# Modeling the Viscoelastic Behavior of Amorphous Shape Memory Polymers at an Elevated Temperature

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## Abstract

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_{g}), (amorphous SMPs exhibit finite deformation and viscoelastic behavior. In this work we develop a model to capture the viscoelastic behavior of the amorphous SMPs at elevated temperatures. The model uses an approach that was initially developed to study non-Newtonian viscoelastic fluids. We accomplish this by developing a multi-branch model based on the theory of multiple natural configurations using the maximization of the rate dissipation to determine the evolution of the natural configurations. We apply our model to study several different deformations at an elevated temperature T = 130 °C and show that this approach is able to capture the viscoelastic behavior of these polymers. The predictions of the theory are then compared with experimental results.

## 1. Introduction

_{g}), the undeformed polymer is in an amorphous state and exhibits viscoelastic behavior (State 1). Upon deformation, the material, and hence the polymer molecules, stretch (State 1 to State 2). If the polymer is now cooled below its glass transition temperature, the polymer molecules will lose their mobility due to glass transition and the modulus of the material increases dramatically (State 2 to State 3). If the material is now unloaded it will not revert back to its original shape but stays in a temporary shape (State 3 to State 4). This temporary shape is between the original shape and deformed shape due to the formation of the glassy phase in the deformed configuration. On heating above the glass transition temperature the original shape is recovered as the material regains its molecular mobility (State 4 to State 1). Based on the above description, it can be seen that in order to characterize the thermo-mechanical behavior of amorphous SMPs it is important to study their viscoelastic behavior at elevated temperatures (in State 1 to State 2). Furthermore, there are a number of polymers which behave in a manner similar to these polymers, and the model developed here can be applied to these types of polymers as well.

_{g}) of the polymers is 98 °C obtained by a dynamic mechanical analysis (DMA) test. To capture the finite-deformation viscoelastic behavior of this polymer at an elevated temperature (T = 130 °C), we develop a multi-branch model. This model has two sets of branches: one equilibrium branch for the hyperelastic response and multiple non-equilibrium branches for the viscoelastic response. We use a Neo-Hookean model for the hyperelastic equilibrium branch, and the model of the viscoelastic non-equilibrium branch is developed using the theory of multiple natural configurations [21]. The material response of materials belonging to many different classes have been modeled using this framework, some of them are: multi-network polymers [22], metal plasticity [23], viscoelastic liquids [24,25], crystallization in polymers [26,27,28], crystallizable SMPs [29,30,31], and light-activated SMPs [32,33]. Classical elasticity and linear viscous fluids arise as simple cases within this theory. Though the problem we are tackling is isothermal, we develop the model in a thermodynamic setting. The amorphous polymer is modeled at elevated temperatures as a viscoelastic solid and the rate of dissipation is always positive. In this approach we have to choose forms for the Helmholtz potential and the rate of dissipation. These are then used in the reduced energy-dissipation equation, which is used to place restrictions on the forms for the stress. We further assume that the rate of dissipation is maximized, here it is associated with viscoelasticity. An identical approach has been used to model a number of materials in which entropy production takes place, for instance in twinning [21] and viscoelastic fluids [24]. We model amorphous Veriflex-E as a viscoelastic solid with two relaxation mechanisms. The model developed is used to solve problems of uniaxial tension, stress relaxation, and loading-relaxation-unloading cycles. The results are then compared against experiment data. The paper is arranged in the following order. Section 2 presents basic continuum theories used to derive our model. Section 3 introduces the finite deformation constitutive model. Section 4 presents the comparisons between the experimental results and the model predictions.

## 2. Preliminaries

## 3. Model Description

#### 3.1. Hyperelsatic Behavior of the Equilibrium Branch

#### 3.2. Viscoelastic Behavior of the Non-Equilibrium Branch

**D**and nothing that only isochoric motions are permissible, it is sufficient to assume the stress has the form:

**.**

#### 3.3. Model Conclusion

## 4. Applications

#### 4.1. Numerical Procedure

#### 4.2. Parameter Identification

#### 4.3. Results and Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Lendlein, A.; Kelch, S. Shape-memory polymers. Angew. Chem. Int. Ed.
**2002**, 41, 2035–2057. [Google Scholar] [CrossRef] - Lendlein, A.; Schmidt, A.M.; Langer, R. AB-polymer networks based on oligo(ε-caprolactone) segments showing shape-menory properties. Proc. Natl. Acad. Sci. USA
**2001**, 98, 842–847. [Google Scholar] [PubMed] - Ge, Q.; Luo, X.; Iversen, C.B.; Mather, P.T.; Dunn, M.L.; Qi, H.J. Mechanisms of triple-shape polymeric composites due to dual thermal transitions. Soft Matter
**2013**, 9, 2212–2223. [Google Scholar] [CrossRef] - Lendlein, A.; Jiang, H.; Jünger, O.; Langer, R. Light-induced shape-memory polymers. Nature
**2005**, 434, 879–882. [Google Scholar] [CrossRef] [PubMed] - Wu, L.; Jin, C.; Sun, X. Synthesis, properties, and light-induced shape memory effect of multiblock polyesterurethanes containing biodegradable segments and pendant cinnamamide groups. Biomacromolecules
**2011**, 12, 235–241. [Google Scholar] [CrossRef] [PubMed] - Lu, H.B.; Huang, W.M.; Yao, Y.T. Review of chemo-responsive shape change/memory polymers. Pigment Resin Technol.
**2013**, 42, 237–246. [Google Scholar] [CrossRef] - Schmidt, A.M. Electromagnetic activation of shape memory polymer networks containing magnetic nanoparticles. Macromol. Rapid Commun.
**2006**, 27, 1168–1172. [Google Scholar] [CrossRef] - Buckley, P.R.; McKinley, G.H.; Wilson, T.S.; Small, W.; Benett, W.J.; Bearinger, J.P.; McElfresh, M.W.; Maitland, D.J. Inductively heated shape memory polymer for the magnetic actuation of medical devices. IEEE Trans. Biomed. Eng.
**2006**, 53, 2075–2083. [Google Scholar] [CrossRef] [PubMed] - Du, H.; Zhang, J. Solvent induced shape recovery of shape memory polymer based on chemically cross-linked poly(vinyl alcohol). Soft Matter
**2010**, 6, 3370–3376. [Google Scholar] [CrossRef] - Liu, C.; Qin, H.; Mather, P.T. Review of progress in shape-memory polymers. J. Mater. Chem.
**2007**, 17, 1543–1558. [Google Scholar] [CrossRef] - Xue, L.; Dai, S.; Li, Z. Biodegradable shape-memory block co-polymers for fast self-expandable stents. Biomaterials
**2010**, 31, 8132–8140. [Google Scholar] [CrossRef] [PubMed] - Wei, Z.G.; Sandström, R.; Miyazaki, S. Shape-memory materials and hybrid composites for smart systems—Part I Shape-memory materials. J. Mater. Sci.
**1998**, 33, 3743–3762. [Google Scholar] [CrossRef] - Yakacki, C.M.; Shandas, R.; Lanning, C.; Rech, B.; Eckstein, A.; Gall, K. Unconstrained recovery characterization of shape-memory polymer networks for cardiovascular applications. Biomaterials
**2007**, 28, 2255–2263. [Google Scholar] [CrossRef] [PubMed] - Eisenhaure, J.D.; Rhee, S.I.; Al-Okaily, A.M.; Carlson, A.; Ferreira, P.M.; Kim, S. The use of shape memory polymers for MEMS assembly. J. Microelectromech. Syst.
**2015**, 25, 69–77. [Google Scholar] [CrossRef] - Gall, K.; Kreiner, P.; Turner, D.; Hulse, M. Shape-memory polymers for microelectromechanical systems. J. Microelectromech. Syst.
**2004**, 13, 472–483. [Google Scholar] [CrossRef] - Ge, Q.; Dunn, C.K.; Qi, H.J.; Dunn, M.L. Active origami by 4D printing. Smart Mater. Struct.
**2014**, 23, 094007. [Google Scholar] [CrossRef] - Yu, K.; Dunn, M.L.; Qi, H.J. Digital manufacture of shape changing components. Extreme Mech. Lett.
**2015**, 4, 9–17. [Google Scholar] [CrossRef] - Liu, Y.; Gall, K.; Dunn, M.L.; Greenberg, A.R.; Diani, J. Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modeling. Int. J. Plast.
**2006**, 22, 279–313. [Google Scholar] [CrossRef] - McClung, A.J.W.; Tandon, G.P.; Baur, J.W. Strain rate- and temperature-dependent tensile properties of an epoxy-based, thermosetting, shape memory polymer (Veriflex-E). Mech. Time Depend. Mater.
**2012**, 16, 205–221. [Google Scholar] [CrossRef] - McClung, A.J.W.; Tandon, G.P.; Baur, J.W. Deformation rate-, hold time-, and cycle-dependent shape-memory performance of Veriflex-E resin. Mech. Time Depend. Mater.
**2013**, 17, 39–52. [Google Scholar] [CrossRef] - Rajagopal, K.R.; Srinivasa, A.R. On the inelastic behavior of solids—Part 1: Twinning. Int. J. Plast.
**1995**, 11, 653–678. [Google Scholar] [CrossRef] - Rajagopal, K.R.; Wineman, A.S. A constitutive equation for nonlinear solids which undergo deformation induced microstructural changes. Int. J. Plast.
**1992**, 8, 385–395. [Google Scholar] [CrossRef] - Rajagopal, K.R.; Srinivasa, A.R. On the thermomechanics of shape memory wires. Z. Angew. Math. Phys.
**1999**, 50, 459–496. [Google Scholar] - Rajagopal, K.R.; Srinivasa, A.R. A thermodynamic frame work for rate type fluid models. J. Non-Newton. Fluid Mech.
**2000**, 88, 207–227. [Google Scholar] [CrossRef] - Karra, S.; Rajagopal, K.R. A thermodynamic framework to develop rate-type models for fluids without instantaneous elasticity. Acta Mech.
**2009**, 205, 105–119. [Google Scholar] [CrossRef] - Rao, I.J.; Rajagopal, K.R. Study of strain-induced crystallization of polymers. Int. J. Solids Struct.
**2001**, 38, 1149–1167. [Google Scholar] [CrossRef] - Rao, I.J.; Rajagopal, K.R. On the modeling of quiescent crystallization of polymer melts. Polym. Eng. Sci.
**2004**, 44, 123–130. [Google Scholar] [CrossRef] - Rao, I.J.; Rajagopal, K.R. A thermodynamic framework for the study of crystallization in polymers. Z. Angew. Math. Phys.
**2002**, 53, 365–406. [Google Scholar] [CrossRef] - Moon, S.; Cui, F.; Rao, I.J. Constitutive modeling of the mechanics associated with triple shape memory polymers. Int. J. Eng. Sci.
**2015**, 96, 86–110. [Google Scholar] [CrossRef] - Barot, G.; Rao, I.J. Constitutive modeling of the mechanics associated with crystallizable shape memory polymers. Z. Angew. Math. Phys.
**2006**, 57, 652–681. [Google Scholar] [CrossRef] - Barot, G.; Rao, I.J.; Rajagopal, K.R. A thermodynamic framework for the modeling of crystallizable shape memory polymers. Int. J. Eng. Sci.
**2008**, 46, 325–351. [Google Scholar] [CrossRef] - Sodhi, J.S.; Rao, I.J. Modeling the mechanics of light activated shape memory polymers. Int. J. Eng. Sci.
**2010**, 48, 1576–1589. [Google Scholar] [CrossRef] - Sodhi, J.S.; Cruz, P.R.; Rao, I.J. Inhomogeneous deformations of light activated shape memory polymers. Int. J. Eng. Sci.
**2015**, 89, 1–17. [Google Scholar] [CrossRef] - Westbrook, K.K.; Kao, P.H.; Castro, F.; Ding, Y.; Jerry, Q.H. A 3D finite deformation constitutive model for amorphous shape memory polymers: A multi-branch modeling approach for nonequilibrium relaxation processes. Mech. Mater.
**2011**, 43, 853–869. [Google Scholar] [CrossRef] - O’Connell, P.A.; McKenna, G.B. Arrhenius-type temperature dependence of the segmental relaxation below Tg. J. Chem. Phys.
**1999**, 110, 11054–11060. [Google Scholar] [CrossRef] - Williams, M.L.; Landel, R.F.; Ferry, J.D. The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. J. Am. Chem. Soc.
**1955**, 77, 3701–3707. [Google Scholar] [CrossRef]

**Figure 4.**Natural configurations associated with the polymer in the amorphous state having two relaxation mechanisms.

**Figure 6.**Plot of true stress versus Hencky strain for uniaxial extension of SMPs at 130 °C with a strain rate of 0.0001/s.

**Figure 7.**Plot of change in engineering stress versus time for stress relaxation after a constant rate extension at 130 °C: (

**a**) relaxation at 40% engineering strain; and (

**b**) stress relaxation at 60% engineering strain. The simulation results are calculated with the equilibrium branch and one non-equilibrium branch.

**Figure 8.**Plot of change in engineering stress versus time for stress relaxation after a constant rate extension at 130 °C: (

**a**) relaxation at 40% engineering strain; and (

**b**) stress relaxation at 60% engineering strain. The simulation results are calculated with the equilibrium branch and two non-equilibrium branches.

**Figure 9.**Plot of true stress versus Hencky strain for uniaxial extension of SMPs at 130 °C with a strain rate of 0.001/s.

**Figure 10.**Plot of true stress versus Hencky strain for uniaxial extension of SMPs at 130 °C with a strain rate of 0.01/s.

**Figure 11.**Plot of engineering stress versus Hencky strain in a loading-relaxation-unloading cycle at 130 °C. The material is stretched to 40% engineering strain with a constant strain rate and held for 60 min. After that the material is unloaded with the same strain rate.

**Figure 12.**Plot of engineering stress versus Hencky strain in a loading-relaxation-unloading cycle at 130 °C. The material is stretched to 60% engineering strain with a constant strain rate and held for 60 min. After that the material is unloaded with the same strain rate.

Description | Parameters | Values |
---|---|---|

Equilibrium Branch Parameter | ||

Shear Modulus (MPa) | μ | 0.43 × 10^{6} |

Non-equilibrium Branch Parameters | ||

Shear Modulus of Branch One (MPa) | μ_{1} | 0.117 × 10^{6} |

Viscosity of Branch One (MPa·s) | η_{1} | 350 × 10^{6} |

Shear Modulus of Branch Two (MPa) | μ_{2} | 0.43 × 10^{6} |

Viscosity of Branch Two (MPa·s) | η_{2} | 4.3 × 10^{6} |

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**MDPI and ACS Style**

Cui, F.; Moon, S.; Rao, I.J.
Modeling the Viscoelastic Behavior of Amorphous Shape Memory Polymers at an Elevated Temperature. *Fluids* **2016**, *1*, 15.
https://doi.org/10.3390/fluids1020015

**AMA Style**

Cui F, Moon S, Rao IJ.
Modeling the Viscoelastic Behavior of Amorphous Shape Memory Polymers at an Elevated Temperature. *Fluids*. 2016; 1(2):15.
https://doi.org/10.3390/fluids1020015

**Chicago/Turabian Style**

Cui, Fangda, Swapnil Moon, and I. Joga Rao.
2016. "Modeling the Viscoelastic Behavior of Amorphous Shape Memory Polymers at an Elevated Temperature" *Fluids* 1, no. 2: 15.
https://doi.org/10.3390/fluids1020015