T-stress extraction

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

where r and $\theta$ 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. ${\mathbf{t}}^o$ is the surface traction on the crack surfaces at the crack tip, and ${\sigma }_{13}^o$ is a constant stress term for ${\sigma }_{13}$ . ${\epsilon }_{33}$ is the extensional strain along the crack front. In plane strain ${\epsilon }_{33}=0$ ; in plane stress the term $[\nu \left(T+t_2^o\right)+E{\epsilon }_{33}]{\delta }_{3i}{\delta }_{3j}$ vanishes. The above expression for the asymptotic stress fields follows directly from the well-known results of Williams (1957), generalized to account for nonzero crack face tractions, ${\mathbf{t}}^o$ , as discussed in the works of Zafosnik et. al. (2005) and Hurtado and Bose (2023).

In Abaqus the T-stress refers to the constant ( $r^0$ ) term in the asymptotic expansion of the stress component ${\sigma }_{11}$ . The T-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 $T<0$ , whereas the path will be unstable and, therefore, will deviate from being straight when $T>0$ (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 T-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.)

The T-stress also plays an important role in elastic-plastic fracture analysis, even though the T-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 T-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 T-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 T-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 T-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 T-stress becomes, the greater the reduction of tensile stress triaxiality. In contrast, a positive T-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 T-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 J-integral; that is, J-dominance will exist. When the tensile stress triaxiality is reduced (indicated by the T-stress becoming more negative), the crack-tip fields will quickly deviate from the HRR solution, and J-dominance will be lost (the asymptotic fields around the crack tip cannot be well characterized by the HRR fields). Thus, using the T-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 J-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 T-stress, we use an auxiliary solution of a line load, with magnitude f, applied in the plane of crack propagation and along the crack line:

The term ${\sigma }_{33}^L=0$ for plane stress.

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

with $\mathbf{M}$ as

In the limit as $r\to 0$ , using the local asymptotic fields,

where $\overline{E}=E$ and $\xi =1$ for plane stress; $\overline{E}=E/\left(1-{\nu }^2\right)$ and $\xi =1+\nu$ for plane strain, axisymmetry, and three dimensions; ${\epsilon }_{33}$ is zero for plane strain and plane stress; $\alpha$ is the thermal expansion coefficient; and $\mathrm{\Delta }\mathrm{\Theta }$ is the temperature difference.

$I_{int}\left(s\right)$ can be calculated by means of the same domain integral method used for J-integral calculation and the stress intensity factor extraction, which has been described in J-integral evaluation, and Stress intensity factor extraction. $I_{int}\left(s\right)$ is doubled if only half the structure is modeled.

The approach used to extract the T-stress was adopted from Nakamura and Parks, 1992.