Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Nonlinear effect of initial stress on crack propagation between similar and dissimilar orthotropic media


Author: M. A. Biot
Journal: Quart. Appl. Math. 30 (1973), 379-406
MSC: Primary 73.35
DOI: https://doi.org/10.1090/qam/400852
MathSciNet review: 400852
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Abstract: The theory of crack propagation in orthotopic media is developed by applying the theory of incremental deformations in the vicinity of a state of initial stress. This is carried out in the context of a new approach to analytical methods and a physical analysis which takes into account plastic deformation under prestress. The state of initial stress is triaxial along the directions of elastic symmetry, and the crack is parallel to these directions. An additional shear component for the initial stress is also taken into account and general conditions are derived for crack propagation, including the case of fluid injection into the crack. The analysis is first carried out for an homogeneous medium. The nonlinear influence of the initial stress appears in two ways: first, through a fundamental purely elastic effect related to the occurrence of surface instability, and second, through the influence of the initial stress on plastic behavior. The particular cases of an isotropic elastic medium with finite initial strain and an orthotropic incompressible medium are discussed. The analysis is extended to a crack between dissimilar orthotropic media with initial stress. The method of analysis leads to a number of simplifications and brings out new properties of the solutions for this type of problem. For incompressible media without initial stress, the typical oscillatory behavior disappears. Uniqueness of the solutions is also derived.


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  • [1] A. A. Griffith, The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. London A 221, 163-198 (1920)
  • [2] A. A. Griffith, The theory of rupture, in Proc. First International Congr. Appl. Mech., Delft, pp. 55-63 (1924)
  • [3] I. N. Sneddon, The distribution of stress in the neighbourhood of a crack in an elastic solid, Proc. Roy. Soc. London. Ser. A. 187 (1946), 229–260. MR 0017160, https://doi.org/10.1098/rspa.1946.0077
  • [4] I. N. Sneddon and H. A. Elliot, The opening of a Griffith crack under internal pressure, Quart. Appl. Math. 4 (1946), 262–267. MR 0017161, https://doi.org/10.1090/S0033-569X-1946-17161-2
  • [5] Ida W. Busbridge, Dual Integral Equations, Proc. London Math. Soc. S2-44, no. 2, 115. MR 1576205, https://doi.org/10.1112/plms/s2-44.2.115
  • [6] R. A. Sack, Extension of Griffith's theory of rupture to three dimensions, Proc. Phys. Soc. London 58, 729-736 (1946)
  • [7] G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture., Advances in Applied Mechanics, Vol. 7, Academic Press, New York, 1962, pp. 55–129. MR 0149728
  • [8] I. N. Sneddon and M. Lowengrub, Crack problems in the classical theory of elasticity, John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0258339
  • [9] R. L. Salganik, The brittle fracture of cemented bodies, P.M.M. 27, 957-962 (1963)
  • [10] J. R. Rice and G. C. Sih, Plane problems of cracks in dissimilar media, J. Applied Mech. E32, 418-423 (1965)
  • [11] F. Erdogan, Stress distribution in bonded dissimilar materials with cracks, J. Applied Mech. E32, 403-410 (1965)
  • [12] M. Gotoh, Some problems of anisotropic plates with cracks along the bonds, Int. J. Fracture Mech. 3, 253-264 (1967)
  • [13] A. H. England, A crack between dissimilar media, J. Appl. Mech. E 32, 400-402 (1965)
  • [14] D. L. Clements, A crack between isotropic and anisotropic media, Quart. Appl. Math. 29, 303-310 (1971)
  • [15] F. Erdogan and G. Gupta, The stress analysis of multilayered composites with a flaw, Int. J. Solids Struct. 7, 39-61 (1971)
  • [16] M. A. Biot, Surface instability in finite anisotropic elasticity under initial stress, Proc. Roy. Soc. Ser. A 273 (1963), 329–339. MR 0152178
  • [17] M. A. Biot, Incremental elastic coefficients of an isotropic medium in finite strain, Appl. Scient. Res. A12, 151-167 (1963)
  • [18] M. A. Biot, Continuum dynamics of elastic plates and multilayered solids under initial stress, J. Math. Mech. 12 (1963), 793–809. MR 0156514
  • [19] M. A. Biot, Continuum theory of stability of an embedded layer in finite elasticity under initial stress, Quart. J. Mech. Appl. Math. 17 (1964), 17–22. MR 0160361, https://doi.org/10.1093/qjmam/17.1.17
  • [20] Maurice A. Biot, Mechanics of incremental deformations. Theory of elasticity and viscoelasticity of initially stressed solids and fluids, including thermodynamic foundations and applications to finite strain, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0185873
  • [21] G. R. Irwin, Fracture dynamics, in Fracturing of metals, pp. 147-166, ASM, Cleveland, Ohio, 1948
  • [22] E. Orowan, Fundamentals of brittle behavior of metals, in Fatigue and fracture of metals (W. M. Murray, ed.), pp. 139-167, John Wiley & Sons, Inc., New York, 1950
  • [23] N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494
  • [24] Th. von Kármán and J. M. Burgers, General aerodynamic theory, in Vol. II Div. E (p. 45) of Aerodynamic theory (ed. W. F. Durand), J. Springer, Berlin, 1936
  • [25] Th. von Kármán, Festigkeitsversuch unter allseitigem Druck, Zeit. Ver. deutsch Ingenieure, 55, 1749-1757 (1911)
  • [26] D. S. Dugdale, Yielding of steel sheets containing slits, J. Mech. Phys. Solids 8, 150-104 (1960)
  • [27] J. N. Goodier and F. A. Field, Plastic energy dissipation in crack propagation, in Proc. Int. Conf. Fracture Solids, Maple Valley, Washington, pp. 103-118, 1962

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DOI: https://doi.org/10.1090/qam/400852
Article copyright: © Copyright 1973 American Mathematical Society


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