Symmetry of tensor functions and spectrum theorem (in Russian), Uspechi Mechaniki (Advances in Mechanics) 11(3) (1988) 77–124
30. J. Rychlewski Unconventional approach to linear elasticity Arch. Mech. 47 1995 149 171
31. K.A. Pericak-Spector S.J. Spector On the representation theorem for linear isotropic tensor functions J. Elasticity 39 1995 181 185
32. C. Sansour On the spatial description in elasticity and the Doyle–Ericksen formula Comput. Methods Appl. Mech. Engrg. 107 1993 239 249
33. J.C. Simo J.E. Marsden On the rotated stress tensor and the material version of the Doyle–Ericksen formula Arch. Rat. Mech. Anal. 86 1984 213 231
34. J.C. Simo R.L. Taylor Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms Comput. Methods Appl. Mech. Engrg. 85 1991 273 310
35. S. Sutcliffe Spectral decomposition of the elastic tensor J. Appl. Mech. 59 1992 762 773
36. −1/2 and more general isotropic tensor functions of C J. Elasticity 15 1985 319 323
37. T.C.T. Ting Invariants of anisotropic elastic constants Quart. J. Appl. Mech. 40 3 1987 431 448
38. R. Trostel Vektor- und Tensoralgebra 1993 Vieweg Wiesbaden
39. C. Truesdell, W. Noll, The nonlinear field theories of mechanics, in: S. Flügge (Ed.), Handbuch der Physik, Vol. III/3, Springer, Berlin, 1965
40. K.C. Valanis R.F. Landel The strain-energy function of a hyperelastic material in terms of the extension ratios J. Appl. Phys. 38 1967 2997 3002
41. L.J. Walpole Fourth-rank tensors of the thirty-two crystal classes: multiplication tables Proc. R. Soc. Lond. A391 1984 149 179
42. Q.-S. Zheng J. Betten On the tensor function representations of 2nd-order and 4th-order tensors Part I. Z. Angew. Math. Mech. 75 1995 269 281