Abstract
Within the framework of finite-strain elastostatics an asymptotic analysis is carried out in order to calculate the singular field near the crack tip in a slab under conditions of plane deformation. A class of Ogden-Ball hyperelastic rubberlike materials and general loading conditions ensuring vanishing tractions on the crack faces near the crack tip are considered. It is shown that the singular deformation field near the crack tip can be specified by applying a rigid body rotation with a subsequent parallel translation to a so-called canonical field. The adjective canonical is adopted here to denote the field with symmetrically opening crack faces, just resembling the displacement field of the symmetric mode in linear elastic fracture mechanics. No analogy with the antisymmetric mode is possible, and the crack equilibrium criterion requires only one stress intensity factor to be determined.
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References
J.R. Rice, Mathematical analysis in the mechanics of fracture. Fracture, Vol. II, Chapter 3. New York: Academic Press (1968).
J.R. Rice and G. F. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material. J. Mech. Phys. Solids 16 (1968) 1–12.
J.W. Hutchinson, Singular behavior at the end of a tensile crack in a hardening material. J. Mech. Phys. Solids 16 (1968) 13–31.
F.S. Wong and R.T. Shield, Large plane deformations of thin elastic sheets of neo-Hookean material. Zeitschrift für angewandte Mathematik und Physik 20 (1969) 176–199.
J.K. Knowles and E. Sternberg, An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack. J. Elasticity 3 (1973) 67–107.
J.K. Knowles and E. Sternberg, Finite-deformation analysis of the elastostatic field near the tip of a crack: reconsideration and higher order results. J. Elasticity 4 (1974) 201–233.
P. J. Blatz and W.L. Ko, Application of finite elastic theory to the deformation of rubbery materials. Trans. Soc. Rheology 6 (1962) 223–251.
J.K. Knowles, The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids. Int. J. Fracture 13 (1977) 611–639.
J.K. Knowles and E. Sternberg, Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example. J. Elasticity 10 (1980) 81–110.
J.K. Knowles and E. Sternberg, Anti-plane shear fields with discontinuous deformation gradients near the tip of a crack in finite elastostatics. J. Elasticity 11 (1981) 129–164.
R.C. Abeyaratne and J.S. Yang, Localized shear discontinuities near the tip of a mode I crack. J. Elasticity 17 (1987) 93–102.
S.A. Silling, Numerical studies of loss of ellipticity near singularities in an elastic material. J. Elasticity 19 (1988) 213–239.
S.A. Silling, Consequences of the Maxwell relation for anti-plane shear deformations of an elastic solid. J. Elasticity 19 (1988) 241–284.
R.A. Stephenson, The equilibrium field near the tip of a crack for finite plane strain of incompressible elastic materials. J. Elasticity 12 (1982) 65–99.
R.W. Ogden, Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. A 326 (1972) 565–584.
R.W. Ogden, Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc. R. Soc. Lond. A 328 (1972) 567–583.
J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337–403.
K.Ch. Le, H. Stumpf and D. Weichert, Variational principles of fracture mechanics. Mitteilungen Institut für Mechanik 64 (1989) 1–65.
H. Stumpf and K.Ch. Le, Variational principles of nonlinear fracture mechanics. Acta Mechanica 83 (1990) 25–37.
C. Truesdell and R. Toupin, The classical field theories. Handbuch der Physik, Vol. III/1, Berlin: Springer (1960).
P.G. Ciarlet, Mathematical Elasticity, Vol. 1. Amsterdam: North-Holland (1988).
J.D. Eshelby, The Continuum Theory of Lattice Defects. Solid State Physics, Vol. 3. New York: Academic Press (1956).
M. Abramowitz and I.A. Stegun (eds.), Handbook of Mathematical Functions. New York: Dover Publications (1965).
E. Kamke, Differentialgleichungen. Leipzig: Geest & Portig K.-G. (1967).
K.Ch. Le, On the singular elastostatic field induced by a crack in a Hadamard material. Quart. J. Mech. Appl. Math. 45 (1992) 101–117.
T.K. Hellen, On the method of virtual crack extensions. Int. J. Numer. Meth. in Engng. 9 (1975) 187–207.
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Le, K.C., Stumpf, H. The singular elastostatic field due to a crack in rubberlike materials. J Elasticity 32, 183–222 (1993). https://doi.org/10.1007/BF00131660
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DOI: https://doi.org/10.1007/BF00131660