# Experimental Study and Numerical Analysis of the Tensile Behavior of 3D Woven Ceramic Composites

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2f}/SiO

_{2}) exhibit high strength, light weight, good heat resistance and excellent dielectric properties. Those characteristics make SiO

_{2f}/SiO

_{2}composites an ideal material for radome, especially for those working under a high temperature environment [1,3].

_{2f}/SiO

_{2}composites are typically manufactured by the silicasol-infiltration-sintering method (solgel method) [4,5]. In this method, as illustrated in Ref. [5], the reinforcement was vacuum impregnated using colloidal silica solution precursor (35 vol% SiO

_{2}) for 0.5 h, then the pressure of the container was increased to 10 bars and maintained for 1 h. This infiltration process was repeated 10 times with each infiltration dried to remove the water content of the gel solution. Finally, the dried reinforcement was sintered in an oven at 450 °C for 2 h in order to remove the coupling agent and bound water. Similarly, Xu el.al used this method to manufacture unidirectional SiO

_{2f}/SiO

_{2}composites, along with other similar work [6]. It is noted that with the solgel method, the manufactured SiO

_{2f}/SiO

_{2}are highly porous.

_{2f}/SiO

_{2}textile composites have been developed, including 2D woven [4], 2.5D woven [7], 3D orthogonal [8], and four-directional [7,9], five-directional braided composites [9]. Compared with 2D woven composites, the 3D woven composites significantly simplify the composite manufacturing by using near-net-shape preforms and removing the 2D woven ply layup process, which is quite labor-intensive. At the same time, due to the presence of third-dimensional yarns, it will be more difficult for 3D woven composites to form delamination under out-of-plane loading as seen in conventional laminates [10,11]. Research on woven SiO

_{2f}/SiO

_{2}composites are focused on their dielectric properties, thermophysical properties [12] and mechanical properties [4,7,8]. For mechanical properties, either failures under flexural loading and shear loading [8] or tensile behaviors of 2D twill woven SiO

_{2f}/SiO

_{2}composites [6] have been studied, while the tensile behaviors of 3D layer-to-layer angle interlock woven SiO

_{2f}/SiO

_{2}composites, in particular the corresponding numerical models, are seldom noticed, which may differ with 2D woven composites as a result of complex fiber architecture in the 3D preforms.

_{2f}/SiO

_{2}composites under warp direction through both experiments and numerical methods. Ceramic composites reinforced by 3D layer-to-layer angle interlock woven preforms were manufactured and tested under warp tension to characterize their mechanical behavior (Section 2). On the other hand, A numerical method is proposed to model the mechanical response of the ceramic composites under tension. X-ray micro-computed Tomography (µCT) was used to provide realistic geometric data for the unit cell model (Section 3.1), and the associated relative displacement boundary conditions for the unit cell are introduced in Section 3.2. The determination of material properties is described in Section 3.3 along with a progressive damage model given in Section 3.4, which has been implemented into a user material subroutine. Finally, the simulation results are compared with the experiment data in Section 4.

## 2. Materials, Manufacture and Testing

_{2f}/SiO

_{2}composite was achieved, obtained by theoretical estimation based on weights and densities of composites and constituent materials.

## 3. Finite Element Modelling

#### 3.1. Geometric Modelling

#### 3.2. Boundary Conditions

#### 3.3. Material Properties

_{f}are considered in the finite element model through the measured yarn cross-sections areas by μCT as below.

_{2f}/SiO

_{2}composites are used here, in which the properties of the porous matrix were obtained by discounting the properties of solid silica. For warp and weft yarns, although their ${V}_{f}^{yarn}$ are slightly different, the strengths are assumed to be identical. All the properties used for the homogenized yarns and porous matrix in the finite element analysis are listed in Table 4.

#### 3.4. Damage Model

_{0}is the original modulus; E is the residual tangent modulus; ω is the damage variable; f is the failure index; m is a material constant.

## 4. Results and Discussion

_{2f}/SiO

_{2}composites under warp tension is compared to the experimental results. Figure 4 shows the comparison of predicted and experimental stress–strain responses for 3D woven SiO

_{2f}/SiO

_{2}composites under warp tension.

_{2f}/SiO

_{2}composites have surface layers that are not in the same topology of the unit cell geometry, their effect on the stiffness is negligible when the thickness of the specimens is significant. The predicted ultimate tensile strength for the 3D woven SiO

_{2f}/SiO

_{2}composites is 76.61 MPa, which agrees well with the experiment, at a strain level of 0.58%. The macroscopic stress–strain curve from the FE model shows that the behavior of the ceramic composites is brittle, similar to the behavior shown in the testing, although local damage has already developed before the ultimate tensile strength.

## 5. Conclusions

_{2f}/SiO

_{2}composites under warp tension was experimentally and numerically investigated. The ceramic composites reinforced by 3D layer-to-layer woven preforms were manufactured through the solgel method and tested. A mesoscale finite element model was proposed to simulate the mechanical behavior of the composites. The model started with the construction of a single layer unit cell for the complex 3D woven preform, assisted by the μCT characterization of the realistic fiber architecture. Damage modelling was based on Tsai–Wu failure criterion in conjunction with a progressive damage approach based on the Matzenmiller model. Predicted results are in good agreement with experimental data in terms of initial failure, brittle stress–strain response and ultimate tensile strength of the material, which demonstrated that the proposed unit cell model is effective in the evaluation of the mechanical behavior of the 3D woven SiO

_{2f}/SiO

_{2}composites at a low computational cost.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Local mesh examples (warp direction cross-sectional view) in the mesh used in convergence study.

Mesh Size | Total Number of Nodes | Stiffness |
---|---|---|

0.12 mm | 12,882 | 14.38 GPa |

0.10 mm | 20,571 | 14.54 GPa |

0.08 mm | 34,250 | 14.40 GPa |

**Figure A2.**Comparison of maximum principal stress contours for warp yarns at a loading strain of 0.013% in the mesh convergence study.

## References

- Kandi, K.K.; Thallapalli, N.; Chilakalapalli, S.P.R. Development of silicon nitride-based ceramic radomes—A review. Int. J. Appl. Ceram. Technol.
**2015**, 12, 909–920. [Google Scholar] [CrossRef] - Mahmoudi, M.; Wang, C.; Moreno, S.; Burlison, S.R.; Alatalo, D.; Hassanipour, F.; Smith, S.E.; Naraghi, M.; Minary-Jolandan, M. Three-dimensional printing of ceramics through “carving” a gel and “filling in” the precursor polymer. ACS Appl Mater. Interfaces
**2020**, 12, 31984–31991. [Google Scholar] [CrossRef] [PubMed] - Miao, X.G.; Qu, Y.R.; Ghezzo, F.; Fang, X.W.; Yue, Y.T.; Zhao, Z.Y.; Liu, R.P. Fused silica ceramics and composites for radome applications. Adv. Mater. Res.
**2014**, 900, 123–129. [Google Scholar] [CrossRef] - Xu, C.-M.; Wang, S.; Huang, X.; Guo, J. Processing and properties of unidirectional SiO
_{2f}/SiO_{2}composites. Ceram. Int.**2007**, 33, 669–673. [Google Scholar] [CrossRef] - Liu, Y.; Zhu, J.; Chen, Z.; Jiang, Y.; Li, C.; Li, B.; Lin, L.; Guan, T.; Chen, Z. Mechanical properties and microstructure of 2.5D (shallow straight-joint) quartz fibers-reinforced silica composites by silicasol-infiltration-sintering. Ceram. Int.
**2012**, 38, 795–800. [Google Scholar] [CrossRef] - Shi, D.; Liu, C.; Cheng, Z.; Li, Z.; Yang, X.; Chen, H. On the tensile behaviors of 2D twill woven SiO
_{2f}/SiO_{2}composites at ambient and elevated temperatures: Mesoscale analysis and in situ experimental investigation. Ceram. Int.**2021**, 47, 12680–12694. [Google Scholar] [CrossRef] - Liu, Y.; Zhu, J.; Chen, Z.; Jiang, Y. Mechanical behavior of 2.5 D (shallow straight-joint) and 3D four-directional braided SiO
_{2f}/SiO_{2}composites. Ceram. Int.**2012**, 38, 4245–4251. [Google Scholar] [CrossRef] - Li, C.; Chen, Z.; Zhu, J.; Liu, Y.; Jiang, Y.; Guan, T.; Li, B.; Lin, L. Mechanical properties and microstructure of 3D orthogonal quartz fiber reinforced silica composites fabricated by silicasol-infiltration-sintering. Mater. Des.
**2012**, 36, 289–295. [Google Scholar] [CrossRef] - Liu, Y.; Chen, Z.; Zhu, J.; Jiang, Y.; Li, B.; Boafo, F.E. Comparison of 3D four-directional and five-directional braided SiO
_{2f}/SiO_{2}composites with respect to mechanical properties and fracture behavior. Mater. Sci. Eng. A**2012**, 558, 170–174. [Google Scholar] [CrossRef] - Aiman, D.; Yahya, M.; Salleh, J. Impact properties of 2D and 3D woven composites: A review. AIP Conf. Proc.
**2016**, 1774, 020002. [Google Scholar] - Zou, X.; Yan, S.; Matveev, M.; Rouse, J.P.; Jones, I.A. Experimental and numerical investigation of interface damage in composite L-angle sections under four-point bending. J. Compos. Mater.
**2020**, 55, 187–200. [Google Scholar] [CrossRef] - Hung, W.-C.; Horng, R.S.; Shia, R.-E. Investigation of thermal insulation performance of glass/carbon fiber-reinforced silica aerogel composites. J. Sol-Gel Sci. Technol.
**2021**, 97, 414–421. [Google Scholar] [CrossRef] - Yan, S.; Zeng, X.; Long, A. Effect of fibre architecture on tensile pull-off behaviour of 3D woven composite T-joints. Compos. Struct.
**2020**, 242, 112194. [Google Scholar] [CrossRef] - Yan, S.; Zeng, X.; Long, A. Experimental assessment of the mechanical behaviour of 3D woven composite T-joints. Compos. Part B Eng.
**2018**, 154, 108–113. [Google Scholar] [CrossRef] [Green Version] - Kaddour, A.S.; Hinton, M.J. Maturity of 3D failure criteria for fibre-reinforced composites: Comparison between theories and experiments: Part B of WWFE-II. J. Compos. Mater.
**2013**, 47, 925–966. [Google Scholar] [CrossRef] - Christensen, R.M. Failure criteria for fiber composite materials, the astonishing sixty year search, definitive usable results. Compos. Sci. Technol.
**2019**, 182, 107718. [Google Scholar] [CrossRef] - Ritman, E.L. Current status of developments and applications of micro-CT. Annu. Rev. Biomed. Eng.
**2011**, 13, 531–552. [Google Scholar] [CrossRef] - Zhang, W.; Yan, S.; Yan, Y.; Li, Y. A parameterized unit cell model for 3D braided composites considering transverse braiding angle variation. J. Compos. Mater.
**2022**, 56, 491–505. [Google Scholar] [CrossRef] - Lin, H.; Brown, L.P.; Long, A.C. Modelling and simulating textile structures using TexGen. Adv. Mater. Res.
**2011**, 331, 44–47. [Google Scholar] [CrossRef] - Tay, T.; Liu, G.; Tan, V.; Sun, X.; Pham, D. Progressive failure analysis of composites. J. Compos. Mater.
**2008**, 42, 1921–1966. [Google Scholar] [CrossRef] - Li, Y.; Yan, S.; Yan, Y.; Zhang, W. Modelling of the compressive behavior of 3D braided tubular composites by a novel unit cell. Compos. Struct.
**2022**, 287, 115303. [Google Scholar] [CrossRef] - Yan, S.; Zeng, X.; Long, A. Meso-scale modelling of 3D woven composite T-joints with weave variations. Compos. Sci. Technol.
**2019**, 171, 171–179. [Google Scholar] [CrossRef] - Llorca, J.; González, C.; Molina-Aldareguía, J.M.; Segurado, J.; Seltzer, R.; Sket, F.; Rodríguez, M.; Sádaba, S.; Muñoz, R.; Canal, L.P. Multiscale modeling of composite materials: A roadmap towards virtual testing. Adv. Mater.
**2011**, 23, 5130–5147. [Google Scholar] [CrossRef] [PubMed] - Lapczyk, I.; Hurtado, J.A. Progressive damage modeling in fiber-reinforced materials. Compos. Part A Appl. Sci. Manuf.
**2007**, 38, 2333–2341. [Google Scholar] [CrossRef] - Nobeen, N.S.; Zhong, Y.; Francis, B.A.P.; Ji, X.; Chia, E.S.M.; Joshi, S.C.; Chen, Z. Constituent materials micro-damage modeling in predicting progressive failure of braided fiber composites. Compos. Struct.
**2016**, 145, 194–202. [Google Scholar] [CrossRef] - Hinton, M.; Kaddour, A.; Pinho, S.; Vyas, G.; Robinson, P.; Huang, Z.; Zhou, Y.; Rotem, A.; Carrere, N.; Laurin, F. Special Issue: The Second World-Wide Failure Exercise (WWFE-II): Part B: Evaluation of Theories for Predicting Failure in Polymer Composite Laminates Under 3-D States of Stress; Comparison with Experiments Preface; Sage Publications Ltd.: London, UK, 2013. [Google Scholar]
- Isart, N.; El Said, B.; Ivanov, D.; Hallett, S.; Mayugo, J.; Blanco, N. Internal geometric modelling of 3D woven composites: A comparison between different approaches. Compos. Struct.
**2015**, 132, 1219–1230. [Google Scholar] [CrossRef] - Green, S.D.; Matveev, M.Y.; Long, A.C.; Ivanov, D.; Hallett, S.R. Mechanical modelling of 3D woven composites considering realistic unit cell geometry. Compos. Struct.
**2014**, 118, 284–293. [Google Scholar] [CrossRef] [Green Version] - Yan, S.; Zeng, X.; Brown, L.; Long, A. Geometric modeling of 3D woven preforms in composite T-joints. Text. Res. J.
**2018**, 88, 1862–1875. [Google Scholar] [CrossRef] [Green Version] - Li, S.; Sitnikova, E. An Excursion into Representative Volume Elements and Unit Cells; Elsevier: Amsterdam, The Netherlands, 2017. [Google Scholar]
- Chamis, C.C. Simplified Composite Micromechanics Equations for Hygral, Thermal and Mechanical Properties. In Proceedings of the 38th Annual Conference of the Society of the Plastics Industry Reinforced Plastics/Composites, Houston, TX, USA, 7–11 February 1983. [Google Scholar]
- Tsai, S.W.; Wu, E.M. A general theory of strength for anisotropic materials. J. Compos. Mater.
**1971**, 5, 58–80. [Google Scholar] [CrossRef] - Li, S.; Sitnikova, E.; Liang, Y.; Kaddour, A.-S. The Tsai-Wu failure criterion rationalised in the context of UD composites. Compos. Part A Appl. Sci. Manuf.
**2017**, 102, 207–217. [Google Scholar] [CrossRef] - Matzenmiller, A.; Lubliner, J.; Taylor, R.L. A constitutive model for anisotropic damage in fiber-composites. Mech. Mater.
**1995**, 20, 125–152. [Google Scholar] [CrossRef]

**Figure 4.**Predicted stress–strain response for the 3D woven SiO

_{2f}/SiO

_{2}composites under warp tension in comparison with test results.

**Figure 5.**Warp yarn full damage contour predicted by the FE model at different loading levels, red color indicates full damage, defined as Equation (7) is greater than 0.9.

**Figure 6.**Transverse stress contour (top, S22) and damage parameter contour (bottom, denoted by SDV5 for Equation (7)) in weft yarns at a loading strain of 0.58% (ultimate strength).

Warp Yarn | Weft Yarn | Number of Measurements | |
---|---|---|---|

Height (SD) in mm | 0.36 (±0.04) | 0.53 (±0.04) | 20 |

Width (SD) in mm | 1.11 (±0.07) | 1.76 (±0.08) | 20 |

Spacing (mm) | 0.16 (±0.03) | 2.53 (±0.10) | 20 |

Directions | Distance Vectors | Boundary Condition Equations |
---|---|---|

ξ-axis | $\mathsf{\Delta}{x}_{\eta}=\left\{\begin{array}{c}{W}_{UC}\\ {L}_{1}\\ 0\end{array}\right\}$ | $\mathsf{\Delta}{u}_{\eta}=\left\{\begin{array}{c}{u}^{\prime}-u\\ {v}^{\prime}-v\\ {w}^{\prime}-w\end{array}\right\}=\left\{\begin{array}{c}{W}_{UC}{\epsilon}_{x}^{0}+{L}_{1}{\gamma}_{xy}^{0}\\ {L}_{1}{\epsilon}_{y}^{0}\\ 0\end{array}\right\}$ |

η-axis | $\mathsf{\Delta}{x}_{\xi}=\left\{\begin{array}{c}{W}_{UC}\\ -{L}_{2}\\ 0\end{array}\right\}$ | $\mathsf{\Delta}{u}_{\eta}=\left\{\begin{array}{c}{u}^{\prime}-u\\ {v}^{\prime}-v\\ {w}^{\prime}-w\end{array}\right\}=\left\{\begin{array}{c}{W}_{UC}{\epsilon}_{x}^{0}-{L}_{2}{\gamma}_{xy}^{0}\\ -{L}_{2}{\epsilon}_{y}^{0}\\ 0\end{array}\right\}$ |

y-axis | $\mathsf{\Delta}{x}_{y}=\left\{\begin{array}{c}0\\ {L}_{UC}\\ 0\end{array}\right\}$ | $\mathsf{\Delta}{u}_{\eta}=\left\{\begin{array}{c}{u}^{\prime}-u\\ {v}^{\prime}-v\\ {w}^{\prime}-w\end{array}\right\}=\left\{\begin{array}{c}{L}_{UC}{\gamma}_{xy}^{0}\\ {L}_{UC}{\epsilon}_{y}^{0}\\ 0\end{array}\right\}$ |

z-axis | $\mathsf{\Delta}{x}_{z}=\left\{\begin{array}{c}0\\ 0\\ {H}_{UC}\end{array}\right\}$ | $\mathsf{\Delta}{u}_{\eta}=\left\{\begin{array}{c}{u}^{\prime}-u\\ {v}^{\prime}-v\\ {w}^{\prime}-w\end{array}\right\}=\left\{\begin{array}{c}{H}_{UC}{\gamma}_{xy}^{0}\\ {H}_{UC}{\gamma}_{xy}^{0}\\ {H}_{UC}{\epsilon}_{z}^{0}\end{array}\right\}$ |

Elastic Constants | Values | |
---|---|---|

Quartz fiber | Elastic modulus | 72–78 GPa |

Poisson’s ratio | 0.25 | |

Tensile strength | 1046 MPa | |

Silica matrix | Elastic modulus | 35–45 GPa |

Poisson’s ratio | 0.26 | |

Tensile strength | 180–220 MPa |

Warp Yarn ${V}_{f}^{yarn}=0.71$ | |||||

${E}_{1}\left(\mathrm{GPa}\right)$ | ${E}_{2}={E}_{3}\left(\mathrm{GPa}\right)$ | ${G}_{12}={G}_{13}\left(\mathrm{GPa}\right)$ | ${G}_{23}\left(\mathrm{GPa}\right)$ | ${\mu}_{12}$$={\mu}_{13}$ | ${\mu}_{23}$ |

52.43 | 12.61 | 6.07 | 4.66 | 0.28 | 0.35 |

Weft yarn ${V}_{f}^{yarn}=0.60$ | |||||

${E}_{1}\left(\mathrm{GPa}\right)$ | ${E}_{2}={E}_{3}\left(\mathrm{GPa}\right)$ | ${G}_{12}={G}_{13}\left(\mathrm{GPa}\right)$ | ${G}_{23}\left(\mathrm{GPa}\right)$ | ${\mu}_{12}$$={\mu}_{13}$ | ${\mu}_{23}$ |

45 | 11 | 5 | 4.07 | 0.29 | 0.35 |

Warp and weft yarn strengths | |||||

${S}_{1t}\left(\mathrm{MPa}\right)$ | ${S}_{2t}={S}_{3t}\left(\mathrm{MPa}\right)$ | ${S}_{1c}\left(\mathrm{MPa}\right)$ | ${S}_{2c}={S}_{3c}\left(\mathrm{MPa}\right)$ | ${S}_{12}={S}_{13}\left(\mathrm{MPa}\right)$ | ${S}_{23}\left(\mathrm{MPa}\right)$ |

200 | 80 | 80 | 80 | 40 | 40 |

Porous Matrix | |||||

${E}_{m}\left(\mathrm{GPa}\right)$ | $\mu $ | ||||

4.5 | 0.35 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lu, H.; Liu, Y.; Yan, S.
Experimental Study and Numerical Analysis of the Tensile Behavior of 3D Woven Ceramic Composites. *Machines* **2022**, *10*, 434.
https://doi.org/10.3390/machines10060434

**AMA Style**

Lu H, Liu Y, Yan S.
Experimental Study and Numerical Analysis of the Tensile Behavior of 3D Woven Ceramic Composites. *Machines*. 2022; 10(6):434.
https://doi.org/10.3390/machines10060434

**Chicago/Turabian Style**

Lu, Hongbo, Yancheng Liu, and Shibo Yan.
2022. "Experimental Study and Numerical Analysis of the Tensile Behavior of 3D Woven Ceramic Composites" *Machines* 10, no. 6: 434.
https://doi.org/10.3390/machines10060434