Constitutive model for shape memory polymer and its thermodynamic responses in finite element analysis

Abstract

BACKGROUND:

As a new intelligent polymer material, shape memory polymer (SMP) was a potential orthodontic appliance material.

OBJECTIVE:

This study aimed to investigate the thermodynamic responses of SMP under different loads via finite element analysis (FEA).

METHODS:

FEA specimens with a specification of 0.1 $\times$ 0.1 $\times$ 1 mm were designed. One end of the specimen was fixed, and the other was subjected to displacement load. Different loading, cooling, and heating rates were separately exerted on the specimen in its shape recovery process and used to observe the responses of the SMP constitutive model. Furthermore, specimens with various tensile elongation and sectional areas were simulated and used to elucidate their effect on shape recovering force.

RESULTS:

The specimens obtained a similar stress of 0.5, 0.44, and 1.07 Mpa for different loading, cooling, and heating rates after a long time. The shape recovering force of specimen increased from 0.0102 to 0.0315 N when the elongation improved from 10% to 40% and to 0.0408 N when the sectional areas were expanded to 0.2 $\times$ 0.2 mm.

CONCLUSION:

The stiffness of SMP was small at a high temperature but large at a low temperature. The effects of the loading, cooling, and heating rates on SMP can be eliminated after a long time. Furthermore, it was possible to increase the recovering force by increasing the elongation or expanding the sectional area of the specimen. The force was quadratically dependent on the elongation ratio.

Keywords

Shape memory polymer constitutive model thermodynamic response finite element analysis

1. Introduction

Oral orthodontics is a specialty focused on adjusting the balance and coordination of facial bones, teeth, and maxillofacial muscles by applying orthodontic force to the misaligned teeth so as to improve facial shape [1, 2]. At present, orthodontic appliances mainly include fixed and invisible appliances. Fixed appliance highly depends on the clinical experience of orthodontist [3, 4]. Archwire selection and bending, as well bracket pasting, require orthodontist’s subjective judgment. On the other hand, invisible appliance is mainly designed for the patient’s teeth crown; however, the accuracy of invisible appliance cannot meet the needs of complex cases [5], and therapeutic schedule frequently changes during the treatment course [6]. In view of the shortcomings of traditional orthodontic appliances, some scholars proposed the application of shape memory polymer (SMP) in orthodontics [7, 8]. SMP is a new intelligent polymer material, it has good mechanical properties and biocompatibility [9, 10, 11] and makes it easy to obtain special shapes via injection molding or extrusion [12]. Most of the SMP products are transparent, making them a potential orthodontic appliance material.

Figure 1 presents the explanation of the shape recovery principle of SMP. In the figure, the SMP product is loaded with external force to change its initial form when its temperature is $T_{h}\geqslant T_{g}$ (glass-transition temperature of SMP). When the temperature is reduced to $T_{l}<T_{g}$ while maintaining the external force, the SMP product will not recover its initial form when the external force is removed; only when the temperature is raised to $T_{g}$ does the SMP product gradually return to its initial form. This is because SMP consists of a fixed hard phase and a reversible soft phase. When the temperature reaches the reversible temperature of the soft phase, the SMP molecules rearrange to their original positions, prompting the initial form recovered.

Before the appliance design and simulation, the constitutive model of SMP material need to be designed. In the early years, Tobushi created a thermodynamic constitutive model of SMP by combining the spring, dashpot, and slip elements together based on the classical viscoelastic theory while considering thermal expansion [13, 14]. The slip unit was used to control energy storage and release during SMP deformation, which was the key component of the model. However, this model was created based on uniaxial tensile experiment, which was only limited to expressing the mechanical behavior of SMP under simple stress but cannot be used to describe its behavior under complex stress. Therefore, our team studied a three-dimensional thermodynamic constitutive SMP model based on Tobushi’s simple model, which can well express the thermodynamic behavior of SMP under complex stress [15, 16]. The SMP responses were affected not only by load but also by environmental temperature and time. To explore the responses of the constructed model under different loads, it was necessary to perform detailed FEA simulations of the shape recovery process of SMP to provide theoretical and technical supports for the subsequent application of SMP in orthodontics.

2. Materials and methods

In this study, the research was mainly based on the constitutive model of SMP, and the mathematical formulas of the constitutive model were employed from our previous studies, as shown in Eqs (2)–(3) [17], where Eq. (2) shows the constitutive relationship of normal stress and Eq. (2) the constitutive relationship of shear stress. The dot on top of the parameters represents their derivative of time in the equations. The material parameters $E$ , $\mu$ , $\lambda$ , and $\alpha$ denote the elastic modulus, viscosity coefficient, relaxation time, and thermal expansion coefficient of the material, respectively; $\sigma$ and $\varepsilon$ , total stress and strain, respectively; $T$ , temperature; $\varepsilon_{v}$ , volume strain; $\varepsilon_{l}$ , critical strain; $C$ , material coefficient related to temperature; $K$ , volume modulus; and $\nu$ , Poisson’s ratio ( $i,j=1,2,3)$ . The use of this constitutive model was based on the interface of user-defined material mechanical behavior (UMAT) subroutine in the FEA software Abaqus 6.13 (Dassault Simulia, Boston, USA). The material parameters used in this constitutive model are listed in Table 1 [17], where $X_{g}$ denotes the value of parameter $X$ at the glass-transition temperature, $T_{g}$ .

$\displaystyle\frac{\lambda}{E}\dot{\sigma}_{ii}+\left({\frac{\lambda}{\mu}-% \frac{C}{E}}\right)\sigma_{ii}=\left({1-C}\right)\varepsilon_{ii}+\lambda\dot{% \varepsilon}_{ii}+\left[{\left({\frac{\lambda}{\mu}-\frac{C}{E}}\right)K+\frac% {1}{3}\left({1-C}\right)}\right]\varepsilon_{v}$

(1) $\displaystyle\qquad\qquad\qquad\qquad\quad\quad\,\,{}+\left({\frac{\lambda}{E}% K-\frac{1}{3}\lambda}\right)\dot{\varepsilon}_{v}+C\varepsilon_{\textit{lii}}-% \lambda\alpha\dot{T}(i=j)$ $\displaystyle\frac{\lambda}{E}\dot{\sigma}_{ij}+\left({\frac{\lambda}{\mu}-% \frac{C}{E}}\right)\sigma_{ij}=({1-C})\varepsilon_{ij}+\lambda\dot{\varepsilon% }_{ij}+C\varepsilon_{\textit{lij}}(i\neq j)$ (2) $\displaystyle K=\frac{E}{3({1-2\upsilon})}$ (3)

Table 1
Material parameters used in the SMP constitutive model

$T_{g}$ /^∘F $E_{g}$ /MPa $\mu_{g}$ /GPa $\cdot$ s $\lambda_{g}$ /s $C_{g}$ $\varepsilon_{lg}$ /% $\alpha$ /K^{- 1} $\nu$

98.24 15.47 2341.7 521 0.112 0.3 0.000116 0.2

Figure 1.
Shape recovery principle of SMP.

The complete shape recovery process of SMP involves four phases: (1) loading (loading at a high temperature), (2) cooling (cooling while maintaining force), (3) unloading (unloading at a low temperature), and (4) recovering (recovering at a high temperature). Hence, the FEA simulations of the recovery process included four main analysis steps and four auxiliary analysis steps after each main step, as presented in Fig. 2. The load and temperature in each main step were diverse, whereas those in the auxiliary steps were inherited from the preceding main steps; the auxiliary steps were used to observe the stress variations in SMP as time went on. The $T_{g}$ of SMP was 36.8^∘C, and the high temperature $T_{h}$ and low temperature $T_{l}$ were set as 56.8^∘C ( $T_{g}+$ 20^∘C) and 16.8^∘C ( $T_{g}-$ 20^∘C), respectively. To simplify the FEA calculation, the SMP specimen was designed as a long strip with a specification of 0.1 $\times$ 0.1 $\times$ 1 mm. One end of the specimen was fixed, and the other was subjected to displacement load. In view of the shape recovery process, the factors that may influence SMP responses included (1) loading rate, (2) temperature in the loading phase, (3) cooling rate, (4) heating rate in the recovering phase, (5) elongation, and (6) sectional area of the specimens. Specific simulations on the aforementioned aspects are necessary to study their influences on SMP responses.

Figure 2.
Displacement and temperature in various analysis steps during the FEA simulation.

In studying the influence of loading rate on SMP response, the temperature of the specimen was set as 36.8^∘C ( $T_{g}$ ), and the loading rates were set as 0.1, 0.01, and 0.001 mm/s, respectively, with the maximum displacement of 0.1 mm. The main analysis step of loading was followed by an auxiliary step with a duration of 300 s. In investigating the effect of temperature on SMP response, the specimen temperatures were set as 16.8^∘C ( $T_{l}$ ), 36.8^∘C ( $T_{g}$ ), and 56.8^∘C ( $T_{h}$ ). The specimen elongation was 0.1 mm, and the loading rate was 0.01 mm/s. The auxiliary analysis step after the loading phase had a duration of 300 s.

In the cooling phase, the time the SMP specimen took to cool from 56.8^∘C ( $T_{h}$ ) to 16.8^∘C ( $T_{l}$ ) was set as 100, 500, and 1000 s, respectively; an auxiliary step with a duration of 3000 s followed after the main step of cooling for observing the effect of cooling rate on SMP specimen. For the loading phase before the cooling, the specimen temperature was 56.8^∘C; loading rate, 0.01 mm/s; and elongation, 0.1 mm; the duration of the auxiliary step after the loading phase was 300 s.

After removing the external load from the SMP specimen, the time the specimen took to heat from 16.8^∘C ( $T_{l}$ ) to 56.8^∘C ( $T_{h}$ ) was set as 10, 100, and 500 s in the recovering phase; an auxiliary step with a duration of 3000 s followed after the main step of recovering for observing the effect of heating rate on SMP specimen. In the loading phase, the specimen temperature was 56.8^∘C; elongation, 0.1 mm; and loading rate, 0.01 mm/s; the duration of the auxiliary step after loading was 300 s. Contrarily, in the cooling phase, the cooling time of specimen was 100 s, and the duration of the auxiliary step after cooling was 300 s.

In the study of the effect of elongation on the specimen’s recovering force, the shape recovery processes of SMP were simulated by setting the elongation as 10%, 20%, 30%, and 40% for the specimens, respectively. The loading rate of the specimen was 0.01 mm/s in the loading phase; cooling time, 100 s in the cooling phase; and heating time, 100 s in the recovering phase. The duration of the auxiliary step after the loading and cooling phases was 300 s, similar to that in the auxiliary step after the recovering phase. The shape recovering force of the specimen was obtained by extracting the reaction force on its fixed end. Furthermore, in the study of the effect of sectional area on the SMP recovering force, another SMP specimen was designed with a sectional area of 0.2 $\times$ 0.2 mm and a length of 1 mm; the elongation of both specimens was 0.1 mm, and the load and time were the same as those in the study of elongation.
3. Results

$T_{g}$ /^∘F	$E_{g}$ /MPa	$\mu_{g}$ /GPa $\cdot$ s	$\lambda_{g}$ /s	$C_{g}$	$\varepsilon_{lg}$ /%	$\alpha$ /K^{- 1}	$\nu$
98.24	15.47	2341.7	521	0.112	0.3	0.000116	0.2

Under the action of load 0.1 mm, the displacement nephogram of the specimen is presented in Fig. 3. Each mesh element in the specimen was uniformly deformed, and the distribution of their stress-strain states was the same. Hence, the thermodynamic responses of SMP can be explained by the states of the nodes in the middle of the specimen.

Figure 3.

Displacement nephogram of SMP specimen under the load of 0.1 mm.

Loading with different loading rates and the variations of von Mises stress in SMP specimen with time are presented in Fig. 4a. The stress gradually increased and tended to level off at the same value of 0.5 Mpa for different loading rates as time extended. The thermodynamic responses of SMP specimen at different temperatures are also presented in Fig. 4b. The specimen tended to obtain greater stress at a low temperature in the loading phase.

The effects of cooling rate on SMP in the cooling phase are presented in Fig. 5a. The von Mises stresses of the specimens tended to a similar value of 0.44 Mpa for different cooling rates after 3000 s, although they may be different at the beginning. The thermodynamic responses of SMP to different heating rates in the recovering phase are presented in Fig. 5b. The relation curves similarly tended to the same stable value of 1.07 Mpa after 3000 s for different heating rates.

The reaction forces at the fixed end of the specimens were recorded during its recovery process, as presented in Fig. 6a. The reaction forces exhibited obvious differences in different elongations, although the tendencies of the relation curves were similar. After 3000 s, the reaction force reached a stable value, which was the shape recovering force of SMP. The values of the shape recovering force were 0.0102, 0.0186, 0.0256, and 0.0315 N for the elongation of 10%, 20%, 30%, and 40%, respectively. Furthermore, the variations of the reaction forces on specimens with different sectional areas were recorded, as presented in Fig. 6b. The recovering forces were 0.0102 and 0.0408 N for the specimens with sectional areas of 0.1 $\times$ 0.1 mm and 0.2 $\times$ 0.2 mm, respectively.

4. Discussion

As a viscoelastic polymer material, the thermodynamic responses of SMP possess both elastic and viscous characteristics under the action of external load [18, 19]. Hence, the spring, dashpot, and slip elements were all considered when the SMP constitutive model was created. In general, the stress increases with the improvement of strain dominated by the spring element but is not necessarily constant when the strain is fixed. On the one hand, the stress may continue to increase due to the slip element that causes slower response of the stress than the strain [20, 21]. On the other hand, it is possible that the stress gradually decreases due to the relaxation effect of the dashpot in the model [22, 23].

Figure 4.

(a) Thermodynamic responses of SMP under different loading rates, (b) thermodynamic responses of SMP at different temperatures.

Figure 5.

(a) Thermodynamic responses of SMP under different cooling rates, (b) thermodynamic responses of SMP under different heating rates.

Figure 6.

(a) Reaction forces of specimens with different elongations, (b) reaction forces of specimens with different sectional areas.

As presented in Fig. 4a, the von Mises stresses of specimens still increased after the load completion, which was mainly due to the hysteresis of stress, the stress had no time to fully responded at the moment of loading, and it was slowly exhibited after the end of the load. The faster the loading rate, the shorter the stress response time, and the lower the stress value obtained in the loading. In the maintaining stage after the loading, the faster the loading rate, the greater the increase in stress, the stresses of specimens gradually tend to the same stable value of 0.5 MPa for different loading rates. It can be concluded that the loading rate only affects the SMP responses in the loading phase, and the same stress value is obtained after a long time. Therefore, the time spent on the deformation stage need not be excessively considered in the SMP application.

The thermodynamic responses of SMP with different loading temperatures are presented in Fig. 4b. SMP tended to generate greater stress under a low loading temperature. This is because the molecular activity of SMP was significantly limited by low temperature and increased its stiffness [24, 25]; thus, the stress was greater than that at a high temperature. Furthermore, the stress attenuation is smaller at a high than at a low temperature, indicating that the hysteresis effect of stress played an important role at a high temperature. In practice, stress hysteresis and stress relaxation both occurred in the deformation process and the variation of SMP stress mainly depended on which one played the decisive role.

In the cooling phase, the specimens achieved the same value under different cooling rates after about 3000 s, as presented in Fig. 5a, implying that the cooling rate had a negligible impact on the SMP responses in the long run. However, the thermodynamic responses for different specimens were discrepant in cooling; the molecular chains of SMP were frozen influenced by temperature reduction, which resulted in the elasticity increased. It showed that the specimen stresses quickly increased when the temperature dropped, although the strain had not changed, and the faster the cooling rate, the quicker the stress increase. When the cooling time was 100 s, the stress of the specimen initially increased and then decreased after the temperature dropped to 16.8^∘C; the initial increase was mainly due to the hysteresis effect of SMP [20, 21]. the stress, which had not been responded in the shot cooling time, showed itself in the subsequent stage. However, this event did not occur in others due to the longer response time provided to the specimens.

The loading and cooling phases were followed by the recovering phase. The thermodynamic responses of SMP to different heating rates are presented in Fig. 5b. The stresses decreased with the improvement of temperature, mainly because the molecular chain activities were reactivated when the temperature of SMP specimens improved to $T_{g}$ , and the elasticity of SMP decreased, resulting in a lower stress under the same strain. The specimen temperature was 36.8^∘C after the recovering phase, and the stresses continually decreased due to the relaxation effect and stabilized at a similar value after a long time for different specimens [22, 23], implying that the heating rate does not need to be significantly considered in the recovering phase.

As presented in Fig. 6a, the greater the specimen elongation, the larger the shape recovering forces. However, the recovering force was not linearly dependent on the elongation; the forces increased by 82.35%, 150.98%, and 208.82% when the elongation improved from 10% to 20%, 30%, and 40%, respectively. The recovering force only increased about two times, whereas the elongation improved four times from 10% to 40%. It showed that increasing the force was easier by improving the elongation when the latter was relatively small; however, this method was not significant when the elongation was already large. The relationship between the elongation ratio and the recovering force can be approximately described by a quadratic polynomial through data fitting, as presented in Eq. (4), where $F_{R}$ denotes the shape recovering force and $x$ the elongation ratio of specimen. Compared with the method of increasing the elongation to obtain larger force, expanding the sectional area of specimen was more direct and significant, as presented in Fig. 6b. The recovering forces increased four times when the sectional areas improved four times, consistent with our understanding.

$\displaystyle F_{R}=-0.0625x^{2}+0.1022x+0.0006$ (4)

The study results have universal applicability, and they can be used as references for similar SMP materials. But, there were also some limitations in this study, one limitation was that only tension was simulated in the investigation of SMP thermodynamic responses. More complex deformations, such as bend, shear, and torsion, were not considered due to the limited article length. Meanwhile, SMP was perceived as an isotropic and homogeneous material in FEA, but its property may be different in various directions influenced by the arrangement of molecular chain. In addition, specific experiments were not conducted on physical specimens limited by experiment condition. Thus, more detailed studies are warranted in the future.

5. Conclusion

To describe the thermodynamic behavior of SMP under different loads, FEA was conducted to simulate the complete shape recovery process. The results indicated that (1) SMP stiffness was small at a high temperature but large at a low temperature; (2) the effects of the loading, cooling, and heating rates on the thermodynamic responses of SMP can be eliminated after a period of stress relaxation; (3) the shape recovering force of SMP does not increase linearly with the improvement of the elongation, and it was easier to increase force by increasing elongation when the latter was relatively small; and (4) the recovering force of SMP was proportional to the sectional area of the specimen, and it can be increased by expanding the sectional area.

Footnotes

Acknowledgments

This work was supported by the Ningbo Municipal Natural Science Foundation of China (Grant No. 2022J181), the Cultivation Project of National Scientific Research Projects of Ningbo Polytechnic in 2022 (Grant No. NZ22GJ001), the National Natural Science Foundation of China (Grant No. 52175280), and the Zhejiang Provincial Natural Science Foundation of China (Grant No. LD22E050013).

Conflict of interest

None declared.

Ethical approval

Not applicable.

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