The terms high and low constraint have been loosely used to distinguish different levels of near tip stress triaxiality in different crack geometries. In this paper, a precise measure of crack tip constraint is provided through a stress triaxiality parameter Q. It is shown that the J-integral and Q are sufficient to characterize the full range of near-tip fracture states. Within this framework J and Q have distinct roles: J sets the size scale over which large stresses and strains develop, while Q scales the near-tip stress distribution relative to a reference high triaxiality state. Specifically, negative (positive) Q values mean that the hydrostatic stress ahead of the crack is reduced (increased) by Qσ0 from the plane strain reference distribution.
The evolution of near-tip constraint as plastic flow progresses from small-scale yielding to fully yielded conditions is examined. It is shown that the Q parameter adequately characterizes the full range of near-tip constraint states in several crack geometries. Through-thickness deformation and stress conditions affect near-tip triaxiality. Stress triaxiality near a three-dimensional crack front is measured by pointwise values of Q.
The J-Q theory provides a framework that allows the toughness locus to be measured and utilized in engineering applications. A method for evaluating Q in fully yielded crack geometries and a scheme to interpolate for Q over the entire range of yielding are presented. Extension of the J-Q theory to creep crack growth is discussed in the concluding section.