Constant strain

Fig. 4. The effect of temperature on the pie2oelectric strain constant, for A, nylon-11 B, nylon-7 and C, poly(vinyhdene fluoride) (PVF2) films (35).

The uniaxial failure envelope developed by Smith (95) is one of the most useful devices for the simple failure characterization of many viscoelastic materials. This envelope normally consists of a log-log plot of temperature-reduced failure stress vs. the strain at break. Figure 22 is a schematic of the Smith failure envelope. Such curves may be generated by plotting the rupture stress and strain values from tests conducted over a range of temperatures and strain rates. The rupture locus moves counterclockwise around the envelope as the temperature is lowered or the strain rate is increased. Constant strain, constant strain rate, and constant load tests on amorphous unfilled polymers (96) have shown the general path independence of the failure envelope. Studies by Smith (97) and Fishman (29) have shown a path dependence of the rupture envelope, however, for solid propellants. 

Values of piezoelectric constants are, however, very scattered among polymers. In the case of oriented poly(y-methyl L-glutamate) film, the piezoelectric strain constant (d-constant) amounts to as much as 10 x 10 8 cgsesu when elongated in a direction at 45° to the draw-axis (Fukada, 1970), which is comparable with d = 6.5 x 10 8 cgsesu for X-cut... 

Fig. 4. Block diagram of the apparatus for measuring complex piezoelectric stress and strain constants of polymer films with varying frequency (Furukawa and...

Fig. 9. Piezoelectric strain constant of uniaxially drawn poly(y-methyl L-glutamate) film (a-helical form) plotted against the angle 6 between draw-axis and stress direction. Draw-ratio = 2. Drawn after Fukada, Date, and Hirai [Nature 211, 1079 (1966)] by permission of Macmillan (Journals) Ltd.

Fig. 11. Complex piezoelectric strain constant (20 Hz), complex Young s modulus (30 Hz), and complex dielectric constant (1kHz) of uniaxially drawn poly(D-propylene oxide) film plotted against temperature. Draw-ratio = 1.5. Degree of crystallinity=40%. Drawn after Furukawa and Fukada [Nature 221,1235 (1969)] by permission of Macmillan (Journals) Ltd.

Fig. 12. Complex piezoelectric strain constant of uniaxially drawn cellulose triacetate film plotted against temperature. Draw-ratio = 2. Plasticizer content = 10%. Frequency = 20 Hz. Drawn after Fukada, Date, and Emura [J. Soc. Mat Sci. Japan 17,335 (1968)] by permission of the Society of Materials Science, Japan...

Fig. 26. Correlation between increment of spontaneous polarization from 80° C to 15° C and piezoelectric strain constant at room temperature for /9-form polarized poly(vinylidene fluoride) films. Poling temperature = 90° C. Poling field = 700 kV/cm (Murayama, 1972)...

Fig. 29. Temperature dependence of complex piezoelectric strain constant of composite film of polyester resin and powdered PZT (50% of the volume) polarized at room temperature under a d.c. field of 100 kV/cm. Reproduced from Fukada and Date [Polymer Journal, 1,410 (1970)] by permission of the Society of Polymer...

Table 3. Piezoelectric strain constant of polymer films at room temperature...

Many types of fatigue tester are used (flexing beams, rotating beams, constant amplitude of cyclic stress or strain, constant rate of increase in amplitude of stress or strain, etc.). 

When a sinusoidal strain is imposed on a linear viscoelastic material, e.g., unfilled rubbers, a sinusoidal stress response will result and the dynamic mechanical properties depend only upon temperature and frequency, independent of the type of deformation (constant strain, constant stress, or constant energy). However, the situation changes in the case of filled rubbers. In the following, we mainly discuss carbon black filled rubbers because carbon black is the most widespread filler in rubber products, as for example, automotive tires and vibration mounts. The presence of carbon black filler introduces, in addition, a dependence of the dynamic mechanical properties upon dynamic strain amplitude. This is the reason why carbon black filled rubbers are considered as nonlinear viscoelastic materials. The term non-linear viscoelasticity will be discussed later in more detail. 

If a material undergoes a sudden infinitesimal shear strain y, the shear stress required to keep that shear strain constant is given by... 

F strain constants reflect only the steric repulsion of the attacking or leaving group of a reaction and contain no additional conformational effects. 

Harley, Hayes, and Smith [18] had measured the zone-center vibron energy ha> 0) of the TmAs04 crystal under external magnetic held. In that crystal, like in TmV04, the dynamic coupling is not zero in the presence of the external magnetic held only. As it is shown on the Fig. 7 the electronic excitation is the soft mode at the absence of the electron-strain interaction only (go = 0, dashed line). However when the electron-strain constant is not zero (all other lines on the Fig. 7) the... 

In stress relaxation experiments, the specimen is rapidly (ideally, instantaneously) extended a given amount, and the stress required to maintain this constant strain is measured as a function of time (Figure 13.4). The stress that is required to maintain the strain constant decays with time. When this stress is divided by the constant strain, the resultant ratio is the relaxation modulus (Ef(t,T), which is a function of both time and temperature. Figure 13.5 shows the stress relaxation curves for PMMA at... 

Piezoelectricity links the fields of electricity and acoustics. Piezoelectric materials are key components in acoustic transducers such as microphones, loudspeakers, transmitters, burglar alarms and submarine detectors. The Curie brothers [7] in 1880 first observed the phenomenon in quartz crystals. Langevin [8] in 1916 first reported the application of piezoelectrics to acoustics. He used piezoelectric quartz crystals in an ultrasonic sending and detection system - a forerunner to present day sonar systems. Subsequently, other materials with piezoelectric properties were discovered. These included the crystal Rochelle salt [9], the ceramics lead barium titanate/zirconate (pzt) and barium titanate [10] and the polymer poly(vinylidene fluoride) [11]. Other polymers such as nylon 11 [12], poly(vinyl chloride) [13] and poly (vinyl fluoride) [14] exhibit piezoelectric behavior, but to a much smaller extent. Strain constants characterize the piezoelectric response. These relate a vector quantity, the electrical field, to a tensor quantity, the mechanical stress (or strain). In this convention, the film orientation direction is denoted by 1, the width by 2 and the thickness by 3. Thus, the piezoelectric strain constant dl3 refers to a polymer film held in the orientation direction with the electrical field applied parallel to the thickness or 3 direction. The requirements for observing piezoelectricity in materials are a non-symmetric unit cell and a net dipole movement in the structure. There are 32-point groups, but only 30 of these have non-symmetric unit cells and are therefore capable of exhibiting piezoelectricity. Further, only 10 out of these twenty point groups exhibit both piezoelectricity and pyroelectricity. The piezoelectric strain constant, d, is related to the piezoelectric stress coefficient, g, by... 

Figure 3.43 Thermomechanical cycle in the tension direction for a specimen of T25C25 (step 1 —> pretension to 25% strain at temperatures above Tg, step 2 —> cooling down to room temperature while holding the pre-strain constant, step 3 —> unloading, which completes the first stage of programming. The Poisson effect is due to the second programming in the transverse direction by compression. Step 4 —> free shape recovery). Source [59] Reproduced with permission from the American Society of Civil Engineers...

For Sample 3, the stress increases linearly in step 1. While holding the strain constant, we see stress relaxation in step 2. Step 3 shows unloading instantly. In step 4, we see structural... 

Figure 5.12 Thermomechanical behavior of SMPFs by both cold and hot tension programmings, (a) Stress-strain-time diagram for Sample 2. Steps 1 to 5 complete programming and Step 6 completes stress recovery, where step 1 is to stretch the fiber bundle to 100% strain at a rate of200 ram/min at 100 °C step 2 is to hold the strain constant for 1 hour step 3 is to cool the fiber to room temperature slowly while holding the pre-strain constant step 4 is to release the fiber bundle from tbe fixture (unloading) step 5 is to relax the fiber in the stress-free condition until the shape is fixed and step 6 is to recover the fiber at 150 °C in the fully constrained condition (adapted from Reference [20]) (b) Stress-strain-time diagram for Sample 3. Steps 1-4 complete programming and step 5 completes stress recovery, where step 1 is to stretch the fiber bundle to 100% strain at a rate of 200 mm/min at room temperature step 2 is to hold the strain constant for 1 hour step 3 is to release the fiber bundle from fixtures (unloading) step 4 is to relax the fiber in the stress-free condition until the shape is fixed and step 5 is to recover the fiber at 150 °C in the fully constrained condition (adapted from Reference [20]) (c) Stress evolution with time for Sample 2 (d) Stress evolution with time for Sample 3.

Figure 5.28 Schematic of stress-strain behavior of SMPFs in the entire thermomechanical cycle by cold-drawing programming (step 1, loading step 2, holding the strain constant for a while (stress relaxation) step 3, unloading step 4, structural relaxation step 5, fiiUy constrained stress recovery)...

It is convenient for users of these elastic constants to get them from tables summarizing all these relations, such as Table 1.4. For isotropic materials, two independent elastic constants are sufficient (as indicated in the Table 1.4) for describing a stress-strain relation. There are different stress-strain constants for various other deformation conditions. 

There are five important figures of merit in piezoelectrics the piezoelectric strain constant d, the piezoelectric voltage constant g, the electromechanical coupling factor k, the mechanical quality factor Qm, and the acoustic impedance Z. These figures of merit are considered in this section. 

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