Tangent delta curve

Figure 4.3 Tangent delta curve for tread compound property prediction [6].

The results of these measurements are listed in Table 5.2 and plotted in Figure 5.8. The tangent delta maximum value due to the glass-rubber transition is clearly visible in the curves measured at 3°C, -9°C and -17°C. 

In contrast with the dielectric constant, the tangent delta/temperature relation between 0°C and 240°C is not (detectably) influenced by these moisture concentrations. The tangent delta/temperature curves of both wet systems show, however, a clear relaxation effect with maxima between -60eC and -70°C. These effects disappear after drying i.e. they stem from the water phase. Such a low-temperature, low-frequency loss process was also detected in dynamic mechanical measurements. Banhegyi et al. reported such an effect due to absorbed water for CaC03 filled polyethylene samples [21]. 

Three of these tangent lines are drawn on the curve in Figure 2.1. You can see qualitatively how to draw them, but you cannot tell by inspection exactly what slope to use. This slope can be found by looking at two points Xo and xq + Ax, where Ax (the separation between the two points, pronounced delta x ) is small. We then determine the amount Ay = f(xr> + Ax ) — f(x) that the height of the curve changes between those two points. The ratio Ay/ Ax approaches the slope of the tangent line, in the limit that Ax is very small, and is called the derivative dy/dx. Another common shorthand is to write the derivative of /(x) as fix). 

Tan Delta n Mathematically expressed as the loss modulus divided by the storage modulus, the tangent of the phase angle between an applied stress and the strain response in a dynamic experiment. NOTE - Tan delta versus temperature curves are commonly reported in dynamic mechanical analysis (DMA) tests. 

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