The assumption of ellipsoidal flaw geometry has been widely used in calculations of the probability of structural failure conditioned on nondestructive (ND) measurements. Clearly, in most cases the flaw geometry is not ellipsoidal and in the particular case of cracks the actual geometry may deviate significantly from a degenerate ellipsoid (i.e., a planar crack with an elliptical plan-view shape). We have investigated the sensitivity of a late stage of the evolution of fatigue failure to model errors of the latter type (i.e., deviations from elliptical shape for planar cracks) by considering two different overall theoretical processes. In the first, we start with a non-elliptical crack and calculate its geometry after a given large number of cycles of uniaxial stress applied perpendicular to the crack plane. In the second process, we start with the same crack but perform a simulated set of ND measurements coupled with an inversion procedure based on the assumption of elliptical geometry and then calculate the geometry of this initially elliptical crack after subjection to the above stress history. A measure of sensitivity to model error is then provided by a comparison of the two terminal geometries. Results for several choices of non-elliptical crack shapes and sets of ND measurements will be discussed

Chang, R.

Fertig, K. W.

Richardson, John M.

Varadan, Vasundara V.

English

The assumption of ellipsoidal flaw geometry has been widely used in calculations of the probability of structural failure conditioned on nondestructive (ND) measurements. Clearly, in most cases the flaw geometry is not ellipsoidal and in the particular case of cracks the actual geometry may deviate significantly from a degenerate ellipsoid (i.e., a planar crack with an elliptical plan-view shape). We have investigated the sensitivity of a late stage of the evolution of fatigue failure to model errors of the latter type (i.e., deviations from elliptical shape for planar cracks) by considering two different overall theoretical processes. In the first, we start with a non-elliptical crack and calculate its geometry after a given large number of cycles of uniaxial stress applied perpendicular to the crack plane. In the second process, we start with the same crack but perform a simulated set of ND measurements coupled with an inversion procedure based on the assumption of elliptical geometry and then calculate the geometry of this initially elliptical crack after subjection to the above stress history. A measure of sensitivity to model error is then provided by a comparison of the two terminal geometries. Results for several choices of non-elliptical crack shapes and sets of ND measurements will be discussed.</p

Richardson, John

Chang, R

Fertig, K

Varadan, Vasundara

Digital Repository @ Iowa State University (ISU)

SENSITIVITY OF FAILURE PREDICTION TO FLAW GEOMETRY J. M. Richardson, R. Chang, and K. w. Fertig, Jr. Rockwell International Science Center Thousand Oaks, California 91360 and V. V. Varadan Ohio State University Columbus, Ohio 43210 ABSTRACT The assumption of ellipsoidal flaw geometry has been widely used in calculations of the probability of structural failure conditioned on nondestructive (ND) measurements. Clearly, in most cases the flaw geometry is not ellipsoidal and in the particular case of cracks the actual geometry may deviate signifi-cantly from a degenerate ellipsoid (i.e., a planar crack with an elliptical plan-view shape). We have investigated the sensitivity of a late stage of the evolution of fatigue failure to model errors of the latter type (i.e., deviations from elliptical shape for planar cracks) by considering two different overall theoretical processes. In the first, we start with a non-elliptical crack and calculate its geometry after a given large number of cycles of uniaxial stress applied perpendicular to the crack plane. In the second process, we start with the same crack but perform a simulated set of NO measure-ments coupled with an inversion procedure based on the assumption of elliptical geometry and then calculate the geometry of this initially elliptical crack after subjection to the above stress history. A measure of sensitivi~ to model error is then provided by a comparison of the two terminal geome-tries. Results for several choices of non-elliptical crack shapes and sets of NO measurements will be discussed. NATURE OF THE PROBLEM As is well known, the calculation of the probabilities of failure, both unconditional and conditioned on NO measurements, is based on a set of mathematical models, most of which are serious-ly oversimplified in several respects. The set consists of models of (a) the measurement process, (b) the failure process (including a model of the stress environment), and (c) the a ~riori statis-tics of defect properties. It is c ear that the modelling of each type of defect underlies all three of the above models and thus the errors in this modelling are a crucial issue. It is thus obvious that the errors in the defect model affect the interpretation of the NO measurements (in terms of an oversimplified state) and the calculation of conditional probability of failure. The former and latter entail the use of measurement and failure models, respectively, and both entail the use of the a priori statistics model. In any case, we may ask if the effects of the defect model errors in the measurement inter-pretation and the failure probability calculation tend to compound or compensate for each other. To throw light on this question we have investigated several "theoretical experiments" involving syn-thetic test data based on defect models that are more complex than the defect model used in the interpretation of NO measurements and the calcu-lation of failure probability. APPROACH Here we give more explicit details of the investigation of the "theoretical experiments" alluded to in the last section. A typical 433 "theoretical experiment" is represented schemat-ically in Fig. 1. Here we show the simulation of both the ideal calculation involving no defect model error and a non-ideal calculation involving a certain type of defect model error. We have explicitly considered the example of a planar SIMULATION OF' PERFECT MEASUREMENT AND INVERSION --Fig. 1 Schematic representation of "theoretical experiment." crack in a metal with a non-elliptical plan view which is incorrectly modelled in the non-ideal calculation as having an elliptical plan view with adjustable parameters. The failure process in each calculation is assumed to be a deterministic process under an assumed cyclic stress based on generally accepted concepts of fatigue crack prop-agation.1 In the non-ideal calculation, a set of scattered waveforms are calculated by applying scattering theory to the assumed non-elliptical crack. The extraction of features (e.g., the low-frequency scattering amplitude and the distance from the geometrical center to the front-face tangent plane perpendicular to the incident wave direction) for each scattering measurement is straightforward- in fact, the calculations could be simplified greatly by deducing the features directly from the assumed non-elliptical crack. The inversion process is limited to elliptical cracks and thus it attempts to find the best elliptical crack in terms of fitting the non-elliptical input features. The last stage of the non-ideal calculation is the prediction of fatigue crack growth starting with the best elliptical crack. The nature of the ideal calculation is readily apparent. In this case, it is assumed that the first three modules of the non-ideal calcualtion are replaced by ideal ones whose final output is exactly the same as the assumed non-elliptical crack. Thus, the prediction of fatigue crack growth starts with the assumed non-elliptical crack. The initial elliptical and non-elliptical cracks will both grow, after a large number of stress cycles, into much larger planar cracks with nearly circular plan views, each having a charac-teristic average radius. The comparison of the angular average radii yielded by the ideal and non-ideal calculations will be used as the measure of the sensitivity to model error. An alterna-tive, and perhaps simpler, comparison procedure involves the consideration of equivalent circular cracks. As shown in Fig. 2, a circular crack is equivalent to a planar crack of arbitrary shape if they both evolve asymptotically (under a given cyclic applied stress) into the same large circular crack. SC81-12894 Fig. 2 Equivalent circular crack. COMPUTATIONAL RESULTS In this section we present computational results for an extreme form of model error, i.e., we consider the model crack to be circular while the actual (in the sense of the theoretical exper-iment discussed above) crack consists of two separate co-planar circular cracks. The partic-ular cases considered are depicted in Fig. 3, 434 Fig. 3 "Theoretical experiment" using the feature d. namely (a) two identical circular cracks, both of radius a, with a distance b between centers and (b) two nonidentical cracks, one having radius a and the other having radius a/2, again with a dis-tance b between centers. Less extreme cases are currently under consideration and the results ensuing from these investigations will be pre-sented in a future communication. We consider mainly two kinds of features, namely Az and d for a set of in-plane pulse echo scattering measurements. The quantity Azw2 is the scattering amplitude in the Rayleigh (i.e., long wavelength) regime where w is the angular fre-quency. The quantity d is the distance from the geometrical center (assuming that this is defined) to the front-face tangent plane (or tangent line in the crack plane). An alternative geometrical property will also be considered. In our computa-tions the different types of features will be con-sidered individually. A study was made with A2 and d as simultaneous inputs to a probabilistic inversionl algorithm, with varying weights reflecting the assumed standard deviations of the experimental errors ascribed to the two fea-tures. The results contained no interesting surprises and will not be reported here. The first series of theoretical experiments involved the single feature Az and a circular crack model. In all cases it was assumed in each case that the equation between the actual cracks was sufficiently large that the quasi-static elastic interactions between the cracks could be neglected in the computation of Az. With this approximation Az consists of the independent con-tributions of the two circular cracks and hence is independent* of the direction of the incident wave. Efforts to compute the effect of inter-action on Az ran into computational difficulties and consequently results are not yet available. The fatigue growth of the model and actual cracks was represented in terms of the concept of equiv-alent circular crack as explained in the last section. In the case of the actual crack the fatigue process was treated both with and without interaction between the separate circular cracks. *It 1s to be emphasized that the wavelength is assumed to be large compared with the total complex scatterers composed of both circular cracks. In Table 1 we present results for the case in which the actual crack consists of two circular cracks with equal radii a1 = a2 = a= 1 and with various values of the center-to-center distance b. The estimate r of the radius of the model circular crack was obtained by observing that A2 for a circular crack is proportional to the cube of its radius and r is the cube root of the sum of the cubes of the radii of the separate circular cracks. The quantity re is the radius of the equivalent circular crac~. As stated before, it is computed with and without interaction. In Table 2 the radii a1 and a2 are different, namely a1 = 1 and a2 = 0.5. The results are of course similar to those with equal radii. Considering the extreme nature of the model error, it is surprising that r and rea are in such close agree-ment. This means that tailure prediction based upon A2 alone is quite insensitive to model error (at least in the case of fatigue crack growth in metals). Table 1. Equal Circles (a 1) Input Feature: A2 Model: Circular Crack b r No Int. 2.5 1.26 3 4.5 6 Interact 1.37 1.35 1.27 No Int. 1.31 1.27 1.18 1.23 1.17 Table 2. Unequal Circles (a1 1, a2 = 0.5) Input Feature: A2 Model: Circular Crack b r No Int. 2 1.04 3 4 5 req Interact No Int. 1.11 1.09 1.09 1.07 1.07 1.05 1.07 1.04 A second series of theoretical experiments was conducted with the d's (the distances from the center to the front face tangent planes) as the sole features. There we have considered only the case in which the radii a1 and a2 are equal with the common value denoted by a which is set equal to 1. In Table 3, we present results in which it is assumed that the incident directions of pulse-echo elastic waves are chosen to be given by multiples of 45° with representative configura-435 tions shown in Fig. 3. The best estimate r of the radius of the circular model crack is the average of all of the d's and these estimates are listed in the second column. The values of r is the same as those given in Table 1. It wif~ be noted that the agreement between r and either of the values of req is poor, especially for the larger values of b. This means that failure prediction based upon the d's along is relatively sensitive to model error. In Table 4, we show that a different geometrical feature, the total area of the crack, yields much greater insensitivity to model error. Table 3. Equal Circles (a 1) Input Feature: d's Model: Circular Crack b r 2.5 1. 76 3 1.91 4.5 2.26 6 2.81 req Interact 1.37 1.35 1.27 1.23 Table 4. Equal Circles (a= 1) Input Feature: Total Crack Area Model: Circular Crack b r req Interact 2.5 1.41 1.37 3 1.35 4.5 1.27 6 1.23 ACKNOWLEDGEMENT No Int. 1.31 1.27 1.18 1.17 No Int. 1.31 1.27 1.18 1.17 This research was sponsored by the Center for Advanced NDE operated by the Rockwell Interna-tional Science Center for the Defense Advanced Research Projects Agency and the Air Force Wright Aeronautical Laboratories under Contract No. Fee615-80-C-5004 REFERENCE 1. R. Chang, "On Crack-Crack Interaction and Coalescence in Fatigue," submitted to Engineering Fracture Mechanics. 

Sensitivity of Failure Prediction to Flaw Geometry

Fertig, K.

Richardson, John M

Fertig, K W

Varadan, Vasundara V

Iowa State University Computer Science Technical Reports

https://dr.lib.iastate.edu/bitstreams/24585793-aabc-43a2-a55c-412bbd384c85/download

Sensitivity of Failure Prediction to Flaw Geometry

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