Performance of the UMAT model

The finite element mesh configuration selected for the analysis is similar to the one used for the UEL model except for the mesh size in the longitudinal direction which is now 0.5mm. Two values of strength were investigated, viz. 7MPa and 9.4MPa which are marginally higher than those used for the UEL model. It was anticipated that the strength needed for the UMAT model must be higher than that employed for the UEL model, to compensate for the influence of compressive stress developing in the cohesive layer upstream of the crack. However, the analysis revealed that the cohesive zone (that part of the cohesive layer which is in tension starting from the crack tip) was sufficiently long (12 elements in length) so that this effect is not significant.

The results obtained for the two values of Omax exhibit the same trend (Fig. 6), but differ in their prediction of crack initiation as seen in the UEL model results. The crack initiation results agree well with UEL model results obtained with comparable values of Omax (Fig. 7). However there is an obvious and a significant difference between the UMAT results and UEL results which is highlighted in Fig. 7. It is seen that the UMAT model predicts that the crack ceases to grow after some growth and the load then begins to gradually increase with the crack opening displacement. There is no reason to suppose that such a prediction has any validity to it and in fact can be shown to be due to an intrinsic deficiency of the model.

Fig. 7 illustrates what happens when an initial crack length lo higher than 51mm is assumed, say 71mm. The load-displacement characteristic has a smaller slope and crack initiation occurs at a point which lies on or close to the unloading characteristic associated with lo = 51mm. But the unloading characteristic associated with lo = 71mm deviates from that associated with an initial crack length of 51mm and lies closer to that given by the experiment and UEL model. However this characteristic too gradually deviates from the correct result. It is clear therefore that the UMAT model, though capable of predicting crack initiation correctly gradually loses its ability to predict continued crack growth.

Fig. 8 shows the magnified image of stress distribution in the close vicinity of the crack tip when the crack opening displacement measured at the point of load application is 25mm, At this instant crack has progressed to a length of 101mm and the next crack tip element is on the point of failure.

Fig. 6. Simulation results from UMAT model

Fig. 7. Comparison between UEL results and UMAT results

Fig. 8. Stress distribution around crack tip

Fig. 9 shows the magnified view of the deformation of the bottom surface of the cohesive elements close to the crack tip for the two cases: (i) The current crack length (lcr ) of 101mm is reached after a crack extension of 50mm from the initial crack length (lo) of 51mm, and (ii) The initial crack length is taken as 101mm and the crack tip element is on the verge of failure ( lcr= lo = 101mm). Note that the downward deflections are plotted above the x-axis in the figure.

There are two features that may be observed in these pictures:

(i) From Fig 8 , it is seen that there is a part of cohesive layer which carries tensile stress in the vicinity of the crack tip, but more significantly this stress gradually diminishes to zero as we travel upstream of the crack and changes to compression.

(ii) In case (i) the depth of the cohesive layer reduces as we travel upstream from the crack tip and a “neck” – a region of slightly reduced depth compared to neighboring regions on either side appears. The bottom surface of the cohesive elements is undulating at the crack tip. This is in contrast to the case (ii) which exhibits a smooth variation of the cohesive layer.

It is clear that item (ii) must be a result of what transpires during the failure of cohesive elements and the accompanying crack extension. As each element fails certain force is released on to the delaminated sheet which rotates so as to compress the cohesive layer downstream. Once a “neck” of certain threshold acuteness is formed, the crack growth is at first seriously inhibited and eventually stops.

Fig. 9. Displacements along the bottom surface of the cohesive elements (when Omax = 7Mpa)

These phenomena are fairly insignificant or virtually absent in the UEL model. Of these, item (i) can have an effect on crack initiation as the compression block upstream can send a wave of transverse compressive stress towards the crack tip thereby delaying the crack initiation. However, as already mentioned this effect appears to be not significant for sandwich members as the cohesive zone is of greater length than in laminated composites – of the order of 6-7mm. The second factor, too, kicks in much later, so much so the UMAT model can be used for a study of delamination in sandwich beams if one is not investigating large crack extensions.