Abstract
An exact expression is derived for a path independent integral surrounding an elliptical hole in an infinite plate with tensile tractions at infinity for plane stress loading conditions. The plate is composed of a non-work-hardening material satisfying the Tresca yield condition under proportional loading and small strain assumptions. This problem may serve as a simple classroom example for the derivation of a relationship between crack opening displacement and path independent integral for a nonlinear elastic material satisfying the Tresca yield condition.
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References
Cherepanov, G.P.: Crack propagation in continuous media. Prikl. Mat. Meh. 31, 476–488 (1967) (in Russian)
Cherepanov, G.P.: Mechanics of Brittle Fracture. McGraw-Hill, New York (1979)
Dugdale, D.S.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960)
Hutchinson, J.W.: Singular behavior at the end of a tensile crack in a hardening material. J. Mech. Phys. Solids 16, 13–31 (1968)
Hutchinson, J.W.: Plastic stress and strain fields at a crack tip. J. Mech. Phys. Solids 16, 337–347 (1968)
Griffith, A.A.: The phenomena of rupture and flow in solids. Trans. R. Soc. Lond. A 221, 163–198 (1920)
Inglis, C.E.: Stresses in a plate due to the presence of cracks and sharp notches. Trans. Inst. Naval Architects 55, 219–241 (1913)
Ivlev, D.D., Ershov, L.V.: Perturbation Method in the Theory of an Elastic–Plastic Body, pp. 160–170. “Nauka,” Glav. red. fiziko-matematicheskoi lit-ry, Moscow (1978) (in Russian)
Lawrence, J.D.: A Catalog of Special Plane Curves, pp. 42–43. Dover, Mineola, NY (1972)
Kachanov, L.M.: Fundamentals of the Theory of Elasticity. Dover, Mineola NY (2004)
Kharchenko, A.P.: Deformation in the vicinity of an elliptical hole in an elastic–plastic solid. Prikl. Meh. (Kiev) 10, 48–55 (1974) (in Russian)
Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium, pp. 510–511. Prentice-Hall, Englewood Cliffs, NJ (1969)
Perlin, P.I.: Approximate method of solution of elastic–plastic problems. Inzhenernyi Sbornik 28, 145–150 (1960) (in Russian)
Rice, J.R.: A path independent integral and the approximate analysis of strain concentrations by notches and cracks. J. Appl. Mech. 35, 379–386 (1968)
Rice, J.R., Rosengren, G.F.: Plane strain deformation near a crack tip in a power-law hardening material. J. Mech. Phys. Solids 16, 1–12 (1968)
Rice, J.R.: Mathematical analysis in the mechanics of fracture. In: Liebowitz, H. (ed.) Fracture: An Advanced Treatise, vol 2, pp. 191–311. Academic, New York (1968)
Sokolnikoff, I.S.: Mathematical Theory of Elasticity. McGraw-Hill, New York (1956)
Sokolovskii, V.V.: Theory of Plasticity, 3rd edn. pp. 228–237. Vyssh. shkola, Moscow (1969) (in Russian)
Timoshenko, S.P.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)
Unger, D.J.: Analytical Fracture Mechanics. Dover, Mineola, NY (2001)
Unger, D.J.: Perfectly plastic caustics for the opening mode of fracture. Theor. Appl. Fract. Mech. 44, 82–94 (2005)
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Unger, D.J. Path Independent Integral for an Elliptical Hole in a Plate under Tension for Plane Stress Deformation Theory. J Elasticity 92, 217–226 (2008). https://doi.org/10.1007/s10659-008-9159-z
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DOI: https://doi.org/10.1007/s10659-008-9159-z