This sub-sampling results inevitably in a sort low-pass filter of the real data, with the stronger gradient and peak values being smoothed. Figure 4.6 show example of results, in terms of the first principal strain, with the grid parallel to the crack (a) and with a rotation of 20 and 3 mm offset in Y direction (b). The virtual markers grid is superimposed for clarity. The missing areas correspond to the elements that are cut by the grid. It is observed that the grid method, because of the pitch size, provides quite accurate results where the strain gradient are small, but the error increases when the strain concentrates, with an underestimation of 2–3 % in the yellow areas; at the crack tip the error is even greater but this region is not considered in this part of the work. The extended optical grid method has been applied to the subsampled nodal data, including the enrichment shape functions given by Eq. (4.2) and adopting the strain difference minimization criterion. In this way, it was possible to assess the strain even in the elements that are cut in two parts by the crack. The corresponding first principal strain maps are reported in Fig. 4.7. Now, only the two elements containing the crack tips are omitted from the computation, with the newly filled areas appearing to provide acceptable results. Since all the data are mapped onto the undeformed configuration, it is possible to compute the error by performing a simple point-by-point subtraction of these data from the real ones given in Fig. 4.5. Moreover, the error due to subsampling depends on the position (offset) of the grid with respect to the crack; for this reason, the grid has been offset in X and Y direction in several positions, obtaining the maximum error distribution shown in Fig. 4.8. One observes that the extension of the measure to the elements with the through crack determines a widening of the area with larger error, but the absolute value of the error doesn’t increase significantly. Fig. 4.5 “Real” strain distribution map as computed by FEM with fine mesh Fig. 4.6 e1 distribution map for a simulated grid method of pitch 4 mm (a) crack parallel, (b) 20 rotated 4 Strain Assessment in Cracked Sheet Metals by Optical Grid Method 43
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