Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Some identities and the structure of $ {\bf N}_i$ in the Stroh formalism of anisotropic elasticity

Author: T. C. T. Ting
Journal: Quart. Appl. Math. 46 (1988), 109-120
MSC: Primary 73C30
DOI: https://doi.org/10.1090/qam/934686
MathSciNet review: 934686
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Stroh formalism of anisotropic elasticity leads to a $ 6 \times 6$ real matrix N that can be composed from three $ 3 \times 3$ real matrices $ {N_i} \left( {i = 1, 2, 3} \right)$. The eigenvalues and eigenvectors of N are all complex. New identities are derived that express certain combinations of the eigenvalues and eigenvectors in terms of the real matrices $ {N_i}$ and the three real matrices H, S, L introduced by Barnett and Lothe. It is shown that the elements of $ {N_1}$ and $ {N_3}$ have simple expressions in terms of the reduced elastic compliances. We prove that $ - {N_3}$ is positive semidefinite and, with this property, we present a direct proof that L is positive definite.

References [Enhancements On Off] (What's this?)

  • [1] A. N. Stroh, Dislocations and cracks in anisotropic elasticity, Phil. Mag. (8) 3 (1958), 625–646. MR 0094961
  • [2] A. N. Stroh, Steady state problems in anisotropic elasticity, J. Math. and Phys. 41 (1962), 77–103. MR 0139306
  • [3] K. Malén and J. Lothe, Explicit expressions for dislocation derivatives, Phys. Status Solidi 39 (1970), 287–296 (English, with German summary). MR 0297189
  • [4] P. Chadwick and G. D. Smith, Foundations of the theory of surface waves in anisotropic elastic materials, Adv. Appl. Mech. 17, 303-376 (1977)
  • [5] D. M. Barnett and J. Lothe, Synthesis of the sextic and the integral formalism for dislocation, Green's functions and surface waves in anisotropic elastic solids, Phys. Norv. 7, 13-19 (1973)
  • [6] J. D. Eshelby, W. T. Read, and W. Shockley, Anisotropic elasticity with applications to dislocation theory, Acta Metall. 1, 251-259 (1953)
  • [7] T. C. T. Ting, Explicit solution and invariance of the singularities at an interface crack in anisotropic composites, Internat. J. Solids Structures 22 (1986), no. 9, 965–983. MR 865545, https://doi.org/10.1016/0020-7683(86)90031-4
  • [8] P. Chadwick and T. C. T. Ting, On the structure and invariance of the Barnett-Lothe tensors, Quart. Appl. Math. 45 (1987), no. 3, 419–427. MR 910450, https://doi.org/10.1090/S0033-569X-1987-0910450-6
  • [9] K. A. Ingebrigtsen and A. Tonning, Elastic surface waves in crystals, Phys. Rev. 184, 942-951 (1969)
  • [10] J. Lothe and D. M. Barnett, On the existence of surface-wave solutions for anisotropic half-spaces with free surface, J. Appl. Phys. 47, 428-433 (1976)
  • [11] H. O. K. Kirchner and J. Lothe, On the redundancy of the N matrix of anisotropic elasticity, Phil. Mag. A 53, L7-L10 (1986)
  • [12] T. C. T. Ting, The critical angle of the anisotropic elastic wedge subject to uniform tractions, J. Elasticity 20 (1988), no. 2, 113–130. MR 965867, https://doi.org/10.1007/BF00040907
  • [13] K. Nishioka and J. Lothe, Isotropic limiting behavior of the six-dimensional formalism of anisotropic dislocation theory and anisotropic Green's function theory. I. Sum rules and their applications, Phys. Status Solidi B 51, 645-659 (1972)
  • [14] D. J. Bacon, D. M. Barnett, and R. O. Scattergood, The anisotropic continuum theory of lattice defects, Progr. Mater. Sci. 23 51-262 (1978)
  • [15] R. J. Asaro, J. P. Hirth, D. M. Barnett, and J. Lothe, A further synthesis of sextic and integral theories for dislocations and line forces in anisotropic media, Phys. Status Solidi B 60, 261-271 (1973)
  • [16] T. C. T. Ting, Effects of change of reference coordinates on the stress analyses of anisotropic elastic materials, Internat. J. Solids and Structures 18 (1982), no. 2, 139–152. MR 639099
  • [17] S. A. Gundersen, D. M. Barnett, and J. Lothe, Rayleigh wave existence theory: a supplementary remark, Wave Motion 9 (1987), no. 4, 319–321. MR 896032, https://doi.org/10.1016/0165-2125(87)90004-7

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73C30

Retrieve articles in all journals with MSC: 73C30

Additional Information

DOI: https://doi.org/10.1090/qam/934686
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society