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.
Keywords
plane stress, mode I crack, perfectly plastic yield
condition, second third invariants deviatoric stress
tensor, differential algebraic equation, DAE
