Three scientists named Hutchinson, Rice and Rosengren independently evaluated the character of crack-tip stress fields in the case of power-law-hardening materials. Hutchinson evaluated both plane stress and plane strain, while Rice and Rosengren considered only plane-strain conditions. Both articles, which were published in the same issue of the Journal of the Mechanics and Physics of Solids, argued that stress times strain varies as 1/r near the crack tip, although only Hutchinson was able to provide a mathematical proof of this relationship. Hutchinson began by defining a stress function Φ for the problem. The governing differential equation for deformation plasticity theory for a plane problem in a Ramberg-Osgood material is more complicated than the linear elastic case:
where the function γ differs for plane stress and plane strain. For the Mode I crack problem, Hutchinson chose to represent Φ in terms of an asymptotic expansion in the following form:
where A,B are constants that depend on θ, the angle from the crack plane. Equation (A2) is analogous to the Williams expansion for the linear elastic case. If s < t, and t is less than all subsequent exponents on r, then the first term dominates as r →0. If the analysis is restricted to the region near the crack tip, then the stress function can be expressed as follows:
where k is the amplitude of the stress function and is a dimensionless function of Φ. Although Equation (A1) is different from the linear elastic case, the stresses can still be derived. Thus the stresses, in polar coordinates, are given by
The boundary conditions for the crack problem are as follows:
In the region close to the crack tip where Equation (A3) applies, elastic strains are negligible compared to plastic strains; only the second term in Equation (A1) is relevant in this case. Hutchinson substituted the boundary conditions and Equation (A3) into Equation (A1) and obtained a nonlinear eigenvalue equation for s. He then solved this equation numerically for a range of n values. The numerical analysis indicated that s could be described quite accurately (for both plane stress and plane strain) by a simple formula:
which implies that the strain energy density varies as 1/r near the crack tip. This numerical analysis also yielded relative values for the angular functions σ_ij. The amplitude, however, cannot be obtained without connecting the near-tip analysis with the remote boundary conditions. The J contour integral provides a simple means for making this connection in the case of small-scale yielding. Moreover, by invoking the path-independent property of J, Hutchinson was able to obtain a direct proof of the validity of Equation (A5).