The singular wedge problem in the nonlinear elastostatic plane stress theory

Tarantino, Angelo Marcello

doi:10.1090/qam/1704455

Author: Angelo Marcello Tarantino
Journal: Quart. Appl. Math. 57 (1999), 433-451
MSC: Primary 74B20; Secondary 74G70
DOI: https://doi.org/10.1090/qam/1704455
MathSciNet review: MR1704455
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Abstract: A finite elastostatic analysis of the singular equilibrium field in the proximity of the apex of a wedge, with clamped-free radial edges and general far-field loading conditions, is performed. The problem is formulated for compressible hyperelastic sheets under a plane stress condition. An asymptotic procedure is proposed to compute the deformation and stress singular fields. Emphasis is placed on the investigation of the dependence of the order of singularity in the asymptotic Piola-Kirchhoff and Cauchy stresses on the wedge angles. The case of a half-plane bounded to a rigid substrate is studied in detail.

M. L. Williams, Stress singularities resulting from various boundary conditions in angular corners of plates in extension, J. Appl. Mech. 19, 526–528 (1952)

C. J. Tranter, Integral transforms in mathematical physics, Methuen and Co. Ltd., London; John Wiley and Sons, Inc., New York, 1956. 2nd ed. MR 0079681
Eli Sternberg and W. T. Koiter, The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity, J. Appl. Mech. 25 (1958), 575–581. MR 0102947

D. B. Bogy and E. Sternberg, The effect of couple-stresses on the corner singularity due to an asymmetric shear loading, Internat. J. Solids and Structures 4, 159–171 (1968)

D. B. Bogy, On the problem of edge-bonded elastic quarter-planes loaded at the boundary, Internat. J. Solids and Structures 6, 1287–1313 (1970)

D. B. Bogy, On the plane elastostatic problem of a loaded crack terminating at a material interface, J. Appl. Mech. 38, 911–918 (1971)

A. E. Green and W. Zerna, Theoretical elasticity, Oxford, at the Clarendon Press, 1954. MR 0064598
M. Hetényi, A method of solution for the elastic quarter-plane, J. Appl. Mech. 27 (1960), 289–296. MR 0111229
Ivo Babuška, Karel Rektorys, and František Vyčichlo, Mathematische Elastizitätstheorie der ebenen Probleme, Akademie-Verlag, Berlin, 1960 (German). In deutscher Sprache herausgegeben von Winfried Heinrich. MR 0115343

D. B. Bogy, Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading, J. Appl. Mech. 35, 460–466 (1968)

J. Dundurs, Discussion of edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading, J. Appl. Mech. 36, 650–652 (1969)

D. B. Bogy and K. C. Wang, Stress singularities at interface corners in bonded dissimilar isotropic elastic materials, Internat. J. Solids and Structures 7, 993–1005 (1971)

V. Hein and F. Erdogan, Stress singularities in a two-material wedge, Internat. J. Fract. Mech. 7, 317–330 (1971)

J. Dundurs and M. S. Lee, Stress concentrations at a sharp edge in contact problems, J. of Elasticity 2, 109–112 (1972)

E. E. Gdoutos and P. S. Theocaris, Stress concentrations at the apex of a plane indenter acting on an elastic half plane, J. Appl. Mech. 42, 688–692 (1975)

P. S. Theocaris and E. E. Gdoutos, Stress singularities at vertices of composite plates with smooth or rough interfaces, Arch. Mech. Stosow. 28, 693–704 (1976)

A. K. Rao, Stress concentrations and singularities at interface corners, Z. angew. Math. Mech. 51, 395–406 (1971)

J. P. Dempsey and G. B. Sinclair, On the stress singularities in the plane elasticity of the composite wedge, J. Elasticity 9 (1979), no. 4, 373–391 (English, with French summary). MR 558884, DOI https://doi.org/10.1007/BF00044615
J. P. Dempsey and G. B. Sinclair, On the singular behavior at the vertex of a bi-material wedge, J. Elasticity 11 (1981), no. 3, 317–327. MR 625955, DOI https://doi.org/10.1007/BF00041942

D. B. Bogy, The plane solution for anisotropic elastic wedges under normal and shear loading, J. Appl. Mech. 39, 1103–1109 (1972)

Angelo Marcello Tarantino, The singular equilibrium field at the notch-tip of a compressible material in finite elastostatics, Z. Angew. Math. Phys. 48 (1997), no. 3, 370–388. MR 1460257, DOI https://doi.org/10.1007/s000330050039

F. S. Wong and R. T. Shield, Large plane deformations of thin elastic sheets of Neo-Hookean material, Z. angew. Math. Phys. 20, 176–199 (1969)

J. K. Knowles and Eli Sternberg, Large deformations near a tip of an interface-crack between two neo-Hookean sheets, J. Elasticity 13 (1983), no. 3, 257–293. MR 719844, DOI https://doi.org/10.1007/BF00042997

P. H. Geubelle and W. G. Knauss, Finite strains at the tip of a crack in a sheet of hyperelastic material: I. Homogeneous case, J. of Elasticity 35, 61–98 (1994)

P. H. Geubelle and W. G. Knauss, Finite strains at the tip of a crack in a sheet of hyperelastic material: II. Special bimaterial cases, J. of Elasticity 35, 99–138 (1994)

P. H. Geubelle and W. G. Knauss, Finite strains at the tip of a crack in a sheet of hyperelastic material: III. General bimaterial case, J. of Elasticity 35, 139–174 (1994)

Angelo Marcello Tarantino, Thin hyperelastic sheets of compressible material: field equations, Airy stress function and an application in fracture mechanics, J. Elasticity 44 (1996), no. 1, 37–59. MR 1417808, DOI https://doi.org/10.1007/BF00042191
J. E. Adkins, A. E. Green, and G. C. Nicholas, Two-dimensional theory of elasticity for finite deformations, Philos. Trans. Roy. Soc. London Ser. A 247 (1954), 279–306. MR 67673, DOI https://doi.org/10.1098/rsta.1954.0019
J. E. Adkins and A. E. Green, Plane problems in second-order elasticity theory, Proc. Roy. Soc. London Ser. A 239 (1957), 557–576. MR 86487, DOI https://doi.org/10.1098/rspa.1957.0062
A. E. Green and J. E. Adkins, Large elastic deformations, Clarendon Press, Oxford, 1970. Second edition, revised by A. E. Green. MR 0269158

W. H. Yang, Stress concentration in a rubber sheet under axially symmetric stretching, J. Appl. Mech. 34, 942–946 (1967)

A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. MR 0010851
Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
Stuart Antman, Existence of solutions of the equilibrium equations for non-linearly elastic rings and arches, Indiana Univ. Math. J. 20 (1970/71), 281–302. MR 266478, DOI https://doi.org/10.1512/iumj.1970.20.20025
S. S. Antman, Regular and singular problems for large elastic deformations of tubes, wedges, and cylinders, Arch. Rational Mech. Anal. 83 (1983), no. 1, 1–52. MR 695361, DOI https://doi.org/10.1007/BF00281086
P. Podio-Guidugli and G. Vergara-Caffarelli, Extreme elastic deformations, Arch. Rational Mech. Anal. 115 (1991), no. 4, 311–328. MR 1120851, DOI https://doi.org/10.1007/BF00375278
Philippe G. Ciarlet and Giuseppe Geymonat, Sur les lois de comportement en élasticité non linéaire compressible, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 295 (1982), no. 4, 423–426 (French, with English summary). MR 695540
K. C. Le, On the singular elastostatic field induced by a crack in a Hadamard material, Quart. J. Mech. Appl. Math. 45 (1992), no. 1, 101–117. MR 1154765, DOI https://doi.org/10.1093/qjmam/45.1.101
Rodney A. Stephenson, The equilibrium field near the tip of a crack for finite plane strain of incompressible elastic materials, J. Elasticity 12 (1982), no. 1, 65–99. MR 651119, DOI https://doi.org/10.1007/BF00043706
K. Ch. Le and H. Stumpf, The singular elastostatic field due to a crack in rubberlike materials, J. Elasticity 32 (1993), no. 3, 183–222. MR 1252075, DOI https://doi.org/10.1007/BF00131660