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3 March 2020

Correction: Bennati et al. An Elastic Interface Model for the Delamination of Bending-Extension Coupled Laminates. Appl. Sci. 2019, 9, 3560

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Department of Civil and Industrial Engineering, University of Pisa, Largo Lucio Lazzarino, I-56122 Pisa, Italy
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Appl. Sci.2020, 10(5), 1711;https://doi.org/10.3390/app10051711
This article belongs to the Special Issue Fatigue and Fracture of Non-metallic Materials and Structures
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We, the authors, wish to make the following corrections to our paper [].
Equations (26), (27), (28), and (41) were affected by some typos and should be substituted by the following ones (corrections are colored in red):
u 1 x = − a 1 + b 1 h 1 f 5 λ 5 2 exp − λ 5 x + f 6 λ 6 2 exp − λ 6 x + b 1 ∑ i = 1 4 f i λ i 3 exp − λ i x + − a 1 + b 1 h 1 f 7 + b 1 f 10 x 2 2 − a 1 f 8 + b 1 f 12 x + f 14 , w 1 x = ∑ i = 1 4 d 1 λ i 2 − c 1 f i λ i 2 exp − λ i x − d 1 h 1 + b 1 f 5 λ 5 3 exp − λ 5 x + f 6 λ 6 3 exp − λ 6 x + + d 1 h 1 + b 1 f 7 + d 1 f 10 x 3 6 + b 1 f 8 + d 1 f 12 x 2 2 − f 16 + c 1 f 10 x + f 15 , and φ 1 x = d 1 ∑ i = 1 4 f i λ i 3 exp − λ i x − d 1 h 1 + b 1 f 5 λ 5 2 exp − λ 5 x + f 6 λ 6 2 exp − λ 6 x + − d 1 h 1 + b 1 f 7 + d 1 f 10 x 2 2 − b 1 f 8 + d 1 f 12 x + f 16 ,
u 2 x = a 2 − b 2 h 2 f 5 λ 5 2 exp − λ 5 x + f 6 λ 6 2 exp − λ 6 x − b 2 ∑ i = 1 4 f i λ i 3 exp − λ i x + + a 2 − b 2 h 2 f 7 + b 2 f 11 x 2 2 + a 2 f 9 + b 2 f 13 x + f 17 , w 2 x = − ∑ i = 1 4 d 2 λ i 2 − c 2 f i λ i 2 exp − λ i x − d 2 h 2 − b 2 f 5 λ 5 3 exp − λ 5 x + f 6 λ 6 3 exp − λ 6 x + + d 2 h 2 − b 2 f 7 − d 2 f 11 x 3 6 − b 2 f 9 + d 2 f 13 x 2 2 − f 19 − c 2 f 11 x + f 18 , and φ 2 x = − d 2 ∑ i = 1 4 f i λ i 3 exp − λ i x − d 2 h 2 − b 2 f 5 λ 5 2 exp − λ 5 x + f 6 λ 6 2 exp − λ 6 x + − d 2 h 2 − b 2 f 7 − d 2 f 11 x 2 2 + b 2 f 9 + d 2 f 13 x + f 19 ,
f 10 = − α 3 B k x α 4 d 2 f 7 , f 11 = α 3 B k x α 4 d 1 f 7 , f 12 = − a 1 + b 1 h 1 d 2 − b 2 − d 2 h 2 b 1 α 4 f 8 − a 2 d 2 − b 2 2 α 4 f 9 , f 13 = a 1 d 1 − b 1 2 α 4 f 8 + a 2 − b 2 h 2 d 1 − b 1 + d 1 h 1 b 2 α 4 f 9 , f 14 = f 17 − h 1 + h 2 f 19 − 1 k x 1 B + α 3 B α 4 c 1 d 2 − c 2 d 1 h 1 f 7 , f 15 = f 18 , and f 16 = f 19 + α 3 B k x α 4 c 1 d 2 − c 2 d 1 f 7 ,
g 10 = − α 3 B k x α 4 d 2 + β 0 b 2 − d 2 h 2 B α 4 g 7 , g 11 = α 3 B k x α 4 d 1 − β 0 b 1 + d 1 h 1 B α 4 g 7 , g 12 = − a 1 + b 1 h 1 d 2 − b 2 − d 2 h 2 b 1 α 4 g 8 − a 2 d 2 − b 2 2 α 4 g 9 , g 13 = a 1 d 1 − b 1 2 α 4 g 8 + a 2 − b 2 h 2 d 1 − b 1 + d 1 h 1 b 2 α 4 g 9 , g 14 = g 17 − h 1 + h 2 g 19 − 1 B k x + α 3 B k x α 4 c 1 d 2 − c 2 d 1 h 1 + β 0 b 1 c 2 − c 1 b 2 + c 1 d 2 h 2 + c 2 d 1 h 1 B α 4 h 1 g 7 , g 15 = g 18 , and g 16 = g 19 + α 3 B k x α 4 c 1 d 2 − c 2 d 1 + β 0 b 1 c 2 − c 1 b 2 + c 1 d 2 h 2 + c 2 d 1 h 1 B α 4 g 7 ,
Furthermore, we observe that the constant terms in the shear stress expressions (18) and (32) (corresponding to Jourawski’s solution for an unbroken beam) should not contribute to the Mode II energy release rate G II . Thus, the peak values of the shear interfacial stress entering Equation (44) should be computed as τ 0 = τ ( 0 ) − f 7 / B and τ 0 = τ ( 0 ) − g 7 / B in the balanced and unbalanced cases, respectively. As a consequence, Equations (45) and (46) should be replaced by the following ones:
G I = H ( σ 0 ) 2 k z B 2 ∑ i = 1 4 f i 2 and G II = 1 2 k x B 2 f 5 + f 6 2
and
G I = H ( σ 0 ) 2 k z B 2 ∑ i = 1 6 g i 2 and G II = 1 2 k x B 2 k x β 0 ∑ i = 1 6 g i μ i ( μ i 2 − α 3 ) 2 .
The corrections do not affect the results and scientific conclusions of the paper. We apologize for any inconvenience caused.

Acknowledgments

We would like to thank Prof. Theodoros Loutas and Mr. Panayiotis Tsokanas of the University of Patras, Greece, for pointing out the typos.

References

  1. Bennati, S.; Fisicaro, P.; Taglialegne, L.; Valvo, P.S. An Elastic Interface Model for the Delamination of Bending-Extension Coupled Laminates. Appl. Sci. 2019, 9, 3560. [Google Scholar] [CrossRef]

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