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A viewpoint of Kaluza–Klein type in elasticity theory
1.H. Lamb, Proc. London Math. Soc. 13, 189 (1882);
1.see also A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Cambridge U.P., London, 1920), p. 281.
2.T. J. I’A. Bromwich, Proc. London Math. Soc. 30, 98 (1889).
3.More general discussions of the effects of gravitation in a sphere of which the material is not incompressible have been given by many researchers. See, for example, Y. Sato, Theory of Elastic Waves (Iwanami‐syoten, Tokyo, 1978), §24.
4.T. Obata and J. Chiba (to be published). Most of this work has been already announced at a symposium held by the Research Committee of Electro‐Magnetic Theory under the auspices of the I.E.E. of Japan, October 1983. A limited number of manuscripts (Manuscript No. EMT‐83‐51) was distributed by the Research Committee.
5.In the case that each specifical axis of minute monocrystals constituting a polycrystal is arrayed along a specifical direction of the polycrystal and other axes are random, the polycrystal is called to be cylindrically symmetric. The symmetry is expressed by the symbol The is equivalent to the in respect to elastic properties. The is a point group. For details, see any standard textbook concerning crystals.
6.T. Obata and J. Chiba (to be published). Most of this work has been already announced at a symposium held by the Research Committee of Electro‐Magnetic Theory under the auspices of the I.E.E. of Japan, October 1983. A limited number of manuscripts (Manus. No. EMT‐83‐52) was distributed by the Research Committee.
7.Th. Kaluza, Sitzungsber. Preuss. Acad. Wiss. Berlin, Math. Phys. K1, 966 (1921);
7.see also P. G. Bergmann, Introduction to the Theory of Relativity (Prentice‐Hall, New York, 1942), §18.
8.F. J. Ernst, Phys. Rev. 167, 1175 (1968).
9.According to Eq. (2.8), υ is the propagation velocity of φ, m is a masslike quantity, and G is a coupling constant. Here M is also a masslike quantity, because differentiation of Eq. (2.7) with respect to yields
10.Any sequences of numbers leading to the order of infinity with yield the limit values (2.9). For example,
11.Under the volumetric strain δ, any infinitesimal volume dVchanges to so the mass density ρ̃ changes to Accordingly,
12.The generalization in Ref. 4 was done for However, as is easily ascertained, any relations in Ref. 4 hold for arbitrary dimensions.
13.Our φ and φ correspond to Bromwich’s and φ, respectively.
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