On extraordinary semisimple matrix for anisotropic elastic materials

Author:
T. C. T. Ting

Journal:
Quart. Appl. Math. **55** (1997), 723-738

MSC:
Primary 73B40; Secondary 73C02, 73D20

DOI:
https://doi.org/10.1090/qam/1486545

MathSciNet review:
MR1486545

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Abstract: The real matrix for anisotropic elastic materials under a two-dimensional steady-state motion with speed is *extraordinary semisimple* when has three identical complex eigenvalues and three independent associated eigenvectors. We show that such an exists when . The eigenvalues are purely imaginary. The material can sustain a steady-state motion such as a moving line dislocation. Explicit expressions of the Barnett-Lothe tensors for are presented. However, cannot be extraordinary semisimple for surface waves. When , can be extraordinary semisimple if the strain energy of the material is allowed to be *positive semidefinite*. Explicit expressions of the Barnett-Lothe tensors and Green's functions for the infinite space and half-space are presented. An unusual phenomenon for the material with positive semidefinite strain energy considered here is that it can support an edge dislocation with zero stresses everywhere. In the special case when is a triple eigenvalue, this material is an *un-pressurable material* in the sense that it can change its (two-dimensional) volume with zero pressure. It is a counterpart of an incompressible material (whose strain energy is also positive semidefinite) that can support pressure with zero volume change.

**[1]**D. M. Barnett and J. Lothe,*Synthesis of the sextic and the integral formalism for dislocations, Green's function and surface waves in anisotropic elastic solids*, Phys. Norv.**7**, 13-19 (1973)**[2]**P. Chadwick and G. D. Smith,*Foundations of the theory of surface waves in anisotropic elastic materials*, Adv. Appl. Mech.**17**, 303-376 (1977)**[3]**J. D. Eshelby, W. T. Read, and W. Shockley,*Anisotropic elasticity with applications to dislocation theory*, Acta Metall.**1**, 251-259 (1953)**[4]**A. N. Stroh,*Steady state problems in anisotropic elasticity*, J. Math. Phys.**41**, 77-103 (1962) MR**0139306****[5]**T. C. T. Ting,*On anisotropic elastic materials that possess three identical Stroh eigenvalues as do isotropic materials*, Quart. Appl. Math.**52**, 363-375 (1994) MR**1276243****[6]**H. O. K. Kirchner and J. Lothe,*On the redundancy of the NÌ matrix of anisotropic elasticity*, Philos. Mag.**A53**, L7-L10 (1986)**[7]**Franz E. Hohn,,*Elementary Matrix Algebra*, Macmillan, New York, 1965 MR**0098753****[8]**T. C. T. Ting,*Barnett-Lothe tensors and their associated tensors for monoclinic materials with the symmetric plane at*, J. Elasticity**27**, 143-165 (1992) MR**1151545****[9]**T. C. T. Ting,*Effects of change of reference coordinates on the stress analyses of anisotropic elastic materials*, Internat J. Solids Structures**18**, 139-152 (1982) MR**639099****[10]**K. Tanuma,*Surface impedance tensors of transversely anisotropic elastic materials*, Quart. J. Mech. Appl. Math.**49**, 29-48 (1996) MR**1379031****[11]**A. N. Stroh,*Dislocations and cracks in anisotropic elasticity*, Philos. Mag.**3**, 625-646 (1958) MR**0094961****[12]**T. C. T. Ting,*Image singularities of Green's functions for anisotropic elastic half-spaces and bimaterials*, Quart. J. Mech. Appl. Math.**45**, 119-139 (1992) MR**1154766****[13]**D. M. Barnett and J. Lothe,*Consideration of the existence of surface wave (Rayleigh wave) solutions in anisotropic elastic crystals*, J. Phys. F.**4**, 671-686 (1974)**[14]**T. C. T. Ting,*Existence of an extraordinary degenerate matrix***N***for anisotropic elastic materials*, Quart. J. Mech. Appl. Math.**49**, 405-417 (1996) MR**1434030****[15]**T. C. T. Ting,*Some identities and the structure of**in the Stroh formalism of anisotropic elasticity*, Quart. Appl. Math.**46**, 109-120 (1988) MR**934686****[16]**D. M. Barnett and J. Lothe,*Free surface (Rayleigh) waves in anisotropic elastic half-spaces: The surface impedance methods*, Proc. Roy. Soc. London**A402**, 135-152 (1985) MR**819916****[17]**T. C. T. Ting,*Surface waves in anisotropic elastic materials for which the matrix***N**(v)*is extraordinary degenerate, degenerate, or semisimple*, Proc. Roy. Soc. London**A453**, 449-472 (1997) MR**1920874**

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DOI:
https://doi.org/10.1090/qam/1486545

Article copyright:
© Copyright 1997
American Mathematical Society