# Evaluating the Residual Stress and Its Effect on the Quasi-Static Stress in Polyethylene Pipes

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Experimental

#### 2.1. Materials and Sample Preparation

#### 2.2. Monotonic Tensile Test

_{eng}) was calculated using the following expression:

#### 2.3. Residual Hoop Stress Measurement

_{res}is the residual hoop stress, C

_{1}and C

_{2}are two constants, and x is the normalized position along the ring specimen wall thickness from the inner surface (x = 0) to the outer surface (x = 1). In order to solve the two constants, C

_{1}and C

_{2}, it is assumed that the total normal force along the pipe cross section should be zero:

_{1}can be expressed as a function of C

_{2}:

_{2}, another assumption, pure bending under the small deformation, is made according to the curved beam theory [17]. Under the small deformation, the arc length along the neutral plane remains constant:

_{n}is the normalized position of the neutral plane along the cross section. Thus, x

_{n}can be calculated by substituting Equation (4) into Equation (6) and is equal to 0.6235. Considering an arbitrary arc, with the radii r and r′ before and after the deformation, along the pipe wall located at a distance y to the neutral axis, the deformation (Δ) can be expressed as:

_{x}has the following expression:

_{t}) is equal to the sum of the moments resulting from the normal stress about the transverse direction, and the elementary forces acted on the section should be zero:

_{in}and R

_{out}represent the radii of the pipe inner surface and outer surface, respectively. The value of R′ can be obtained by substituting the R

_{in}and R

_{out}after the deformation into Equation (18). It should be also noted that the change in the outer diameter due to the release of residual stress, ΔD, can be approximated as twice the change in the radius of the neutral axis (ΔD = 2(R − R′)) for the small deformation. Alternatively, Equation (14) can be rewritten as:

_{t}= M

_{r}, the constant C

_{2}can be calculated as:

_{0}and h are measured values, while R and R’ can be calculated based on Equation (18). The constitutive equation for the stress–strain relationship before yielding, as described in [18,19], can be expressed as:

#### 2.4. Finite Element (FE) Simulation

## 3. Results and Discussion

_{1}and C

_{2}based on Equations (4) and (21), the distribution of the residual hoop stress along the pipe wall thickness was obtained. Figure 7a summarizes the outer diameter changes after cutting off an arc section of 120° from each specimen. The error bars were calculated based on the results of the three duplicates. Figure 7b sets the PE #5 pipe as an example to show the reproducibility of the results on the three duplicate tests. The coincidence of the three curves in Figure 7b indicates good reproducibility of the test results. Hereafter, we chose the average of the three duplicate tests for the analysis.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Measurement of the outer diameter change due to the ring slitting: (

**a**) a 3D-printed mold for the slitting; and (

**b**) the setup of the optical comparator for the measurement of the diameter change. (After ref. [7], with kind permission from Wiley.)

**Figure 5.**The engineering stress–stroke curves of the six PE pipes. (The red solid triangles mark the peak stress.)

**Figure 7.**(

**a**) Summary of the outer diameter changes after the cutting process and (

**b**) the PE #5 pipe as an example of the reproducibility of the results on the three duplicates.

**Table 1.**Material characteristics of the PE pipes used in the study, which are identical to those in ref. [16].

Material | Pipe Code | Density (g/cc) | Resin Yield Strength (MPa) | Hydrostatic Design Basis (MPa) at 23 °C | Melt Index (g/10 min) at 190 °C/2.16 kg |
---|---|---|---|---|---|

#1 u-MDPE | PE2708 | 0.940 | 19.3 | 8.62 | 0.2 |

#2 u-HDPE | PE3408 | 0.944 * | 22.8 * | 11.03 | 0.08 |

#3 b-MDPE | PE2708 | 0.940 | 19.3 | 8.62 | >0.15 |

#4 b-HDPE | PE4710 | 0.949 | 24.8 | 11.03 | 0.08 |

#5 u-MDPE | PE2708 | 0.940 | 19.3 | 8.62 | 0.2 |

#6 b-HDPE | PE4710 | 0.949 | >24.1 | 11.03 | 0.08 |

Material | Pipe Code | Elastic Modulus (MPa) |
---|---|---|

#1 u-MDPE | PE2708 | 570 |

#2 u-HDPE | PE3408 | 600 |

#3 b-MDPE | PE2708 | 560 |

#4 b-HDPE | PE4710 | 795 |

#5 u-MDPE | PE2708 | 560 |

#6 b-HDPE | PE4710 | 795 |

Material | Quasi-Static Stress at DB Transition (MPa) | Hydrostatic Design Basis at 23 °C (MPa) | Long-Term Hydrostatic Strength (MPa) |
---|---|---|---|

#1 u-MDPE | 8.18 | 8.62 | 8.27 to 10.55 |

#2 u-HDPE | 10.25 | 11.03 | 10.55 to 11.93 |

#3 b-MDPE | 7.25 | 8.62 | 8.27 to 10.55 |

#4 b-HDPE | 10.86 | 11.03 | 10.55 to 11.93 |

#5 u-MDPE | 7.25 | 8.62 | 8.27 to 10.55 |

#6 b-HDPE | 9.88 | 11.03 | 10.55 to 11.93 |

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**MDPI and ACS Style**

Tan, N.; Lin, L.; Deng, T.; Dong, Y.
Evaluating the Residual Stress and Its Effect on the Quasi-Static Stress in Polyethylene Pipes. *Polymers* **2022**, *14*, 1458.
https://doi.org/10.3390/polym14071458

**AMA Style**

Tan N, Lin L, Deng T, Dong Y.
Evaluating the Residual Stress and Its Effect on the Quasi-Static Stress in Polyethylene Pipes. *Polymers*. 2022; 14(7):1458.
https://doi.org/10.3390/polym14071458

**Chicago/Turabian Style**

Tan, Na, Liyang Lin, Tao Deng, and Yongwu Dong.
2022. "Evaluating the Residual Stress and Its Effect on the Quasi-Static Stress in Polyethylene Pipes" *Polymers* 14, no. 7: 1458.
https://doi.org/10.3390/polym14071458