Issue 27

L. Marsavina et alii, Frattura ed Integrità Strutturale, 27 (2014) 13-20; DOI: 10.3221/IGF-ESIS.27.02 15 This technique was further developed on mode II fatigue cracks, Stanley and Chan, [9]. (a) (b) (c) Figure 2 : A typical set of data points from a thermoelastic scan of a crack (Stanley and Chan [2]). A study of simulated inclined edge and centre cracks by Stanley and Dulieu-Smith [10] showed that the isopachic contour in the crack tip region generally took the form of a cardioid curve, centred on the crack tip. By determining the area and orientation of a typical cardioid,  K I and  K II were calculated. Using Eq. (2) the following equation was derived: 2 2 2 2 (1 cos( 2 )) I II K K r A S          (4) where  = tan -1 (  K I /  K II ). For a constant value of S , Eq. (4) represents a closed curve in ( r,  ) co-ordinates which represent a cardioid or “apple” shape. In the experimental investigation a series of line scans parallel to the crack line were used to construct a curve by recording the location(s) on each line scan where a particular value of thermoelastic signal was measured. Hand fitting a cardioid shape to experimental data did not result in a perfectly symmetrical curve and scatter in the results was noted. The agreement between experimental and theoretical  K I /  K II values was concluded to be no better than moderate. In a later paper the omission of the non-singular stress term in the calculation of stress intensity factors in the original work of Stanley and Dulieu-Smith [10] was discussed. A method was suggested for including this factor and determining its value from a map of thermoelastic data. However, no results were presented to indicate whether experimental values were closer to theoretical predictions for  K I and  K II when account was taken of the non-singular stresses. The cardioid curve method was extended to allow inclusion of all the data from around the crack tip (Fulton et al. [11], Dulieu-Barton et al. [12]). A computer program is used to calculate the area and orientation of nine cardioid curves of constant thermoelastic signal. A potential source of error in the method was highlighted as being its sensitivity to an exact knowledge of pixel size in the map of thermoelastic data. This technique was further developed to greater accuracy by Dulieu-Barton and Worden, [13] using a curve fitting routine based on a genetic algorithm to generate the cardioid curves from thermoelastic data, which were subsequently used to determine the stress intensity factor. An alternative approach for determining stress intensity factors for cracks subject to mixed-mode loading which had previously been used in photoelastic analysis, was developed by Tomlinson et al. [7]. A Newton-Raphson iteration combined with a least squares approach was used to fit the equations describing the stress field around the crack tip, based on Mushkelishvili’s approach, to the experimental data. This approach allows different applied stress fields to be described which may include non-uniform stress fields. The computerised analysis method required the map of data obtained from each SPATE scan to be interrogated at a number of points on lines radiating from the crack tip between an inner and an outer radius in the "singularity dominated zone". The co-ordinates of these points and the SPATE signal value were input into a new computer program which calculates  K I and  K II . The mean and variance of the least-squares fit of the solution to the data points are also calculated in order to give an indication of the accuracy of the results. A further data array method was developed by Lesniak [6] where the Airy stress function was fit to the array of thermoelastic data using a least squares method. The method uses the complete stress function, rather than just the singularity term, and therefore can account for the boundaries of a component. It is also very fast, since multiple terms of the stress field can be least-squares fit in seconds, which gives the potential to perform live crack growth analyses. The procedure gave errors of 3% for mode I cases when compared to analytical solutions. A mixed-mode example has also been published but with less accurate results, Lesniak et al. [14].