# Experimental and Numerical Investigation of the Mesoscale Size Effect in Notched Woven Composites

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material and Mechanical Testing

#### 2.2. Numerical Model

#### 2.3. Data-Driven Calibration of Material Properties

#### 2.3.1. Optimization of the Elastic Properties

_{xx}, ε

_{yy}, and ε

_{xy}are transverse, longitudinal, and plane shear strain fields, respectively; the FE and DIC superscripts indicate numerical and experimental/DIC grid-interpolated strain fields; RMS is the root mean square value of the strain field; and ∑ indicates the summation over all the strain field grid points. To obtain better results, it is useful to weigh the strain field differences over their respective experimental RMS values, since strains in the loading direction were an order of magnitude greater than those in the transverse direction. During the optimization stage, the model was loaded imposing the experimental load curve. For the validation and stress concentration study, this was changed to a constant displacement rate, to better duplicate the experimental tests.

#### 2.3.2. Failure Properties

- 1.
- The stresses causing failure are longitudinal stress ${\sigma}_{yy}$ and shear stress ${\tau}_{xy}$;
- 2.
- In vertical tows (“fiber direction”), failure occurs by fiber breakage. The failure criterion is thus:

- 3.
- In horizontal tows (“matrix direction”), failure occurs by matrix failure due to tensile stresses combined with shear stresses. The failure criterion is thus:

## 3. Results

#### 3.1. Experimental Results

_{nom}is the nominal tensile strength calculated over the gross (unnotched) section, while UTS

_{net}was obtained considering the reduction in section due to the hole. UTS

_{nom}exhibits a monotonic decreasing trend with increasing hole sizes. The specimens with the smallest hole size show the highest UTS

_{net}and low variability; larger hole specimens have lower values but much more significant dispersion in the results. Overall, however, dispersion in the results appears low, with a maximum coefficient of variation of 2.81%.

#### 3.2. Elastic Properties

#### 3.3. Failure Properties

- 1.
- Stress concentration. The affected volume scales up with the hole diameter. This leads to an increased probability of encountering a weak point within the structure that causes premature failure, and, consequently, a decrease in load-bearing capability. Indeed, as observed in Table 3, the nominal strength of the notched specimen, which is the ratio between the maximum force and the nominal notched cross-section, decreases with the hole diameter. This effect can also be quantified by the ratio of the remote longitudinal stress ${\sigma}_{1}^{\infty}$ —defined earlier—to the maximum longitudinal stress in the entire specimen ${\sigma}_{1}^{MAX}$.
- 2.
- Stress localization. Close to the hole, stress sharply rises in a very limited volume. This leads to an observed local increase in material strength. This effect can be described by the ratio between the longitudinal tensile strength of the material in the notched specimens ${S}_{1}$ —determined as per Section 2.3.2—to the same property in the unnotched specimen, indicated as ${S}_{1}^{0}$.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Hild, F.; Roux, S. Digital Image Correlation: From Displacement Measurement to Identification of Elastic Properties—A Review. Strain
**2006**, 42, 69–80. [Google Scholar] [CrossRef][Green Version] - Lanza Di Scalea, F.; Hong, S.S.; Cloud, G.L. Whole-Field Strain Measurement in a Pin-Loaded Plate by Electronic Speckle Pattern Interferometry and the Finite Element Method. Exp. Mech.
**1998**, 38, 55–60. [Google Scholar] [CrossRef] - Pierron, F.; Green, B.G.; Wisnom, M.R. Full-Field Assessment of the Damage Process of Laminated Composite Open-Hole Tensile Specimens. Part I: Methodology. Compos. Part A Appl. Sci. Manuf.
**2007**, 38, 2307–2320. [Google Scholar] [CrossRef] - Caminero, M.A.; Lopez-Pedrosa, M.; Pinna, C.; Soutis, C. Damage Monitoring and Analysis of Composite Laminates with an Open Hole and Adhesively Bonded Repairs Using Digital Image Correlation. Compos. B Eng.
**2013**, 53, 76–91. [Google Scholar] [CrossRef] - Bazant, Z.P. Size Effect on Structural Strength: A Review. Arch. Appl. Mech.
**1999**, 69, 703–705. [Google Scholar] - Pagnoncelli, A.P.; Tridello, A.; Paolino, D.S. Modelling Size Effects for Static Strength of Brittle Materials. Mater. Des.
**2020**, 195, 109052. [Google Scholar] [CrossRef] - Awerbuch, J.; Madhukar, M.S. Notched Strength of Composite Laminates: Predictions and Experiments—A Review. J. Reinf. Plast. Compos.
**1985**, 4, 3–159. [Google Scholar] [CrossRef] - Batista, A.; Tinô, S.; Fontes, R.; Nóbrega, S.; Aquino, E. Analytical, Experimental and Finite Element Analysis of the Width/Diameter Hole Ratio Effect in Vinylester/Carbon Hybrid Twill Weave Composites. Compos. Part C Open Access
**2020**, 2, 100033. [Google Scholar] [CrossRef] - Tinô, S.R.L.; Fontes, R.S.; De Aquino, E.M.F. Theories of Failure Average Stress Criterion and Point Stress Criterion in Notched Fiber-Reinforced Plastic. J. Compos. Mater.
**2014**, 48, 2669–2676. [Google Scholar] [CrossRef] - Fontes, R.S.; Bezerra, H.A.D.; De Batista, A.C.M.C.; Tinôa, S.R.L.; De Aquino, E.M.F. Failure Theories and Notch Type Effects on the Mechanical Properties of Jute-Glass Hybrid Composite Laminates. Mater. Res.
**2019**, 22, e20180269. [Google Scholar] [CrossRef] - Sattar, S.; Pedrazzoli, D.; Zhang, M.; Kravchenko, S.G.; Kravchenko, O.G. Notched Tensile Strength of Long Discontinuous Glass Fiber Reinforced Nylon Composite. Compos. Part A Appl. Sci. Manuf.
**2022**, 163, 107217. [Google Scholar] [CrossRef] - Ma, Z.; Chen, J.; Yang, Q.; Li, Z.; Su, X. Progressive Fracture Analysis of the Open-Hole Composite Laminates: Experiment and Simulation. Compos. Struct.
**2021**, 262, 113628. [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] - Jočić, E.; Marjanović, M. Progressive Failure Analysis of Open-Hole Composite Laminates Using FLWT-SCB Prediction Model. Int. J. Mech. Sci.
**2022**, 227, 107407. [Google Scholar] [CrossRef] - Anzelotti, G.; Nicoletto, G.; Riva, E. Mesomechanic Strain Analysis of Twill-Weave Composite Lamina under Unidirectional in-Plane Tension. Compos. Part A Appl. Sci. Manuf.
**2008**, 39, 1294–1301. [Google Scholar] [CrossRef] - Bruno, L. Mechanical Characterization of Composite Materials by Optical Techniques: A Review. Opt. Lasers Eng.
**2018**, 104, 192–203. [Google Scholar] [CrossRef] - Réthoré, J.; Muhibullah; Elguedj, T.; Coret, M.; Chaudet, P.; Combescure, A. Robust Identification of Elasto-Plastic Constitutive Law Parameters from Digital Images Using 3D Kinematics. Int. J. Solids. Struct.
**2013**, 50, 73–85. [Google Scholar] [CrossRef] - Réthoré, J. A Fully Integrated Noise Robust Strategy for the Identification of Constitutive Laws from Digital Images. Int. J. Numer. Methods Eng.
**2010**, 84, 631–660. [Google Scholar] [CrossRef] - He, T.; Liu, L.; Makeev, A. Uncertainty Analysis in Composite Material Properties Characterization Using Digital Image Correlation and Finite Element Model Updating. Compos. Struct.
**2018**, 184, 337–351. [Google Scholar] [CrossRef] - Ogierman, W.; Kokot, G. Analysis of Strain Field Heterogeneity at the Microstructure Level and Inverse Identification of Composite Constituents by Means of Digital Image Correlation. Materials
**2020**, 13, 287. [Google Scholar] [CrossRef][Green Version] - ASTM D5766/D5766M-11(2018); Standard Test Method for Open-Hole Tensile Strength of Polymer Matrix Composite Laminates. American Society for Testing and Materials: West Conshohocken, PA, USA, 2011; 8p. [CrossRef]
- ASTM D3039/D3039M-17; Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials. American Society for Testing and Materials: West Conshohocken, PA, USA, 2011; 13p. [CrossRef]
- Schreier, H.; Orteu, J.J.; Sutton, M.A. Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications; Springer Science & Business Media: New York, NY, USA, 2009; ISBN 9780387787466. [Google Scholar]
- Gutmann, H.M. A Radial Basis Function Method for Global Optimization. J. Glob. Optim.
**2001**, 19, 201–227. [Google Scholar] [CrossRef]

**Figure 1.**Images of one of the notched specimens (C22) on the test fixture, captured with one of the cameras used for DIC: (

**a**) reference image used to build the model; (

**b**) speckle image taken during testing. The DIC area-of-interest is overlayed on both images.

**Figure 2.**Progressive abstraction of the 2 × 2 twill weave unit cell: (

**a**) imaging of the specimen during testing; (

**b**) schematic representation, highlighting the vertical (warp) and horizontal (weft) fiber directions; (

**c**) simplified model, with two materials to represent the two orientations and two layers to represent the weave. Global (in black) and local (in orange) material directions are shown. The tensile loading direction is $y$; 1 is the “fiber direction”, 2 is the “matrix direction”.

**Figure 3.**Stages in the process of building the specimen material map: (

**a**) raw .tiff image captured with DIC cameras, (

**b**) after K-means clustering and filtering, (

**c**) centroid of the tows calculated from the clusters overlaid on the initial image, (

**d**) fitting through optimization of a geometric grid (red circles) to the centroids (blue dots), (

**e**) selection of the vertical tow area, (

**f**) material orientation map used to prepare the FEM model via a Python script.

**Figure 5.**(

**a**) Load-displacement plot for the four types of specimens. (

**b**) Surface average stress and strains of the tested specimen. Engineering stress value is computed using the gross section of the specimen, and strain value is the average over the whole DIC area of interest. Refer to Table 1 for an explanation of the labels.

**Figure 6.**DIC-determined strain fields: (

**a**) longitudinal strain; (

**b**) transverse strain; (

**c**) shear strain.

**Figure 7.**(

**a**) Comparison between experimentally and numerically determined stress and strain curves; (

**b**) Example of a comparison between experimentally (left) and numerically determined longitudinal strain maps for Specimen C22.

**Figure 8.**Maps showing the percentage of the volume of material per element in the two orientations, the longitudinal stress of which surpasses the remote one, in the proximity of the hole. Remote stress was calculated as the maximum stress in a portion of the specimen away from the stress concentration.

**Figure 9.**Plots of the stress state values at the most heavily loaded elements at failure on (

**a**) the ${\sigma}_{yy}$-${\tau}_{xy}$ plane (fiber direction in the material); (

**b**) the ${\sigma}_{xx}$-${\tau}_{xy}$ plane (matrix direction in the material). The grey cloud contains all stress states of the elements in the plain specimen at failure.

**Figure 10.**The two effects on stress ratios for increasing hole diameters. (

**a**) Curves are the local tensile strength ${S}_{1}$ versus unnotched strength ${S}_{1}^{0}$, remote stress versus maximum stress, their superposed effect, and the macro-level trend of ultimate nominal tensile stress; (

**b**) close-up of the last two curves.

Specimen | l [mm] | w [mm] | D [mm] |
---|---|---|---|

C20 | 250 | 24 | Unnotched |

C21 | 250 | 24 | 2 |

C22 | 250 | 24 | 4 |

C23 | 250 | 24 | 8 |

Specimen | UTS_{nom} [MPa] | UTS_{net} [MPa] | ||||
---|---|---|---|---|---|---|

Mean | St. Dev. | CoV% | Mean | St. Dev. | CoV% | |

C20 | 732 | 16.7 | 2.28 | — | — | — |

C21 | 646 | 3.0 | 0.46 | 704 | 3.2 | 0.45 |

C22 | 543 | 14.4 | 2.65 | 652 | 17.4 | 2.67 |

C23 | 445 | 12.5 | 2.81 | 668 | 18.8 | 2.81 |

**Table 3.**Average material properties from three successive optimization runs. These values were used in the remainder of the study.

Property | ${\mathrm{E}}_{1}$ [GPa] | ${\mathrm{E}}_{2}$ [GPa] | ${\mathrm{G}}_{12}$ [GPa] | ${\mathsf{\upsilon}}_{12}$ |
---|---|---|---|---|

Mean | 102.388 | 15.519 | 1.891 | 0.050 |

St. dev. | 2.504 | 0.905 | 0.093 | 0.001 |

CoV% | 2.45 | 5.83 | 4.93 | 1.15 |

Specimen | ${\mathrm{S}}_{1}$ [MPa] | ${\mathrm{S}}_{2}$ [MPa] | ${\mathrm{S}}_{12}$ [MPa] |
---|---|---|---|

C20 | 2131 | 370 | 117 |

C21 | 3623 | 616 | 125 |

C22 | 3676 | 586 | 170 |

C23 | 3922 | 631 | 190 |

**Table 5.**Volumes affected by the stress concentration in vertical tows (Volume 1), in horizontal tows (Volume 2), and in the whole specimen.

Specimen | Volume 1 [mm^{3}] | Volume 2 [mm^{3}] | Total Volume [mm^{3}] |
---|---|---|---|

C21 | 5.65 | 4.33 | 9.99 |

C22 | 15.51 | 8.99 | 24.50 |

C23 | 82.61 | 60.96 | 143.57 |

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**MDPI and ACS Style**

Ferrarese, A.; Boursier Niutta, C.; Ciampaglia, A.; Ciardiello, R.; Paolino, D.S.; Belingardi, G. Experimental and Numerical Investigation of the Mesoscale Size Effect in Notched Woven Composites. *Appl. Sci.* **2023**, *13*, 4300.
https://doi.org/10.3390/app13074300

**AMA Style**

Ferrarese A, Boursier Niutta C, Ciampaglia A, Ciardiello R, Paolino DS, Belingardi G. Experimental and Numerical Investigation of the Mesoscale Size Effect in Notched Woven Composites. *Applied Sciences*. 2023; 13(7):4300.
https://doi.org/10.3390/app13074300

**Chicago/Turabian Style**

Ferrarese, Andrea, Carlo Boursier Niutta, Alberto Ciampaglia, Raffaele Ciardiello, Davide S. Paolino, and Giovanni Belingardi. 2023. "Experimental and Numerical Investigation of the Mesoscale Size Effect in Notched Woven Composites" *Applied Sciences* 13, no. 7: 4300.
https://doi.org/10.3390/app13074300