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On the Scale of the Nonlinear Effect in a Crack Problem

Knowles, J. K. and Rosakis, A. J. (1986) On the Scale of the Nonlinear Effect in a Crack Problem. Journal of Applied Mechanics, 53 (3). pp. 545-549. ISSN 0021-8936. doi:10.1115/1.3171808.

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When crack problems are analyzed on the basis of nonlinear theories, such as finite elasticity or deformation theory of plasticity, it is inevitable that nonlinear effects will predominate near a crack-tip, even if the loads are small. The most favorable circumstance in this regard occurs when the loads are so small that the zone of significant nonlinearity lies within the region of validity of the near-tip approximation to the global solution of the associated linearized crack problem. This situation - called small-scale yielding for crack problems in plasticity - permits simplifications in analysis which are often decisive; see, e.g., Knowles (1977) and Rice (1968). Insofar as we know, there are no analytical estimates available of the level of load below which nonlinear effects are guaranteed to be small-scale in the above sense. Indeed, even a precise version of the question seems to be lacking. In the present note we formulate and answer such a question for an especially cooperative crack problem; that corresponding to finite anti-plane shear of an infinite medium containing a crack of finite length for an elastic material of Neo-Hookean type. The associated boundary value problem is a linear one for Laplace's equation and thus can be solved globally. Nevertheless, there is a significant nonlinear effect of Kelvin type in the stress field. We give a condition under which this nonlinear response occurs on a small scale near the crack tips.

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Rosakis, A. J.0000-0003-0559-0794
Additional Information:© 1986 American Society Mechanical Engineers. Manuscript received by ASME Applied Mechanics Division, December 16, 1985; final revision February 11, 1986. Paper No. 86-WA/APM-30. Contributed by the Applied Mechanics Division for presentation at the Winter Annual Meeting, Anaheim, CA, December 7-12, 1986, of the American Society of Mechanical Engineers. The authors gratefully acknowledge discussions with C. O. Horgan and E. Sternberg. The second author would also like to acknowledge the support of the National Science Foundation through the 1985 Presidential Young Investigator Award (NSF Grant No. MSM-8451204).
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Official Citation:Knowles J. K., Rosakis A. J. On the Scale of the Nonlinear Effect in a Crack Problem J. Appl. Mech. 53, 545-549 (1986) (5 pages); doi:10.1115/1.3171808
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:51154
Deposited By: Tony Diaz
Deposited On:03 Nov 2014 20:55
Last Modified:10 Nov 2021 19:06

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