This section is segregated into four subsections, reporting the dynamic thermomechanical responses of FFF and VPP printed materials followed by synthesizing the data in the context of mechanics of time-dependent materials.
3.1 Dynamic Thermomechanical Behavior of FFF Printed Materials
Figure 3 reported the storage and loss moduli of three materials printed using the FFF process, including TPU (Fig. 3a), ABS (Fig. 3b), and carbon-reinforced nylon (Fig. 3c), as a function of temperature ranging from − 100°C to the respective terminal point. The figures also show the \({tan}\delta\) (=\(E"/E{\prime }\)) as a measure of the dampening behavior of these materials (Youssef 2021). As mentioned above, the mechanical responses in Fig. 3 were collected at a frequency of 0.1 Hz while heating the sample at 3°C/min. The results in Fig. 3 indicate three noteworthy and synergetic observations: dynamic response, operating range, and temperature sensitivity. In general, these thermomechanical responses are typical of homo-, co-, and ter-polymers, irrespective of reinforcement. It is worth noting that thermal transitions in the forthcoming discussion are conventionally defined based on the peaks from the \({tan}\delta\)-temperature plots.
TPU reported a cryogenic glass transition temperature, consistent with its room temperature rubbery behavior, leading to a mechanically compliant response at room temperature and up to the terminal testing temperature. TPU reported a single thermal transition, i.e., glass transition, congruent with its molecular structure being homopolymer. The glass transition temperature (Tg) of TPU was found to be ca. -46°C, limiting the glassy regime to cryogenic temperatures. The effective modulus (\({E}_{eff}=\sqrt{{{E}^{{\prime }}}^{2}+{{E}^{{\prime }{\prime }}}^{2}}\)) of glassy TPU at -100°C was 2960 ± 101 MPa (average of three samples), descending to 51 ± 7 MPa at room temperature (23°C) and plateauing thereafter at a slightly higher level (e.g., \({E}_{eff}\)= 117 ± 8 MPa at 100°C). The latter observation (i.e., \({E}_{eff}\)= 89 ± 13 MPa for 23°C < T < 100°C) indicates that TPU exhibits an extended rubbery plateau and can be integrated into applications requiring a broad temperature operating range with high compliance, as discussed in a forthcoming section. Since the value of the loss factor (\({tan}\delta\)) is well below unity, it indicates the relative degree of energy dissipation is lower than comparable materials with \({tan}\delta\) > 1; hence, TPU might be suitable for high cushioning requirements. Here, the overall mechanical behavior of TPU is associated with local and global chain motions such that the energy-dissipating viscous elements have lower contributions to the dynamic properties of TPU. Notably, the thermomechanical spectrum shown in Fig. 3a agrees with the reported melting temperature of thermoplastic polyurethane (185–220°C) (Stribeck et al. 2011; Frick and Rochman 2004) and the quasi-static modulus compared to the effective modulus at room temperature (Kim et al. 2017; Yanagihara et al. 2015). This foreseen utility is further substantiated by the peak of \({tan}\delta\), spreading over a broad temperature range from approximately − 80°C to 2°C, evidencing relatively extended cushioning efficacy within the leathery and rubbery regimes. TPU continues to maintain the same values of the loss factor above 0°C, showing nearly constant \({tan}\delta\) at 0.054 ± 0.003 up to the test terminal temperature.
It is imperative to highlight the notable dip in the \({tan}\delta\) plot of TPU for the temperature range 0°C < T < 50°C that transpired, by definition, from similar localized minima in the storage and loss moduli. The sudden reduction in the resistance to deformation metrics, i.e., moduli or \({tan}\delta\), is generally attributed to gradual sliding and slippage of the primary backbone as a function of temperature, owning to its linear polymer molecular structure, due to the dominance of the thermal relaxation energy over mechanical strain energy dissipation. Remarkably, this trend coincides with the oscillating strain response shown in the inset of Fig. 3a. The strain-temperature plot indicates a pronounced valley, where the strain started to descend at 0°C as a function of temperature, reaching a local minimum at ~ 34°C before assuming a reversing trend reaching a plateau at ~ 55°C. The reduction in the strain percentage ascertains the governance of macromolecular motions by the input thermal energy rather than the induced strain energy from the applied deflection amplitude.
Figure 3b exemplifies the thermomechanical behavior of 3D printed samples using ABS, extending from − 100°C to the testing terminal temperature of 150°C. The latter was extended specifically for ABS to capture additional thermal transitions characteristic of this material (Kraus, Childers, and Gruver 1967). The loss factor of ABS (Fig. 3b) shows two thermal transitions at -74°C and 108°C, corresponding to butadiene and styrene blocks, respectively (Camelio et al. 1998; Kraus, Childers, and Gruver 1967; Cotten and Schneider 1963). Notably, the glass transition of copolymers of acrylonitrile was reported to range between 105°C and 140°C based on the molecular weight and processing conditions (Cotten and Schneider 1963); hence, it might have been slightly obscured or subsided by the thermal softening of ABS at the testing terminal temperature. The thermal transition at 108°C is associated with the styrene monomer, which is expected to be at higher concentrations than other copolymers, i.e., acrylonitrile and butadiene (Kraus, Childers, and Gruver 1967). The butadiene glass transition was found to be slightly higher than previous reports, probed at -74°C herein while shown to be < -85°C in (Kraus, Childers, and Gruver 1967), which is attributed to the block length of the butadiene chains and their respective molecular weight within the ABS structure. The difference in Tg of butadiene is further substantiated by the faint peak in the \({tan}\delta\) plot, evidencing a low concentration of this monomer within the overall ABS macromolecule. In general, the results of glass transitions reported for ABS abide by the forecasts of the energy, volume, and mass model (Camelio et al. 1998). Furthermore, the storage modulus of ABS, shown in Fig. 3b, is in excellent agreement with previous reports of the dynamic behavior of this terpolymer (Dakshinamurthy and Gupta 2018; Yin, Fan, and Wang 2008). For example, it is well established that the operating temperature of ABS is extended up to ca. 80°C, which is apodictic from the values of the storage modulus ranging from 3603 ± 992 MPa in the glass regime at -100°C to 1.5 ± 0.8 MPa at the end of the testing range. The overshoot in the storage modulus before the glass transition of the styrene block is ascribed to stress relief, which was induced in the samples during the FFF 3D printing process (Menard and Menard 2020). The loss factor for ABS exemplifies a material with a higher relative degree of energy dissipation, indicating activation of viscous attenuation mechanisms. Hence, ABS might be more suitable for applications requiring higher dampening properties than TPU, which might be more fit for cushioning.
Finally, Fig. 3c shows the thermomechanical behavior of chopped carbon fiber reinforced nylon, which was 3D printed using the FFF printing process. Expectedly, the storage modulus, a measure of the material stiffness, is the highest of all tested samples, exceeding 5587 ± 787 MPa at the onset testing temperature. The effect of adding chopped carbon fibers on the mechanical behavior also transpired in the loss modulus and loss factors as a function of temperature, similar to their influence on the storage modulus. It is worth noting that the thermomechanical behavior of carbon-reinforced nylon herein trumps the results recently reported by Alarifi (Alarifi 2022). The differences between the results shown in Fig. 3c and (Alarifi 2022) are attributed to three possible sources: (1) the ratio of the reinforcing carbon fibers (~ 8% herein vs. ~11% in (Alarifi 2022)), (2) processing parameters (e.g., printing speed and temperature), and print orientation. The difference in the carbon fiber reinforcing ratio and the length of the carbon fiber (414 µm herein vs. 119 µm in (Alarifi 2022)) contributes to the change in the material properties, as predicted by micromechanics models (Curtis, Bader, and Bailey 1978; Christensen and Waals 1972). In (Alarifi 2022), Alarifi explored three printing orientations leaning towards unidirectionality, while the samples printed herein (also shown in Fig. 1) relied on alternating bidirectional printing to enhance mechanical performance. Otherwise, the thermomechanical performance reported in Fig. 3c agrees with the results of Alarifi (Alarifi 2022). However, since the measurements done herein were over a broad range of temperatures (-100°C to 100°C), the loss factor plot captures two thermal transitions at -79°C and 49°C.
The thermal transitions of nylon have attracted assiduous research since its early stages of development using thermal (e.g., differential scanning calorimetry) and thermomechanical (e.g., dynamic mechanical analysis) methods (Forster 1968; Greco and Nicolais 1976; Gordon 1971). The manifestation of dual thermal transitions in nylons puzzled early researchers due to the amorphous molecular structure of nylons encompassing nonpolar hydrocarbons and amide groups with strong polarity (Gordon 1971; Boyer 1963). It is well-accepted that the glass transition of nylons is between 40°C and 70°C, depending on the molecular arrangements and the processing conditions (Greco and Nicolais 1976; Gordon 1971; Forster 1968). The latter results in shifting glass transition towards the higher bound due to interactions between the polar amide segments and the hydrogen bonding between the chains, especially after thermal annealing. Notably, thermal annealing is a byproduct of the FFF printing process, where the filaments were heated up to 270°C (well above melting temperature for nylon to achieve proper follow of the carbon reinforced polymer through the printing nozzle), then cooled down rapidly upon deposition on the print plate that was heated to only 60°C during the printing process. Therefore, 50°C is taken as the glass transition of nylon, even though at lower temperatures, the polymer exhibits pseudo-rubber behavior as reported previously in (Gordon 1971; Greco and Nicolais 1976). Based on differential thermal analysis, Gordon postulated that the pseudo-rubber behavior below the glass transition temperature is associated with hydrogen bonding between the hydrocarbon chains being under dynamic equilibrium, giving rise to relative chain mobility and monotonic descent in the storage modulus as a function of temperature (Gordon 1971). The arising of an early thermal transition at -79°C is then ascribed to the localized rearrangement of the hydrogen bonding (i.e., a second-order transition that only involves changes in the heat capacity without latent heat) (Gordon 1971; Youssef 2021).
3.2 Dynamic Thermomechanical Behavior of VPP Printed Materials
Figure 4 plots the dynamic mechanical behavior of thermosetting ABS-like and rubber-like 3D printable, photocurable resins, including the storage modulus, the loss modulus, and the \({tan}\delta\) as function of temperature (-100°C < T < 100°C). The commercial notation of ‘ABS-like’ and ‘rubber-like’ is based on the mechanical properties at room temperature, where the former imitates the mechanical properties of ABS (e.g., Fig. 3b) while the latter exhibits a relatively high loss modulus and loss factor, emulating rubbers. Indeed, the samples fabricated with the ABS-like resin reported a storage modulus of \({E}^{{\prime }}\) = 1837 ± 339 MPa, a loss modulus of \({E}^{{\prime }{\prime }}\) = 158 ± 12 MPa, and a loss factor of \({tan}\delta\) = 0.096 ± 0.022 at room temperature, which are comparable to the respective values for FFF printed counterparts (Fig. 3b). However, the dampening properties of ABS-like photocurable resins are inferior to FFF printed ABS counterparts since the loss factor is less than unity. The glass transition temperature for the ABS-like resin is 65°C, limiting the operating range since the samples experienced a significant loss of stiffness after that. The storage modulus shows a monotonic descent below the defined glass transition, indicating the sensitivity of photo-polymerized macromolecule to a temperature that dynamically induces localized conformational mobility. The latter transpired in the loss modulus of the investigated resins, evidencing a dip in the glass regime due to a decrease in dissipated energy as the chains align (Silver, Gonzalez-Mercedes, and Mesica 2022). These results agree with recent reports on the viscoelastic properties of printable photocurable resins (Casado et al. 2022). Generally, the dynamic behavior of the rubber-like resin is consistent with the mechanistic expectations of rubbers as a function of temperature, resulting in a glass transition of 32°C and rapid descent in the storage modulus in the denoted glassy regime. The viscoelastic response of rubber-like photocurable resin trumps FFF-printed TPU, exemplified by extended temperature operating range. At the outset, it is imperative to note that while the dynamic properties of printable resins could be comparable to the FFF printed counterparts, the formers are plagued with their sensitivity to ultraviolet radiation, limiting their application to internal components or indoor usages.
3.3 Preliminary Mechanics of 3D Printed Materials
The storage moduli of all 3D printed materials, irrespective of additive manufacturing methods, are fitted into a robust stiffness-temperature model (Mahieux and Reifsnider 2001). The motivation for exploring this phenomenological model stems from (1) the pronounced dependence of the storage modulus of polymers on the operating temperatures, evidencing several orders of magnitude reduction between the glassy and rubber regimes (Menard and Menard 2020; Lakes 2017; Youssef 2021), and (2) the facilitation of future engineering applications seeking robust mechanics relationships to be integrated into design analyses (Alasfar et al. 2022; Mahieux and Reifsnider 2001; Gunter 2006). As shown previously in Fig. 3 and Fig. 4, the storage modulus-temperature response exhibits multiple thermal transitions induced by molecular motions stemming from primary and secondary inter- and intra-molecular bonds (Mahieux and Reifsnider 2001; Ashby and Jones 1986; Platzer 1969; Miller 1966). Mahieux and Reifsnider postulated a series of equations (modified in Eq. 1) based on simple spring-dashpot response, time-temperature equivalency in thermorheologically simple materials, the activation energies responsible for primary and secondary molecular bonds, and the Weibull distribution of the bond rupture associated with the latter.
$$E\left(T\right)= \sum _{i=1}^{n} ({E}_{i}-{E}_{i-1}){e}^{\left({-\left(\frac{T}{{T}_{i}}\right)}^{{m}_{i}}\right)}+{E}_{T}$$
1
The storage modulus, \(E\left(T\right)\), as a function of temperature of the investigated 3D printed polymers was fitted into one of the above listed equations based on the number of thermal transitions (\({T}_{i}\)). \({E}_{i}\) represents the storage modulus at the plateau region before a given thermal transition, \({E}_{T}\) is the terminal modulus prior to the mechanical or thermal flow region, and \({m}_{i}\) is the Weibull coefficient corresponding to the statistics of bond breakage at each transition, directly related to the degree of restriction in the molecular mobility. It is also imperative to note that the experimental characterization intentionally avoided the melt transition since this research emphasizes mechanical applications. Nonetheless, this approach can readily be extended to included additional thermal transitions, e.g., melting point, in future research to further optimize the additive manufacturing process or explore utility of these polymers close to the thermal flow behavior. Figure 5a is a demonstrative schematic of the physical correspondence of the fitting parameters with the experimental storage modulus as a function of temperature. Figure 3 and Fig. 4 include the model results, showing reasonable agreement between the curve fitting and the experimental data while providing insights into the contributions of the molecular motions based on the resulting values of \({m}_{i}\).
Table 3
Stiffness-Temperature Model Parameters for Each Printing Material.
Material | Transitions | \({\varvec{E}}_{1}\) (MPa) | \({\varvec{T}}_{1}\) (°C) | \({\varvec{m}}_{1}\) | \({\varvec{E}}_{2}\) (MPa) | \({\varvec{T}}_{2}\) (°C) | \({\varvec{m}}_{2}\) | \({\varvec{E}}_{3}\) (MPa) | \({\varvec{T}}_{3}\) (°C) | \({\varvec{m}}_{3}\) | \({\varvec{E}}_{\varvec{T}}\) (MPa) |
FFF-TPU | 1 | 3182 | -60 | 10.73 | - | - | - | - | - | - | 97 |
FFF-ABS | 3 | 3871 | -75 | 17.96 | 2809 | 2 | 5.96 | 797 | 105 | 191.33 | 12 |
FFF-CFN | 3 | 5953 | -86 | 10.43 | 4664 | -47 | 10.14 | 3514 | 40 | 19.54 | 775 |
VPP-ABS-like resin | 2 | 5347 | -57 | 7.48 | 3024 | 39 | 14.85 | - | - | - | 64 |
VPP-F69 resin | 2 | 6033 | -61 | 8.86 | 3215 | 9 | 18.14 | - | - | - | 29 |
Table 2 collates the curve fitting parameters, including the plateau moduli defining the onset stiffness preceding each transition, the transition temperature, and respective Weibull coefficients, deciphering the bond relaxation processes based on the chain participation and bond strength and stiffness. Mahieux and Reifsnider discussed the broad range of Weibull coefficients (\({m}_{i}\)) based on the available conformational changes as a function of temperature (Mahieux and Reifsnider 2001). They foresaw a large value of the Weibull coefficient for amorphous polymers due to the lack of large restrictions of molecular motion since the bond strength is bounded within a narrow distribution (Mahieux and Reifsnider 2001). Alternatively, Mahieux and Reifsnider argued that mi assumes small values when conformational mobility is restricted by crosslinking or localized crystallinity (Mahieux and Reifsnider 2001). These mechanistic insights transpired herein for the investigated additively manufactured polymers based on the resolved values of the Weibull coefficients.
The resolved values of the Weibull distribution coefficients are plotted in Fig. 5b as a function of the corresponding transition temperatures, showing a correlation between the increase in temperature and the values for \({m}_{i}\). The latter observation persisted, irrespective of the number of transitions, printing process, and type of polymer. The results corroborate the presumptions of Mahieux and Reifsnider, where low values of \({m}_{i}\) coincided with low temperature, while the opposite was also confirmed (Mahieux and Reifsnider 2001). The low values of Weibull distribution coefficients indicate significant molecular restrictions given the glassy state of the polymers, where the molecular motions are confined to localized vibrations, bending and stretching, and side group perturbations (Mahieux and Reifsnider 2001). On the other hand, liberated molecular motions are associated with higher values of \({m}_{i}\), evidencing a greater level of chain participation in an overall reduction in resistance to deformation (i.e., low storage moduli) due to loosening the bond strength and stiffness at higher temperatures, increasing the available free volume, and allowing gradual and significant chain motions. The results in Table 3 and Fig. 5b are further synthesized as a function of printing materials to reconcile the values of the storage moduli and transition temperatures with the data discussed in § 3.1 and § 3.2. For example, the curve fitting parameters for FFF-CFN exemplify higher stiffness in the glassy regime substantiated by restrictive molecular motions based on the relatively low value of the Weibull coefficient. However, higher chain mobility coincides with increased values of the Weibull coefficient as the temperature increases (i.e., the value of m3 for the third transition) such that FFF-CFN transitions away from the glassy region and into the rubbery regime. Finally, the terminal modulus of FFF-CFN is relatively higher than all other tested materials due to the reinforcement of the stiff, short carbon fibers. Similar conclusions are evidenced for the remaining printed materials, irrespective of the additive manufacturing process, giving rise to the mechanics of these polymers as a function of temperature and their corresponding molecular structures (discussed in § 3.1 and § 3.2).
3.4 Case Study
The performance of walking and running shoes relies on the materials and structures of the soles, which comprises insole in contact with the foot providing cushioning and conformability, midsole to mitigate the ground reaction force, and the outsole to reduce abrasion and increase traction (Uddin et al. 2020; Zahir Uddin et al. 2024). Figure 6a shows the anatomy of the shoe soles, showing the in-, mid-, and out-sole layers. Recently, 3D printing was used to mass produce walking and running shoes by prominent shoe manufacturers, seeking customization and tailor-ability by designing the different sole layers with unique structures and materials (Zolfagharian et al. 2021; Chhikara et al. 2023; Schwaar 2023). Selecting a specific structure depends on the manufacturing process, including resolution, speed, cost, and repeatability, which is a focus of a different research direction. However, matching materials for a sole layer can be demonstrated herein by considering the 3D printable polymers discussed above. In such consideration, walking and running shoes can be deployed in a wide range of temperatures, ranging from 0°C (winter conditions), 23°C (spring conditions), and 50°C (hot summer conditions). Needless to say, the stiffness requirements hinge on the weight of the subject since body reaction force is a function of body weight and moving speed (Zolfagharian et al. 2021). Hence, the storage moduli and loss factors for the additively manufactured polymers at three temperatures (0°C, 23°C, and 50°C) assist in matching materials with loading and operating conditions. Figure 6b summarizes the storage moduli and loss factors at the above-mentioned temperatures, evidencing a broad range of mechanical properties suitable for the design of walking and running shoes. The requirements of mechanical compliance with a relatively higher loss factor, i.e., rubber-like polymer, indicate VPP-F69 resins can be suitable for the fabrication of outsoles. It is imperative to note that additional measurements of the wear resistance of VPP-F69 as a function of temperature and terrain are necessary to further substantiate the suitability of this material for outsoles. On the other hand, FFF-printable thermoplastic polyurethane, providing balance between mechanical compliance and dampening (i.e., better cushioning) can be more suitable for insoles, leveraging these desirable mechanics and agility of the FFF printing process. Finally, the midsole requires significant stiffness to provide rebounding and energy absorption, pointing to either FFF-CFN or VPP-ABS-like resin based on the level of structural tailoring, where the former might be conducive for simplified geometries while the latter is suitable for higher degrees of customization, e.g., lattice or mathematically generated structures. Future research will emphasize the construction of multilayer soles and submitting these 3D printed structures to simulated and deployment conditions to ascertain the utility of additive manufacturing of these polymers in developing custom orthotics.