Thermomechanical Modeling of Amorphous Glassy Polymer Undergoing Large Viscoplastic Deformation: 3-Points Bending and Gas-Blow Forming
Abstract
:1. Introduction
2. Results
- the thermal deformation is assumed to be an isotropic thermal expansion, i.e., , where denotes the second-order identity tensor.
2.1. Hyperelasticity
2.2. Viscoplastic Flow Rule
2.3. Strain Softening
2.4. Strain Hardening
2.5. Criterion for Viscoplasticity
2.6. Temperature Evolution
3. Calibration of the Model Parameters
3.1. 1D Version of the Constitutive Equations
3.2. Model Parameter Calibration
- The reference elastic modulus was determined from the experimental stress-strain response at the reference temperature . The model parameter was determined from the experimental stress-strain response at different temperatures. Poisson’s ratio and the thermal expansion coefficient were obtained from the literature [26].
- The model parameters , , m, and control the dependence of the initial yield stress on the strain rate and temperature. The model parameter represents the initial yield stress at the minimum strain rate and the reference temperature , and it was determined approximately from the experimental stress-strain response at the strain rate of 0.0005 s−1 and the temperature of 25 °C. The model parameters and m were determined by fitting the exponential curve relationship between the yield stress and the logarithmic strain rate over the strain rate from 0.0005 s−1–8400 s−1. The model parameter was determined by fitting the linear relationship between the yield stress and the temperature over the temperature from −60 °C–120 °C. See the literature [25] for more details.
- The strain softening-associated model parameters , , , , and control the evolution of the internal variable . According to Equation (9), and are used to compute the state variable , while and are used to compute the state variable . Figure 1a,b shows how the parameters and affect the strain softening behavior of amorphous polymers. , , and were determined from the experimental stress-strain response at the reference temperature , while and were determined from the experimental stress-strain response at different temperatures.
- The strain hardening-associated model parameters , , , , and control the evolution of the internal variable . According to Equation (11), and are used to compute the state variable , while and are used to compute the state variable . Figure 1c,d shows how the parameters and affect the strain hardening behavior of amorphous polymers. , , and are determined from the experimental stress-strain response at the reference temperature , while and are determined from the experimental stress-strain response at different temperatures.
- The specific heat capacity c, the thermal conductivity k, and the mass density were obtained from the literature [26]. The Taylor–Quinney factor was determined from the temperature change experimentally measured by the thermal imager. The reference temperature took the room temperature of 25 °C.
3.3. Model Validation
4. Three-Point Bending
5. Gas-Blow Forming
6. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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Elasticity: , , , ; |
Yielding: , , m, ; |
Softening: , , , , ; |
Hardening: , , , , ; |
Heat: c, k, , , . |
Categories | Parameters | Constants | Values |
---|---|---|---|
Elasticity | E (MPa) | (MPa) | 2271 |
(MPa/K) | −5.4 | ||
0.4 | |||
(K−1) | |||
Yielding | (MPa) | 35.16 | |
(s) | 0.746 | ||
m | 0.118 | ||
(K−1) | |||
Softening () | (MPa) | 130.45 | |
0.036 | |||
(K−1) | 10.24 | ||
0.105 | |||
(K−1) | 26.86 | ||
Hardening () | (MPa) | 82.36 | |
−0.012 | |||
(K−1) | 3.12 | ||
1.45 | |||
(K−1) | 6.46 | ||
Heat | c (J/(kg · K)) | ||
k (W/(m · K)) | 0.19 | ||
(kg/m3) | |||
0.86 | |||
(K) | 298 |
Gas Pressure | Major Axis | Minor Axis |
---|---|---|
Pressurization 1.0 MPa | | |
Pressurization 2.0 MPa | | |
Pressurization 3.0 MPa | | |
Depressurization 0.0 MPa | | |
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Wang, J.; Xu, Y.; Zhang, W.; Ren, X. Thermomechanical Modeling of Amorphous Glassy Polymer Undergoing Large Viscoplastic Deformation: 3-Points Bending and Gas-Blow Forming. Polymers 2019, 11, 654. https://doi.org/10.3390/polym11040654
Wang J, Xu Y, Zhang W, Ren X. Thermomechanical Modeling of Amorphous Glassy Polymer Undergoing Large Viscoplastic Deformation: 3-Points Bending and Gas-Blow Forming. Polymers. 2019; 11(4):654. https://doi.org/10.3390/polym11040654
Chicago/Turabian StyleWang, Jun, Yingjie Xu, Weihong Zhang, and Xuanchang Ren. 2019. "Thermomechanical Modeling of Amorphous Glassy Polymer Undergoing Large Viscoplastic Deformation: 3-Points Bending and Gas-Blow Forming" Polymers 11, no. 4: 654. https://doi.org/10.3390/polym11040654