Abstract
The present paper is concerned with an infinite slab containing a crack and deformed at infinity to a state of finite simple shear. The material of the slab is taken to be homogeneous, isotropic, elastic, and incompressible, and is further assumed to belong to a class of materials which admit nontrivial states of anti-plane shear. The analysis is carried out for the fully nonlinear equilibrium theory of finite elasticity. The stress field near the crack-tips is studied in detail; one of the special materials considered is such that the shear stresses near a crack tip remain bounded, despite the presence of unbounded displacement gradients. An analogy between the crack problem in finite anti-plane shear and the problem of transonic flow of a gas past a flat plate is pointed out and discussed.
Résumé
Le présent mémoire est relatif à une plaque infinie comportant une fissure et déformée à l'infini dans un état de cisaillement fini simple. Le matériau de la plaque est considéré comme homogène, isotrope, élastique et incompressible, et il est en outre supposé appartenir à une classe de matériau qui admet des états non triviaux de cisaillement anti-planaire. L'analyse est effectuée suivant la théorie d'équilibre complðement non linéaire de l'élasticité finie. Le champ de contrainte au voisinage des extrémités de fissure est étudié dans le détail; un des matériaux spéciaux considérés est tel que les forces de cisaillement au voisinage de l'extrémité d'une fissure demeurent liées en dépit de la présence de gradiants de déplacement non liés. Une analogie entre le problème de fissuration dans une situation de cisaillement anti-planaire et le problème de l'écoulement transonique d'un gaz au-delà d'une tôle plane est mise en avant et discutée.
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References
J.R.Rice and G.F.Rosengren, Journal of the Mechanics and Physics of Solids, 16 (1968) 1.
J.W.Hutchinson, Journal of the Mechanics and Physics of Solids, 16 (1968) 13.
J.W.Hutchinson, Journal of the Mechanics and Physics of Solids, 16 (1968) 337.
J.R.Rice, Journal of Applied Mechanics, 34 (1967) 287.
J.A.H. Hult and F.A. McClintock, Elastic-plastic stress and strain distribution around sharp notches under repeated shear, Proceedings of the Ninth International Congress of Applied Mechanics, Brussels, Belgium, 8 (1956) 51.
J.C.Amazigo, International Journal of Solids and Structures, 10 (1974) 1003.
J.C.Amazigo, International Journal of Solids and Structures, 11 (1975) 1291.
K.K. Lo, Finite-deformation crack in an infinite body under anti-plane simple shear, to appear.
F.S. Wong and R.T. Shield, Zeitschrift für angewandte Mathematik and Physik, 20 (dy1969) pp176.
J.K.Knowles and E.Sternberg, Journal of Elasticity, 3 (1973) 67.
J.K.Knowles and E.Sternberg, Journal of Elasticity, 4 (1974) 201.
J.K.Knowles and E.Sternberg, Journal of Elasticity, 5 (1975) 341.
A.E.Green and J.E.Adkins, Large Elastic Deformations, Clarendon Press, Oxford (1960).
C.Truesdell and W.Noll, The nonlinear field theories of mechanics, Handbuch der Physik, vol. III/1, Springer, Berlin (1965).
J.E.Adkins, Proceedings of the Cambridge Philosophical Society, 50 (1954) 334.
L.R.G.Treloar, The Physics of Rubber Elasticity, Clarendon Press, Oxford (1958).
C. Truesdell, Instabilities of isotropic perfectly elastic materials in simple shear, Proceedings of the Eleventh International Congress of Applied Mechanics, Munich, Germany (1964) 139.
R.Courant and D.Hilbert, Methods of Mathematical Physics, Vol. II, Interscience, N. Y. (1962).
J.K.Knowles and T.A.Pucik, Journal of Elasticity, 3 (1973) 155.
F.B.Hildebrand, Advanced Calculus for Applications, Prentice-Hall, Englewood Cliffs (1962).
I.Sneddon and M.Lowengrub, Crack Problems in the Classical Theory of Elasticity, Wiley, New York (1969).
R.Courant and K.O.Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York (1948).
W.R.Sears, editor, General Theory of high speed aerodynamics, Vol. VI of High Speed Aerodynamics and Jet Propulsion, J.C.Charyk, general editor; Princeton University Press, Princeton (1954).
L.Bers, Mathematical aspects of subsonic and transonic gas dynamics, Surveys in Applied Mathematics, Vol. 3, John Wiley and Sons, New York (1958).
J.D.Eshelby, The continuum theory of lattice defects, Solid State Physics, 3, Academic Press, New York (1956).
J.R.Rice, Journal of Applied Mechanics, 35 (1968) 379.
J.K.Knowles and E.Sternberg, Archive for Rational Mechanics and Analysis, 44 (1972) 187.
J.D.Cole, Perturbation Methods in Applied Mathematics, Ginn-Blaisdell, Waltham (1968).
K.Oswatitsch and G.Kuerti, Gas Dynamics, Academic Press, New York (1956).
C.Ferrari and F.G.Tricomi, Transonic Aerodynamics (English translation by R.H. Cramer). Academic Press, New York (1968).
J.D.Cole, SIAM Journal on Applied Mathematics, 29 (1975) 763.
T.Y.Thomas, Plastic Flow and Fracture in Solids, Academic Press, New York, 1961.
J.K. Knowles, Journal of the Australian Mathematical Society, Series B, to appear.
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The results communicated in this paper were obtained in the course of an investigation supported under Contract N00014-75-C-0196 between the California Institute of Technology and the Office of Naval Research.
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Knowles, J.K. The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids. Int J Fract 13, 611–639 (1977). https://doi.org/10.1007/BF00017296
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DOI: https://doi.org/10.1007/BF00017296