Two general representation theorems for arbitrary-order-tensor-valued isotropic and anisotropic tensor functions of vectors and second order tensors. (English) Zbl 0879.15032
It is shown that irreducible representations for \(r\)th order-tensor-valued isotropic on anisotropic tensor functions depending on an arbitrary number of vectors and second order tensors as variable may be formed by the union of irreducible representations for certain \(r\)th order-tensor-valued isotropic or anisotropic tensor functions merely depending on not more than three variables (for \(r\geq1\)) or four variables (for \(r=0\)) and hence that the representation problem for the former may be reduced to that for the latter.
The paper contains the sections: Criteria for functional bases and generating sets, Representation theorem for tensor-valued functions, and A representation theorem for scalar-valued functions.
The paper contains the sections: Criteria for functional bases and generating sets, Representation theorem for tensor-valued functions, and A representation theorem for scalar-valued functions.
Reviewer: Y.Kuo (Knoxville)
MSC:
| 15A72 | Vector and tensor algebra, theory of invariants |
| 15A90 | Applications of matrix theory to physics (MSC2000) |
| 70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |
| 20H15 | Other geometric groups, including crystallographic groups |
Keywords:
skewsymmetric and symmetric tensors; irreducible representations; isotropic on anisotropic tensor functionsReferences:
| [1] | Pipkin, Arch. Rat. Mech. Analysis Appl. 12 pp 420– (1963) |
| [2] | Wineman, Arch. Rat. Mech. Analysis Appl. 17 pp 184– (1964) |
| [3] | Wang, Arch. Rat. Mech. Analysis Appl. 36 pp 166– (1970) |
| [4] | Arch. Rat. Mech. Analysis Appl. 43 pp 392– (1971) |
| [5] | Smit, Internat. J. Eng. Sci. 9 pp 899– (1971) |
| [6] | : Theory of invariants. In: (ed.): Continuum physics. Academic Press, New York 1971, Vol. I, pp. 239–353. |
| [7] | Boehler, ZAM 57 pp 323– (1977) |
| [8] | Boehler, ZAMM 59 pp 157– (1979) |
| [9] | (ed): Applications of tensor functions in solid mechanics. CISM Courses and Lectures No. 292. Springer-Verlag, Wien, New York 1987. · doi:10.1007/978-3-7091-2810-7 |
| [10] | Liu, Internat. J. Eng. Sci. 20 pp 1099– (1982) |
| [11] | Pennisi, Internat. J. Eng. sci. 25 pp 1059– (1987) |
| [12] | Zhang, Arch. of Mech. 42 pp 267– (1990) |
| [13] | Rychlewski, Advances in Mechanics 14 pp 75– (1991) |
| [14] | Betten, Advances in Mechanics 14 pp 79– (1991) · Zbl 0745.51009 |
| [15] | Pennisi, Internat. J. Eng. Sci. 30 pp 679– (1992) |
| [16] | Zheng, Internat. J. Eng. Sci. 31 pp 679– (1993) |
| [17] | Zheng, Internat. J. Eng. Sci. 31 pp 1013– (1993) |
| [18] | Zheng, Internat. J. Eng. Sci. 31 pp 1399– (1993) |
| [19] | ; : A general representation theorem for anisotropic invariants. In: Chien, W. Z. et al. (eds.): Proc. 2nd Internat. Conf. Nonlin. Mech., Beijing, August 23–26, Peking University Press, Beijing 1993; pp. 206–210. |
| [20] | Xiao, ZAMM |
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