Explicit expressions of Barnett-Lothe tensors and their associated tensors for orthotropic materials
Authors:
Changsong Dongye and T. C. T. Ting
Journal:
Quart. Appl. Math. 47 (1989), 723-734
MSC:
Primary 73B40
DOI:
https://doi.org/10.1090/qam/1031687
MathSciNet review:
MR1031687
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Abstract: The three Barnett-Lothe tensors S, H, L, appear very often in the solutions to two-dimensional anisotropic elasticity problems. So do their associated tensors $\hat S\left ( \theta \right ), \hat H\left ( \theta \right ), \hat L\left ( \theta \right )$ for line forces and dislocations and $S\left ( \upsilon \right ), H\left ( \upsilon \right ), L\left ( \upsilon \right )$ in the problem of surface waves. Explicit expressions of the components of these tensors are derived and presented for orthotropic materials in which the planes of material symmetry coincide with the coordinate planes. With minor modifications, the results for S, H, L and $\hat S\left ( \theta \right ), \hat H\left ( \theta \right ), \hat L\left ( \theta \right )$ can be applied to orthotropic materials in which only the ${x_3} = 0$ plane coincides with one of the planes of material symmetry.
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D. M. Barnett and J. Lothe, An image force theorem for dislocations in anisotropic bicrystals, J. Phys. F. 4, 1618–1635 (1974)
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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)
A. N. Stroh, Dislocations and cracks in anisotropic elasticity, Phil Mag. 3, 625–646 (1958)
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)
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)
D. M. Barnett and J. Lothe, An image force theorem for dislocations in anisotropic bicrystals, J. Phys. F. 4, 1618–1635 (1974)
D. M Barnett and J. Lothe. Line force loadings on anisotropic half-spaces and wedges, Phys. Norv. 8, 13–22 (1975)
P. Chadwick and G. D. Smith, Foundations of the theory of surface waves in anisotropic elastic materials, Adv. in Appl. Mech. 17, 303–376 (1977)
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P. Chadwick and D. A. Jarvis, Surface waves in a pre-stressed elastic body, Proc. Roy Soc. London A 366, 517–536 (1979)
T. C. T. Ting. Line forces and dislocations in anisotropic elastic composite wedges and spaces, Phys. Status Solidi, B 146, 81–90 (1988)
T. C. T. Ting, Explicit solution and invariance of the singularities at an interface crack in anisotropic composites, Int. J. Solids Structures 22, 965–983 (1986)
T. C. T. Ting The critical angle of the anisotropic elastic wedge subject to uniform tractions, J. Elasticity 20, 113–130 (1988)
Qianqian Li and T. C. T. Ting, Line inclusions in anisotropic elastic solids, J. Appl. Mech., in press (1989)
Chyanbin Hwu and T. C. T. Ting, Two-dimensional problems of the anisotropic elastic solids with an elliptic inclusion, Quart. J. Mech. Appl. Math., in press (1989)
H. O. K. Kirchner and J. Lothe, Displacements and tractions along interfaces, Phil. Mag. A 56, 583–594 (1987)
P. Chadwick and N. J. Wilson, Surface waves in orthotropic and cubic elastic materials, to appear, (1989)
P. Chadwick, Wave propagation in transversely isotropic elastic media. I Homogeneous plane waves. II Surface waves. III The special case ${a_s} = 0$ and the inextensible limit, Proc. Roy. Soc. London A 422, 23–121 (1989)
P. Chadwick, private communications
J. D. Eshelby, W. T. Read, and W. Shockley, Anisotropic elasticity with applications to dislocation theory, Acta Metallurgica 1, 251–259 (1953)
T. C. T. Ting and Chyanbin Hwu, Sextic formalism in anisotropic elasticity for almost non-semisimple matrix N, Internat. J. Solids Structures 24, 65–76 (1988)
H. O. K. Kirchner and J. Lothe, On the redundancy of the N matrix of anisotropic elasticity, Phil. Mag. A 53, L7–L10 (1986)
P. Chadwick and T. C. T. Ting, On the structure and invariance of the Barnett-Lothe tensors, Quart. Appl. Math. 45, 419–427 (1987)
T. C. T. Ting, The eigenvectors of S matrix and their relations with line dislocations and forces in anisotropic elastic solids, Micromechanics and Inhomogeneity, The Toshio Mura Anniversary Volume, edited by G. J. Weng, M. Taya, and H. Abe, Springer-Verlag, New York, 1989, pp. 449–467
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Article copyright:
© Copyright 1989
American Mathematical Society
