Issue 19

G. Bolzon et alii, Frattura ed Integrità Strutturale, 19 (2012) 20-28; DOI: 10.3221/IGF-ESIS.19.02 26 C ALIBRATION OF THE FRACTURE PROPERTIES arameters F max , F 1, F plat , D 1 and D 2 (introduced above) characterize the response of the equi−distanced springs, which allow to simulate the fracture propagation in the aluminium laminate. Their calibration is based on the data collected during the tensile tests, namely: the load−displacement curve visualised in Fig. 2 and the progressive mismatch between paper and aluminium boundary, deduced from the analysis of the digital images and partly reported in the graph sequence of Fig. 9. Since snapshots are acquired at a fixed frequency and the loading velocity is also known, a correspondence can be established between each snapshot and the points along the overall force−displacement curve. The sought parameters can be recovered by an inverse analysis procedure [14] based on the minimization of the quadratic discrepancy function  , which accounts for two contributions: C  relevant to the crack opening profiles deduced from snapshots 7 − 41 (see e.g. Fig. 9); L  defined by force levels sampled over a grid of 100 equally spaced points along the displacement axis of the load − elongation curve (Fig. 2): 1 2           d d d d T T C C L L C L    In the above expression, d C , d L represent dimensionless (normalized with respect the units) residual vectors, evaluated as the difference between physical quantities measured in the experiment and computed by the simulation, as a function of the sought parameters: d C collects opening displacements originally evaluated in mm and d L forces originally evaluated in N. The minimization of the discrepancy function ω is performed by a gradient − based algorithm, starting from different initialization vectors, summarized in Tab. 1, to single out the likely occurrence of local minimum points. The parameter values obtained at convergence are also listed in Tab. 1. These numerical results are conventionally rounded to the third decimal figure, regardless to the uncertainty of the estimation. The most recurrent among the different solutions, graphically represented in Fig. 11 (left), correspond to almost equivalent local minimum points while the lowest (optimum) value  opt of the discrepancy function is recovered from the second initialization. However, notice that all solutions practically return the same estimation of the parameter F max , which reflects the overall material strength. The graphs in Fig. 11 (right) compare the experimental output with the load − displacement curves reconstructed on the basis of the converged parameter values listed in Tab. 1. Notice that all computed material responses constitute a reasonable approximation of the real behaviour. The informative content of the DIC measurements can be appreciated from the results listed in Tab. 2 and graphed in Fig. 12, which refer to identification results recovered by including the contribution only in the discrepancy function. Notice that the parameter F max is identified with the same accuracy as in Tab. 1 and that the experimental load − elongation curve is closely reproduced in the only significant portion that corresponds to the exploited information range: in fact, the last considered digital image (41) corresponds to about 3 mm overall displacement. F max [N] F 1 [N] F plat [N] D 1 [mm] D 2 [mm] ω opt ( , C L   ) Trial 1 Initialization 0.6 0.54 0.2 1.52 5 63.29 (39.31+23.98) Converged 1.129 1.054 0.427 0.847 3.510 Trial 2 Initialization 0.75 0.483 0.45 1.6 9 48.64 (32.19+16.45) Converged 1.150 0.575 0.316 2.375 10.064 Trial 3 Initialization 0.55 0.44 0.05 1.4 3.4 63.28 (39.33+23.95) Converged 1.126 1.046 0.427 0.894 3.504 Trial 4 Initialization 0.85 0.1 0.02 1.1 5.5 65.95 (41.60+24.35) Converged 1.150 0.924 0.427 1.236 3.434 Table 1 : Initialization vectors and parameter values returned by the minimization of the discrepancy function ω including contributions from both the crack opening profile ( C  ) and from the load – elongation curve ( L  ). P