Selection of Parameters: General
Calculations have indicated that the SERR in the opening mode is dominant and thus the shear mode is virtually inactive. The critical strain energy release rate as measured by Li (Li, X. et al, 2000) is 1.270N/mm. The next in order of importance is the selection of maximum stress, Omax. This determines the point at which delamination growth initiates. Consider for example the performance of the UEL model (Fig. 4) or the UMAT model (Fig. 6) for various levels of Omax. The most striking feature of the results is that the unloading paths for a given model are parallel to each other; only the crack initiation point – which is almost coincident with the maximum load attained – is different. Thus
crack propagation rate is virtually unaffected by the choice of Omax. Similar observations have been made by the authors elsewhere (Li, Y. et al, 2005 and Alfano, G. et al, 2001).
The following considerations are germane to the selection of the Omax:
(i) The maximum stress must be representative of the tensile strength of the material in which the crack propagates.
(ii) The more physically based cohesive stress-deformation laws involve a rising path till a maximum stress is attained and an unloading path taking the stress to zero when failure is deemed to be complete. On the contrary in the present study the cohesive stress is constant for the most part. If the crack opening displacement is the critical parameter indicating failure, it follows the Omax in our model must be about one half of the actual strength of the material (used in the conventional models).
(iii) At crack initiation, the failure occurs at the crack tip by micro-cracking at an angle into the core as it were. Since we postulate crack growth parallel to the interface, the stress needed to initiate such a crack is probably greater than the actual tensile strength of the material.
The tensile strength of the core material – H200 Divinycell foam – is about 6MPa (Abot, J. L., 2000). The core material is likely to be reinforced due to the presence of a stiff adhesive (of strength 35MPa, Abot, J. L., 2000) bonding the facing sheet with the core for a small distance from the crack interface. It is difficult, therefore to assess precisely the strength of the material; suffice it to say together the effect of the adhesive and the crack kinking at crack initiation (item (iii) above) will put the effective strength significantly more than the actual strength of 6MPa. However in view of item (ii), it is thought appropriate to use the actual strength of 6MPa or values in that range for Omax.
Performance of the UEL model
The finite element model consisted of 4-noded plane strain elements with reduced integration (CPE4R) throughout and the sizes of the element chosen are indicated in Fig. 3. The typical size of the element in the direction of the crack is 0.25mm and thus in one layer there were 840 elements end to end. (A trial run with a coarser mesh with 0.50mm size gave essentially the same results.) Loading was introduced by prescribing the deflection at the point of application of the load and computing the reaction thereof. A total deflection of 30mm was reached in 1000 increments.
Fig. 3. Finite Element Configuration (not to scale)
The values of the maximum cohesive stress, Omax, selected were 3, 6 and 9MPa respectively. Of these the first and the last are 50% less and 50% more respectively from the nominal strength of the core material, viz. 6MPa. The other parameters of the model: Tmax = 20.0Mpa, 8J0 = 1.e-2, S20 = 1.e-3, GIc = 1.270N/mm, GIIc is set at an arbitrarily high value.
Li (Li, X., 2000) studied the problem experimentally by a sequence of loading up to incipient crack growth and unloading fully and reloading as before. Fig.4 displays the key points taken from these experimental results which the authors have taken the liberty of re-plotting ensuring the characteristics passed through the origin (elimination of zero error) but maintaining the slopes of the reloading characteristics and the loads recorded at the crack initiation at the end of each loading phase.
As already mentioned, the prediction of crack initiation point depends on the Omax selected – the value of 3MPa is clearly too small for the corresponding behavior is too compliant in the vicinity of crack initiation and a value of 9MPa overestimates the load corresponding to crack initiation. Thus as far as foam core sandwich members are concerned, we may conclude that reasonable results can be obtained by selecting for Omax the actual strength of the core for delamination predictions using the UEL model. A typical stress distribution around the crack tip is shown in Fig. 5 where it is seen that the model preserves the features of a sharp crack with a highly localized stress concentration with little damage upstream of the crack.
Fig. 4. Simulation results from UEL model vs. experimental results