高应变速率下的橡胶物理机械行为（英）

MECHANICAL BEHAVIOR OF RUBBER AT HIGH STRAIN RATES C. M. ROLAND* CHEMISTRY DIVISION, CODE 6120 NAVAL RESEARCH LABORATORY WASHINGTON, DC 20375-5342 ABSTRACT Methods to obtain the mechanical response of rubber at high rates of strain are reviewed. These techniques include the extrapolation of low strain, low strain rate data, the limitations of which are discussed, extrapolations to elevated hydrostatic pressure, and direct determinations using split Hopkinson bar and drop weight testers, as well as miscella- neous methods. Some applications involving rubber at strain rates sufficient to induce a transition to the glassy state are described. CONTENTS I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .429 II. Extrapolation of Low Strain Rate Measurements . . . . . . . . . . . . . . . . . . .431 A. Time-Temperature Superpositioning . . . . . . . . . . . . . . . . . . . . . . . .431 B. Temperature-Volume Superpositioning . . . . . . . . . . . . . . . . . . . . . .435 C. Temperature-Pressure Superpositioning . . . . . . . . . . . . . . . . . . . . .438 III. Direct Measurement at High Rates and Large Strains . . . . . . . . . . . . . . . .441 A. Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .441 B. Servohydraulic Test Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . .443 C. Split Hopkinson Bar (SHB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .443 D. Drop Weight and Pendulum Testers . . . . . . . . . . . . . . . . . . . . . . . . .446 E. Expanding Ring Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .451 F. Catapult Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .452 IV. Strain-Induced Glass Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .452 A. Wet Skid Resistance of Tires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .453 B. Sound Transmission and Damping . . . . . . . . . . . . . . . . . . . . . . . . .453 C. Impact Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .454 V. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .455 VI. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .456 VII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .456 I. INTRODUCTION This review discusses the mechanical behavior of rubber at high strain rates, a topic that taken to the extreme is a contradiction in terms. Rubber refers to any amorphous, flexible chain high-polymer having a sub-ambient glass transition temperature, Tg. Since effectively Tg is defined as the temperature at which the material response (i.e., the local segmental dynamics) becomes significantly slower than the experimental time scale (i.e., Deborah number >> 1), the glass transition is rate-dependent. This means rubber being strained very rapidly can behave as a glass, so that the rubbery state may not persist at high strain rates. In fact, as discussed herein in Section IV, the transition of rubber to a glass underlies some applications for rapidly strained rub- ber. The other phase change induced by the deformation of rubber, strain-induced crystallization, is also affected by high strain rates. In high speed tension tests on natural rubber, Mitchell and Meier1 found that strain-crystallization required 45 to 65 ms at ambient temperature. This result was corroborated in subsequent studies by Glaser and Eirich.2 Lake et al.3 reported that when 429 * Ph: 220-767-1719; Fax: 202-767-0594; email: roland@nrl.nay.mil Downloaded from http://polymerphysics.netDownloaded from http://polymerphysics.net crack growth rates in natural rubber exceed about 1 cm/s, there is insufficient time for strain crys- tallization at the crack tip. Accurate testing of rubber at high rates of strain can be difficult. For linear measurements (modulus independent of strain amplitude), dynamic mechanical spectroscopy provides charac- terization over a wide range of rates (~ 5 decades); however, the upper frequency is usually only 10 – 100 Hz. Specialized instruments extend the range to ~104 Hz4 but these are not in common use. Atomic force microscopy (“nanoindenters”) yield indirectly the mechanical properties of surfaces,5,6 and these can be operated at rates as high as 1 MHz.7,8 However, the strains are low and only the surface is probed. The frequency range of conventional mechanical spectroscopy can be extended by invoking the time-temperature superposition principle.9 First demonstrated by Tobolsky and Andrews,10 this method is illustrated in Figures 1 and 2, which show respectively strain to failure11 and fric- tion12 measurements on rubber at high rates and velocities. The data obtained at various temper- atures superpose to form a master curve; however, at the lowest temperatures (highest reduced rates) there is no overlap of measurements at different temperatures. Thus, while there is no indi- cation of a breakdown of the superposition principle, the validity of the master curves for high reduced frequencies cannot be judged from the data per se. The curves were constructed assum- ing superpositioning to be valid, which relies in turn on one of two assumptions: that the molec- ular motions relevant to the property being tested remain the same at all test temperatures or if not, that all modes have the same temperature dependence. 430 RUBBER CHEMISTRY AND TECHNOLOGY VOL. 79 FIG. 1. – The elongation at break versus negative logarithm of the reduced extension rate for an SBR. The actual tests covered as much as 3 decades in extension rate.11 In this review we scrutinize the practice of time-temperature shifting to characterize the mechanical properties of rubber at high rates, and also discuss approaches to predict the response at elevated pressure for rates beyond those actually measured. Various methods of directly test- ing rubber at high strains and high strain rates are reviewed. Finally, we describe some applica- tions involving rapidly deformed rubber. The focus herein is on the stress/strain behavior rather than the failure properties of elastomers at high rates of strain. II. EXTRAPOLATION OF LOW STRAIN RATE MEASUREMENTS A. TIME-TEMPERATURE SUPERPOSITIONING In Figure 3 are shown master curves of the dynamic shear moduli for uncrosslinked cis-1,4- polyisoprene (synthetic natural rubber, PI).13 There is apparently good superpositioning of the data, which were measured over a range of temperatures from Tg (= -71 °C) to 80 °C. Vertical arrows on the figure denote the terminal relaxation time (onset of flow), the longest Rouse relax- ation time (onset of entanglement constraints demarcating the rubbery plateau), and the local segmental relaxation time (involving intramolecularly correlated motion of a few backbone bonds14,15). As frequency increases, successively shorter length scales are involved in the under- lying motions, and eventually no polymeric modes contribute to the response. Note that in Figure 3, measurements at only one temperature are shown in the transition zone; thus, any breakdown in time-temperature superpositioning cannot be detected. Horizontal shifting of the (relatively featureless) curves will cause their overlap, particularly with the usual small adjustments in ordi- nate values. However, if one compares the loss tangent, tan δ, for these same data (Figure 3 inset), there is a marked change in shape with temperature, revealing a breakdown of superposi- tioning in the transition zone. This breakdown is due to the difference in temperature dependence MECHANICAL BEHAVIOR OF RUBBER AT HIGH STRAIN RATES 431 FIG. 2. – Friction coefficient of natural rubber on silicon carbide paper versus the log of the reduced sliding velocity in cm/s. The reference temperature was 20 °C and the actual temperatures were -58 to 90 °C. The material was isomerized to suppress crystallization.12 of the local segmental motion and the chain (polymeric) dynamics, a phenomenon first discov- ered in polystyrene more than 40 years ago.16,17 (For unentangled linear polymers, these chain motions are described as Rouse modes; for higher molecular weight polymers, the long-time processes include both Rouse modes involving chain units between entanglements and the ter- minal chain modes described, for example, by reptation models.9,17,18) Clearly this thermorheo- logical complexity can only be observed by measurements extending over a broad enough range of frequencies. If isothermal data are taken over only a few decades, results can be combined apparently successfully to yield master curves, as shown in Figure 3. Of course, for spectro- scopies that probe only the segmental motions, such as dielectric relaxation of “type-B” dipoles (transverse to polymer chain so that the normal modes are dielectrically inactive;), any departure from time-temperature equivalence becomes moot. PI has type-A dipoles (parallel to the backbone) and thus dielectric spectroscopy can be used to probe the motions of the chain end-to-end vector. By combining mechanical and dielectric results, shift factors, a(T), for both modes can be obtained over a wide, overlapping range, and the stronger T-dependence of the local modes is made evident (Figure 4). In the more usual experiment, the segmental mode is measured at low temperatures and the chain modes at high temperatures, with the collected shift factors forming a smooth, continuous curve; however, enor- 432 RUBBER CHEMISTRY AND TECHNOLOGY VOL. 79 FIG. 3. – Master curves for the dynamic storage and loss moduli of PI (M w =500 kg/mol) at a reference temperature = -10 °C.13 The arrows denote (from left to right) the frequency associated with the terminal chain mode (onset of flow), the slowest Rouse mode (onset of entanglement effects) and local segmental relaxation (onset of the glass transition). The inset shows the loss tangent over a temperature range from -48 °C to -66 °C, corresponding to the transition zone. In the master curves, data measured at only one temperature is used for the transition zone in order to obtain ostensibly satisfactory superpositioning. mous errors would result from extrapolation using such results. Similar behavior to Figure 4 is also seen in broadband dielectric measurements on lower molecular weight PI.19 The phenomenon shown for PI is completely general. For every polymer for which sufficient measurements have been made, the local and chain modes exhibit different time-temperature shift factors. This is shown in Figure 5 for atactic polypropylene,20 wherein results from mechan- ical, dielectric, light scattering and NMR measurements are combined to yield relaxation times, τ, and a(T) encompassing more than 14 decades of frequency. The timescale for the segmental dynamics changes more with a given change in temperature than do the chain modes. MECHANICAL BEHAVIOR OF RUBBER AT HIGH STRAIN RATES 433 FIG. 4. – Time/temperature shift factors for PI from mechanical and dielectric spectroscopies.13 Typically segmental modes are measured at low T and chain modes at high T, leading to large extrapolation errors. Although the breakdown of time-temperature superpositioning is usually seen as a decrease in the height of the loss tangent peak with increasing temperature or poor overlap in the transi- tion zone of different isothermal data sets, for polyisobutylene (PIB) the effect is more dramat- ic: there is a large change in the shape of the loss tangent peak with changing temperature (Figure 6).21 This peculiar behavior is believed to arise from the viscoelastic contribution of “sub-Rouse” modes,17,22 which are chain segments too short to exhibit Gaussian statistics (and so cannot be regarded as Rouse modes) but too long to participate in the segmental dynamics associated with structural relaxation. It is ironic that one of the most marked examples of deviation from the superposition principle is seen in the polymer largely responsible for the universal acceptance of the very concept.23 434 RUBBER CHEMISTRY AND TECHNOLOGY VOL. 79 FIG. 5. – Local segmental relaxation times and time/temperature shift factors for atactic polypropylene determined by the indicated methods.20 The ordinate values for a(T) of the chain modes are shifted arbitrarily (the relaxation times for polymeric modes are always larger than the segmental relaxation times when compared at the same temperature). B. TEMPERATURE-VOLUME SUPERPOSITIONING The previous section describes the problems associated with invoking an equivalence between time and temperature effects in order to extrapolate dynamic properties to frequencies beyond those actually measured. An analogous approach can be used to generate master curves of moduli measured at different pressures9 but the data are scarce and interpretation of the results problematic. Dielectric measurements at elevated pressure are much more common, since the absence of moving parts facilitates immersion of the sample in a pressurizing fluid.24 From such data it has been found that local segmental relaxation times can be superposed when plotted as a function of temperature times the specific volume (V ≡ inverse of mass density), with the latter raised to a power of γ.25-27 The exponent γ is a material constant, usually determined by empiri- cal superpositioning of relaxation times measured at various combinations of temperature and pressure and plotted versus TVγ. The exponent can be related to the intermolecular repulsive potential, leading to a connection of γ to the Grüneisen parameter and to the pressure coefficient of Tg.28 If the relaxation times are expressed as a function of the configurational entropy, an expression can be derived showing that τ will be a function of TVγ.29 This seems to support an entropy basis for the arrest of molecular motions associated with the glass transition. Results for several rubbers are shown in Figure 7.24,26 Analogous behavior is observed for simple (small molecule) liquids in the supercooled regime; their structural relaxation properties, involving molecular reorientations and translations, parallel the local segmental dynamics of polymers.24 In fact, it is not possible to distinguish a polymer from a molecular liquid based on relaxation measurements near Tg. MECHANICAL BEHAVIOR OF RUBBER AT HIGH STRAIN RATES 435 FIG. 6. – Mechanical loss tangent for polyisobutylene measured at different temperatures.21 The two peaks reflect two viscoelastic mechanisms having different temperature dependences. The particular sample was the same as that used in Reference 23. As mentioned above, type-A polymers have a dipole moment parallel to the backbone and thus exhibit dielectrically-active normal modes, reflecting global motion of the chain. The chain modes not only have a different dependence on temperature than the local segmental modes (Figures 4 - 6), but also differ in pressure- and volume-dependences.30,31 Interestingly, however, the normal mode relaxation times superpose when plotted versus the same function TVγ using the same value of γ. Results are shown in Figure 8 for polyoxybutylene,32 Figure 9 for PI,33 and Figure 10 for polypropyleneglycol.33 Note that in each case, the variation of τ with TVγ is steep- er for the segmental mode than for the normal mode, consistent with the stronger T-, P-, and V- dependences of the former. 436 RUBBER CHEMISTRY AND TECHNOLOGY VOL. 79 FIG. 7. – Segmental relaxation times measured by dielectric spectroscopy for 1,2-polybutadiene (1,2-PB), polyvinylmethylether (PVME), polymethylphenysiloxane (PMPS), and polymethyltolylsiloxane (PMTS) versus the reciprocal of the product of temperature times specific volume with the latter raised to the indicated power. The different symbols for each rubber correspond to different measurement conditions: varying temperature at 0.1 MPa (!!) and varying pressure at different temperatures. MECHANICAL BEHAVIOR OF RUBBER AT HIGH STRAIN RATES 437 FIG. 8. – Scaled plots of the dielectric relaxation times for polyoxybutylene (M w = 5.3 kg/mol) measured at various conditions of T and P.32 Although the same value of γ (= 2.65) superposes both the local modes and the chain modes, the latter have a weaker dependence on TVγ. FIG. 9. – Scaled plots of the dielectric relaxation times for 1,4-polyisoprene (M w = 11.1 kg/mol) measured at various conditions of T and P33 (original data from Floudas et al.)30 Although the same value of γ (= 3.5) superposes both the local modes and the chain modes, the latter have a weaker dependence on TVγ. This scaling of relaxation times is not useful for extrapolating beyond frequencies actually measured; however, since data at elevated pressure can usually be obtained only over a limited frequency range,24,34 scaling enables extrapolation of high pressure data to the frequencies meas- ured at ambient pressure. Moreover, the scaling exponent γ can be obtained directly from pres- sure-volume-temperature data.28 This means that the variation of relaxation times with P and V can be determined without carrying out any relaxation measurements beyond ambient pressure. Since the deformation of rubber at very high strain rates often involves high pressures (e.g., the production of shock waves in rubber), characterization of the mechanical response may entail quantifying the effect of pressure. C. TEMPERATURE-PRESSURE SUPERPOSITIONING Sometimes the local segmental relaxation function (i.e., time-dependence of modulus or dielectric loss) does not change shape when pressure is varied at fixed temperature; an example is polymethylphenylsiloxane, shown in Figure 11.35 More usually there is some broadening as pressure increases, similar to the increase in breadth of the relaxation function with isobaric cool- ing towards Tg. The relaxation time determined for a given material at ambient pressure can be maintained constant with increases of both temperature and pressure; that is, various combina- tions of P and T can be found for which the frequency of the maximum in the mechanical or dielectric loss remains the same. 438 RUBBER CHEMISTRY AND TECHNOLOGY VOL. 79 FIG. 10. – Scaled plots of the dielectric relaxation times for polypropyleneglycol (M w = 4.0 kg/mol) measured at various conditions of T and P.33 Although the same value of γ (= 2.55) superposes both the local modes and the chain modes, the latter have a weaker dependence on TVγ. Recently it was discovered that at a fixed value of local segmental (or structural) relaxation time, the relaxation function is constant, independent of thermodynamic conditions.36-38 This means that T-P superpositioning applies to the dispersion in the mechanical or dielectric loss. Examples are shown in Figure 12 for 1,2-polybutadiene39 and 13 for poly(ethyl-co-vinyl acetate).40 The only exception to this behavior appears to be strongly hydrogen-bonded liquids, for which changes in T and P change the material itself (i.e., the degree of H-bonding), not just the dynamics.36 The fact that the shape of the relaxation function depends solely on the relax- ation time means that measurements at ambient pressure suffice to characterize the local seg- mental function for any T,P combination. The value of τ(T,P) can be determined by invoking the scaling procedure described in the preceding section, in combination with the equation of state (PVT relation). MECHANICAL BEHAVIOR OF RUBBER AT HIGH STRAIN RATES 439 FIG. 11. – Dielectric loss curves for segmental relaxation of PMPS measured at the indicated pressures.35 The data were shifted to superimpose on the curve for P = 42.4 MPa. The steep rise toward lower frequency is due to conductivity from mobile ions within the polymer.