Four different poly(vinyl chloride) (PVC) materials varying in plasticizer content were studied in a combined experimental and analytical investigation including uniaxial compression tests at strain rates ranging from 10−3 to 104 s−1 at room temperature, and temperatures ranging from −115 to 100°C at a rate of 10−2 s−1. Additional tests using a dynamic mechanical and thermal analyser were conducted on each PVC material to give a more detailed analysis of temperature and rate dependence. Adjusting the plasticizer content allows the temperature at which transitions, specifically the α-transition, or ‘glass transition’, and β-transition, occur to be moved in order to better examine the interplay of temperature and strain rate dependence. This program of research is then extended to time–temperature superposition where a novel application and interpretation of an established superposition method is presented.
(a) Rate and temperature dependence
The mechanical properties of polymers are studied because of their widespread scientific and industrial importance, as seen in the automotive, aerospace, military and medical industries. Most polymers (including poly(vinyl chloride) (PVC) and its plasticized counterparts) exhibit time-dependent mechanical behaviour, as evidenced by rate-dependent elastic moduli, yield strength and post-yield behaviour. This rate and temperature sensitivity is seen to change in different temperature and rate regimes (with sensitivity increasing at higher rates of strain) depending on different molecular mobility mechanisms being activated. In several polymers, the sensitivity is understood to increase at higher rates and/or lower temperatures because of a lack of secondary molecular mobility (β-motions), which causes increased strength and stiffness [1–4]. Alternatively, the increased strength in more rubbery or elastomeric polymers is accounted for by the change in molecular mobility during the glass transition (or α-transition). These transitions are usually seen within polymers at lower temperatures, but can be shifted in temperature by changes in strain rate, as the transition increases in temperature with increasing strain rate [5–8].
The capability to alter these molecular-level transitions through the incorporation of additives (i.e. plasticizers) offers new opportunities to understand the mechanical properties of polymers. In many applications in which PVC is used, it is blended with a plasticizer based on a phthalate ester compound, such as diisononyl phthalate (DINP) . The plasticizer is understood to increase the free volume in the polymer  and is used to enhance ductility and decrease properties such as the yield stress and stiffness. By changing the amount of plasticizer, the temperature and rate dependence are potentially changed, allowing one to better analyse and compare the mechanisms that govern these dependencies. Thus, materials such as PVC have the potential to offer great insight into understanding the interplay between temperature and rate dependence of polymers. This paper describes a combination of experimental and analytical research which were conducted with attention to the extensive theory and understanding that has been established in the described literature. A wide range of experimental investigations were pursued in studying the rate and temperature dependence of four different PVC materials ranging in plasticizer content.
(b) Time–temperature superposition
Several models have attempted to capture large strain time–temperature equivalence in amorphous polymers, which are commonly obtained via an adaptation of the Ree–Eyring multi-process rate-activated approach (e.g. [5,10,11]). A demonstration of equivalent stress states as a function of strain rate and temperature is presented in figure 1. In this figure, similar to that seen in Furmanski et al. , the parameter D relates the strain rate and temperature sensitivities during deformation in units of temperature per decade of strain rate (or strain rate on a logarithmic scale). A pragmatic way to do this is using an equation developed by Siviour et al.  1.1where D is a mapping parameter that quantifies the interaction between rate and temperature and maps from a temperature T to a new temperature T0, while mapping the strain rate to a new strain rate . For example, this mapping could be used to relate data obtained at a fixed temperature over different strain rates to data obtained at a fixed strain rate over different temperatures.
Bauwens-Crowet [5,13,14] and colleagues are understood to be the first to describe large strain behaviour and yield stress, over a wide range of temperatures and strain rates, using a time–temperature equivalence method. The group achieved a master curve of superposed rate and temperature-dependent data, presented in figure 2. The master curve spanned strain rates from below 10−4 s−1 up to 106 s−1, derived from experimental data spanning 10−4 to 100 s−1 and temperatures −20 to 100°C.
Richeton et al.  carried out uniaxial compression stress–strain tests on three amorphous polymers, including polycarbonate (PC), poly(methyl methacrylate) (PMMA) and polyamideimide, at temperatures ranging from −40 to 180°C and strain rates ranging from 0.0001 to 5000 s−1. The Richeton–Ahzi cooperative model was used to fit experimental time–temperature superposition data, as well as to validate the rate/temperature superposition principle for the reduced yield stress. The Richeton–Ahzi model differentiates from the two-process Ree–Eyring model (shown in figure 2), in that below 10−3 s−1, the cooperative model predicts an approximately constant yield stress independent of strain rate, whereas the Ree–Eyring model calculates that the yield stress will continue to decrease with decreasing rate.
Siviour et al. [6,8] were the first since Bauwens-Crowet and partners to investigate time–temperature equivalence in large strain response. PC and polyvinylidene difluoride (PVDF) were investigated over a wide range of strain rates (10−4 to 104 s−1) at constant temperature (room temperature), and a wide range of temperatures (−50 to 150°C) at constant strain rate (103 s−1). The authors proposed an empirical formula (equation (1.1)) for mapping the yield (or ‘peak’) stress dependence on temperature to the dependence on strain rate, which agreed well with experimental data and comparisons in literature; results for PC are presented in figure 3. The formula employed a linear inter-dependence of temperature and strain rate, using a reference strain rate and reference temperature as experimental constants, as well as the experimentally determined mapping parameter D. The parameter D has been termed the temperature–strain-rate equivalence parameter by other authors who have used the formula (e.g. ) and relates the strain rate and temperature sensitivities during deformation. This parameter can be found from the rate sensitivity seen in dynamic mechanical and thermal analysis (DMTA) data, or alternatively via temperature- and rate-dependent yield data. Siviour's formula was able to capture several changes in deformation mechanisms during changes in yield stress, including the inflection in data which involves the glass transition in PVDF, and the inflection which is understood as the beginning of the β-transition in PC.
Other authors have applied the same formula to PTFE and high-density polyethylene (HDPE), ultra-high molecular weight polyethylene (UHMWPE) and cross-linked polyethylene [15,16]. Both found that the formula gave good agreement between their temperature- and rate-dependent yield stress data. Williamson et al. [17,18] were able to successfully implement the linear time–temperature superposition formula to composites such as polymer-bonded explosives, finding that a D parameter of −13.1 K/decade of strain rate fit their datasets. These different works presented the wide agreement of time–temperature superposition in a range of dissimilar polymers via the single-parameter empirical model.
Cady et al.  mentioned the possibility of simulating or predicting the high-rate response of polymers by low-rate/low-temperature tests. An investigation such as this would experimentally apply the phenomena described by the Williams et al.  and Siviour et al.  formulae in time–temperature superposition. Furmanski et al. [12,21] found a linear empirical formulation of time–temperature equivalence in HDPE while investigating HDPE and UHMWPE, when using jump-rate compression tests to investigate isothermal high-rate response with the absence of adiabatic heating. More recently, Kendall & Siviour  have taken this further by simulating not only the increased yield stress at high rates, using time–temperature superposition, but also the effects of adiabatic heating in the deforming material. This enabled them to reproduce the high-rate stress–strain response of a PVC in quasi-static experiments with excellent agreement. More recently, this technique has been applied to PC and PMMA .
This paper will present a combined experimental and analytical investigation in uniaxial compression and DMTA testing of four different PVC materials varying in plasticizer. Adjusting the plasticizer content allows for lower order transitions (i.e. the α-transition, or ‘glass transition’, and β-transition) to be moved in order to better examine the interplay of temperature and strain rate dependence in each material. This study is then extended to time–temperature superposition where a novel application and interpretation of a superposition method is presented.
2. Material and methods
Four different PVCs with the same base polymer were chosen for this research and were manufactured and supplied by Solvin. As seen in table 1, the four PVCs examined range from an unplasticized form, ‘PVC’, listed at the top, to the most plasticized form, ‘PPVC-6’, listed at the bottom, with a high plasticizer level of 60 wt%. DINP was used as the plasticizing agent. The PVC came as pressed plates with dimensions of 11.5×20 cm and a thickness of 2.5 mm. These plates were machined to the specified specimen geometries.
3. Experimental method
(a) Uniaxial compression
Uniaxial compression tests were performed over rates ranging from 10−3 to 14 000 s−1. Tests at 10−3, 10−2 and 10−1 s−1 were performed using an Instron testing machine. Force measurements were taken from the load cell and displacement measurements from a clip gauge extensometer attached to the loading platens. Quasi-static experiments were conducted over the temperature range −115 to +100°C using a standard Instron environmental chamber. Temperature readings from the environmental chamber were confirmed by the use of thermocouples attached to the platens adjacent to the specimen–platen interface. Additionally, when the environmental chamber is used, its size necessitated a very long set of anvils between the cross-head and specimen, has the potential to cause problems with anvil alignment: in order to correct this, a local fixture was added for better alignment. This local fixture is a smaller compression device with two faces with inserts for compression platens of choice, and four rods which surround the inserts in order to help better guide the compression.
Experiments at rates ranging from 1 to 100 s−1 were conducted on a custom-built hydraulic load frame. Linear variable differential transformers were used to measure the displacements of the anvils, and a strain gauge-based load cell measured the force supported by the specimen. These data were used to calculate the stress–strain relationship in the sample.
High-rate tests were performed from 550 to 14 000 s−1 on a split-Hopkinson pressure bar apparatus using magnesium alloy (AZM), titanium alloy (Ti-6Al-4V) and steel bars. The striker bar, incident bar and transmitted bars all had diameters 12.7 mm, with lengths of 0.5, 1.0 and 0.5 m, respectively. The strain gauges were placed 0.5 m from the specimen–bar interface on the incident bar and 50 mm from the specimen–bar interface on the transmitted bar. The standard analysis was used to calculate stress–strain curves, and dynamic equilibrium was confirmed after each test by calculating force–time profiles for both specimen–bar interfaces .
(b) Dynamic mechanical and thermal analysis
DMTA testing was conducted in order to understand the rate and temperature dependence of the molecular mobility transitions. These experiments were carried out on a TA Instruments Q800 DMTA. The rectangular specimens had dimensions of 2.5 mm height, 10 mm width and 35 mm length and were used in the dual-cantilever mode of the DMTA machine with a set amplitude of 20 μm. Tests were performed at temperatures ranging from −150 to 100°C, and frequencies of 1, 10 and 100 Hz.
In order to compare these data to those obtained in uniaxial experiments, the test frequency may be converted to a strain rate by examining one-quarter of a cycle in the sinusoidal load program. The time duration of this quarter cycle is known from the test frequency, and the strain amplitude achieved during this time can be approximated from the prescribed displacement amplitude and the known specimen gauge length. The increase in strain over this time is approximated to be linear, and thus an average strain rate can be calculated as 3.1where d0 is the displacement amplitude and lg is the specimen gauge length .
4. Results and discussion
(a) Uniaxial compression
Figures 4–12 summarize the rate- (at 20°C) and temperature-dependent (at 10−2 s−1) response of each material tested. The figures provide a summary of the representative low to medium to high-rate responses of each material. Each material shows a distinct response to rate and temperature. In this study, the yield stress is recognized as the peak stress that the polymer reaches around yield: the intrinsic yield described by Bowden . When the observed behaviour is more rubbery, i.e. the more plasticized materials or at elevated temperatures, the yield is taken as the true stress at 5% true strain, which is approximately where the polymers are observed to yield when below their glass transition temperature.
The PVC material demonstrates a response to uniaxial deformation consistent with that seen in previous literature (as discussed in the Introduction) in both its temperature- and rate-dependent responses. Its glassy polymer response is characterized by initial linear visco-elasticity, a nonlinear shift to yield, then strain softening followed by hardening (presented in figures 4a and 5a). Within the rate-dependent yield data (figure 4b), PVC exhibits an approximately linear dependence in the low-to-medium rate regime and transitions to a second linear dependence in the high-rate regime. The slope of the second linearity in the high-rate regime is steeper and corresponds to a material transition which has a combined effect of the β- and α-transitions. These transitions become clear upon analysing DMTA data which will be presented in the following section. In the temperature-dependent data, this slope increase (reading right to left) of the second linearity can be seen at approximately −50°C.
The behaviour of PPVC-2 in uniaxial compression is summarized in figures 6a and 7a. The addition of plasticizer (20 wt% for PPVC-2) allows for an increase in strain softening. For example, in the highest rate of loading, the yield is ca 160 MPa, dropping to 105 MPa after softening, a drop of 55 MPa compared with 25 MPa in PVC. PPVC-2 also exhibits strong rate dependence, with much less post-yield softening at low rates when compared with medium and high-rate responses. Soong et al.  blended a 20 wt% dioctyl phthalate (Sigma-Aldrich) plasticizer with PVC (PPVC-2), and unlike the results presented here, found an increase in strain hardening compared with (unplasticized) PVC as strain rate increased in value. The increase in strain softening is investigated via multiple temperature measurement methods which have confirmed that the softening is due to adiabatic (or at least non-isothermal) conditions during high-rate testing.
During deformation, a fraction of work is converted into heat. The rate determines the amount of time available for the heat to escape the specimen, where an appropriately high rate would not allow any of the heat to escape the specimen during the time scale of testing (approx. 150 μs) causing thermal strain softening of the specimen. This increase in strain softening with increasing rate of deformation has also been seen in PMMA by Arruda et al.  and is investigated in more detail for the PPVC-2 material in Kendall et al. . The increased strain softening of PPVC-2 relative to PVC is expected to be due to the larger rate and temperature dependence of the flow stress of PPVC-2, so that the effect of adiabatic heating is larger. The importance of adiabatic heating in high-rate response is also seen when comparing the low-temperature quasi-static data, showing significant strain hardening at higher strains, to high-rate room temperature data, in which this is not observed. Research by Kendall et al.  shows that the removal of strain hardening at higher rates can be fully explained by adiabatic heating effects.
Finally, it is instructive to compare the stress–strain curves for PPVC-2 to PVC. In PPVC-2, the shape of the yield in the low-temperature response is much sharper than that of PVC. This sharpness is a result of the flow stress of PPVC-2 being more sensitive to temperature and log(strain rate) than that of PVC. The temperature-dependent data of PPVC-2 are in contrast to the rate-dependent data, the isothermal quasi-static behaviour demonstrates an increase in the gradient of strain hardening with decreasing temperature, while the rate-dependent data demonstrate large amounts of thermal strain softening at these same true strain values. Additionally, when compared to PVC, the shape of the yield is much sharper in the temperature-dependent true stress–true strain response of PPVC-2, compared with a much broader yield in PVC data. This sharpness is an additional illustration of the greater sensitivity of the flow stress of the plasticized PPVC-2 compared to PVC. This analysis of flow stress is similar to the analyses of Richeton et al.  when comparing amorphous polymers (e.g. PC and PMMA).
The yield stress dependence upon strain rate and temperature of PPVC-2 is shown in figures 6b and 7b. The rate-dependent data illustrate a transition in material response at approximately 150 s−1, which is demonstrated in figure 8. The transition is understood to correspond with the addition of the β-transition (or a lack of β motions) to the previously single activated process (α) and is more subtly seen in the temperature-dependent data at −50°C, and the DMTA data in the following section.
PPVC-4 and PPVC-6 present similar responses to each other which are summarized in figures 9–12. Both have glass transition temperatures well-below room temperature. The glass transition of PPVC-4 is observed in the temperature-dependent yield data (figure 10b) at approximately 0°C, while the glass transition of PPVC-6 is recognizable at −30°C (figure 12b). Both of these material transition temperatures have been confirmed by DMTA results in the next section. These transitions are also visible in the stress–strain curves in both the temperature- and rate-dependent data.
PPVC-4 displays glassy polymer behaviour in both its temperature- and rate-dependent stress–strain responses. Interestingly, the rate-dependent response of PPVC-4 presents strain-hardening, versus an absence of strain hardening in the temperature-dependent response for specimens with similar yield stresses. More specifically, this behaviour is seen in figures 9a and 10a, when comparing high strain rates (2000–14 000 s−1) to low-temperature data at −35°C where similar stresses are reached in both sets of data.
The temperature-dependent stress–strain behaviour of PPVC-6 (figure 12a), like PPVC-4 in figure 10a, presents three regions in its behaviour, moving from a rubbery response at room temperature (and above) and going through a leathery response up until its glass transition temperature (at −30°C) where a glassy polymer response is seen in the shape of the yield and post-yield strain softening and subsequent strain hardening. The response of PPVC-4 presents several similarities to that of PPVC-6, although differences are seen in the glassy response at high rates in the rate-dependent stress–strain data of PPVC-4 versus the leathery-glassy response at high rates in the rate dependency of PPVC-6, illuminating key differences from the increase in DINP-plasticizer content. The PPVC-6 rate-dependent stress–strain behaviour between 100 and 1770 s−1 presents a transition from a leathery response to a response that combines the leathery response and the glassy response—a glassy polymer yield with post-yield strain softening is not visible. It is worth noting the similarity of the PPVC-6 stress–strain response at −20°C and the high-rate responses of PPVC-6 centred at 2000 s−1, connections such as these highlight the interplay between temperature and strain rate which were exploited and used as the foundation for the simulation method presented previously by Kendall & Siviour [22,23].
(b) Dynamic mechanical and thermal analysis
Characteristic storage and loss modulus curves for all tested frequencies are shown in figures 13–16, for PVC—PPVC-6, respectively. The test frequencies are converted to their approximate strain rates using equation (3.1).
In figure 13, the PVC loss modulus has two large peaks corresponding to two material transitions: the α- and the β-transitions. PVC has a clear and relatively sharp glass (α) transition centred at 82°C which agrees with literature on neat (or ‘unplasticized’) PVC . The α-transition is visible in the storage modulus data with a drop in modulus from 2000 MPa to a very small value. In figures 14–16, this transition is seen to shift to lower temperatures with increasing amounts of plasticizer. PPVC-2 has an α-peak located at 45°C, PPVC-4 at 0°C and PPVC-6 at −20°C. The loss modulus peak corresponding to this transition also broadens with the addition of plasticizer: for PVC, it begins at 60°C and ends just above 100°C, spanning 40°C, while PPVC-6 spans up to 100°C (PPVC-4 also spans 100°C, and PPVC-2 spans roughly 80°C). This transition is understood as constrained translations and rotations of large segments of the polymer main chains—this agrees with the addition of DINP, the plasticizing agent, which is considered to create more ‘free volume’ than the unplasticized PVC (which has a relatively simple chemical structure that would take much more energy to rotate/translate main chain segments with the smaller amount of free volume) .
The loss modulus peak relating to the β-transition is within the −100°C range with the unplasticized PVC as seen in figure 13, although for the plasticized PPVCs, this is not the case. In the PVC curve, the β-peak, at approximately −45°C, overlaps slightly with the peak corresponding to the glass transition, although there is enough spacing for the effect of the two transitions to be observed in the storage modulus response. In order to locate the β-peak and better understand the movement of the peak corresponding to increases in plasticizer content, additional DMTA tests ranging from −150 to 30°C were conducted (figures 14b–16b).
With the addition of plasticizer, a much more distinct β-transition, distinct from the α-transition, is seen not only in the loss modulus but even more clearly in the storage modulus when comparing PPVCs to PVC. To quantify this separation, we see that neat PVC has a separation between α- and β-peaks of approximately 125°C versus 195°C for PPVC-2, 150°C for PPVC-4 and 130°C for PPVC-6. Interestingly, the β-transition appears to be in the same temperature location (approx. −150°C) regardless of the amount of plasticizer added. These results will be investigated further in the discussion in the next section.
(c) Initial time–temperature superposition results
In figures 17–20, the time–temperature equivalence yield data fitted with the Siviour et al.  method (equation (1.1)). Each mapping parameter was fitted manually and gave very good agreement between rate and temperature equivalence for all PVC materials (PVC, PPVC-2, 4 and 6). The rate or temperature-dependent data are converted to an equivalent test temperature or rate test using different mapping parameters or ‘temperature–strain-rate equivalence factors’ for each material.
Owing to relatively weaker superposition results of PPVC-2 (figure 18), the time–temperature equivalence method was further improved upon in order to better understand the time–temperature behaviour of this material. A key point of this polymer is that the rate-dependent data obtained encompasses both the α- and β-transitions, whereas the others encompass either α (PPVC-4, PPVC-6) or β (PVC) only.
5. Time–temperature superposition
(a) Discussion of dynamic mechanical and thermal analysis and uniaxial compression
Figure 21 presents the similarities in yield stress of the temperature-dependent quasi-static (10−2 s−1) data and the rate- and temperature-dependent elastic modulus data from DMTA testing by superimposing the two sets of curves. These data present clear similarities between the increases in elastic modulus (from DMTA data) and yield stress (from compression data) with decreasing temperature. The minor differences in temperature values for α- and β-transitions (comparing yield data versus modulus data) are attributed to the broadness of these transitions (seen in loss modulus behaviour).
The dependence of PVC and PPVC(-2, -4, -6)'s yield strengths upon strain rate and temperature are summarized in §4a. These curves make clear the material transitions which occur at high strain rate/low temperature. A comparison of these data and DMTA data is useful in predicting the transitions and the rate at which they occur. For PVC and PPVC(-2, -4, -6)'s yield stresses dependence upon temperature, there are clear qualitative similarities to the shapes of the curves produced in DMTA when measuring the dependence of the elastic modulus with respect to temperature (figure 21) illustrating the consistency of and confirming the α- and β-transitions in the uniaxial compression data and the DMTA data.
(b) Extension of time–temperature superposition discussion
(i) Separation of α and β processes in PPVC-2
The DMTA results for PPVC-2 were further investigated for a better understanding of the interplay between temperature and rate dependence. This eventually led to an extension of the rate equivalence formula of Siviour et al. .
Typically, authors have obtained the shift factor for this equivalence by fitting the two sets of yield stress data (versus temperature and log(strain rate)). However, instead DMTA data can be used. Initially, the DMTA results in §4b were used to obtain a shift factor of 6°C/decade strain rate, and the result of using this is shown in figure 22. This notably improves the fit at low to moderate rates (10−3 to 85 s−1) and moderate to high temperatures (0–100°C), but the correspondence at high rates/low temperatures is still poor. This departure can be attributed to the β-transition in the material. The DMTA data confirm observations in the literature that the rate dependence of the β-transition temperature is greater than that of the α-transition. This has been explicitly explored by Mulliken & Boyce , who have developed a method for predicting high-rate elastic moduli by separating the α- and β-contributions, shifting them and reconstructing at higher rates (the so-called DSR method). In this case, the β-transition may be observed at room temperature in high-rate data, even if it is not apparent until very low temperatures in quasi-static experiments.
Therefore, the DMTA data were again consulted to obtain a second shift factor of 13°C/decade strain rate for the β-transition. To implement this into a framework for understanding the yield stresses, the α-transition shift factor 6°C/decade strain rate is used for low-rate/high-temperature data and the β shift, 13°C/decade strain rate for high-temperature low-rate data, to obtain the excellent fits in figure 23.
Hence, this time–temperature equivalence methodology is given as 5.1and 5.2These different values recognize that the β- and α-transitions have different temperature–strain rate equivalence factors. The use of the independently determined shift factors from DMTA shows exceptional agreement in time–temperature equivalence of the yield stress data. This good, and very interesting, agreement highlights the fact that several processes involved in determining an amorphous polymer's elastic modulus and yield stress overlap with one another. Furthermore, this presents a new addition to the methodologies of time–temperature equivalence and offers opportunities in standardization, automation and a more scientific method for finding shift factors (Dα and Dβ) for time–temperature equivalence from independently obtained DMTA data.
(ii) Application to other poly(vinyl chloride) materials
The dual-process time–temperature superposition method (via DMTA shift values) was also applied to PVC1 which is presented in figure 24, demonstrating a very good agreement in the new addition to time–temperature equivalence methodologies. The two other materials (PPVC-4 and PPVC-6) do not experience the effects of the β-transition at the rates tested.
DMTA-derived shift factors were determined for all PVC-based materials in table 2. The data in table 2 compare shift factors obtained by fitting yield stress data in §4c to those from the DMTA data. It is clear that for PVC, where the glass transition is above room temperature, the time–temperature equivalence for yield stress is driven by the β-transition. Similarly for PPVC-2, the single-parameter yield stress fit is poor because both transitions play a role, and hence two values of D presented in the previous section. For PPVC-4 and PPVC-6, where the glass transition is below room temperature, it is driven by the glass (α) transition.2
In addition, upon examining the measured shift factor values for the α- and β-transitions, it is noted that Dβ is much larger than Dα. Macroscopically, this difference demonstrates the higher rate sensitivity of the β-transition compared with the α-transition. On the molecular scale, there is a difference between the amount of energy that is necessary to enable the movement associated with the much smaller and more localized scale polymer chain (and/or side groups attached to them) motion, β-motions, versus the larger and cooperative chain motion associated with the glass transition, α-motions .
Coupling this analysis with DMTA results, in examining the DMTA curves of the PVC materials, the β-transition appears to shift to a much lower temperature once plasticizer is added (in this case 20 wt%). Interestingly, regardless of the amount of further plasticizer added (e.g. 20 wt% for PPVC-2 versus 60 wt% for PPVC-6), the β-transition stays in approximately the same position at the lowest rate (0.035 s−1 or 1 Hz): the amount of plasticizer does not shift the approximate location of the β-transition, even though the α-transition shifts to a lower temperature once plasticizer is added (as expected). Thus, the addition of more phenyl rings (or DINP), which is understood to create additional free volume in the material [7,31,32], does not affect the location of the β-transition. Hence, the β-transition must include more than just the behaviour of phenyl rings (i.e. a π-flip), if making comparisons to materials such as PC with and/or without additives (e.g. ), and therefore there must be some intra- or intermolecular cooperativity included.
Referring back to table 2, it is noted that the addition of plasticizer increases both shift factors (α and β), or increases the sensitivity of the location of the transitions to the rate of deformation. Namely, the more plasticizer added to (or rubbery behaviour of) a polymer, the polymer needs lower rates of deformation to observe the shift of the β-transition to room temperature. More specifically, to stop the polymer chain or side group motion which occurs on a much more localized scale or ‘intramolecular skeletal modes of vibration’ , a much shorter time scale of deformation is required. The shift factor of materials determines the rate dependence of the material and how easily these transitions are seen in material response at ambient temperature.
Four PVCs with different plasticizer levels have been extensively characterized through uniaxial compression experiments over a broad range of strain rates and temperatures.
Moreover, DMTA characterization and analysis has been performed on all materials. In each material, the effects of polymer transitions on rate dependence are clear: the glass transition in all of the materials and the β-transition in the unplasticized and 20% plasticized (PPVC-2). Superimposition of elastic modulus curves from DMTA testing and yield stress dependence upon temperature curves help confirm the robust interplay of temperature and rate dependences of several different types of polymer (whether an amorphous polymer such as PVC, or a rubber such as PPVC-6).
In addition, the accuracy of a time–temperature equivalence method was investigated using PPVC-2. Building upon the linear formulation of time–temperature equivalence , the new addition to the methodology was to separate the β and α influences in high-rate and low-temperature data. This emphasized the advantages of an alternative method for finding shift factors (Dα and Dβ) experimentally via DMTA data, and a better understanding was obtained of the driving factors in time–temperature equivalence for describing amorphous, and rubbery, polymer time–temperature behaviour.
The electronic supplementary material consists of the true stress–true strain and DMTA data which was used in this study.
7. Funding statement
Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant no. FA8655-09-1-3088.
The authors thank Dr J. L. Jordan and Dr J. R. Foley for their support of the research, and R. Froud and R. Duffin for ongoing technical support. The authors would also like to thank Solvay for providing the materials tested in this paper. M.J.K. thanks Dr Igor Dyson for invaluable help with low-rate experiments. The US Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright notation thereon. The authors thank Dr M. Snyder and Dr R. Pollak for their ongoing support.
↵1 The formula was not applied to PPVC-4 or PPVC-6, as the temperatures and strain rates tested did not reach the β transitions of either material. Therefore, the experimentally fitted shift values (data which only included α processes) agreed well with the Dα found from DMTA data.
↵2 tanδ or loss modulus peaks may be used to find the temperature locations of the material transitions (α and β). These values are also used to find the shift factors (Dα andDβ). Differences in this thesis are typically approximately 1°C/decade strain rate when using tanδ rather than the loss modulus. Loss modulus peaks were used in all cases in this paper.
- Received January 5, 2014.
- Accepted March 31, 2014.
- © 2014 The Author(s) Published by the Royal Society. All rights reserved.