Shear modulus relaxation

Here is tire tensile stress and = lL-/L-, where is tire initial lengtli of tire sample and AL is tire sample elongation. In shear experiments, tire shear relaxation modulus G(t) is defined as where... 

The shear relaxation modulus of the bead-spring system is given by an... 

For an experiment carried out in shear it is possible to define a shear relaxation modulus at some time t as... 

The shear relaxation modulus can in general be written as an integral over the relaxation time spectrum H. At the same time Equation 3 can also be used. Thus, we have... 

Figure 5. Shear relaxation modulus of NR as a function of crosslink density at 25°C.

The configurational response to flow depends upon which of the normal modes interact frictionally with the flow field. In simple shear the distribution envelope in the flow direction alone is altered, and only the N normal modes associated with the flow direction are active. The polymer contribution to the shear relaxation modulus for a system with v chains per unit volume is ... 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

Figure 1.31 Shear relaxation modulus for a chlorosulfonated polyethylene at various pressures.

FIG. 13.57 Logarithmic plot of shear relaxation modulus vs. time at 25 °C and various pressures from 1 to 4600 bars as indicated for a chlorosulfonated polyethylene. From Fillers and Tschoegl (1977). Courtesy The American Institute of Physics. 

Figure 3.30 (a) The nonlinear shear relaxation modulus G (r, y)// as a function of time for various... 

As already noted, the measured nonlinear shear relaxation modulus, for linear molecules with little polydispersity, is in excellent agreement with the Doi-Edwards model at long times. However, for melts or concentrated solutions of very high molecular weight (e.g., 10 for polystyrene, where 0 is the polymer volume fraction), the measuredfiamping function, h(y), is drastically lower than the Doi-Edwards prediction (Einaga et al. 1971 Vrentas and Graessley 1982 Larson etal. 1988 Morrison and Larson 1992). This anomalous... 

Figure 5.8 Double logarithmic plot of the shear relaxation modulus in the time domain.

The combination of Eqs. (6.12) and (6.46) leads to the important relationship between the zero shear viscosity and the shear relaxation modulus (2,8,9),... 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

Other highly crystallinity polymers such as polyisopropylene and poly-oxymethylene also exhibit in order of increasing temperature the y, P, and a relaxation processes. It is worth noting that while polyisopropylene exhibits a well-developed P absorption, polyoxymethylene, like HDPE, exhibits two prominent a and y relaxations and a small P relaxation whose intensity seems to increase as the degree of crystallinity decreases (43). This behavior is illustrated in Figure 12.33, where both the shear relaxation modulus and the logarithmic decrement of polyoxymethylene are plotted against temperature. 

Find the temperature at which the storage shear relaxation modulus of PMMA at 10 Hz will be the same as at 1 Hz and 18°C. Activation energy for the P relaxation in this region is 75 kJ/mol. 

Because the shear relaxation modulus G t) with both its storage G (d) and loss parts G"(ffl) of the complex modulus are connected by Fourier transform it follows... 

If the rubber exhibits linear mechanical behavior, defined by proportionality between stress and strain and time invariance of the response, the Boltzmann superposition equation can be used to predict the stress for any strain history (Boltzmann, 1874 Larson, 1988). In terms of the shear relaxation modulus, the equation is... 

For the shear relaxation modulus, the Rouse model gives... 

The shear relaxation modulus Gs t) and the first normal-stress difference function G i(t), both normalized on a per-segment basis and with kT set to 1, are obtained from the constitutive equation of the Rouse model (Eq. (7.55) with Sp replaced by Tp) as... 

Tobolsky and his coworkers made extensive efforts to characterize the stress relaxation characteristics of elastomers, notably polyisobutylene. The stress would decay over time to zero at a rate dependent on temperature and molecular weight (Fig. 2). They expressed the relaxation through a series of exponentials or a spectrum of relaxation times. Consider the shear stress decay a t) following an shear imposed strain yo- This may be used to define a shear relaxation modulus G i) through... 

This constitutive equation includes only one material function G(t)> called the shear relaxation modulus. 

Shear relaxation modulus Shear stress relaxation modulus Shear loss modulus Complex shear modulus Enthalpy Henry... 

The sample as a whole is characterized by a shear-relaxation modulus G(f) that is equal to the instantaneous stress induced in the sample divided by the fixed shear deformation. It is convenient to represent the actual shear-relaxation modulus in terms of a distribution of relaxation times p(x). The relationship can then be expressed as ... 

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