Issue 33

A. Winkler et alii, Frattura ed Integrità Strutturale, 33 (2015) 262-288; DOI: 10.3221/IGF-ESIS.33.32 280 Let us assume, for the sake of argument, that we have a material point of interest, directly connected to the outside world with a film condition. In this case we take   /     a dq dx k , where  a is the ambient temperature and k the effective conductivity in Watts per cubic meter per Kelvin. Thus, the amount of heat removed is simply a linear function of the temperature difference between the material point and some arbitrary environment. In reality this is extremely difficult to justify, since the correct heat flux will be a function of the geometry and the actual heat removed from the specimen , and not the material point, and it is a complex function of environmental conditions. As for the heat generation term, this is the energy dissipation J which we already computed. To see whether we reach an equilibrium condition in terms of temperature or not, we need to set the rate of temperature change to 0, which yields     2 2 1 2 2 2 2 0 1 0 1 0 4              a a E k E E E E Applying a further simplification (we just can’t seem to stop ourselves), we assume that only  is a function of temperature, and that the elasticity will remain the same. This is of course not the case, as the stiffness will typically also drop as the temperature rises. At this point we need to introduce the temperature dependence of the viscosity. Common choices for doing so are the Arrhenius model or the Williams-Landel-Ferry (WLF) model [43]. The Arrhenius model tells us that when the viscosity is dominated by molecular kinetics that 0 exp           Q R In this equation, Q is the activation energy and R is the gas constant. The value 0  is a reference viscosity. The Arrhenius model is essentially the same as the Reynolds exponential model. The WLF model is more suitable for polymer melts, and would therefore be our primary choice. In this case we have   1 0 0 2 0 exp                   A A Where 0  is a reference temperature and 1 A and 2 A are constants. Trying to solve these equations analytically is not possible for either Arrhenius or the WLF form. We will thus have to resort to a numerical scheme to give us a feeling for the numbers. Note that this is a numerical scheme applied to an already simplified problem. The numerical approach would become much more involved if we were dealing with all the actual (multi-axial) behaviours. WLF Parameters 1 C [-] 2 C [C] 0 θ [C] 1 51.6 -15 Stiffness Properties 0 E [MPa] 1 E [MPa]  [MPa s] 7150 2285 4100 Heat Properties  [kg m -3 ] p c [J kg -1 K -1 ] k [W m -1 ] 1400 3000 126 Load Settings  a  a f 23 50 1 Table 2 : Properties for the equivalent uniaxial calculation. We will now invoke some numbers. Base properties are listed in Tab. 2. We are going to modify the loading settings to observe the influence pertaining to the temperature over time. The simplified analyses are going to take into account the dissipation, change of viscosity as a function of temperature, and convection of heat to the environment. Applying the simple model at this stage, together with the heat equation, we can observe the behaviour as given in Fig. 23 and Fig. 24. Reviewing the figures, it becomes evident that the rise in temperature is strongly dependent on the frequency with which the load is being applied. We note that for this simple model, even at 1Hz, we see a distinct change in