Abstract
The classical solution for an isotropic elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2θ0 satisfies the equation tan 2θ0 = 2θ0. This is the critical wedge angle which also renders infinite solutions for other types of loadings. In this paper, we study the associated problem for the anisotropic elastic wedge. We first present uniform stress solutions which are possible for symmetric loadings. For antisymmetric loadings, a uniform stress solution is in general not possible and we present a non-uniform stress solution in which the stress depends on θ but not on r. The non-uniform stress solution breaks down at a critical angle. We present an equation for the critical angle which depends on the elastic constants. The Stroh formalism is employed in the analysis. An integral representation of the solution is obtained by using new identities which are derived in the paper.
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References
S.P. Timoshenko and J.N. Goodier: Theory of Elasticity, McGraw-Hill, Inc., 3rd edn. (1970) p. 141.
E.Sternberg and W.TKoiter: The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity. J. Appl. Mech. 25 (1958) 575–581.
J.P.Dempsey: The wedge subjected to tractions: a paradox resolved. J. Elasticity 11 (1981) 1–10.
T.C.T.Ting: The wedge subjected to tractions: a paradox re-examined. J. Elasticity 14 (1984) 235–247.
A.N.Stroh: Dislocations and cracks in anisotropic elasticity. Phil. Mag. 7 (1958) 625–646.
A.N.Stroh. Steady state problems in anisotropic elasticity. J. Math. Phys. 41 (1962) 77–103.
K.Malen and JLothe: Explicit expressions for dislocation derivatives. Phys. Status Solidi 39 (1970) 287–296.
D.M.Barnett and J.Lothe: Synthesis of the sextic and the integral formalism for dislocation, Green's function and surface waves in anisotropic elastic solids. Phys. Norv. 7 (1973) 13–19.
D.M.Barnett and J.Lothe: Line force loadings on anisotropic half-spaces and wedges. Phys. Norv. 8 (1975) 13–22.
P.Chadwick and G.D.Smith: Foundations of the theory of surface waves in anisotropic elastic materials. Adv. Appl. Mech. 17 (1977) 303–376.
T.C.T.Ting: Explicit solution and invariance of the singularities at an interface crack in anisotropic composites. Int. J. Solids Structures 22 (1986) 965–983.
E.Reissner: Note on the theorem of the symmetry of the stress tensor. J. Math. Phys. 23 (1944) 192–194.
T.C.T.Ting: Elastic wedge subjected to anti-plane shear tractions—a paradox explained. Q. J. Mech. Appl. Math. 38, Part 2 (1985) 245–255.
J.D.Eshelby, W.T.Read and W.Shockley: Anisotropic elasticity with applications to dislocation theory. Acta Metall. 1 (1953) 251–259.
T.C.T.Ting: Effects of change of reference coordinates on the stress analyses of anisotropic elastic materials. Int. J. Solids Structures 18 (1982) 139–152.
P.Chadwick and T.C.T.Ting: On the structure and invariance of the Barnett-Lothe tensors. Q. Appl. Math. 45 (1987) 419–427.
T.C.T.Ting: Some identities and the structure of N i in the Stroh formalism of anisotropic elasticity. Q. Appl. Math. 46 (1988) 109–120.
T.C.T.Ting and S.C.Chou: Edge singularities in laminated composites. Int. J. Solids Structures 17 (1981) 1057–1068.
H.O.K.Kirchner and J.Lothe: On the redundancy of the −N matrix of anisotropic elasticity. Phil. Mag. A, 53 (1986) L7-L10.
P. Chadwick, private communications.
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Ting, T.C.T. The critical angle of the anisotropic elastic wedge subject to uniform tractions. J Elasticity 20, 113–130 (1988). https://doi.org/10.1007/BF00040907
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DOI: https://doi.org/10.1007/BF00040907