# Stress Evolution of Amorphous Thermoplastic Plate during Forming Process

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## Abstract

**:**

_{g}of the PEI. The second stage, corresponding to the plate cooling from above T

_{g}to below T

_{g}, contributes a large portion of the residual stress in a short time. The final residual stress, the magnitude of which is affected by the cooling rate and plate thickness, shows a parabolic distribution through the thickness of the plate. These important conclusions are beneficial for improving the quality of an amorphous thermoplastic plate, while allowing highly efficient production.

## 1. Introduction

## 2. Theory

#### 2.1. Polyetherimide

#### 2.2. Thermal Analysis

#### 2.3. Mechanical Analysis

_{0}and B are calculated from Equations (5)–(7):

_{g}.

## 3. Finite Element Analysis

#### 3.1. Model

#### 3.2. Program Validation

## 4. Thermally-Induced Mechanical Change

#### 4.1. Temperature Profile

_{g}to room temperature T

_{room}, typically 400–20 °C. In this model the maximum temperature and room temperature were set as 240 °C and 180 °C, respectively. Although this temperature range is small, it reflects the actual physical processes as it covers all the material phases, and exhibits all material changes. The temperature decreases linearly from 240 to 180 °C in 150 s, and then remains constant up to 200 s, as shown in Figure 5.

#### 4.2. Volume Shrinkage

_{g}, the rate of strain is high and approximately linear with a value of −6 × 10

^{−5}s

^{−1}. From 75 to 150 s, the rate of strain decreases to −6.6 × 10

^{−6}s

^{−1}albeit with linear slope when the temperature is lower than T

_{g}. After 150 s, no more shrinkage occurs since the temperature keeps constant:

#### 4.3. Modulus and Shift Factor

_{p}, its Young’s modulus is small as the PEI is in its rubbery region. When the temperature is lower than T

_{p}, but higher than T

_{g}(in the range of approximately 220 °C–210 °C), the PEI enters the leathery region, and its Young’s modulus increases significantly. After 75 s, when the temperature is less than T

_{g}, its Young’s modulus remains approximately constant at a large value of 3.4 GPa. Moreover, the PEI stress relaxation time is also influenced by the temperature, and its temperature-dependent shift factor is shown in Figure 6b at a reference temperature of 180 °C. The shift factor increases exponentially when the temperature decreases linearly to T

_{g}. Below T

_{g}, the rate of increase of the shift factor decreases, and the value of shift factor approaches 1. As a result, the stress relaxes in an extremely short time at high temperatures, e.g., greater than T

_{g}. In contrast, the PEI at low temperatures is analogous to a purely elastic material. Both Young’s modulus and the shift factor have significant impacts on the stress evolution.

## 5. Stress Evolution

#### 5.1. Residual Stress Distribution

_{xx}and σ

_{zz}exhibit similar distributions. However, the out-of-plane stress σ

_{yy}differs, with a smaller value. As the in-plane stresses are dominant, we focus on σ

_{zz}in the following discussion.

_{zz}on the X–Y, X–Z, and Y–Z planes in Figure 7c–e clearly show the boundary effect. For example, in Figure 7e, σ

_{zz}exhibits a different pattern near the free boundary, but a uniform pattern in the areas away from the free boundary, including the monitoring position. Typically, the length and width of a plate-like plastic product are much greater than its thickness, so the monitoring position showing the uniform pattern of σ

_{zz}is suitable to be used in the following analyses in order to avoid the boundary effect.

#### 5.2. Temperature Distribution Through Plate Thickness

_{g}at 75 s. The plate then cools further until the bottom surface reaches T

_{g}at 100 s. The temperature of the entire plate decreases further, and approaches T

_{room}. These four curves divide the whole process into three stages, which can also be clearly identified in the temperature-versus-time curves in Figure 8b. In the actual manufacturing process, the duration of stages 1 and 3 will be significantly longer due to the larger temperature ranges used. The duration of stage 2 is affected by both the cooling rate and plate thickness, as it represents the time required by a plate to completely cool down from above T

_{g}to below T

_{g}, indicated by the gray-shadowed area in Figure 8.

#### 5.3. Thermal Strain and Rate of Strain

_{g}, respectively. The rate of strain gradually changes in stage 2 from −6.03 × 10

^{−5}to −6.55 × 10

^{−6}s

^{−1}.

#### 5.4. Stress in Stage 1

_{zz}in stage 1. The top surface of the plate shows tensile stress, while the bottom surface shows compressive stress. As the PEI closest to the top surface cools down first, ideally, that part will typically shrink before other parts. However, because of the identical displacement, the PEI inside, which is hotter and, thus, has a greater volume, prevents the outside cooler part from shrinking at the ideal rate of strain. Consequently, the nodes on the top surface show tensile stress and the nodes on the bottom surface show compressive stress. However, the values are negligible due to the low stiffness and short relaxation time of the material. When the PEI is in the rubbery region, its Young’s modulus is approximately two thousand times smaller than that in the glassy region. Thus, the formed stress will be negligible, as stress equals strain times Young’s modulus. In addition, even if a certain level of stress is formed, it will relax in a very short time when the temperature exceeds T

_{g}.

#### 5.5. Stress in Stage 2

_{zz}in stage 2. In general, the top surface shows compressive stress while the bottom surface shows tensile stress. In addition, there is a knee point showing the maximum tensile stress on each curve, indicated by the black circles. Figure 12 correlates the stress and temperature distribution at 92 s, a typical time in stage 2, and clearly shows that the knee point in the stress curve corresponds to the position of temperature T

_{g}. This knee point divides the stress curve into two parts. The relatively hot material has a temperature greater than T

_{g}, and is close to the bottom surface. The relatively cold material has a temperature lower than T

_{g}, and is close to the top surface. The relative hot PEI that is in the leathery region ideally shrinks more than that in the glassy region, as the ideal rate of strain of the PEI decreases when the temperature is less than T

_{g}. However, the relatively cold PEI obstructs the fast shrinkage of the hot PEI, i.e., the hot PEI promotes the shrinkage of the cold PEI, because of the identical thermal strain. As a result, the PEI exhibits an uneven stress distribution through the thickness. The value of the stress is determined by the rapidly increasing Young’s modulus when the PEI is in its leathery region. Furthermore, as the PEI in the glassy region has a lengthy stress relaxation time, the stress remains, and then accumulates to form a parabolic shape.

#### 5.6. Stress in Stage 3

_{zz}in stage 3. From 100 to 150 s, there is the same rate of strain through the entire plate as the temperature is lower than the T

_{g}throughout, and the bottom and top surfaces cool at the same rate. Thus, neighboring areas do not restrict interaction. Both the tensile and compressive stresses then decrease due to the stress relaxation. After 150 s, the PEI near the top surface no longer shrinks, as its temperature remains constant at T

_{room}. However, the PEI near the bottom surface continues to shrink marginally more. Therefore, the tensile and compressive stresses increase slightly. However, as the PEI is in the glassy region with high modulus and long relaxation times, and also because the temperature difference between the top and bottom surfaces are small in this simulation case, the residual stress change is insignificant in stage 3.

#### 5.7. Stress Evolution Process

## 6. Forming Mechanism of Residuals Stress

## 7. Conclusions

- Stage 1:
- When the temperature of the whole plate is greater than T
_{g}, stress barely forms. - Stage 2:
- When the plate cools to T
_{g}from one side to the other side, a large portion of the residual stress forms in a relatively short time, although the duration of this stage is typically short. - Stage 3:
- Until the whole plate cools to room temperature, the residual stress changes further, and finally a parabolic-shaped residual stress forms.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

T | Temperature, K |

T_{g} | Glass transition temperature, 483 K |

T_{room} | Room temperature, 298 K |

T_{t} | Pressure-dependent T_{g} |

T | Time, s |

V | Volume, m^{3} |

V_{0} | Reference volume, m^{3} |

P | Pressure, MPa |

ρ | Density, kg/m^{3} |

Cp | Specific heat, kJ/(kg·K) |

k | Thermal conductivity, W/(m·K) |

B | Material coefficients |

C | Constant in Tait equation, 0.0894 |

A_{T} | Shift factor |

E | Young’s modulus, MPa |

τ | Stress relaxation time, s |

ε | Strain |

σ | Stress, MPa |

b | Fitting parameter |

## Appendix A. Cooling Rate

**Figure A1.**Temperature-versus-position curves of fast cooling rate case at four different times also divided the whole process into three stages.

_{zz}also exhibits three stages. Stages 1 and 3 are analogous to those of the slow cooling case. The stress evolution in stage 2 also contributed to the major portion of the residual stress, but the stress-versus-thickness curve differs slightly. Figure A2 shows the correlation of stress and temperature through the thickness of the plate at 34 s, a typical time in stage 2. The knee point in the stress curve is a peak, which means the stress increases from the bottom surface to the peak point. This differs from Figure 11 and Figure 12, where the stresses at positions between the knee point and bottom surface have approximately similar values. It can be seen in Figure A2 that the position where stress begins to develop is where the temperature reaches T

_{p}. When the temperature is lower than T

_{p}, the stress increases primarily because of the rapid increase in material stiffness.

**Figure A2.**Knee point in stress curve corresponds to T

_{g}, and point where stress begins to develop corresponds to T

_{p}.

## Appendix B. Plate Thickness

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**Figure 1.**Material properties. (

**a**) Specific heat capacity; (

**b**) thermal conductivity; (

**c**) specific volume; and (

**d**) Young’s modulus.

**Figure 3.**One-side cooling of PEI plate. (

**a**) Exploded view of the schematic diagram of the experimental setup; (

**b**) one-side cooling caused by cold aluminum plate; (

**c**) temperature profiles on top and bottom surfaces; and (

**d**) photograph of final warpage.

**Figure 4.**Simulation of one-side cooling of PEI plate. (

**a**) Temperature profiles at different positions through thickness; and (

**b**) simulated final warpage.

**Figure 6.**Mechanical property changes during cooling process. (

**a**) Young’s modulus; and (

**b**) shift factor.

**Figure 7.**Contour of stress distribution on different planes. (

**a**–

**c**) σ

_{xx}, σ

_{yy}, and σ

_{zz}on the X-Y plane; (

**d**,

**e**) σ

_{zz}on the X-Z and Y-Z planes.

**Figure 8.**Temperature distribution through plate thickness at different times. (

**a**) Temperature-versus-position curves at five different times; and (

**b**) temperature-versus-time curves at three different positions.

**Figure 15.**Difference between ideal shrinkage and actual shrinkage. (

**a**) Ideal thermal strain at different positions and actual strain; and (

**b**) the ideal rate of strain and actual rate of strain.

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**MDPI and ACS Style**

Wu, Q.; Ogasawara, T.; Yoshikawa, N.; Zhai, H.
Stress Evolution of Amorphous Thermoplastic Plate during Forming Process. *Materials* **2018**, *11*, 464.
https://doi.org/10.3390/ma11040464

**AMA Style**

Wu Q, Ogasawara T, Yoshikawa N, Zhai H.
Stress Evolution of Amorphous Thermoplastic Plate during Forming Process. *Materials*. 2018; 11(4):464.
https://doi.org/10.3390/ma11040464

**Chicago/Turabian Style**

Wu, Qi, Tomotaka Ogasawara, Nobuhiro Yoshikawa, and Hongzhou Zhai.
2018. "Stress Evolution of Amorphous Thermoplastic Plate during Forming Process" *Materials* 11, no. 4: 464.
https://doi.org/10.3390/ma11040464