Frequency dependence shear stress relaxation

We tiu-n from the frequency-dependent shear modulus and viscosity considered above to the time-dependent relaxation modulus. As mentioned in the previous section, we focus on the linear viscoelastic domain, in which the shear stress a t) depends linearly on the velocity gradient g(t) [2]. The relaxation modulus G(t) is now implicitly introduced through a relation between... 

The classical viscoelastic properties are the dynamic shear moduli, written in the frequency domain as the storage modulus G ( y) and the loss modulus G a>), the shear stress relaxation function G t), and the shear-dependent viscosity j (k). Optical flow birefringence and analogous methods determine related solution properties. Nonlinear viscoelastic phenomena are treated briefly in Chapter 14. 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

Figure 3.15 The frequency-dependent in-phase and out-of-phase components of the dynamic viscosity, rj and rj in small-amplitude oscillatory shear, along with the shear-rate dependence of the first normal stress coefficient hi (y) for a 0.05 wt% solution of polystyrene of molecular weight 2.25 X 10 in a solvent of oligomeric styrene. The lines through the data show the predictions of the Zimm theory for r and 2r)"f(o and the Zimm theory for hi(y) modified to account for finite extensibility, as discussed in Section 3.6.2.2.I. The dashed lines are the contributions of the individual Zimm relaxation modes to 2rj"((o) /

The Bird-Carreau model is an integral model which involves taking an integral over the entire deformation history of the material (Bistany and Kokini, 1983). This model can describe non-Newtonian viscosity, shear rate-dependent normal stresses, frequency-dependent complex viscosity, stress relaxation after large deformation shear flow, recoil, and hysteresis loops (Bird and Carreau, 1968). The model parameters are determined by a nonlinear least squares method in fitting four material functions (aj, 2, Ai, and A2). 

The shift factor ap can be used to combine time-dependent or frequency-dependent data measured at different pressures, exactly as ap is used for different temperatures in Section A above, and with a shift factor ar,p data at different temperatures and pressures can be combined. It is necessary to take into account the pressure dependence of the limiting values of the specific viscoelastic function at high and low frequencies, of course, in an analogous manner to the use of a temperature-dependent Jg and the factor Tp/Topo in equations 19 and 20. The pressure dependence of dynamic shear measurements has been analyzed in this way by Zosel and Tokiura. A very comprehensive study of stress relaxation in simple elongation, with the results converted to the shear relaxation modulus, of several polymers was made by Fillers and Tschoegl. An example of measurements on Hypalon 40 (a chlorosulfonated polyethylene lightly filled with 4% carbon black) at pressures from 1 to 4600 bars and a constant temperature of 25°C... 

This square-root dependence on tw is a fundamental featme of linear chains in the Rouse model. The shear modulus at intermediate frequencies is a signature of the internal, intra-chain dynamics, which is determined by the topology of the GGS. As stressed before, the viscoelastic relaxation forms can be expressed through the relaxation spectrum H r), see Eq. 27. Here one finds [3] ... 

The amount of the width-dependent shift can also be calculated. For each point of the wire cross section, the strain was minimized with respect to the nearest neighbors by applying the finite-element method [220-222]. These calculations take into account only the relaxed edges in the y direction. Every point of the wire cross section is then characterized by a particular phonon wave number, dependent on the stress in this point. More detailed calculations of the strain field in short wires or dots result in shear strains at the corners of the wires, which have to be also considered in the calculation of the wave-number shift [223]. The calculated position dependent frequencies show a strong inhom-... 

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