Temperature-Dependent E-Modulus

Gibson et al. [8] presented a temperature-dependent E-modulus model in 2004. Mechanical properties were assumed to degrade during the glass transition as described by Eq. 5.3 ... 

A theoretical model for a temperature-dependent E-modulus was developed by Mahieux and Reifsnider [9,10]. In this model, WeibuU-type functions were used to describe the modulus change, over the full range of transition temperatures. 

An empirical model temperature-dependent E-modulus was proposed by Gu and Asaro in 2005 [11] ... 

Substituting the theoretical results of and into Eq. 5.6, and taking Eg = 12.3 GPa as the original modulus (modulus of glassy state) and Bj. = 3.14GPa as the modulus at approximately 250 °C (modulus of leathery or rubbery state) from DMA experiments, the temperature-dependent E-modulus can be obtained. 

On the basis of the model developed in Chapter 5, the temperature-dependent B-modulus, E(T), determined by Eq. (5.6) normalized by the value at ambient temperature (glassy state), E, is calculated as ... 

However, when a material is a structural composite of resin and continnons reinforcement such as carbon fiber or woven E-glass, the task of representation by finite element methods is considerably more complicated. Regardless of this difficulty, there are a considerable number of workers who specialize in these types of thermo-mechanical models, [64-88], who have published modeling woik specifically in this area for many years. The key concept used in such models is the idea of temperature-dependent material modulus, and how this can be calculated in tandem with through-thickness solid-temperature models generated in the Henderson tradition. 

For a semi-crystalline polymer the E-modulus shows between Tg and (in which region it is already lower than below Tg), a rather strong decrease at increasing T, whereas with amorphous polymers, which are used below Tg, the stiffness is not much temperature dependent (apart from possible secondary transitions). The time dependency, or the creep, shows a similar behaviour. 

One of the simplest criteria specific to the internal port cracking failure mode is based on the uniaxial strain capability in simple tension. Since the material properties are known to be strain rate- and temperature-dependent, tests are conducted under various conditions, and a failure strain boundary is generated. Strain at rupture is plotted against a variable such as reduced time, and any strain requirement which falls outside of the boundary will lead to rupture, and any condition inside will be considered safe. Ad hoc criteria have been proposed, such as that of Landel (55) in which the failure strain eL is defined as the ratio of the maximum true stress to the initial modulus, where the true stress is defined as the product of the extension ratio and the engineering stress —i.e., breaks down at low strain rates and higher temperatures. Milloway and Wiegand (68) suggested that motor strain should be less than half of the uniaxial tensile strain at failure at 0.74 min.-1. This criterion was based on 41 small motor tests. 

Fig. 10. Temperature dependence of swelling ratio X and modulus G (g cm 2) for the poly(N,N -diethylacrylamide/sodium methacrylate) gels in water. The networks A, B, C, D, E, F, G and H were prepared with molar fractions xMNa = 0, 0.0045, 0.0095, 0.0157, 0.0234, 0.0310, 0.0432 and 0.0667, respectively (O) X values, ( ) G values. From Ilavsky et al. [18]...

In the case of PPO (Fig. 11), CDA, and CTA (Fig. 12), d decreases with increasing temperature in the region of mechanical relaxations and d" exhibits peaks in the same region. On the other hand, as illustrated in Fig. 13, d in polypeptides has a maximum and a minimum, and d" accordingly changes its sign with temperature. This maximum and minimum of d may, however, be mostly ascribed to the temperature dependence of the elastic modulus, and e shows a similar behavior to that in PPO and other polymers. 

In addition to knowing the temperature shift factors, it is also necessary to know the actual value of ( t ) at some temperature. Dielectric relaxation studies often have the advantage that a frequency of maximum loss can be determined for both the primary and secondary process at the same temperature because e" can be measured over at least 10 decades. For PEMA there is not enough dielectric relaxation strength associated with the a process and the fi process has a maximum too near in frequency to accurately resolve both processes. Only a very broad peak is observed near Tg. Studies of the frequency dependence of the shear modulus in the rubbery state could be carried out, but there... 

In this paragraph comparatively much attention will be paid to the curve in which tensile stress is plotted out in relation to relative elongation, because important properties can be inferred from this curve. One of these is the elastic modulus, a material property which was briefly discussed in chapter 9. This E-modulus often depends on the temperature and this relationship is represented in a log E-T curve. Next properties of the three groups of materials are compared in a table and finally some attention will be paid to processing and corrosion . 

Fig. 1 Typical example of the temperature dependence of the mechanical loss modulus, E", showing a, ft and y transitions...

For amorphous polymers, the glass-rubber transition is usually referred as the a transition, the solid state transitions (frequently called sub-Tg or secondary transitions) being designated by /3, y, S,... A typical example is shown in Fig. 1 with the temperature dependence of the mechanical loss modulus, E", which exhibits several peaks corresponding to the various transitions. 

Fig. 90 Temperature dependence and reproducibility of the loss modulus, E"y for the 1.81 and 1.8 T copolyamides (from [60])...

Fig. 105 Temperature dependence for PMMA at 1 Hz of a the loss modulus, E" and b the loss compliance, / (from [75])...

However, it is interesting to perform a more direct comparison of the experimental results to check whether some differences between the mechanical and dielectric behaviours could exist as a function of temperature. The appropriate quantity is E" for the mechanics and, for the dielectric response, it is the dielectric loss modulus, m" (defined as e"/ sa + s"2)). Figure 112 shows the temperature dependence of E" and m" at 1 Hz, obtained by superposing the low-temperature part of the j3 transition. 

Fig. 127 Temperature dependence of the loss modulus, E", at 1 Hz, for PMMA and various CMIMx copolymers (from [79])...

The ft transition of a series of MGIMx copolymers has been investigated by dynamic mechanical analysis [80]. The temperature dependence of the loss modulus, E"y at 1 Hz is shown in Fig. 138. In the region of the a transition, when increasing the MGI content, a shift towards a higher temperature is observed. 

The temperature dependence of the yield stress, ay, of PMMA obtained at a strain rate, s = 2 x 10-3 s-1, is shown in Fig. 18. A sigmoidal curve is observed, which looks like the temperature dependence of the Young s modulus, E. When increasing the strain rate a similar behaviour is observed. 

In order to check whether the temperature dependence of oy would reflect the change of modulus only, the ratio cry/E is plotted in Fig. 19. It is clear that the modulus does not normalise the yield stress behaviour, the latter decreasing more than the modulus when temperature increases. 

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