A Mechanical Model for Stress Relaxation of Polylactic Acid/Thermoplastic Polyurethane Blends

Abstract

Polylactic acid (PLA) is considered a promising biodegradable polymer alternative. Due to its high brittleness, composite materials made by melt blending thermoplastic polyurethane (TPU) with PLA can enhance the toughness of PLA. To understand the forced aging caused by stress relaxation in polymer materials, this study explains the stress relaxation experiments of PLA/TPU blends with different mass ratios under applied strain through mechanical model simulations. The Kelvin representation of the standard linear solid model (SLSM) is used to analyze the stress relaxation data of TPU/PLA blends, successfully explaining that the Young’s moduli (E₁ and E₂) of springs decrease with increasing temperature and TPU content. The viscosity coefficient of the PLA/TPU blends decreases with increasing temperature, and its reciprocal follows the Arrhenius law. For TPU/PLA blends with increased concentration of TPU, the activation energy for stress relaxation shows a linear decrease, confirmed by the glass transition point measured by DMA, indicating that it does not involve chemical reactions.

1. Introduction

Aging is an undesirable process for polymeric materials, but it is inevitable. Stress relaxation of polymeric material is a kind of forced aging, which is often simulated by the mechanical model. The mechanical model consisting of springs and dashpots is often used to understand the deformation mechanism of the long-term forced aging problem of polymeric materials. The simplest mechanical models are the Maxwell and Voigt models, where the former and the latter are spring and dashpot connected in series and parallel, respectively. More complex models are the standard linear solid model (SLSM) with three elements [1] and the Burgers model with four elements [2]. Burgers and SLSM models described the creep recovery, and stress relaxation behaviors, respectively. Furthermore, the real polymeric materials cannot be described by the model with a few dashpots and springs [3]. The Maxwell–Weichert model with Maxwell elements connected in parallel was proposed to explain the stress relaxation experiment [4,5]. In addition to the mechanical model, an empirical equation was also applied to analyze the stress relaxation phenomenon of polymeric materials. Fancey [6] utilized the Kohlrausch–Williams–Watts (KWW) function based on the Weibull distribution function to represent the viscoelastic change of the stress relaxation experiment. His idea arises from the Weibull function, often used to explain mechanical failure [7].

Polylactic acid (PLA) has been considered a promisingly biodegradable alternative to traditional petro-polymers due to its attractive mechanical properties, high renewability, good biodegradability, and relatively low cost [8]. Pelouze [9] was the first to prepare low molecular weight PLA using the condensing L-lactic acid and removing water constantly. Sosnowski et al. [10] obtained high molecular weight PLA via ring-opening polymerization (ROP) of lactide with protic compounds as initiators and tin (II) octoate (Sn(Oct)₂) as a catalyst. The properties of PLA are affected more by the chiral carbon in lactic acid than by the molecular weight of PLA. PLA with different concentrations of L-Isomer and D-Isomer can be semi-crystalline or amorphous, which has a melting point ca. 175 °C and a glass transition point ca. 55–60 °C [11].

Although PLA has better mechanical properties than commodity polymers such as polystyrene, polyethylene, and polyethylene terephthalate [12], it is brittle, has low impact strength, and has a low crystallization degree. PLA-based composites and PLA-based polymer blends are the solutions to enhance the mechanical properties. Raquez et al. [13] classified nanocomposites as plate-like nanofillers (1D), nanofibers (2D), and nanoparticles (3D). Tokoro et al. [14] mixed the PLA with bamboo fibers to form a PLA/bamboo composite, which has greater bending strength and impact strength than pristine PLA at 25 °C. Metta et al. [15] used Hakee Rheomix to prepare a PLA and polycaprolactone (PCL) blend. They found that the PLA/PCL blend increases the elongation, impact strength, and loss factor, and decreases the modulus and strength of the PLA/PCL blend by increasing the concentration of PCL up to 20% PCL. Furthermore, Takayama and Todo [16] used lysine tri-isocyanate (LTI) as an additive to PLA/PCL blends to enhance the miscibility of PLA/PCL so that PLA/PCL has better tensile strength and impact strength. Ho et al. [17] prepared the thermoplastic polyolefin elastomer-graft-polylactide by reacting polylactide with maleic anhydride-functionalized TPD (TPD-MAH) in the presence of 4-dimethylaminopyridine (DMP). The formed PLA/TPO blend has a larger elongation at break, tensile toughness, and Izod impact strength than pristine PLA, but lower tensile strength and tensile modulus than the latter. Tsai et al. [18] prepared the PLA/PMMA (polymethyl methacrylate) blend by hot-press method at 190 °C under compression stress of 1.96 MPa. They studied the effect of UV irradiation on physical aging via DSC.

Thermoplastic polyurethane (TPU), owning excellent elastic properties, high transparency, great toughness, and biocompatibility is a potential candidate for implanted devices [19]. Li and Shimizu [20] found that melt blending of TPU with PLA can enhance the toughness of PLA. Jhao et al. [2] reported that when temperature is high, the activation energies for the steady-state creep and transient creep of PLA/TPU blends decrease linearly with the increase in the concentration of TPU. In addition to the creep, when the temperature is high enough, the stress relaxation behavior also occurs frequently in the polymeric materials. This prompted us to investigate the effect of TPU concentration on the stress relaxation behavior of PLA/TPU blends. We proposed the Kelvin representation of the standard linear solid model to describe the stress relaxation of PLA/TPU blends. The existing models are also discussed.

2. Experimental

The PLA 4032D with Tg ca. 55–60 °C and TPU 8785A with 300-grade were purchased from Nature Works LLC (Minnetonka, MN, USA) and Bayer Co, Ltd. (Leverkusen, Germany), respectively. The former has a melting point of 155–170 °C, and the latter melting point is 160–180 °C. The PLA was a semi-crystalline polymer with 98% L-isomer and 2% D-isomer. The chemical structures of polylactide (PLA) and thermoplastic polyurethane (TPU) are shown in Figure 1a,b, respectively.

Pristine PLA and TPU, and PLA/TPU blends with 30/70, 50/50, and 70/30 mass ratios were fabricated via a melting blend method at 200 °C [2]. Briefly, the PLA and TPU mixture was dried in an oven at 80 °C for 4 h and moved to an injection molding machine at the injection molding point of 200 °C. The PLA/TPU blend was injected into a mold to form a thin plate where the molding temperature and pressure were 30 °C and 2.94 MPa, respectively. The dumb-bell-shape specimens with a gauge length of 9.7 mm, width of 1.5 mm, and thickness of 0.7 mm were laser-cut from the thin plate. After grinding on 400, 800, 1200, and 2500 grit CarbiMet papers and polishing with 1 µm alumina slurries, the specimens were annealed at 50 °C for one day. Then, the furnace was cooled down to 25 °C to relax the residual stress arising from the sample preparation.

The stress relaxation was performed with a TA 800 dynamic mechanical analyzer (DMA) (TA Instrument Co., Ltd., New Castle, DE, USA). Because of instrument limitations, the stress relaxations of PLA/TPU blends of different ratios of PLA to TPU were operated under different applied strains. The specimens were maintained at a preset temperature for at least 5 min to reach thermal equilibrium, stretched to a specific elongation, and kept the same strain for 60 min. The stress as a function of time was recorded. The glass transition temperature was performed at a temperature range from −80 °C to 110 °C at a heating rate of 5 °C/min with a frequency of 1 Hz using a TA 800 dynamic mechanical analyzer (DMA) (TA Instrument Co., New Castle, DE, USA).

3. Results

Using DMA, we measure the glass transition temperatures of the prepared PLA/TPU blends. Figure 2 shows the curves of tan(δ) versus temperature for PLA/TPU blends with different ratios of PLA to TPU. The temperatures of the right and left peaks in the figure correspond to the glass transition temperatures of PLA and TPU, respectively. From Figure 2, we obtained the glass transition temperatures of the PLA/TPU blends listed in Table S1. The glass transition temperature of PLA decreases slightly with increasing the TPU concentration of the PLA/TPU blend, but the trend of the glass transition temperature of TPU versus temperature is the opposite.

Figure 3 shows the stress relaxation of PLA/TPU blends with 70/30 for the mass ratio of PLA to TPU under different strains at temperatures of 10, 20, 30, and 40 °C. For a given strain and temperature, the stress decreases rapidly with time, and its slope changes slowly until it reaches a steady state. For a given time and strain, the stress increases with increasing temperature. Figure S1–S4 in Supplementary Information illustrate the stress relaxation of PLA/TPU blends with 100/0, 50/50, 30/70, and 0/100 for the mass ratio of PLA to TPU under different applied strains at 10–40 °C, respectively. They have a similar trend to the PLA/TPU blend with 70/30 for the mass ratio of PLA to TPU. For the same applied strain and temperature, the stress at a given time is larger for the specimen with the smaller mass ratio of PLA to TPU.

Here, R² is a confidential interval composed of an upper bound and a lower bound denoting the range within which the estimate would be expected to fall if resampled.

4. Discussion

The Kelvin representation of the standard linear solid model (SLSM) is used to analyze the stress relaxation data of TPU/PLA blends. The Kelvin representation of SLSM consists of spring 1 with Young’s modulus E₁ connected with the Kelvin–Voigt element of Young’s modulus E₂ and viscosity coefficient ${\eta }_2$ in series, as shown in Figure 4. The stress σ(t) of Kelvin representation of SLSM for a given strain ε is written as

$$\sigma \left(t\right)=\frac{E_1{E}_2\epsilon }{E_1+E_2}+\frac{E_1^2\epsilon }{E_1+E_2}\exp (-\frac{(E_1+E_2)t}{{\eta }_2})$$

(1)

The solid lines in Figure 3 and Figures S1–S4 in Supplementary Information are obtained using Equation (1) with the mechanical constants E₁, E_2, and ${\eta }_2$ listed in

Table 1. SLSM mechanical constants E₁, E₂, and η₂ of PLA/TPU blend with 70/30 for the mass ratio of PLA to TPU at different temperatures.

Temperature	$10 \mathbf{°C}$	$20 \mathbf{°C}$	$30 \mathbf{°C}$	$40 \mathbf{°C}$
E1 (GPa)	1.53 $\pm$ 0.07	1.37 $\pm$ 0.07	1.22 $\pm$ 0.08	1.12 $\pm$ 0.04
E2 (GPa)	10.2 $\pm$ 0.4	4.53 $\pm$ 0.24	2.08 $\pm$ 0.09	0.722 $\pm$ 0.31
$\mathsf{\eta }$2 (GPa·min)	160 $\pm$ 10	71.6 $\pm$ 2.6	33.3 $\pm$ 1.2	13.9 $\pm$ 0.7
R2	0.974	0.982	0.991	0.980

and Tables S2–S5 in Supplementary Information. E₁ and E₂ decrease with increasing temperature for all PLA/TPU blends. E₁ and E₂ decrease with increasing concentration of TPU in PLA/TPU blend. The viscosity coefficient decreases with increasing temperature, and its reciprocal follows the Arrhenius plot, as shown in Figure 5. The viscosity coefficient increases with the TPU content of the PLA/TPU blend. From the slope of Figure 5, we obtain the activation energy of stress relaxation of the PLA/TPU blend, as shown in Figure 6. According to Figure 6, the activation energy of stress relaxation decreases linearly with the increasing TPU content of the TPU/PLA blend, which is not involved in the chemical reaction, indicating that the two components of the system are immiscible and consistent with the reports in the literature [21,22,23]. This phenomenon was supported by the glass transition point measured by DMA. Table S1 in Supplementary Information shows the glass transition points Tg of the PLA and TPU are 68.3 and –23.2 °C, respectively. The two Tg of PLA/TPU polymer blend are shown in Table S1 in Supplementary Information. One is near 68.3 °C, and the other –23.2°C. That confirms the immiscibility between PLA and TPU.

According to probability and statistics, the Weibull distribution function is a continuous probability distribution commonly applied to material reliability and failure analysis. Fancey [24] used the Kohlrausch–Williams–Watts (KWW) function based on the Weibull distribution to analyze the stress relaxation data of nylon 6, 6 fiber. The KWW function is empirical, which is expressed as

$$\sigma \left(t\right)=E_{KWW1}\epsilon +E_{KWW2}\epsilon \exp [-{(t/{\tau }_{KWW })}^{{\beta }_{KWW}}]$$

(2)

where $E_{KWW1}$ and $E_{KWW2}$ are the Young’s moduli, and $\epsilon$ is the applied strain. $\tau$ and $\beta$ are relaxation time and shape parameter, respectively.

Equation (2) equals Equation (1) if the following equations are satisfied.

$$E_{KWW1}=\frac{E_1{E}_2}{E_1+E_2}$$

(3)

$$E_{KWW2}=\frac{{E_1}^2}{E_1+E_2}$$

(4)

$$\frac{1}{{\tau }_{KWW}}=\frac{E_{KWW2}}{{\eta }_{KWW2}}=\frac{E_1+E_2}{{\eta }_2} \mathrm{if} {\beta }_{KWW}=1$$

(5)

Equations (3) and (4) are obtained from the equality of Equations (1) and (2) at time infinity and the initial time, respectively, and Equation (5) is from that of the exponential terms in Equations (1) and (2).

The solid lines in Figure 7 were obtained using Equation (2) with the mechanical parameters listed in

Table 2. The mechanical parameters of the KWW model to fit the stress relaxation of PLA/TPU blend with 50/50 for the mass ratio of PLA to TPU under the applied strain of 1.2%.

Temperature	10 °C	20 °C	30 °C	40 °C
EKWW1 (MPa)	836	644	433	268
EKWW2 (MPa)	176	257	305	397
ηKWW2 (MPa·min)	533	1311	1586	1826
τKWW (min)	3.03	5.1	5.20	4.6
βKWW	0.91	0.76	0.80	0.79
R2	0.943	0.969	0.980	0.968

, where the symbols represent the stress relaxation data of the PLA/TPU blend with 50/50 for the mass ratio of PLA to TPU under an applied strain of 1.2% at different temperatures. The solid lines in Figure S2c in Supplementary Information were obtained using Equation (1) with the mechanical parameters listed in Table S3 in Supplementary Information, where the experimental data are the same as those in Figure 7. According to Table 2, the confidential interval R² of the KWW function is comparable to that of SLSM. However, the former uses four mechanical parameters to curve fit, and the latter uses three. The difference in the confidential intervals using the SLSM and KWW models is due to β_KWW being 0.91~0.79, not 1. Furthermore, the parameter β_KWW in Equation (2) has no physical meaning. The value of $E_{KWW1}$ decreases with increasing temperature, but that of $E_{KWW2}$ has the opposite trend of decreasing temperature. Young’s modulus with an increasing temperature function violates the concept of polymeric materials [6,25]. Therefore, we do not recommend the KWW model to simulate the polymer’s stress relaxation behavior.

Heuchel et al. [4] used the modified Maxwell–Weichert model to describe the stress relaxation phenomenon of amorphous shape-memory polymers. The modified Maxwell–Weichert model consists of spring 1 and Maxwell elements 2 and 3 connected in parallel and can be expressed as

$$\sigma \left(t\right)=E_{MW1}\epsilon +E_{MW2}\epsilon \exp \left(-t{/\tau }_{MW2}\right)+E_{MW3}exp\left(-t{/\tau }_{MW3}\right)$$

(6)

where $E_{MW1}$, $E_{MW2}$, and $E_{MW3}$ are the Young’s moduli of springs 1, 2, and 3, and $\epsilon$ is the applied strain. ${\tau }_{MW2}$ and ${\tau }_{MW3}$ are the relaxation times of Maxwell elements 2 and 3. Note ${\tau }_{MW2}$ and ${\tau }_{MW3}$ are related to viscosity coefficients as

$${\tau }_{MWi}={\eta }_{MWi}/E_{MWi}$$

(7)

where $i$ can be 2 and 3 to represent dashpots of Maxwell elements 2 and 3, respectively. Comparing Equation (1) with Equation (6), we find that

$$\frac{E_1{E}_2}{E_1+E_2}=E_{MW1}$$

(8)

$$E_1=E_{MW1}+E_{MW2}+E_{MW3}$$

(9)

Note that Equations (8) and (9) are obtained from the equality of Equations (1) and (6) at time infinity and the initial time, respectively.

The solid curves in Figure 8 were obtained using Equation (6) with the mechanical parameters listed in

Table 3. The mechanical parameters of the Maxwell–Weichert model to fit the stress relaxation of PLA/TPU blend with 50/50 for the mass ratio of PLA to TPU under the applied strain of 1.2%.

Temperature	10 °C	20 °C	30 °C	40 °C
EMW1 (MPa)	847	642	442	267
EMW2 (MPa)	149	252	222	342
ηMW2 (MPa·min)	978	1189	1460	1594
EMW3 (MPa)	102	79	102	137
ηMW3 (MPa·min)	33.7	15.0	47.4	37.0
τMW2 (min)	6.56	4.72	6.58	4.66
τMW3 (min)	0.33	0.19	0.46	0.27
R2	0.997	0.992	0.998	0.997

, where the symbols represent the stress relaxation data of the PLA/TPU blend with 50/50 for the mass ratio of PLA to TPU under the applied strain of 1.2% at different temperatures. It can be seen from Table 2 and Table 3 that $E_1$, $E_2$, $E_{KWW1}$, $E_{KWW2}$, $E_{MW1}$, $E_{MW2}$, and $E_{MW3}$ follow Equations (3), (4), (8), and (9). The Maxwell–Weichert model has the largest confidential interval R² among the above three models because it has five mechanical parameters to curve-fit the experimental data. According to Table 3, the value of E_MW1 decreases with increasing temperature, but the values of E_MW2 and E_MW3 have a random relation with the temperature. Young’s modulus as a random relation with temperature violates the concept of polymeric materials [6,25]. Therefore, the Maxwell–Weichert model is unsuitable for simulating the polymer’s stress relaxation behavior.

Instead of the Kelvin representation of SLSM, the Maxwell representation of SLSM can also fit the stress relaxation data of PLA/TPU blends with different mass ratios of PLA to TPU. The Maxwell representation of SLSM consists of spring 1 connected with the Maxwell element 2 in parallel. Ju et al. [1] obtained the solution of stress relaxation of polymeric materials using the Maxwell representation of SLSM as

$$\sigma \left(t\right)=E_{M1}\epsilon +E_{M2}\epsilon \exp (-E_{M2}t/{\eta }_{M2})$$

(10)

$${\tau }_M={\eta }_{M2}/E_{M2}$$

(11)

where $E_{M1}$ and $E_{M2}$ are Young’s moduli of springs 1 and 2, respectively. ${\eta }_{M2}$ is the viscosity coefficient of dashpot 2 and ${\tau }_M$ is the relaxation time of Maxwell representation. Equation (10) is identical to Equation (1) if the following equations are satisfied.

$$E_{M1}=\frac{E_1{E}_2}{E_1+E_2}$$

(12)

$$E_{M2}=\frac{{E_1}^2}{E_1+E_2}$$

(13)

$${\eta }_{M2}={\left(\frac{E_1}{E_1+E_2}\right)}^2{\eta }_2$$

(14)

Equations (12) and (13) are obtained from the equality of Equations (1) and (10) at time infinity and the initial time, respectively, and Equation (13) is from that of the exponential terms in Equations (1) and (10).

Using Equations (12)–(14) and Table S2 in Supplementary Information, we obtain the $E_{M1}$, $E_{M2,}$ and ${\eta }_{M2}$ of PLA/TPU blends with 50/50 for the mass ratio of PLA to TPU, listed in

Table 4. The mechanical parameters of Maxwell representation of the SLSM for the stress relaxation of PLA/TPU blend with 50/50 for the mass ratio of PLA to TPU.

Temperature	$10 \mathbf{°C}$	$20 \mathbf{°C}$	$30 \mathbf{°C}$	$40 \mathbf{°C}$
EM1 (GPa)	0.823 $\pm$ 0.038	0.634 $\pm$ 0.027	0.444 $\pm$ 0.023	0.278 $\pm$ 0.019
EM2 (GPa)	0.176 $\pm$ 0.011	0.257 $\pm$ 0.011	0.305 $\pm$ 0.012	0.398 $\pm$ 0.021
ηM2 (GPa·min)	1.31 $\pm$ 0.06	1.50 $\pm$ 0.08	1.61 $\pm$ 0.07	1.80 $\pm$ 0.08

. According to Table 4, $E_{M1}$ decreases with increasing temperature, but $E_{M2}$ increases with increasing temperature. Young’s modulus of spring 2 (Maxwell element) increases with increasing temperature, violating the polymeric material concept. Lagakos et al. [25] and Ferry [6] illustrated that the amorphous polymer has the trend of Young’s modulus decreasing temperature function. Therefore, the Maxwell representation of SLSM cannot be used to describe the stress relaxation behavior of the PLA/TPU blend.

The above analyses reveal that the KWW model, the Maxwell–Weichert model, and the Maxwell representation of SLSM led to incompetent trend, which is physically unsound. Such a result likely indicates that these methods do not “correctly” connect with the microstructure of the PLA/TPU blends and the use of the relaxation parameters (times and index). In general, it requires the correct formulation to catch up the main deformation mechanisms if possible.

5. Summary and Conclusions

Stress relaxation experiments were conducted on PLA/TPU blends with varying mass ratios under applied strain. This study utilized the Kohlrausch–Williams–Watts (KWW) function of the Weibull distribution and the Maxwell expression of the standard linear solid model (SLSM) to analyze the stress relaxation data at different PLA to TPU mass ratios, finding that the Young’s modulus increased with temperature. However, if converted to the Maxwell–Weichert model, the relationship between Young’s modulus and temperature appeared random. The Young’s modulus of the three models contradicts the concept of polymer materials. Compared to the Kelvin representation of the SLSM used for analyzing the stress relaxation data of TPU/PLA blends, Young’s moduli (E₁ and E₂) of all PLA/TPU blends decreased with the temperature. Additionally, E₁ and E₂ decreased as the concentration of TPU in the PLA/TPU blends increased. The viscosity coefficient of the PLA/TPU blends decreased with increasing temperature, and its reciprocal followed the Arrhenius law. The viscosity coefficient increased with the TPU concentration in the PLA/TPU blends. As the TPU content in the TPU/PLA blends increased, the activation energy for stress relaxation linearly decreased, a phenomenon not involving chemical reactions, supported by DMA measurements of the glass transition point, confirming the incompatibility between PLA and TPU.