On the Ultrasonic Evaluation of Chaotically Weakened Welds

The ultrasonic evaluation of the structural integrity of welds is investigated by employing the generic model of a weakened weld across a planar surface. The weakening of the weld is considered to be caused by a large number of chaotically sized defects, which for computational convenience are taken to be penny-shaped micro-voids or micro-cracks. The weakened weld is insonified by an ultrasonic beam which is decomposed into plane longitudinal or shear waves. The purpose of this study is to examine the effect of different chaotic defect maps on the reflection of the ultrasonic beam. To this end, the micro-defect diameter, d, is generated by a Mandelbrot-like delayed map of the form, d(n)=abs(d(n-m))^k-c, where d(n) is the diameter of the nth defect, k and c are positive parameters and m is a positive integer defining delay. This way, all the defect sizes are obtained given the sizes of the first m defects. Depending on the choice of the parameters k, c, m the produced defect sizes are either periodic or chaotic. For each m=1, 2, or 3, two-dimensional curves of the parameters k and c are obtained in this study which yield chaotic defect size sequences. Then, for a large number of such k and c values, the reflection of the ultrasonic beam is computed for the weakened weld of two similar isotropic metals. The weld compliance is subsequently readily obtained as is proportional to the reflection coefficient. It is noted that for a quasi-static ultrasound insonification and non-interacting defects, the weld compliance and the ultrasound reflection coefficient are determined by the average cubed micro-crack diameter. For each value of the parameter k, the weld compliance exhibits a highly non-linear dependence on the parameter c with the appearance of multiple extrema for the values of c that yield chaotic defect sizes.