Framework for quantifying crack tip constraint

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.