The effect of crack-tip interactions on the curve-fitting of mixed-mode isopachics

Thermoelastic Stress Analysis (TSA) is a full-field technique for experimental stress analysis that has proved to be extremely effective for studying stress fields in the vicinity of crack-tips. An understanding of such fields is vital to the development of effective diagnosis and prognosis algorithms for Non-Destructive Testing (NDT) and Structural Health Monitoring (SHM). The key to crack-tip studies using TSA is the observation that the stress-sum contours (isopachics) in the vicinity of the tip take the form of a simple curve ? the cardioid. This was exploited in [1] in order to estimate the Stress Intensity Factors (SIFs) for crack-tips in mode 1 and mixed-mode opening. The analysis [1] made use of the cardioid nature of the isopachics by deriving expressions for the SIFs in terms of the cardioid area and the positions of certain tangents to the curve.
Recent work by the authors has allowed the estimation of crack-tip Stress Intensity Factors (SIFs) by curve-fitting a cardioid form to measured isopachics from Thermoelastic Stress Analysis (TSA). Both Genetic Algorithms (GAs) [2] and Differential Evolution (DE) [3] proved successful for the actual parameter estimation, but some of the curve-fits indicated that the cardioid form was inappropriate for the base model.
A possible explanation for the poor curve-fit is that the cardioid form is only theoretically suitable for an isolated crack-tip stress field, as derived from the Westergaard equations. The effect of the other crack-tip in a central crack hac been neglected from previous analyses. Further work has [4] considered a mode 1 central crack, placed in a plate, which therefore had two interacting crack-tips. Figure 1 shows the analytically derived stress sum field around both an isolated mode 1 crack and a twin crack-tip. The curves for an isolated crack-tip are shown as solid lines ? these curves are true cardioids. The curves for the twin crack-tip case are shown as dashed lines. It is clear that these curves are not cardioids, although the level of distortion is quite small for the inner curves.
The object of the current paper is to determine numerically, the stress field for a mixed-mode crack system and to quantify the effect of any interactions on the curve-fitting procedure and compare with experimental data for a 30o crack.