The paper deals with constructing the anisotropic orientation distribution function of internal structure elements in nonhomogeneous material. This function is necessary for estimating the average mechanical properties of the materials. For the specifying of this function, we propose to utilize the partial sums of Fourier series on three-dimensional rotation group. We show that these partial sums can be pre- sented in coordinate free tensorial form, thereby reducing the anisotropy of distribution function to the anisotropy of tensors possessing some specific properties. By way of illustration, we consider a simple example in case of cubic symmetry.
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References
1.
[1] Adams, B. L., Boehier, J. P., Guidi, M., and Onat, E. T.: Group theory and representation of microstructure and mechanical behaviour of polycrystals. J. Mech. Phys. Solids, 40, 723-723 (1992).
2.
[2] Gel'fand, I. M., Minlos, R. A., and Shapiro, Z. Ya.: Representations of the Rotation and Lorentz Groups and their Applications, Pergamon, Oxford, 1963.
3.
[3] Tamuzh, V. P. and Lagzdin' A. Zh.; A version of phenomenological fracture theory. Polymer Mechanics4, 493-493 (1968).
4.
[4] Lagzdin' A. Zh.: Fourier expansion of vector field on the unit sphere in generalized spherical functions by means of tensors. La2. Mat. Ezhegodnik1(in Russian) 9, 117-117 (1971).
5.
[5] Lagzdin' A. Zh.: On construction of partial sums of Fourier series on the rotation group SO(3) by means of tensor components. Latv. Mat. Ezhegodnik (in Russian) 23, 13-13 (1979).
6.
[6] Lagzdias, A., TamuBs, V.Teters, G., and KrOgers, A.: Orientational Averaging in Mechanics of Solids. Longman Scientific & Technical, London, 1992.
7.
[7] Lokhin, V. V., and Sedov, L. I.: Nonlinear tensor functions of several tensor variables. Prikladnaya Mat. Mech. (in Russian) 27, 393-393 (1963).
