# Mixed-Mode Delamination Growth Prediction in Stiffened CFRP Panels by Means of a Novel Fast Procedure

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

^{cr}and displacement u

^{cr}). Then, the buckling load and shape due to a propagation of the initial delamination are obtained by means of a second eigenvalue buckling analysis (location A’ in Figure 2, characterized by a critical load F

^{cr}’ and displacement u

^{cr}’). Once the local buckling has occurred, the residual contribution to the global stiffness of the buckled thinner sub-laminate can be neglected. Hence, considering an unknown propagation displacement u* beyond the critical displacements u

^{cr}and u

^{cr}’, the amount of energy E(u*) (area A’ACB in Figure 2), released by the propagation of the delamination, can be evaluated by using two linear static analyses to determine the stiffness K

^{A}and K

^{A+}

^{ΔA}.

^{m}(u*) equates the critical value G

_{IC}.

_{glo}, two cases can be considered.

_{glo}). This case is representative of mode I dominant propagation mode. Therefore, the theory developed for pure mode I is still effective, as shown in Figure 3. In addition, a coefficient α, which correlates the contribution of G

_{IC}and G

_{IIC}to the total ERR, is introduced, which ranges between zero (pure mode II) and one (pure mode I). Hence, the total amount of energy released during the propagation can be evaluated as:

_{I}

^{max}is the maximum value of G

_{I}found by non-linear VCCT analyses along the delamination front.

_{glo}). In this case, the structure experiences the global buckling of the skin before delamination growth initiation takes place. Indeed, after the global buckling of the skin, the structure resulting stiffness can be assumed equal to the stiffness of the stringers, since the skin loses all the capabilities to further carry compressive load. Hence, the mode II released energy is related to both local and global critical load and the unknown propagation displacement u*. Therefore, according to Figure 4, the additional energy related to mode II can be evaluated as:

_{I}and G

_{II}distributions along the delamination front can be assumed dependent on the local nodal out-of-plane and radial displacements, respectively:

^{i}and Δr

^{i}are, respectively, the opening and sliding displacements at location i of the delamination front.

## 3. Numerical Applications and Discussion

#### 3.1. Test Case 1: Validation and Applicability of the Mode I Fast Procedure

_{3s}, [+45°/−45°/0°/0°/90°]

_{s}, and [+45°/−45°/0°/0°/90°]

_{2s}. A 20 mm radius delamination between the third and fourth ply (90°/−45° interface) has been considered.

^{cr}, the critical compressive load F

^{cr}, and the linear nodal opening Δw

^{i}trend on the delamination front with a size A are evaluated. Hence, the node with maximum ERR is identified, as shown in Figure 8a.

^{cr}’ and u

^{cr}’.

^{0}of undamaged structure, the post-buckling stiffness K

^{A}of the structure with delamination size A, and the post-buckling stiffness K

^{A+}

^{ΔA}of the structure with delamination size A + ΔA are evaluated by means of static linear analyses. In order to evaluate the stiffness K

^{A}and K

^{A+}

^{ΔA}, the thinner sub-laminate has been removed from the finite element model to take in account the stiffness lost related to the delamination buckling. Finally, the applied loads and displacements at growth initiation and the ERR on the delamination front are evaluated.

_{buckling}and the compressive delamination initiation strains ε

_{del}and loads F

_{del}between the proposed linear approach and VCCT can be found as well, as reported in Table 2. Moreover, the advantage of the Fast procedure in terms of computational efforts, respect to the standard VCCT approach, can be appreciated by comparing the computational times reported in Table 2.

#### 3.2. Test Case 2: Validation and Applicability of the Mixed Mode I and II Fast Procedure

_{4}]

_{S}and [(−45°/+45°/0°)

_{2}]

_{S}

_{2}.

_{II}distribution probably because the strain energy release rate distribution undergoes significant variations from the delamination buckling condition to delamination growth condition, eventually, passing through the global buckling of the skin.

_{II}distribution.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Seon, G.; Makeev, A.; Nikishkov, Y.; Lee, E. Effects of defects on interlaminar tensile fatigue behavior of carbon/epoxy composites. Compos. Sci. Technol.
**2013**, 89, 194–201. [Google Scholar] [CrossRef] - Andrews, S.D.; Ochoa, O.O.; Owens, S.D. The Effects of Fastener Hole Defects. J. Compos. Mater.
**1993**, 27, 2–20. [Google Scholar] [CrossRef] - Elder, D.J.; Thomson, R.S.; Nguyen, M.Q.; Scott, M.L. Review of delamination predictive methods for low speed impact of composite laminates. Compos. Struct.
**2004**, 66, 677–683. [Google Scholar] [CrossRef] - Geubelle, P.H.; Baylor, J.S. Impact-induced delamination of composites: A 2D simulation. Compos. Part B Eng.
**1998**, 29, 589–602. [Google Scholar] [CrossRef] - Parlapalli, M.R.; Soh, K.C.; Shu, D.W.; Ma, G. Experimental investigation of delamination buckling of stitched composite laminates. Compos. Part A Appl. Sci. Manuf.
**2007**, 38, 2024–2033. [Google Scholar] [CrossRef] - Craven, R.; Iannucci, L.; Olsson, R. Delamination buckling: A finite element study with realistic delamination shapes, multiple delaminations and fibre fracture cracks. Compos. Part A Appl. Sci. Manuf.
**2010**, 41, 684–692. [Google Scholar] [CrossRef] - Karihaloo, B.L.; Stang, H. Buckling-driven delamination growth in composite laminates: Guidelines for assessing the threat posed by interlaminar matrix delamination. Compos. Part B Eng.
**2008**, 39, 386–395. [Google Scholar] [CrossRef] - Ashizawa, M. Fast Interlaminar Fracture of a Compressively Loaded Composite Containing a Defect. In Proceedings of the Fifth DOD/NASA Conference on Fibrous Composites in Structural Design; Report No. NADC-81096-60; Naval Air Development Center: Warminster, PA, USA, January 1981; pp. 1–1269. [Google Scholar]
- Ramkumar, R.L. Fatigue Degradation in Compressively Loaded Composite Laminates; NASA CR-165681; NASA Langley Research Center: Hampton, VA, USA, 1981.
- Ramkumar, R.L. Performance of a Quantitative Study of Instability-Related Delamination Growth; NASA CR-166046; NASA Langley Research Center: Hampton, VA, USA, 1983.
- Byers, B.A. Behaviour of Damaged Graphite/Epoxy Laminates under Compression Loading; NASA CR-159293; NASA Langley Research Center: Hampton, VA, USA, 1980.
- Chai, H.; Knauss, W.G.; Babcock, C.D. Observation of Damage Growth in Compressively Loaded Laminates. Exp. Mech.
**1983**, 23, 329–337. [Google Scholar] [CrossRef] - Shivakumar, K.N.; Whitcomb, J.D. Buckling of a Sublaminate in a Quasi-Isotropic Composite Laminate. J. Compos. Mater.
**1985**, 19, 2–18. [Google Scholar] [CrossRef] [Green Version] - Whitcomb, J.D.; Shivakumar, K.N. Strain-Energy Release Rate Analysis of a Laminate with a Postbuckled delamination. In Numerical Methods in Fracture Mechanics; NASA TM-89091; NASA Langley Research Center: Hampton, VA, USA, 1987. [Google Scholar]
- Kim, H.J.; Hong, C.S. Buckling and Postbuckling Behaviour of Composite Laminates with an Embedded Delamination. Compos. Sci. Technol.
**1997**, 57, 557–564. [Google Scholar] [CrossRef] - Singh, K.L.; Dattaguru, B.; Ramamurthy, T.S.; Mangalgiri, P.D. Delamination Tolerance Studies in Laminated Composite Panels. Sadhana
**2000**, 25, 409–422. [Google Scholar] [CrossRef] - Shahwan, K.; Waas, A.M. Unilateral Buckling of rectangular Plates. Int. J. Solids Struct.
**1994**, 31, 75–89. [Google Scholar] [CrossRef] - Shahwan, K.; Waas, A.M. Buckling of Unilaterally Constrained Plates: Application to the Study of Delaminations in Layered Structures. J. Frankl. Inst.
**1998**, 335, 1009–1039. [Google Scholar] [CrossRef] - Whitcomb, J.D. Analysis of a Laminate with a Postbuckled Embedded Delamination, Including Contact Effects. J. Compos. Mater.
**1992**, 26, 1523–1535. [Google Scholar] [CrossRef] - Whitcomb, J.D. Three Dimensional Analysis of a Postbuckled Embedded Delamination. J. Compos. Mater.
**1989**, 23, 862–889. [Google Scholar] [CrossRef] - Köllner, A.; Völlmecke, C. Post-buckling behaviour and delamination growth characteristics of delaminated composite plates. Compos. Struct.
**2018**, 203, 777–788. [Google Scholar] [CrossRef] [Green Version] - Kutlu, Z.; Chang, F.-K. Modeling Compression Failure of laminated Composites Containing Multiple Through-the-Width Delaminations. J. Compos. Mater.
**1992**, 26, 350–387. [Google Scholar] [CrossRef] - Orifici, A.C.; de Zarate Alberdi, I.O.; Thomson, R.S.; Bayandor, J. Compression and post-buckling damage growth and collapse analysis of flat composite stiffened panels. Compos. Sci. Technol.
**2008**, 68, 3150–3160. [Google Scholar] [CrossRef] [Green Version] - Krueger, R. The Virtual Crack Closure Technique: History, Approach and Applications; NASA/CR-2002-211628; ICASE Report; NASA Langley Research Center: Hampton, VA, USA, 2002.
- Liu, P.F.; Hou, S.J.; Chu, J.K.; Hu, X.Y.; Zhou, C.L.; Liu, Y.L.; Zheng, J.Y.; Zhao, A.; Yan, L. Finite element analysis of postbuckling and delamination of composite laminates using virtual crack closure technique. Compos. Struct.
**2011**, 93, 1549–1560. [Google Scholar] [CrossRef] - Gudmundson, P. Micromechanically based constitutive models for damage evolution in composite laminates. Int. J. Damage Mech.
**2000**, 9, 29–39. [Google Scholar] [CrossRef] - Turon, A.; Dávila, C.G.; Camanho, P.P.; Costa, J. An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Eng. Fract. Mech.
**2007**, 74, 1665–1682. [Google Scholar] [CrossRef] - Riccio, A.; Linde, P.; Raimondo, A.; Buompane, A.; Sellitto, A. On the use of selective stitching in stiffened composite panels to prevent skin-stringer debonding. Compos. Part B Eng.
**2017**, 124, 64–75. [Google Scholar] [CrossRef] - Riccio, A.; Russo, A.; Sellitto, A.; Raimondo, A. Development and application of a numerical procedure for the simulation of the “Fibre Bridging” phenomenon in composite structures. Compos. Struct.
**2017**, 168, 104–119. [Google Scholar] [CrossRef] - Hallett, S.R.; Green, B.G.; Jiang, W.G.; Wisnom, M.R. An experimental and numerical investigation into the damage mechanisms in notched composites. Compos. Part A Appl. Sci. Manuf.
**2009**, 40, 613–624. [Google Scholar] [CrossRef] - Hughes, T.; Cottrell, J.; Bazilevs, Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng.
**2005**, 194, 4135–4195. [Google Scholar] [CrossRef] [Green Version] - Dimitri, R.; Fantuzzi, N.; Tornabene, F.; Zavarise, G. Innovative numerical methods based on SFEM and IGA for computing stress concentrations in plates with discontinuities. Int. J. Mech. Sci.
**2016**, 118, 166–187. [Google Scholar] [CrossRef] - Nguyen, V.P.; Nguyen-Xuan, H. High-order B-splines based finite elements for delamination analysis of laminated composites. Compos. Struct.
**2013**, 102, 261–275. [Google Scholar] [CrossRef] - Guo, Y.; Nagy, A.P.; Gürdal, Z. A layerwise theory for laminated composites in the framework of isogeometric analysis. Compos. Struct.
**2014**, 107, 447–457. [Google Scholar] [CrossRef] - Dimitri, R.; Zavarise, G. Isogeometric treatment of frictional contact and mixed mode debonding problems. Comput. Mech.
**2017**, 60, 315–332. [Google Scholar] [CrossRef] - Dimitri, R.; Tornabene, F. Numerical Study of the Mixed-Mode Delamination of Composite Specimens. J. Compos. Sci.
**2018**, 2, 30. [Google Scholar] [CrossRef] - Dimitri, R.; Tornabene, F.; Zavarise, G. Analytical and numerical modeling of the mixed-mode delamination process for composite moment-loaded double cantilever beams. Compos. Struct.
**2018**, 187, 535–553. [Google Scholar] [CrossRef] - Riccio, A.; Damiano, M. A Fast Numerical Methodology for Delamination Growth Initiation Simulation. In Damage Growth in Aerospace Composites; Springer: Berlin/Heidelberg, Germany, 2015; pp. 199–220. ISBN 978-3-319-04003-5. [Google Scholar]
- Riccio, A.; Damiano, M.; Raimondo, A.; Di Felice, G.; Sellitto, A. A fast numerical procedure for the simulation of inter-laminar damage growth in stiffened composite panels. Compos. Struct.
**2016**, 145, 203–216. [Google Scholar] [CrossRef] - Riccio, A.; Zarrelli, M.; Giordano, M. A Linear Numerical Approach to Simulate the Delamination Growth Initiation in Stiffened Composite Panels. J. Compos. Mater.
**2010**, 44, 1841–1866. [Google Scholar] [CrossRef] - ABAQUS Manual; Revision 6.5-1; Theory; Dassault Systèmes Simulia Corp.: Providence, RI, USA, 2004.
- Sellitto, A.; Borrelli, R.; Caputo, F.; Riccio, A.; Scaramuzzino, F. Methodological approaches for kinematic coupling of non-matching finite element meshes. Procedia Eng.
**2011**, 10, 421–426. [Google Scholar] [CrossRef]

**Figure 1.**Delamination buckling scenarios: (

**a**) Local buckling of the thinner sub-laminate and (

**b**) Mixed buckling of both sub-laminates.

**Figure 2.**Stiffened composite panel stiffness: K

^{0}: Before delamination buckling; K

^{A}: After the local delamination buckling; and K

^{A+}

^{ΔA}: After the propagated delamination buckling.

**Figure 3.**Schematic rapresentation of energy relase rate evaluation considering mode I and mode II interaction: u* < u

_{glo}.

**Figure 4.**Schematic rapresentation of energy relase rate evaluation considering mode I and mode II interaction: u* > u

_{glo}.

**Figure 7.**Deformed shape of linear buckling analysis for the stiffened panel with delamination size A.

**Figure 9.**Influence of the element discretization in the radial direction on the critical propagation strain.

**Figure 10.**Influence of the element discretization in the tangential direction on the critical propagation strain.

**Figure 11.**Comparison of proposed approach and virtual crack closure technique (VCCT) in terms of normalized energy release rate (ERR) distribution along delamination front.

**Figure 17.**Comparison of strain energy release rate (SERR) distribution for SS#1 damage configuration.

**Figure 20.**SS#1-comparison between mixed mode I and II fast procedure and VCCT of SERR distribution.

E_{1}(GPa) | E_{2} = E_{3}(GPa) | G_{12} = G_{13}(GPa) | G_{23}(GPa) | ν_{12} = ν_{13}(-) | ν_{23}(-) | G_{IC}(J/m ^{2}) | G_{IIC}(J/m ^{2}) | t_{p}(mm) |
---|---|---|---|---|---|---|---|---|

147.0 | 9.0 | 5.0 | 3.0 | 0.3 | 0.42 | 175 | 532 | 0.188 |

Approach | ε_{buckling} (με) | ε_{del} (με) | F_{del} (kN) | Computational Time (s) |
---|---|---|---|---|

Fast Procedure | 980.3 | 1700.7 | 366.6 | 295 |

VCCT | 1024.8 | 1780.0 | 382.9 | 4349 |

E_{1}(GPa) | E_{2} = E_{3}(GPa) | G_{12} = G_{13}(GPa) | G_{23}(GPa) | ν_{12} = ν_{13}(-) | ν_{23}(-) | G_{Ic}(J/m ^{2}) | G_{IIc}(J/m ^{2}) | t_{p}(mm) |
---|---|---|---|---|---|---|---|---|

155.0 | 8.57 | 7.4 | 4.8 | 0.33 | 0.52 | 280 | 519 | 0.125 |

Delamination Interface | Delamination Depth | ε_{cr} | Error | Analysis Method |
---|---|---|---|---|

4th–5th(90°/+45°) | 12.5% | 2415.33 | VCCT | |

2887.44 | 19.5% | Mode I Fast | ||

3104.18 | 22.2% | Mixed mode I and II Fast | ||

5th–6th(+45°/−45°) | 15.62% | 2678.00 | VCCT | |

2645.89 | 1.2% | Mode I Fast | ||

2788.06 | 3.9% | Mixed mode I and II Fast | ||

6th–7th(−45°/0°) | 18.7% | 3607.00 | VCCT | |

3284.56 | 9.8% | Mode I Fast | ||

3620.72 | 0.4% | Mixed mode I and II Fast | ||

7th–8th(0°/90°) | 21.8% | 3938.00 | VCCT | |

3266.97 | 20.5% | Mode I Fast | ||

3528.79 | 11.6% | Mixed mode I and II Fast | ||

8th–9th(90°/+45°) | 25% | 3752.00 | VCCT | |

3474.52 | 8.0% | Mode I Fast | ||

3602.93 | 4.1% | Mixed mode I and II Fast |

Delamination Interface | Delamination Depth | ε_{cr} | Error | Analysis Method |
---|---|---|---|---|

4th–5th(90°/+45°) | 12.5% | 2890.67 | VCCT | |

2843.92 | 1.6% | Mode I Fast | ||

3007.27 | 3.9% | Mixed mode I and II Fast | ||

5th–6th(+45°/−45°) | 15.62% | 3288.00 | VCCT | |

3065.76 | 7.2% | Mode I Fast | ||

3165.35 | 3.9% | Mixed mode I and II Fast | ||

6th–7th(−45°/0°) | 18.7% | 4804.00 | VCCT | |

4361.94 | 10.1% | Mode I Fast | ||

4509.78 | 6.5% | Mixed mode I and II Fast | ||

7th–8th(0°/90°) | 21.8% | 5500.00 | VCCT | |

4734.76 | 16.2% | Mode I Fast | ||

4954.13 | 11.0% | Mixed mode I and II Fast | ||

8th–9th(90°/+45°) | 25% | 5926.00 | VCCT | |

5838.88 | 1.5% | Mode I Fast | ||

5874.92 | 0.9% | Mixed mode I and II Fast |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sellitto, A.; Saputo, S.; Damiano, M.; Russo, A.; Riccio, A.
Mixed-Mode Delamination Growth Prediction in Stiffened CFRP Panels by Means of a Novel Fast Procedure. *Appl. Sci.* **2019**, *9*, 4761.
https://doi.org/10.3390/app9224761

**AMA Style**

Sellitto A, Saputo S, Damiano M, Russo A, Riccio A.
Mixed-Mode Delamination Growth Prediction in Stiffened CFRP Panels by Means of a Novel Fast Procedure. *Applied Sciences*. 2019; 9(22):4761.
https://doi.org/10.3390/app9224761

**Chicago/Turabian Style**

Sellitto, Andrea, Salvatore Saputo, Michele Damiano, Angela Russo, and Aniello Riccio.
2019. "Mixed-Mode Delamination Growth Prediction in Stiffened CFRP Panels by Means of a Novel Fast Procedure" *Applied Sciences* 9, no. 22: 4761.
https://doi.org/10.3390/app9224761