Summary
In this paper we return to the asymptotic analysis, within the nonlinear equilibrium theory of compressible elastic solids, of the deformations and stresses near the tip of a traction-free crack in a slab of all-around infinite extent under conditions of plane strain. As before, the loading at infinity is taken to be one of uniform uni-axial tension at right angles to the faces of the crack. We show that once a restrictive assumption introduced at the start of our earlier asymptotic treatment of the problem is relinquished, certain perplexing anomalies encountered in the previous analysis no longer arise. The present reconsideration of the problem leads to modifications in the dominant-order results for the “secondary” deformations and stresses, while those pertaining to the physical quantities of primary interest remain unaffected. Furthermore, this investigation encompasses some higher-order considerations, which supply an essential clarification and improvement of the lowest-order asymptotic solution.
Zusammenfassung
In dieser Arbeit wird die asymptotische Ermittelung, im Rahmen der nichtlinearen Elastizitätstheorie ebener Verformungen, von dem elastostatischen Feld am Ende eines Risses in einer allseitig unendlichen Scheibe wieder aufgenommen und fortgesetzt. Wir zeigen hier, dass gewisse physikalisch unerklärbare Resultate in unserer früheren Behandlung des Problems beseitigt werden wenn man eine unserer ursprünglichen Annahmen über die asymptotische Struktur der Lösung verallgemeinert. Obwohl diese Verallgemeinerung keinen Einfluss auf die wichtigsten. vorher gefundenen Resultate hat, führt sie zu wesentlichen Änderungen in den sekundären Spannungen und in der sekundären Verschiebung. Die vorliegende Arbeit enthält überdies auch asymptotische Betrachtungen zweiter Ordnung die für das Verständnis der Lösung niedrigster Ordnung unentbehrlich sind.
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References
Knowles, J. K. and Sternberg, Eli, An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack, Journal of Elasticity, 3 (1973) 2, 67
Wong, F. S. and Shield, R. T., Large plane deformations of thin elastic sheets of neo-Hookean material, Zeitschrift für angewandte Mathematik und Physik, 20 (1969) 2, 176
Knauss, W. G., Stresses near a crack in a rubber sheet, Journal of the Society for Experimental Stress Analysis, 8 (1968) 177
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The results communicated in this paper were obtained in the course of an investigation conducted under Contract N00014-67-A-0094-0020 of the California Institute of Technology with the Office of Naval Research in Washington, D. C.
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Knowles, J.K., Sternberg, E. Finite-deformation analysis of the elastostatic field near the tip of a crack: Reconsideration and higher-order results. J Elasticity 4, 201–233 (1974). https://doi.org/10.1007/BF00049265
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DOI: https://doi.org/10.1007/BF00049265