# Creep Response of Neat and Carbon-Fiber-Reinforced PEEK and Epoxy Determined Using a Micromechanical Model

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## Abstract

**:**

## 1. Introduction

_{2s}coupons. The authors reviewed the effects of temperature, moisture, and stress levels on the time-dependent shear modulus for two carbon epoxy laminate systems. This study has shown that a moisture concentration of about 1% may be considered as the critical limit for the investigated system. In other words, the viscoelastic deformation occurs much more rapidly beyond this limit. It was further demonstrated that for the above laminate, shear stresses in excess of 50% of the ultimate tensile strength could lead to irreversible deformation during creep.

## 2. Micromechanical Model

- Continuous fibers, circular in cross section and extended in the X
_{1}-direction, are arranged in a rectangular array in the X_{2}–X_{3}plane. - The fibers are supposed to be linearly elastic and anisotropic while the polymeric matrix is nonlinearly viscoelastic but isotropic.
- The interaction between fibers and matrix is merely mechanical and no crack or holes may develop under load.

_{1}, ${x}_{2}^{(\lambda )}$, ${x}_{3}^{(\lambda )}$) with its origin located as shown in Figure 4 is introduced. Furthermore, the displacement in each of the two subcells is expressed in a first-order linear expansion as a function of the distance from the origin to the local coordinate axes [27,28]. This means that the displacement at any point in the subcell is:

_{1}is the direction of anisotropy and the X

_{2}–X

_{3}plane is the plane of isotropy.

## 3. Calculation of the Average Stresses in the Unit Cell

_{1}, X

_{2}, X

_{3}) and has a volume V. The problem to be discussed now is that of computing the average stress ${\overline{\sigma}}_{ij}$ in the representative volume cell. This is a sample of the whole composite which contains a sufficient number of the constituent materials and, at the same time, is representative of the bulk material, as far as the structural point of view is concerned. Therefore, the representative volume cell retains the same properties as those of the entire composite. The volume average of the stress distribution can be written as:

## 4. Determination of the Continuity Conditions

## 5. Calculation of the Average Strains in the Unit Cell

- the six subcell stresses ${\overline{S}}_{11}^{(\lambda )}$, ${\overline{S}}_{22}^{(\lambda )}$, and ${\overline{S}}_{33}^{(\lambda )}$;
- the four micro-variables ${\xi}_{2}^{(\lambda )}$ and ${\zeta}_{3}^{(\lambda )}$;
- and the three composite strains ${\overline{\epsilon}}_{11}$, ${\overline{\epsilon}}_{22}$, and ${\overline{\epsilon}}_{33}$.

_{2}, X

_{3}) plane, the relations which can be applied in the X

_{2}direction are

_{3}direction:

_{2}or X

_{3}, the relation corresponding to the direction with no applied load should be replaced accordingly by either:

_{l}was supplied by the manufacturer. Since the transverse properties of the fibers (i.e., E

_{t}, ${\upsilon}_{lt},{\upsilon}_{tt}$) are not available, their values were computed based on an inverse technique.

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The primary, secondary, and tertiary creep (adapted after [2]).

**Figure 6.**Comparison between micromechanical and Finite Element prediction for ${\epsilon}_{22}$ of glass epoxy composite subjected to ${\sigma}_{22}=103.4\text{}\mathrm{MPa}$ and ${\sigma}_{33}=34.5\text{}\mathrm{MPa}$

**Figure 7.**The creep strain ${\epsilon}_{22}$ for neat PEEK resin subjected to ${\sigma}_{22}=26\hspace{0.17em}\mathrm{MPa}$ at 23 °C.

**Figure 8.**The creep strain ${\epsilon}_{22}$ for neat PEEK resin subjected to ${\sigma}_{22}=13\mathrm{MPa}$ at 80 °C.

**Figure 9.**The creep strain ${\epsilon}_{22}$ for carbon/PEEK subjected to ${\sigma}_{22}=26\mathrm{MPa}$ at 80 °C.

**Figure 10.**The creep strain ${\epsilon}_{22}$ for carbon/PEEK subjected to ${\sigma}_{22}=47\mathrm{MPa}$ at 80 °C.

**Figure 11.**The creep strain ${\epsilon}_{22}$ for carbon/epoxy subjected to ${\sigma}_{22}=30\mathrm{MPa}$ at 80 °C.

**Figure 12.**The creep strain ${\epsilon}_{22}$ for carbon/epoxy subjected to ${\sigma}_{22}=36\mathrm{MPa}$ at 80 °C.

Material | E_{l} (MPa) | E_{t} (MPa) | v_{l} | v_{t} |
---|---|---|---|---|

T-800 Carbon Fiber | 294.00 | 13.46 | 0.336 | 0.281 |

IM-6 Carbon Fiber | 272.00 | 15.95 | 0.356 | 0.261 |

6373 Epoxy Resin | 3.79 | 3.79 | 0.32 | 0.32 |

APC-2 PEEK Resin | 4.54 | 4.54 | 0.34 | 0.34 |

Glass Fiber | 77 | 77 | 0.2 | 0.2 |

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**MDPI and ACS Style**

Katouzian, M.; Vlase, S.
Creep Response of Neat and Carbon-Fiber-Reinforced PEEK and Epoxy Determined Using a Micromechanical Model. *Symmetry* **2020**, *12*, 1680.
https://doi.org/10.3390/sym12101680

**AMA Style**

Katouzian M, Vlase S.
Creep Response of Neat and Carbon-Fiber-Reinforced PEEK and Epoxy Determined Using a Micromechanical Model. *Symmetry*. 2020; 12(10):1680.
https://doi.org/10.3390/sym12101680

**Chicago/Turabian Style**

Katouzian, Mostafa, and Sorin Vlase.
2020. "Creep Response of Neat and Carbon-Fiber-Reinforced PEEK and Epoxy Determined Using a Micromechanical Model" *Symmetry* 12, no. 10: 1680.
https://doi.org/10.3390/sym12101680