Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Boundary value problems of holomorphic vector functions and applications to anisotropic elasticity

Authors: M. Z. Wang and G. P. Yan
Journal: Quart. Appl. Math. 55 (1997), 231-241
MSC: Primary 73B40; Secondary 73V35
DOI: https://doi.org/10.1090/qam/1447576
MathSciNet review: MR1447576
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Abstract: Using Stroh's formalism, plane problems of anisotropic elasticity are turned into the boundary value problems of holomorphic functions. A general method is presented for solving the boundary value problems. The displacement and the stress boundary value problems of an anisotropic body in an elliptical region are solved.

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  • [1] J. D. Eshelby, W. T. Read, and W. Shockley, Anisotropic elasticity with applications to dislocation theory, Acta Metallurgica 1, 251-259 (1953)
  • [2] A. N. Stroh, Dislocations and cracks in anisotropic elasticity, Phil. Mag. (8) 3 (1958), 625–646. MR 0094961
  • [3] A. N. Stroh, Steady state problems in anisotropic elasticity, J. Math. and Phys. 41 (1962), 77–103. MR 0139306
  • [4] 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)
  • [5] D. M. Barnett and J. Lothe, An image force theorem for dislocations in anisotropic bicrystals, J. Phys. Fluids 4, 1618-1635 (1974)
  • [6] D. M. Barnett and J. Lothe, Line force loadings on anisotropic half-spaces and wedges, Phys. Norv. 8, 13-22 (1985)
  • [7] D. M. Barnett and J. Lothe, Free surface (Rayleigh) waves in anisotropic elastic half-spaces: the surface impedance method, Proc. Roy. Soc. London Ser. A 402 (1985), no. 1822, 135–152. MR 819916
  • [8] 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)
  • [9] P. Chadwick and G. D. Smith, Foundations of the theory of surface waves in anisotropic elastic materials, Adv. Appl. Mech. 17, 303-376 (1977)
  • [10] H. O. K. Kirchner and J. Lothe, Displacements and tractions along interfaces, Philos. Mag. A 56, 583-594 (1987)
  • [11] 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
  • [12] T. C. T. Ting, Line forces and dislocations in anisotropic elastic composite wedges and spaces, Phys. Status Solidi B 146, 81-90 (1988)
  • [13] T. C. T. Ting, The anisotropic elastic wedge under a concentrated couple, Quart. J. Mech. Appl. Math. 41 (1988), no. 4, 563–578. MR 980217, https://doi.org/10.1093/qjmam/41.4.563
  • [14] P. Chadwick, Wave propagation in transversely isotropic elastic media. I. Homogeneous plane waves, Proc. Roy. Soc. London Ser. A 422 (1989), no. 1862, 23–66. MR 990852
  • [15] Chyan Bin Hwu and T. C. T. Ting, Two-dimensional problems of the anisotropic elastic solid with an elliptic inclusion, Quart. J. Mech. Appl. Math. 42 (1989), no. 4, 553–572. MR 1033702, https://doi.org/10.1093/qjmam/42.4.553
  • [16] Qianqian Li and T. C. T. Ting, Line inclusions in anisotropic elastic solids, J. Appl. Mech. 56, 556-563 (1989)
  • [17] Jianmin Qu and Qianqian Li, Interfacial dislocation and its application to interface crack in anisotropic materials, J. Elasticity 26, 169-195 (1991)
  • [18] Zhigang Suo, Singularities, interfaces and cracks in dissimilar anisotropic media, Proc. Roy. Soc. London Ser. A 427 (1990), no. 1873, 331–358. MR 1039790
  • [19] T. C. T. Ting, Some identities and the structure of 𝑁ᵢ in the Stroh formalism of anisotropic elasticity, Quart. Appl. Math. 46 (1988), no. 1, 109–120. MR 934686, https://doi.org/10.1090/S0033-569X-1988-0934686-3
  • [20] 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
  • [21] 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)
  • [22] T. C. T. Ting, On the orthogonal, Hermitian and positive definite properties of the matrices 𝑖𝐵⁻¹\overline{𝐵} and -𝑖𝐴⁻¹\overline{𝐴} in anisotropic elasticity, J. Elasticity 30 (1993), no. 3, 277–284. MR 1220167, https://doi.org/10.1007/BF00041146
  • [23] T. C. T. Ting, Barnett-Lothe tensors and their associated tensors for monoclinic materials with the symmetry plane at 𝑥₃=0, J. Elasticity 27 (1992), no. 2, 143–165. MR 1151545, https://doi.org/10.1007/BF00041647
  • [24] T. C. T. Ting and G. P. Yan, The anisotropic elastic solid with an elliptic hole or rigid inclusion, Internat. J. Solids Structures 27, 1879-1894 (1991)
  • [25] T. C. T. Ting and M. Z. Wang, Generalized Stroh formalism for anisotropic elasticity for general boundary conditions, Acta Mech. Sinica 8, 193-207 (1992)
  • [26] M. Z. Wang, T. C. T. Ting, and Gong Pu Yan, The anisotropic elastic semi-infinite strip, Quart. Appl. Math. 51 (1993), no. 2, 283–297. MR 1218369, https://doi.org/10.1090/qam/1218369
  • [27] S. G. Lekhnitskii, Anisotropic Plate, Gordon and Breach Science Publishers, 1968
  • [28] N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending, Translated from the Russian by J. R. M. Radok, P. Noordhoff, Ltd., Groningen, 1963. MR 0176648
  • [29] Huajian Gao, Stress analysis of holes in anisotropic elastic solids: conformal mapping and boundary perturbation, Quart. J. Mech. Appl. Math. 45 (1992), no. 2, 149–168. MR 1176729, https://doi.org/10.1093/qjmam/45.2.149

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DOI: https://doi.org/10.1090/qam/1447576
Article copyright: © Copyright 1997 American Mathematical Society

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