Complex moduli

This behavior is usually analy2ed by setting up what are known as complex variables to represent stress and strain. These variables, complex stress and complex strain, ie, T and y, respectively, are vectors in complex planes. They can be resolved into real (in phase) and imaginary (90° out of phase) components similar to those for complex modulus shown in Figure 18. 

Fig. 18. Resolution of the complex modulus G into two vectors, G the storage modulus, and G the loss modulus the phase angle is 5.

The elastic and viscous modulus properties of mechanical strength are used to calculate the complex modulus and the loss compliance of compounds. These two parameters are used to assess dry traction and handling. Higher values relate directiy to dry traction and handling (cornering) within limits. 

Snow and wet traction are highly dependent on the tread pattern. Although the tread pattern overwhelms the compound properties in significance, the latter can play a role in optimizing snow traction. Compounds using polymers with low glass-transition temperature, T (—40 to —OS " C), remain more flexible at low temperatures. Tread compounds with low complex modulus at 0—20°C have better snow traction. 

We could thus represent these two moduli on a phasor diagram as shown in Fig. 2.54. El leads 2 by 90° (tt/2 radians) and from this diagram it is possible to define a complex modulus, E where... 

Fig. 2.54 Phasor diagram showing complex modulus ( ) relative to loss ( 2) and storage ( 1)...

In calculation the authors of the model assume that the cube material possesses the complex modulus EX and mechanical loss tangent tg dA which are functions of temperature T. The layer of thickness d is composed of material characterized by a complex modulus Eg = f(T + AT) and tg <5B = f(T + AT). The temperature dependences of Eg and tg SB are similar to those of EX and tg <5A, but are shifted towards higher or lower temperatures by a preset value AT which is equivalent to the change of the glass transition point. By prescibing the structural parameters a and d one simulates the dimensions of the inclusions and the interlayers, and by varying AT one can imitate the relationship between their respective mechanical parameters. 

During dynamic measurements frequency dependences of the components of a complex modulus G or dynamic viscosity T (r = G"/es) are determined. Due to the existence of a well-known analogy between the functions r(y) or G"(co) as well as between G and normal stresses at shear flow a, seemingly, we may expect that dynamic measurements in principle will give the same information as measurements of the flow curve [1],... 

This exchange process leads to a decrease in the modulus of elasticity and is therefore a dissipative contribution to the complex modulus [46] ... 

The value of this latter parameter is proportional to the energy dissipated as heat per cycle, and is known as the loss modulus. The former quantity, Gj, is proportional to the recoverable energy, and is called the storage modulus. The two are combined to form the complex modulus, G related by the equation... 

FIGURE 30.9 Modeling the strain dependence of complex modulus. 

According to test protocols described above, RPA-FT test were performed at 100°C, 1 Hz on all samples additional tests at 100°C, 0.5 Hz were performed on IMA TR, IMA FM, and IMA-AG samples. Essentially three types of data will be discussed hereafter The complex modulus G (as derived from the main torque component in the FT torque spectrum), the corrected total torque harmonic component, i.e., cTTHC, and the Q1/Q2 ratio. 

Data in Table 30.1 clearly show that, whatever the position of the test sample along the compounding line, there is a substantial difference between run 1 and run 2 data, particularly in what the linear modulus data are concerned. However, G is an extrapolated value and quite unrealistic values are obtained on certain samples, e.g., TR and AA, and it might be safer to consider modulus variations along the compounding line by using the (recalculated) complex modulus at 10% strain (Figure 30.13). 

FIGURE 30.12 Ethylene-propylene-diene monomer (EPDM) compounding complex modulus strain dependence samples TR, EM, and AG. 

Ethylene-Propylene-Diene Monomer (EPDM) Compounding RPA-FT Results at 100°C Complex Modulus Dependence on Strain Eit Parameters of Equation 30.3... 

FIGURE 30.20 Mixing siUca-fiUed compound Complex modulus strain dependence at selected position along the mixing hue. 

FIGURE 30.22 Mixing silica-filled compound Complex modulus versus strain amplitude as modeled with Equation 30.3 typical changes in nonlinear viscoelastic features along the mixing hne. 

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 6 Loss tangent (<5) plotted as a function of complex modulus G for a series of syndiotactic and isotactic polypropylene. Reproduced with permission from Rojo et al. [32]. Copyright Wiley-VCH Verlag GmbH Co. KgaA. iPP, isotactic PP sPP, syndiotactic PP Met-iPP, metallocene-catalyzed PP (see Ref. [32] for full details).

The ratio of the stress to the strain is the complex modulus G (a>). We can rearrange this expression to give the complex modulus and the frequency, and using Equation (4.14) we have ... 

This expression describes the variation of the complex modulus with frequency for a Maxwell model. It is normal to separate the real and imaginary components of this expression. This is achieved by multiplying through by (1 — icut) to give... 

The storage modulus increases from zero to a maximum value in fact as the frequency tends to infinity the complex modulus and G (oS) equate, i.e. 

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