J-integrals and mode mixity by the method of 2ESLs

Reddy

Figure 7.2: Distribution of the ERRs and mode mixity along the delamination front, problem a in Figure 5.1, case I. Plate widths: b=100 mm (a),b=160 mm (b) (Compare to Fig. 7.10).

The ERR and the mode mixity are presented through Figures 7.2 and 7.5 for problem a in Figure 5.1a and through Figures 7.6-7.9 for problem b in Figure 5.1b. In each ﬁgure

Reddy

Figure 7.3: Distribution of the ERRs and mode mixity along the delamination front, problem a in Figure 5.1, case II. Plate widths: b=100 mm (a),b=160 mm (b) (Compare to Fig. 7.11).

Reddy

Figure 7.4: Distribution of the ERRs and mode mixity along the delamination front, problem a in Figure 5.1, case III. Plate widths: b=100 mm (a),b=160 mm (b) (Compare to Fig.7.12).

CHAPTER 7. ENERGY RELEASE RATES AND MODE MIXITY

Reddy

Reddy Reddy

Reddy

Figure 7.5: Distribution of the ERRs and mode mixity along the delamination front, problem a in Figure 5.1, case IV. Plate widths: b=100 mm (a),b=160 mm (b) (Compare to Fig.7.13).

Reddy

Figure 7.6: Distribution of the ERRs and mode mixity along the delamination front, problem b in Figure 5.1, case I. Plate widths: b=60 mm (a), b=90 mm (b) (Compare to Fig. 7.14).

Reddy

Figure 7.7: Distribution of the ERRs and mode mixity along the delamination front, problem b in Figure 5.1, case II. Plate widths: b=60 mm (a),b=90 mm (b) (Compare to Fig. 7.15).

Reddy

Figure 7.8: Distribution of the ERRs and mode mixity along the delamination front, problem b in Figure 5.1, case III. Plate widths: b=60 mm (a),b=90 mm (b) (Compare to Fig. 7.16).

CHAPTER 7. ENERGY RELEASE RATES AND MODE MIXITY

Reddy

0 1000 2000 3000 4000 5000 0 900 1800 2700 3600 4500

Figure 7.9: Distribution of the ERRs and mode mixity along the delamination front, problem b in Figure 5.1, case IV. Plate widths: b=60 mm (a),b=90 mm (b) (Compare to Fig. 7.17).

G_T =G_II +G_III is the total ERR. The solution by the VCCT, Reddy TSDT (Szekr´enyes (2014c)), SSDT (Szekr´enyes (2015)) and the corresponding FSDT (Szekr´enyes (2013c)) results are compared to each other. The material is the same as that in Chapter 6 (Table 6.1). In Figure 7.2a it can be seen that for case I the FSDT and SSDT solutions underpredict G_II, moreover the FSDT agrees quite well with the Reddy TSDT in the case ofG_III, in this respect the SSDT is worst if b = 100 mm. On the contrary, the Reddy TSDT agrees excellently with the numerical results for both components. Figure 7.2b presents the results for the wider plate, this time the SSDT performs better, even though the Reddy TSDT and FSDT provide good agreement, as well. It is important to highlight that the agreement between analysis and numerical calculation is the worst at and nearby the edges (ify= 0 or y = b), it is clear that the analytical models do not take the edge eﬀects into account. So the agreement is investigated along the delamination front except for the edge regions. In case II, presented in Figure 7.3, the same conclusions hold. Based on Figures 7.4 and 7.5 for cases III and IV (i.e. when the bottom plate thickness is larger) it is shown that the FSDT overpredicts, the SSDT underpredicts/overpredicts signiﬁcantly the mode-III ERR, simultaneously, the mode-II ERR by Reddy TSDT agrees better with the numerical results.

The major diﬀerence between the FSDT, SSDT and Reddy TSDT solutions is the shear strain continuity at the interface plane and the satisfaction of the dynamic B.C. in the latter case. That is the reason for the diﬀerences presented in Figures 7.4 and 7.5. In accordance with Figures 7.4 and 7.5, the FSDT seems to be inaccurate in cases III and IV for both plate widths. Eventually, the SSDT and especially the Reddy TSDT approach quite well both ERR components for each plate width in case III (Figure 7.4), but if b=160 mm, then the mode-II ERR is dissimilar to the FE solution at the edges. Compared to the VCCT results, the mode-III ERR is approximated very well by Reddy TSDT, on the contrary the SSDT solution becomes inaccurate in case IV (Figure 7.5). The fracture is mode-III dominated in

problem a .

Figures 7.6-7.9 demonstrate the ERR and mode mixity distributions for problem b . Similar results were obtained to those presented in Figures 7.2-7.5, however the fracture is mode-II dominated, and so the mode-II ERR agrees better with the numerical results.

Running through on cases I, II, III and IV, the conlusions are that the FSDT gives correct results only in cases I and II, in case III the overprediction of the mode-III ERR becomes moderate, while in case IV the estimation of G_III is not acceptable. The SSDT and Reddy TSDT perform very well compared to the FSDT. Considering all of the cases the Reddy TSDT is deﬁnitely the best solution for problem a with b = 100 mm, however if b = 160 mm the overall performance of the SSDT solution is better.

The ﬁnal conclusion is that the FSDT is applicable only to those cases, when the delam-ination is not far from the midplane of the plate. The SSDT solves the problem better in case IV than Reddy TSDT, however, considering all of the cases the Reddy TSDT would be the best choice among the models based on the method of 2ESLs. However, we should not forget about the locking eﬀect taking place in the deﬂection and shown in Figure 6.2.

Also, it is clear, that the presence of the delamination induces complex deformations along the delamination front, which could be better captured by higher-order plate theories than FSDT. This consequence has already been conﬁrmed in recent papers (Szekr´enyes (2013b, 2014b,d)). It can be seen based on the results, that case IV is the critical case, when the accuracy of the method of 2ESLs is not satisfactory. Therefore in the sequel the results of the method of 4ESLs are presented.

In document Nonsingular delamination modeling in orthotropic composite plates by semi-layerwise analysis (Pldal 96-101)