On anisotropic elastic materials that possess three identical Stroh eigenvalues as do isotropic materials

Author:
T. C. T. Ting

Journal:
Quart. Appl. Math. **52** (1994), 363-375

MSC:
Primary 73B40; Secondary 73C02

DOI:
https://doi.org/10.1090/qam/1276243

MathSciNet review:
MR1276243

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Abstract: For anisotropic elastic materials for which the displacements depend on and only, a general solution for depends on one variable where is an eigenvalue of the fundamental elasticity tensor of Stroh. There are six 's which consist of three pairs of complex conjugates. For isotropic materials, are the eigenvalues of multiplicity three. We point out trivial cases in which a completely anisotropic material has the eigenvalues and has the solutions to two-dimensional elasticity problems that are identical to the solutions for isotropic materials. Excluding these trivial cases, we show that can be the eigenvalues of multiplicity three for monoclinic materials with the symmetry plane at , at , or at any plane that contains the -axis. If the symmetry plane is at , then occur only when the material is transversely isotropic with the axis of symmetry at the -axis. We also consider the general case in which the eigenvalues are arbitrary and are of multiplicity three. The eigenrelation associated with the triple eigenvalues is nonsemisimple for all cases studied here. There are only two independent eigenvectors associated with the triple eigenvalues.

**[1]**J. D. Eshelby, W. T. Read, and W. Shockley,*Anisotropic elasticity with applications to dislocation theory*, Acta Metall.**1**, 251-259 (1953)**[2]**A. N. Stroh,*Dislocations and cracks in anisotropic elasticity*, Philos. Mag.**3**, 625-646 (1958)**[3]**S. G. Lekhnitskii,*Theory of elasticity of an anisotropic body*, Mir Pub. Moscow, 1981**[4]**T. C. T. Ting,*Barnett-Lothe tensors and their associated tensors for monoclinic materials with the symmetry plane at*, J. Elasticity**27**, 143-165 (1992)**[5]**S. C. Cowin and M. M. Mehrabadi,*On the identification of material symmetry for anisotropic elastic materials*, Quart J. Mech. Appl. Math.**40**, 451-476 (1987)**[6]**F. E. Hohn,*Elementary Matrix Algebra*, Macmillan, New York, 1964, pp. 337-340**[7]**D. M. Barnett and J. Lothe,*Synthesis of the sextic and the integral formalism for dislocations*, Greens functions, and surface waves in anisotropic elastic solids, Physica Norvegica**7**, 13-19 (1973)**[8]**P. Chadwick and G. D. Smith,*Foundations of the theory of surface waves in anisotropic elastic materials*, Adv. Appl. Mech.**17**, 303-376 (1977)**[9]**T. C. T. Ting,*Effects of change of reference coordinates on the stress analysis of anisotropic elastic materials*, Internat. J. Solids Structures**18**, 139-152 (1982)**[10]**T. C. T. Ting,*The Stroh formalism and certain invariances in two-dimensional anisotropic elasticity*in Modern Theory of Anisotropic Elasticity and Applications (J. J. Wu, T. C. T. Ting, and D. M. Barnett, eds.), SIAM Proc. Ser., SIAM, Philadelphia, 1991, pp. 3-32**[11]**D. M. Barnett and P. Chadwick,*The existence of one-component surface waves and exceptional subsequent transonic states of type*2, 4*and E*1*in anisotropic elastic media*in Modern Theory of Anisotropic Elasticity and Applications (J. J. Wu, T. C. T. Ting, and D. M. Barnett, eds.), SIAM Proc. Ser., SIAM, Philadelphia, 1991, pp. 199-214**[12]**D. M. Barnett, P. Chadwick, and J. Lothe,*The behavior of elastic surface waves polarized in a plane of material symmetry*. I.*Addendum*, Proc. Roy. Soc. London Ser. A**433**, 699-710 (1991)**[13]**T. C. T. Ting,*The motion of one-component surface waves*, The P. Chadwick Symposium Volume, J. Mech. Phys. Solids**40**, 1637-1650 (1992)

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DOI:
https://doi.org/10.1090/qam/1276243

Article copyright:
© Copyright 1994
American Mathematical Society