applied fracture mechanics

204Applied Fracture Mechanics10, 1000, 1000000, with the scatter of 10%, the summed differences are of the order ofmagnitude 1, 100, 100000, and hence, dynamic changes in the total value of the sum Sdepend on values of differences – as a quadratic function it is characterized by a linearfunction of the derivative, which also means that for differences close to zero (e.g. 10 -5 ,10 -8 , etc.) this dynamic change is much smaller than for differences of highermagnitudes, which influences the „flexibility” of the performed approximation;- if the test data significantly differ from each other in magnitude (e.g. from 1 to 100000 orfrom 10 -8 to 10 -2 ), the approximated values near the lower threshold contribute muchless to the total sum S than approximated values near the upper threshold; this meansthat, e.g. tens or hundreds of test data with differences in magnitude of 100% fromvalue 1 are less significant in performing the approximation than one or a few datapoints which differ by 1% from value 100000.According to the above stated example, the approximation is “asymmetric” since betterapproximation will be achieved for higher values of test data, neglecting differences aroundsmaller values – an example of such approximation is shown in Fig. 4b, where one can seea good fit of theoretical description of 3 curves for large values of da/dN (over 10 -4 mm/cycle)while there is an evident misfit for smallest values (below 10 -5 mm/cycle). The presentedapproximation has been achieved by satisfying the LSM criterion, i.e. the minimum value ofthe sum S. When the test data are within a wide range of values, e.g. 5 orders of magnitude,i.e. from 10 -2 to 10 -7 mm/cycle, then differences between the highest values and theapproximating function will have the largest effect on the square sum S of deviations whiledifferences for small values, sometimes of 2-3 orders of magnitude, do not contribute muchto the total sum S.Hence, the misfit of the approximating function for low values of da/dN, practically forvalues lower by only 1-2 orders of magnitude than the maximum values of da/dN. Withinthis range the theoretical description is rather random and has rather no effect on the valueof the sum S, which indicates that this criterion is rather useless for this type of analysis.It seems reasonable to use one of the following criterion modifications, which will allow toremove the above stated problems:- changing the form of the criterion, or- using logarithmic values of da/dN,nn i i i i2 2 (16)S log y log y or S log y log y .i1 i1In the present study the first variant has been examined (see section 3.3) due to the fact thatit is more general since it does not limit itself only to positive values of predicted yi, which isa requirement in the second variant. In the case of crack propagation test data all the da/dNvalues are positive; therefore the second variant could also be used.Since the criterion for fitting the theoretical description to the test data in the form ofequation (15) or (16), or any other, is closely connected with the number of approximated