Modulus isochronous

ISOCHRONOUS STRESS - STRAIN CURVE CREEP MODULUS - TIME CURVE... 

Figure 9.10. Presentation of creep data sections through the creep curves at constant time and constant strain give curves of isochronous stress-strain, isometric stress-log (time) and creep modulus-log (time). (From ICI Technical Service Note PES 101, reproduced by permission of ICI...

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. 

The only unknown on the right hand side is a value for modulus E. For the plastic this is time-dependent but a suitable value may be obtained by reference to the creep curves in Fig. 2.5. A section across these curves at the service life of 1 year gives the isochronous graph shown in Fig. 2.13. The maximum strain is recommended as 1.5% so a secant modulus may be taken at this value and is found to be 347 MN/m. This is then used in the above equation. 

From the 3 year isochronous curve for polypropylene, the initial modulus, E =... 

The maximum stress or strain is not specified so an iterative approach is needed. From the 1 year isochronous for PP the initial modulus is 370 MN/m2... 

From the 1 year isochronous curve, the initial modulus = 370 MN/m2... 

Having suggested the connections between relaxation descriptors and the data it is now important to realize that here is sufficient information in isochronal scans that, with numerical analysis now readily carried out by computer, detailed parameters that describe relaxation can be determined jointly. Analysis is most conveniently carried out with the aid of a parameterized empirical phenomenological function. The method as implemented by us uses for each relaxation peak a Cole- Cole -like function (4) to represent the complex modulus,... 

Figure 4 shows stress-strain curves measured at an extension rate of 94% per minute on the TIPA elastomer at 30°, —30°, and —40°C. With a decrease in temperature from 30° to -40°C, the ultimate elongation increases from 170% to 600%. The modulus Ecr(l), evaluated from a one-minute stress-strain isochrone, obtained from plots like shown in Figure 1, increases from 1.29 MPa at 30°C to only 1.95 MPa at —40°C. This small increase in the modulus and the large increase in the engineering stress and elongation at fracture results from viscoelastic processes. 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

The most obvious problem of non-linearity is the definition of a modulus. For a linear viscoelastic material we need to define not only a real and an imaginary modulus but also a spectrum of relaxation times if we are fully to describe the material - although it is more usual to quote either an isochronous modulus or a modulus at a fixed frequency. We must, for a full description of a non-linear material give the moduli (and relaxation times) as a function of strain as well this will not usually be practicable so we satisfy ourselves by quoting the modulus at a given strain. The question then arises as to whether this... 

A coincidence between r and Tf, accompanied by the formation of a maximum of energy losses (peak of the loss modulus G" or of the loss factor tan <5 = G7G ), can be achieved by varying either the period Tf at constant temperature (isothermal experiment) or t (by changing the temperature) at the constant period Tf (isochronous experiment). The latter procedure is experimentally easier to implement and is therefore more frequently used. 

In order to know how is the variation of the mechanical properties of the polymers with temperature it is necessary to know the time of the measurements. In fact, E and D values obtained at different temperatures are comparable themselves if the time considered for the experiment is the same. Therefore the comparison of the experiments at different temperatures at the same time are isochrones [1-7,15-20], It is interesting to analyze the effect of the temperature on the elastic modulus. The classical schematic representation of this behaviour is shown on Fig. 2.4 ... 

Dynamic Mechanical Properties. Figure 15 shows the temperature dispersion of isochronal complex, dynamic tensile modulus functions at a fixed frequency of 10 Hz for the SBS-PS specimen in unstretched and stretched (330% elongation) states. The two temperature dispersions around — 100° and 90°C in the unstretched state can be assigned to the primary glass-transitions of the polybutadiene and polystyrene domains. In the stretched state, however, these loss peaks are broadened and shifted to around — 80° and 80°C, respectively. In addition, new dispersion, as emphasized by a rapid decrease in E (c 0), appears at around 40°C. The shift of the primary dispersion of polybutadiene matrix toward higher temperature can be explained in terms of decrease of the free volume because of internal stress arisen within the matrix. On the other... 

Figure 15. Temperature dispersion of isochronal complex, dynamic tensile modulus function at a fixed frequency of 10 Hz, observed for the SBS-PS specimen at unstrethed and stretched (330% elongation) states...

The isochrones showing the temperature dependence of the components of the complex relaxation modulus are rather insensitive to the crystallinity... 

The isochronal modulus will be the slope of the stress-strain response, and thus equal to the term in the brackets. As t - 0, the exponential can be expanded to show that the slope is simply ... 

Creep data is usually obtained for a number of different stresses, as creep modulus will only be independent of stress over limited ranges. It may also be important to obtain data as a function of temperature. Commonly, isochronous stress-strain curves are derived from the creep curves at different stress levels as a useful way of displaying the information. 

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