Curve master

At least four loading times are necessary for at least one temperature, and at least two loading times are necessary for other test temperatures, to construct the isotherms of the stiffness in terms of loading times and from that to derive the master curve. 

The stiffness modulus at any combination of temperature and loading time is derived from the master curve determined. 

The derivation of master curve from isotherms is explained in detail in CEN EN 12697-26 (2012), Annex G. 

For the determination of stiffness modulus, the conditions of temperature and loading time typically used are 15°C and 0.02 s, respectively (CEN EN 13108-20/AC 2008). 

5 Stress-Based Fatigue Analysis High Cycle Fatigue 

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The final composite curve is called the master curve at the temperature of the unmoved isotherm. 

The isothermal curves of mechanical properties in Chap. 3 are actually master curves constructed on the basis of the principles described here. Note that the manipulations are formally similar to the superpositioning of isotherms for crystallization in Fig. 4.8b, except that the objective here is to connect rather than superimpose the segments. Figure 4.17 shows a set of stress relaxation moduli measured on polystyrene of molecular weight 1.83 X 10 . These moduli were measured over a relatively narrow range of readily accessible times and over the range of temperatures shown in Fig. 4.17. We shall leave as an assignment the construction of a master curve from these data (Problem 10). 

Note that subtracting an amount log a from the coordinate values along the abscissa is equivalent to dividing each of the t s by the appropriate a-p value. This means that times are represented by the reduced variable t/a in which t is expressed as a multiple or fraction of a-p which is called the shift factor. The temperature at which the master curve is constructed is an arbitrary choice, although the glass transition temperature is widely used. When some value other than Tg is used as a reference temperature, we shall designate it by the symbol To. 

Williams and Ferryf measured the dynamic compliance of poly(methyl acrylate) at a number of temperatures. Curves measured at various temperatures were shifted to construct a master curve at 25°C, and the following shift factors were obtained ... 

Taking the concept of a master curve a step further, the relationships in Table 10.1 can also be incorporated into such curves so that graphs of z versus a characteristic dimension relative to X are plotted for various geometries. 

This allows the production of master curves, which can be used to estimate changes ia modulus or other properties over a long period of time by shorter tests over different temperatures. 

Master curves can also be constmcted for crystalline polymers, but the shift factor is usually not the same as the one calculated from the WLF equation. An additional vertical shift factor is usually required. This factor is a function of temperature, partly because the modulus changes as the degree of crystaHiuity changes with temperature. Because crystaHiuity is dependent on aging and thermal history, vertical factors and crystalline polymer master curves tend to have poor reproducibiUty. 

Master curves can be used to predict creep resistance, embrittlement, and other property changes over time at a given temperature, or the time it takes for the modulus or some other parameter to reach a critical value. For example, a mbber hose may burst or crack if its modulus exceeds a certain level, or an elastomeric mount may fail if creep is excessive. The time it takes to reach the critical value at a given temperature can be deduced from the master curve. Frequency-based master curves can be used to predict impact behavior or the damping abiUty of materials being considered for sound or vibration deadening. The theory, constmction, and use of master curves have been discussed (145,242,271,277,278,299,300). 

Curves for the viscosity data, when displayed as a function of shear rate with temperature, show the same general shape with limiting viscosities at low shear rates and limiting slopes at high shear rates. These curves can be combined in a single master curve (for each asphalt) employing vertical and horizontal shift factors (77—79). Such data relate reduced viscosity (from the vertical shift) and reduced shear rate (from the horizontal shift). 

Extensive tests have shown that if the final creep strain is not large then a graph of Fractional Recovery against Reduced Time is a master curve which... 

Thus all the different temperature related data in Fig. 2.58 could be shifted to a single master curve at the reference temperature (7 ). Alternatively if the properties are known at Tref then it is possible to determine the property at any desired temperature. It is important to note that the shift factor cannot be applied to a single value of modulus. This is because the shift factor is on the horizontal time-scale, not the vertical, modulus scale. If a single value of modulus 7, is known as well as the shift factor ar it is not possible to... 

The term r Vf in Equation (3.71) can be interpreted as a reduced fiber-volume fraction. The word reduced is used because q 1. Moreover, it is apparent from Equation (3.72) that r is affected by the constituent material properties as well as by the reinforcement geometry factor To further assist in gaining appreciation of the Halpin-Tsai equations, the basic equation. Equation (3.71), is plotted in Figure 3-39 as a function of qV,. Curves with intermediate values of can be quickly generated. Note that all curves approach infinity as qVf approaches one. Obviously, practical values of qV, are less than about. 6, but most curves are shown in Figure 3-39 for values up to about. 9. Such master curves for various vaiues of can be used in design of composite materiais. 

For a monolayer film, the stress-strain curve from Eqs. (103) and (106) is plotted in Fig. 15. For small shear strains (or stress) the stress-strain curve is linear (Hookean limit). At larger strains the stress-strain curve is increasingly nonlinear, eventually reaching a maximum stress at the yield point defined by = dT Id oLx x) = 0 or equivalently by c (q x4) = 0- The stress = where is the (experimentally accessible) static friction force [138]. By plotting T /Tlx versus o-x/o x shear-stress curves for various loads T x can be mapped onto a universal master curve irrespective of the number of strata [148]. Thus, for stresses (or strains) lower than those at the yield point the substrate sticks to the confined film while it can slip across the surface of the film otherwise so that the yield point separates the sticking from the slipping regime. By comparison with Eq. (106) it is also clear that at the yield point oo. 

MWDs p x) are plotted against reduced size x = 1/ L) of the chains for a number of densities (f) and are seen to collapse nicely on a single master curve, Fig. 5(a). The exponential decay, expected from Eq. (16) at high densities, is clearly observed in contrast to the indicated exp(-7x) behavior. This finding is in agreement with the simulations in Id [62], but it contradicts the predictions of Gujrati [15] according to whom the Shulz distribution, Eq. (16b), holds independently of the overlap. 

It turns out that a rather simple description of this nonlinear relaxation in terms of a single relaxation time,, depending on the final average chain length Loo, is suggested by a scaling plot of L t) for different L o, as shown in Fig. 18 for an initial exponential MWD. It is evident from Fig. 18 that the response curves, L o — L t), for different L o may be collapsed onto a single master curve, 1 - L t)lLoo = /(V Loo) measured in units of a... 

In fact, the variable x /Gi controls the "crossover" from one "universality class" " to the other. I.e., there exists a crossover scaling description where data for various Gi (i.e., various N) can be collapsed on a master curve Evidence for this crossover scaling has been seen both in experiments and in Monte Carlo simulations for the bond fluctuation model of symmetric polymer mixtures, e.g Fig. 1. One expects a scaling of the form... 

First, we will demonstrate from simple geometric arguments that the existence of a strain rate distribution across the tube gives further support to the previously described existence of a master curve (Fig. 42). 

A flow field analysis at fixed conical angle and varying orifice diameters confirmed that, all the strain rate distribution functions are exactly superpos-able onto a single curve when plotted against the dimensionless parameters s, = exx/(v0/r0) and x = x/r0. Three such master curves for different angles are... 

In order to produce the master curve illustrated, each section would have been completed in the time range 10 -10" s, but at different temperatures. Combining the sections then produces the overall master curve. 

Figure 7.10 Stress relaxation master curve at a given temprature...

FIGURE 1.7 Construction of a master curve of dynamic modulus /a versus log (frequency) by lateral shifting of experimental results made over a small frequency range but at several different temperatures. 

Dynamic mechanical measurements for elastomers that cover wide ranges of frequency and temperature are rather scarce. Payne and Scott [12] carried out extensive measurements of /a and /x" for unvulcanized natural mbber as a function of test frequency (Figure 1.8). He showed that the experimental relations at different temperatures could be superposed to yield master curves, as shown in Figure 1.9, using the WLF frequency-temperature equivalence, Equation 1.11. The same shift factors, log Ox. were used for both experimental quantities, /x and /x". Successful superposition in both cases confirms that the dependence of the viscoelastic properties of rubber on frequency and temperature arises from changes in the rate of Brownian motion of molecular segments with temperature. 

FIGURE 1.9 Master curves of in-phase fjJ) and out-of-phase (/r") components of the complex shear modulus of uncross-linked natural ruhher versus log (frequency) at Tg. (From Payne, A.R. and Scott, J.R., Engineering Design with Rubber, Interscience Publishers, New York, 1960.)... 

FIGURE 1.12 Master curve of tear energy Gc versus rate R of tear propagation at Tg for three cross-linked elastomers polybutadiene (BR, Tg — —96°C) ethylene-propylene copolymer (EPR, Tg — —60°C) a high-styrene-styrene-butadiene rubber copolymer (HS-SBR, Tg — —30°C). (From Gent, A.N. and Lai, S.-M., J. Polymer Sci., Part B Polymer Phys., 32, 1543, 1994. With permission.)... 

FIGURE 24.4 Master curves of the local segmental relaxation times for 1,4-polyisoprene (-y = 3.0) 1,2-polybutadiene (7=1.9) polyvinylmethylether (7 = 2.55) polyvinylacetate (7 = 2.6) polypropylene glycol (7 = 2.5) polyoxybutylene (7 = 2.8) poly(phenyl glycidyl ether)-co-formaldehyde (7 = 3.5) polymethylphe-nylsiloxane (7 = 5.6) poly[(o-cresyl glycidyl ether)-co-formaldehyde] (7 = 3.3) and polymethyltolylsiloxane (PMTS) (7 = 5.0) [15 and references therein]. Each symbol for a given material represents a different condition of T and P. 

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