Vol. 3, No. 4, 2008

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A plane stress perfectly plastic mode I crack problem for a yield condition based on the second and third invariants of the deviatoric stress tensor

David J. Unger

Vol. 3 (2008), No. 4, 795–807

A statically admissible solution for the opening mode of fracture under plane stress loading conditions is obtained for a yield condition containing both the second and third invariants of the deviatoric stress tensor. This yield locus lies approximately midway between the Mises and Tresca yield loci in the principal stress plane. The crack problem addressed is analogous to an earlier one investigated by John W. Hutchinson for the Mises yield condition. A stress function approach to the present problem results in a differential algebraic equation rather than an ordinary differential equation as in the former case. It is found that a reduction of order is possible for this second order differential equation of the sixth degree through a simple transformation which generates a Clairaut equation. This equation can be integrated analytically to obtain the general solution of the governing second order differential equation for uniform states of stress. This general solution is applicable to two of three distinct sectors of the plane crack problem. The remaining sector in the plane is governed by the singular solution of this Clairaut equation. The first integral of the singular solution, which is the envelope of general solution, is found through the use of a contact transformation. This transformation aids in reduction of this equation to that of a first order differential equation of the thirtieth degree. The primitive of this first order differential algebraic equation is obtained by numerical solution. An approximate analytical solution to the problem is also provided. These results are compared to those obtained previously for the analogous crack problem under the Mises yield condition.

plane stress, mode I crack, perfectly plastic yield condition, second third invariants deviatoric stress tensor, differential algebraic equation, DAE
Received: 11 November 2007
Revised: 29 February 2008
Accepted: 6 March 2008
Published: 1 June 2008
David J. Unger
Department of Mechanical and Civil Engineering
University of Evansville
1800 Lincoln Avenue
Evansville, IN 47722
United States