# Estimating Low- and High-Cyclic Fatigue of Polyimide-CF-PTFE Composite through Variation of Mechanical Hysteresis Loops

^{1}

^{2}

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## Abstract

**:**

^{3}(10%) for the composite, whereas they reduced from 30 down to 25 J/m

^{3}(17%) for neat PI in the low-cycle fatigue mode. For high-cycle fatigue, energy losses decreased from 10 to 9 J/m

^{3}(10%) and from 17 to 14 J/m

^{3}(18%) for neat PI and composite, respectively. For this reason, the changes of the energy losses due to hysteresis are of prospects for the characterization of both neat PI and the reinforced PI-based composites.

## 1. Introduction

## 2. Materials and Methods

^{3}in size (Figure 1) and the samples were cut using a milling machine.

^{3}. The notches with a depth of 2 mm and a curvature radius of 0.25 ± 0.01 mm were formed using a double-tooth disk cutter with diamond cutting edges and a ‘GT-7016-A3’ V-notch sampling machine (Gotech Testing Machines, Taichung City, Taiwan). Copper films about 10 nm thick were deposited on the fracture surfaces using a “JEOL JEE-420” vacuum evaporator (JEOL USA, Inc., Peabody, MA, USA). A “LEO EVO 50” scanning electron microscope (Carl Zeiss, Oberkochen, Germany) was employed at an accelerating voltage of 20 kV.

^{3}. The required surface quality was ensured by further polishing with sandpapers of various grit sizes up to 1000 grit. The tests were carried out using a “Biss Nano 15 kN” servo-hydraulic testing machine (Bangalore Integrated System Solutions Pvt Ltd., Bengaluru, India), (Figure 2).

- 1.
- The dynamic modulus (${E}_{dyn}$) was determined as a fitting line slope between two points of a hysteresis loop. The dynamic modulus was calculated by the ratio of the stress range over the strain one (Equation (1) and illustrated in Figure 1):$${E}_{dyn}=\left|{E}^{\ast}\right|=\frac{\Delta \sigma}{\Delta \epsilon}=\frac{{\sigma}_{max}-{\sigma}_{min}}{{\epsilon}_{max}-{\epsilon}_{min}}$$
_{dyn}is the dynamic modulus which is equal to the absolute value of the complex modulus (|E^{∗}|); the latter characterizes the viscoelastic behavior of polymer materials. Due to viscous nature, a strain rate of viscoelastic materials depends on the time and exhibits hysteresis variation pattern. - 2.
- The second parameter was a hysteresis loop area (Energy loss), which corresponds to energy loss at each cycle (Figure 1). The energy dissipation is associated with heating as well as structure rearrangements [38]. The hysteresis loop area is a measure of energy losses under cyclic loading. In this paper, the E
_{dyn}and energy loss were employed to characterize the structural state of the cyclically loaded neat PI and the PI-based composite. - 3.
- The strain energy was calculated as the area under the loading curve from its beginning to the point of reaching the maximum value (Figure 3).
- 4.
- To assess the damping capacity of the studied materials, the ratio of energy loss over the total strain energy in a cycle was determined. The relative damping value was estimated as:$$\mathsf{\psi}=100\frac{Energyloss}{StrainEnergy}\%,$$

## 3. Results

#### 3.1. Structural Study

#### 3.2. Static Loading

_{f}was predictably lower for the “PI/PTFE/MCF” composite than that for neat PI (both in absolute and relative terms). This was directly related to the ability to re-arrange the inner structure of the polymer, which was significantly suppressed by loading with MCF and partially by PTFE particles.

#### 3.3. Fatigue Tests

_{max}and stress

**σ**

_{max}levels), as well as the number of cycles prior to failure. The tests were carried out in the displacement (strain) control mode. The number of specimens tested under three different strain levels for each type of material was equal to seven.

**σ**/2) stress level in a cycle. In the LCF region (10

^{3}–10

^{4}cycles), the difference in the amplitude (Δ

**σ**/2) of the withstand stresses was about 10 MPa that corresponded to the variations of the yield strength of the materials. Thus, the increase in fatigue life for the “PI/PTFE/MCF” composite was more pronounced in the LCF range due to a smaller proportion of plastic strains. The stress fatigue limit was higher for the “PI/PTFE/MCF” composite, which was predictable and related to its higher yield/strength level.

^{4}cycles). The second was a transitional one between the LCF and HCF modes; its middle part corresponded to the yield point. This section was characterized by a slight increase in durability with a decrease in the cyclic load amplitude. The third section began at ≈10

^{5}cycles and had a gentle slope of the curves that corresponded to the HCF mode. In this section, even small changes in the amplitude levels caused very significant variations in durability. The fatigue curves (Figure 7b), drawn after recalculating the applied stresses relative to the yield point, were also similar but located farther apart.

#### 3.4. Fractographic Studies

_{0.1}= 1.3, the fracture surface exhibited three characteristic zones (Figure 8, middle column): ones of stable and accelerated crack propagation (Figure 8d,e), as well as a final rupture zone (Figure 8f). The stable crack propagation zone was characterized by typical round shaped structural elements, as well as fatigue striations which had been formed when crack transited to accelerated propagation (Figure 8e). The relief formed in the final rupture zone was associated with the development of plastic strains and was accompanied by local stretching of the polymer upon the crack propagation by the normal opening mechanism. The final rupture zone had a similar appearance with that after the static test, but possessed a less-developed fracture surface relief.

_{0.1}= 0.7, the specimens withstood more than 10

^{5}cycles at the load amplitude below the yield point. For this reason, the fracture surface appeared quite smooth and uniform. As in the LCF case, three fracture zones were distinguished: the stable and accelerated crack propagation ones (Figure 8g,h), and the final rupture zone (Figure 8h,i). In the stable crack propagation zone, the relief was similar to that for the LCF mode. In the unstable crack propagation zone, which was characterized by the rather rapid process development, the fractographic relief as a whole followed the pattern of neat PI supermolecular structure (both in appearance and in the size of structural elements).

## 4. Analysis of the Mechanical Hysteresis Loops

^{3}for neat PI in the LCF mode and ~10 J/m

^{3}in HCF, whereas these levels were ~26 and ~13 J/m

^{3}for the composite (Figure 10a; curves 1, 3 and Figure 10b; curves 1, 3). The composite possessed a 30% higher energy loss per cycle compared to neat PI. In the LCF mode, similar trends towards energy loss were evident for both materials with enlarging the number of cycles (by ~3 J/m

^{3}for neat PI and by ~5 J/m

^{3}for the composite). Energy losses almost did not change in the HCF mode, thus the fitting straight lines on the graphs, have nearly zero inclination angles to the horizontal axis.

_{dyn}modulus was generally lower at the LCF mode (ε/ε

_{0.1}= 1.3), which was associated with the possibility of inelastic strain development. In this case, the E

_{dyn}modulus variation was three times greater than that in the HCF mode. For the composite, the E

_{dyn}modulus decreased by ~60 MPa in the LCF mode, whereas by ~20 MPa in the HCF (Figure 10a; curve 2 and Figure 10b; curve 2). In turn, neat PI showed lower E

_{dyn}rising: by ~30 MPa in the LCF mode and by ~10 MPa in the HCF (Figure 10a; curve 4 and Figure 10b; curve 4). Thereby, the ranges of both Energy loss and E

_{dyn}levels were significantly reduced when transiting from the LCF to the HCF mode due to the lowering intensity of ongoing structural changes. This fact could be interpreted as a decrease in the sensitivity of these parameters to the development of strain processes, including the accumulation of scattered damages. The scatter of both Energy loss and E

_{dyn}values reflected the instability of the damage development process in the bulk material.

_{dyn}plots could be an informative parameter for characterizing the fatigue properties and kinetics of both neat PI and its triple composite.

_{dyn}modulus under cyclic loads was characteristic of neat PI.

## 5. Discussion

_{dyn}growth), but reduces the cyclic fracture toughness (which was reflected through the reduction in the Energy loss). The critical state (typically interpreted as the exhaustion of the relaxation capacity) could be considered the moment, when the crucial fracture energy is below the strain energy. This corresponds to the initiation and subsequent propagation of a crack.

## 6. Conclusions

_{dyn}modulus decreased by 60 MPa (2.5% of the initial value) in the LCF mode for the composite, whereas by 20 MPa (0.8%) in the HCF one. In turn, neat PI showed lower E

_{dyn}rising: by 30 MPa (2.4%) in the LCF mode and by ~10 MPa (0.8%) in the HCF. The variations of the dynamic modulus did not enable to clearly distinguish the damage accumulation mechanisms because of the negligible decrease in its level.

^{3}to 18 J/m

^{3}(10%) for the composite and from 30 to 25 J/m

^{3}(17%) for neat PI in the LCF mode. In the HCF case, energy losses decreased from 10 to 9 (10%) and from 17 to 14 J/m

^{3}(18%) for neat PI and the composite, respectively. For this reason, the changes in the energy losses due to hysteresis are promising for the characterization of both neat PI and the reinforced PI-based composites.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 5.**The stress–strain diagrams: 1 (black)—neat PI; 2 (red)—“PI/MCF” composite; 3 (green)—“PI/PTFE” composite; 4 (blue)—“PI/PTFE/MCF” composite.

**Figure 6.**The fatigue curves for neat PI and the “PI/PTFE/MCF” composite according to the strain (

**a**) and stress (

**b**) amplitudes.

**Figure 7.**The strain (

**a**) and stress (

**b**) fatigue curves for neat PI and the “PI/PTFE/MCF” composite normalized over the yield point.

**Figure 8.**The SEM micrographs of the neat PI fracture surfaces: left column—under static tension, middle column—under LCF, right column—under HCF: (

**a**) fracture initiation region, (

**b**) “cellular” relief pattern, (

**c**) final rupture zone, (

**d**) fracture initiation region, (

**e**) transition from stable crack propagation zone to its accelerated propagation one, (

**f**) final rupture zone, (

**g**) fracture initiation region, (

**h**) transition from accelerated crack propagation zone to its final rupture one, (

**i**) final rupture zone.

**Figure 9.**The SEM micrographs of the “PI/PTFE/MCF” composite fracture surfaces: left column—under static tension, middle column—under the LCF, right column—under the HCF: (

**a**–

**c**) high degree heterogeneity relief, (

**d**–

**f**) signs of fracture by the normal opening mechanisms, (

**g**–

**i**) MCF “depleted” regions in on the fracture surface.

**Figure 10.**Changes in the dynamic modulus (black) and energy losses (red) vs. the number of cycles for neat PI and the “PI/PTFE/MCF” composite; LCF mode with

**σ**= 1.3 of the yield point (

**a**) and HCF mode with

**σ**= 0.7 of the yield point (

**b**). Curves designations are the following (1)—Energy loss for “PI/PTFE/MCF”; (2)—E

_{dyn}for “PI/PTFE/MCF”; (3)—Energy loss for neat PI; (4)—E

_{dyn}for neat PI.

**Figure 11.**The total strain energy, energy losses and damping capacity for neat PI and the “PI/PTFE/MCF” composite; LCF mode with

**σ**= 1.3 of the yield point (

**a**) and HCF mode with

**σ**= 0.7 of the yield point (

**b**).

Material | σ_{UTS} (MPa) | ε (%) | E (GPa) | σ_{0.1} (MPa) | ε_{0.1} (%) | ε_{f} (%) ^{1} |
---|---|---|---|---|---|---|

Neat PI | 104 ± 4 | 6.5 ± 0.4 | 3.08 ± 0.15 | 43 ± 3 | 1.52 ± 0.13 | 70 |

“PI/PTFE” composite | 89 ± 5 | 5.6 ± 0.2 | 2.75 ± 0.18 | 42 ± 4 | 1.64 ± 0.09 | 71 |

“PI/MCF” composite | 105 ± 4 | 3.0 ± 0.3 | 5.41 ± 0.21 | 63 ± 3 | 1.26 ± 0.11 | 58 |

“PI/PTFE/MCF” composite | 97 ± 3 | 2.0 ± 0.2 | 6.9 ± 0.3 | 61 ± 4 | 1.00 ± 0.06 | 50 |

^{1}ε

_{f}is the proportion (in percent) of inelastic (irreversible) strains in the total strains before fracture and found as ε

_{f}= (ε − ε

_{0.1})/ε 100%.

ε/ε_{0.1} | ε_{max} (%) | σ_{max} (MPa) | N_{f} |
---|---|---|---|

Neat PI | |||

1.3 | 2.0 | 61 ± 6 | 9000 ± 4000 |

1.0 | 1.5 | 52 ± 3 | 50,000 ± 8000 |

0.7 | 1.1 | 41 ± 2 | 110,000 ± 10,000 |

“PI/PTFE/MCF” composite | |||

1.2 | 1.2 | 80 ± 6 | 6000 ± 3000 |

1.0 | 1.0 | 63 ± 4 | 50,000 ± 10,000 |

0.7 | 0.7 | 53 ± 3 | 130,000 ± 20,000 |

LCF | HCF | ||
---|---|---|---|

Strain Energy for the first cycle (Pa = J/m ^{3}) | Neat PI | 228 | 85 |

“PI + PTFE + MCF” | 168 | 59 | |

Energy loss for the first cycle (J/m^{3}) | Neat PI | 20 | 10 |

“PI + PTFE + MCF” | 30 (+50%) | 17 (+70%) | |

Total Strain Energy (MJ/m^{3}) | Neat PI | 0.67 | 11.00 |

“PI + PTFE + MCF” | 1.02 | 10.57 | |

Total Energy loss (MJ/m^{3}) | Neat PI | 0.02 | 0.53 |

“PI + PTFE + MCF” | 0.08 | 1.12 | |

Damping capacity ψ (%) | Neat PI | 3% | 5% |

“PI + PTFE + MCF” | 8% | 11% | |

Fatigue life (Cycles ×10^{3}) | Neat PI | 3 | 120 |

“PI + PTFE + MCF” | 6 (+100%) | 180 (+50%) |

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**MDPI and ACS Style**

Panin, S.V.; Bogdanov, A.A.; Eremin, A.V.; Buslovich, D.G.; Alexenko, V.O.
Estimating Low- and High-Cyclic Fatigue of Polyimide-CF-PTFE Composite through Variation of Mechanical Hysteresis Loops. *Materials* **2022**, *15*, 4656.
https://doi.org/10.3390/ma15134656

**AMA Style**

Panin SV, Bogdanov AA, Eremin AV, Buslovich DG, Alexenko VO.
Estimating Low- and High-Cyclic Fatigue of Polyimide-CF-PTFE Composite through Variation of Mechanical Hysteresis Loops. *Materials*. 2022; 15(13):4656.
https://doi.org/10.3390/ma15134656

**Chicago/Turabian Style**

Panin, Sergey V., Alexey A. Bogdanov, Alexander V. Eremin, Dmitry G. Buslovich, and Vladislav O. Alexenko.
2022. "Estimating Low- and High-Cyclic Fatigue of Polyimide-CF-PTFE Composite through Variation of Mechanical Hysteresis Loops" *Materials* 15, no. 13: 4656.
https://doi.org/10.3390/ma15134656