The asymptotic expansion of the stress field near a sharp crack in a linear elastic body with respect to , the distance from the crack tip, is

(Williams, 1957), where and are the in-plane polar coordinates centered at the crack tip. The local axes are defined so that the 1-axis lies in the plane of the crack at the point of interest on the crack front and is perpendicular to the crack front at this point; the 2-axis is normal to the plane of the crack (and thus is perpendicular to the crack front); and the 3-axis lies tangential to the crack front. is the extensional strain along the crack front. In plane strain ; in plane stress the term vanishes.

The -stress represents a stress parallel to the crack faces. It is a useful quantity, not only in linear elastic crack analysis but also in elastic-plastic fracture studies.

The -stress usually arises in the discussions of crack stability and kinking for linear elastic materials. For small amounts of crack growth under Mode I loading, a straight crack path has been shown to be stable when , whereas the path will be unstable and, therefore, will deviate from being straight when (Cotterell and Rice, 1980). A similar trend has been found in three-dimensional crack propagation studies by Xu, Bower, and Ortiz (1994). Hutchinson and Suo (1992) also showed how the advancing crack path is influenced by the -stress once cracking initiates under mixed-mode loading. (The direction of crack initiation can be otherwise predicted using the criteria discussed in “Prediction of the direction of crack propagation,” Section 2.16.4.)

The -stress also plays an important role in elastic-plastic fracture analysis, even though the -stress is calculated from the linear elastic material properties of the same solid containing the crack. The early study of Larsson and Carlsson (1973) demonstrated that the -stress can have a significant effect on the plastic zone size and shape and that the small plastic zones in actual specimens can be predicted adequately by including the -stress as a second crack-tip parameter. Some recent investigations (Bilby et al., 1986; Al-Ani and Hancock, 1991; Betegón and Hancock, 1991; Du and Hancock, 1991; Parks, 1992; and Wang, 1991) further indicate that the -stress can correlate well with the tensile stress triaxiality of elastic-plastic crack-tip fields. The important feature observed in these works is that a negative -stress can reduce the magnitude of the tensile stress triaxiality (also called the hydrostatic tensile stress) ahead of a crack tip; the more negative the -stress becomes, the greater the reduction of tensile stress triaxiality. In contrast, a positive -stress results only in modest elevation of the stress triaxiality. It was found that when the tensile stress triaxiality is high, which is indicated by a positive -stress, the crack-tip field can be described adequately by the HRR solution (Hutchinson, 1968; Rice and Rosengren, 1968), scaled by a single parameter: the -integral; that is, -dominance will exist. When the tensile stress triaxiality is reduced (indicated by the -stress becoming more negative), the crack-tip fields will quickly deviate from the HRR solution, and -dominance will be lost (the asymptotic fields around the crack tip cannot be well characterized by the HRR fields). Thus, using the -stress (calculated based on the load level and linear elastic material properties) to characterize the triaxiality of the crack-tip stress state and using the -integral (calculated based on the actual elastic-plastic deformation field) to measure the scale of the crack-tip deformation provides a two-parameter fracture mechanics theory to describe the Mode I elastic-plastic crack-tip stresses and deformation in plane strain or three dimensions accurately over a wide range of crack configurations and loadings.

To extract the -stress, we use an auxiliary solution of a line load, with magnitude , applied in the plane of crack propagation and along the crack line:

The term for plane stress.

The interaction integral used is exactly the same as that for extracting the stress intensity factors:

with as

In the limit as , using the local asymptotic fields,

where for plane stress and for plane strain, axisymmetry, and three dimensions. is zero for plane strain and plane stress.

can be calculated by means of the same domain integral method used for -integral calculation and the stress intensity factor extraction, which has been described in “J-integral evaluation,” Section 2.16.1, and “Stress intensity factor extraction,” Section 2.16.2.