Representation of vector-valued hemitropic functions of a symmetric tensor and a vector. (English) Zbl 07920565
Summary: We revisit the question of finding the most general representation of vector-valued functions such that \({\mathbf{f}}({\mathbf{Q}}{\mathbf{A}}{\mathbf{Q}}^T,{\mathbf{Q}}{\mathbf{u}}) = {\mathbf{Q}} {\mathbf{f}}({ \mathbf{A}},{\mathbf{u}})\) for all rotation tensors \({\mathbf{Q}}\), all symmetric tensors \({\mathbf{A}}\), and all vectors \({\mathbf{u}}\). Seven scalar invariants and three vectors are identified that form the basis for the functional dependence in its simplest form. Six of the seven scalar invariants are required in the case of isotropy with respect to the full orthogonal group. The results are relevant to hemitropic vector-valued functions in continuum mechanics, such as the thermoelastic heat flux.
MSC:
| 74A20 | Theory of constitutive functions in solid mechanics |
References:
| [1] | Cowin, S. C.; Mehrabadi, M. M., On the identification of material symmetry for anisotropic elastic materials, Q. J. Mech. Appl. Math., 40, 451-476, 1987 · Zbl 0625.73014 · doi:10.1093/qjmam/40.4.451 |
| [2] | Zheng, Q.-S., On transversely isotropic, orthotropic and relative isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Part I: two dimensional orthotropic and relative isotropic functions and three dimensional relative isotropic functions, Int. J. Eng. Sci., 31, 10, 1399-1409, 1993 · Zbl 0780.73008 · doi:10.1016/0020-7225(93)90005-f |
| [3] | Nowacki, W., Theory of Asymmetric Elasticity, 1985, Oxford: Pergamon Press, Oxford |
| [4] | Healey, T. J., Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids, 7, 4, 405-420, 2002 · Zbl 1090.74610 · doi:10.1177/108128028482 |
| [5] | Dill, E. H., Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity, 2007, Boca Raton: CRC, Boca Raton |
| [6] | Green, A. E.; Adkins, J. E., Large Elastic Deformations, 1960, Oxford: Clarendon Press, Oxford · Zbl 0090.17501 |
| [7] | Koh, S. L.; Eringen, A. C., On the foundations of non-linear thermo-viscoelasticity, Int. J. Eng. Sci., 1, 199-229, 1963 · doi:10.1016/0020-7225(63)90034-X |
| [8] | Liu, I.-S., Continuum Mechanics, 2002, Berlin: Springer, Berlin · Zbl 1058.74004 · doi:10.1007/978-3-662-05056-9 |
| [9] | Noll, W., Representations of certain isotropic tensor functions, Arch. Math., 21, 1, 87-90, 1970 · Zbl 0194.06401 · doi:10.1007/BF01220884 |
| [10] | Del Piero, G., Representation theorems for hemitropic and transversely isotropic tensor functions, J. Elast., 51, 11, 43-71, 1998 · Zbl 0922.73002 · doi:10.1023/A:1007485908989 |
| [11] | Zheng, Q.-S., Theory of representations for tensor functions—a unified invariant approach to constitutive equations, Appl. Mech. Rev., 47, 11, 545-587, 1994 · Zbl 0999.74501 · doi:10.1115/1.3111066 |
| [12] | Xiao, H., On anisotropic scalar functions of a single symmetric tensor, Proc. R. Soc. A, 452, 1950, 1545-1561, 1996 · Zbl 0869.73011 · doi:10.1098/rspa.1996.0082 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
