Stress-strain relation isochronal

Figures 29 A and B show the original data from general biaxial extension measurements on the NR sample. Here the measured stresses at and a2 at a series of fixed Xx are plotted against X2. All these values are isochronal (10 min). The graphs have been displayed to illustrate the accuracy of our measurements. In Fig. 30, the observed at are compared with the predictions from other stress-strain relations.

At very small strains, the term in exp(0.9o ) becomes negligible, and the isochronous stress-strain relation is linean... 

As indicated above, the stress-strain presentation of the data in isochronous curves is a format which is very familiar to engineers. Hence in design situations it is quite common to use these curves and obtain a secant modulus (see Section 1.4.1, Fig. 1.6) at an appropriate strain. Strictly speaking this will be different to the creep modulus or the relaxation modulus referred to above since the secant modulus relates to a situation where both stress and strain are changing. In practice the values are quite similar and as will be shown in the following sections, the values will coincide at equivalent values of strain and time. That is, a 2% secant modulus taken from a 1 year isochronous curve will be the same as a 1 year relaxation modulus taken from a 2% isometric curve. 

From isochronous stress-strain curves relating to endurance times of, say, 1, 10, 100, and 1000 hours, and so on, the magnitude of the stress to give the critical strain at each duration of loading can be easily deduced. This procedure can be repeated for different selected levels of strain. In general, the more critical the application and the longer the time... 

One way to obtain long-term information is through the use of the time-temperature-superposition principle detailed in Chapter 7. Indeed, J. Lohr, (1965) (the California wine maker) while at the NASA Ames Research Center conducted constant strain rate tests from 0.003 to 300 min and from 15° C above the glass transition temperature to 100° C below the glass transition temperature to produce yield stress master curves for poly(methyl methacrylate), polystyrene, polyvinyl chloride, and polyethylene terephthalate. It should not be surprising that time or rate dependent yield (rupture) stress master curves can be developed as yield (rupture) is a single point on a correctly determined isochronous stress-strain curve. Whether linear or nonlinear, the stress is related to the strain through a modulus function at the yield point (mpture) location. As a result, a time dependent master curve for yield, rupture, or other failure parameters should be possible in the same way that a master curve of modulus is possible as demonstrated in Chapter 7 and 10. 

In determining creep properties, a series of specimens are subjected to static loads at different levels and their increase in strain over time is measured. Data may be either presented directly as creep strain on a logarithmic time scale, as creep modulus c(t)= isochronous stress—strain diagrams where the relation between stress on the axis of ordinates over strain on the axis of abscissa is presented for different time levels. The schematic approach to converting creep data into an isochronous stress-strain diagram is illustrated in Fig. 34.7. 

Tensile test at different temperatures (23°C, 40°C, 60°C, 80°C) and strain rates (1 %/h to 100 %/h) as well as creep and relaxation experiments were accomplished according to DIN EN ISO 53455 and DIN 53444 respectively. Based on creep experiments isochronous stress-strain-diagrams were constructed to describe the relation between stress and strain at defined times. 

Fig. 12 Isochronous stress vs. volumetric strain curves for a LDPE foam of 60 kg/m density for different creep times (hours). The slope of the linear fits is related to the pressure...

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