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Group theory and representation of microstructure and mechanical behavior of polycrystals. (English) Zbl 0760.73058
Summary: We review the group theoretical basis of the result that the observed mechanical behavior of a material can be represented by constitutive (differential) equations which govern the evolution of state variables and that these variables are even-rank irreducible tensors. On the other hand, microscopic observations of the internal structure of a polycrystal produce functions that are defined on “curved” objects such as the unit sphere of directions or the set of distinct orientations of a cube, etc. We show, in terms of an example (the crystallite orientation distribution function for a macroscopically homogeneous polycrystal composed of grains of a cubic crystalline solid), that representations of such functions give rise to Fourier coefficients that are also irreducible tensors. The tensorial state variables will be related to these tensorial Fourier coefficients. A major problem of the mechanics of materials is to develop methods that enable one, for a given material and for a given purpose, to extract tensorial state variables and the laws for their evolution from the knowledge obtained from the studies of the microstructure and behavior of the material.

MSC:
74A60 Micromechanical theories
74M25 Micromechanics of solids
74E15 Crystalline structure
74E10 Anisotropy in solid mechanics
74A20 Theory of constitutive functions in solid mechanics
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
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References:
[1] Adams, B.L.; Field, D.P., Acta metall. mater., 39, 2405, (1991)
[2] Backus, G., Rev. geophys. spacephys., 8, 633, (1970)
[3] Bröcker, T.; Tom Dieck, T., Representation of compact Lie groups, (1985), Springer New York · Zbl 0581.22009
[4] Bunge, H.J., Z. metallkunde, 56, 872, (1965)
[5] Bunge, H.J., Texture analysis in materials science, (1982), Butterworths London
[6] Courant, R.; Hilbert, D., ()
[7] Geary, J.E.; Onat, E.T., Oak ridge national laboratory report, (1974), ORNL-TM-4525.
[8] Gel’fand, I.M.; Minlos, R.A.; Shapiro, Z.Ya., Representations of the rotation and Lorentz groups and their applications, (1963), Pergamon Press Oxford · Zbl 0108.22005
[9] Guidi, M.; Adams, B.L.; Onat, E.T., Textures microstruct., (1991), To appear in
[10] Harren, S.V.; Asaro, R.J., J. mech. phys. solids, 37, 191, (1989)
[11] Molinari, A.; Canova, G.R.; Ahzi, S., Acta metall., 35, 2983, (1987)
[12] Onat, E.T., Proc. IUTAM symposium, Vienna, (), 252
[13] Onat, E.T., Engng fracture mech., 25, 605, (1986)
[14] Onat, E.T., (), 85
[15] Roe, R.J., J. appl. phys., 36, 2024, (1965)
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