Stress relaxation after cessation of steady-state flow

Stress Relaxation after Cessation of Steady-State Flow 

following steady shear flow of a viscoelastic liquid at a shear rate y, the flow is abruptly halted, the stress (initially equal to r o7) decays with a time dependence given by  

Comparison of this with equation 19 shows that the longer relaxation times are weighted relative to stress relaxation after sudden strain. Equation 64 can be inverted to obtain H from by approximation procedures (Chapter 4). Alternatively, in terms of the relaxation modulus G t), as given in equation 13 of Chapter 1, 

The steady-state compliance can also be related to r (l) by an equation related to equation 54  

These equations can be applied to elongational flow in the linear regime with appropriate substitutions. 

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At very low shear rates, the normal stress coefficients 1,0 and 2,0 are also independent of 721 i.e., the normal stress differences are proportional to 721. At higher shear rates, 1 and 2 are observed to decrease. The course of stress relaxation after cessation of steady-state flow and the magnitude of the steady-state compliance J° are also strongly affected at high shear rates. In general, description of these phenomena requires more complicated constitutive equations than the single-integral models mentioned above. 

In the glassy polymer, V, there is very little stress relaxation over many decades of logarithmic time, since no backbone contour changes occur in the densely crystalline polymer, VIII, there is some relaxation at very long times through whatever mechanism is responsible for the creep which also occurs in this region. Examples of stress relaxation after cessation of steady-state flow (Fig. 1-5) are not included here because of the limited applicability of this type of experiment. Such stress relaxation is of particular interest in connection with nonlinear phenomena, however, and will be illustrated in Section C below. 

There are other more complicated experimental situations where viscoelastic behavior can also be predicted in terms of the relaxation and retardation spectra or other functions. These include deformations at constant rate of strain and constant rate of stress increase, stress relaxation after cessation of steady-state flow, and creep recovery or elastic recoil, all of which were mentioned in Chapter 1, as well as nonsinusoidal periodic deformations. In referring to stress a, strain y, and rate of strain 7, the subscript 21 will be omitted here although it is understood that the discussion applies to shear unless otherwise specified. 

In Section F of Chapter 3, some more complicated viscoelastic experiments were formulated. The stress-strain or strain-stress curves can provide spectra through first differentiating to give the transient functions (equations 59 and 63 of Chapter 3) and then applying formulas such as equations 4 to 11. The stress relaxation after cessation of steady-state flow can provide the spectrum H through the following approximation relation ... 

In a similar test, the applied shear rate is suddenly reduced to zero. This process is called stress relaxation after cessation of steady-state flow. The decaying shear stress is measured as a function of time ... 

In stress relaxation after cessation of steady shear flow, the elastic dumbbells give no dependence of the relaxation process on the steady-state shear rate, but the rigid dumbbells do. In addition the elastic dumbbells show the shear and normal stresses relaxing with exactly the same... 

Stress Relaxation after Cessation of Steady-State Non-Newtonian Flow... 

FIG. 2-13. Relaxation of shear stress and primary normal stress difference after cessation of steady state flow, for a 4% solution of polystyrene with molecular weight 1.8 X 10 in chlorinated diphenyl as described in the text. Numbers refer to the shear rate preceding cessation of flow. 

The relaxation of the primary normal stress difference after cessation of steady-state flow at strain rate 7 can also be expressed in terms of linear viscoelastic properties by these models. For example, in terms of the relaxation spectrum, the rubberlike liquid theory of Lodge ° provides ... 

The primary normal stress coefficient during steady-state shear flow is also related to the relaxation of shear stress after cessation of steady-state flow, as shown by combining equations 67 and 74 ... 

For polymers with sharp molecular weight distribution, a terminal relaxation time T] can usually be determined experimentally from the flnal stages of stress relaxation either after sudden strain or after cessation of steady-state flow the latter kind of experiment weights the desired parameter more strongly as can be shown by equations 19 and 64 of Chapter 3, when expressed in terms of a discontinuous set of relaxation times rather than a continuous spectrum. Alternatively, it can be obtained from the constant Ag (the ratio of G /(tP at very low frequencies). Since the very narrow distribution of relaxation times in the terminal zone is close to a single terminal time t, which may be approximately identified with t , of the Graessley theory or of the Doi-Edwards theory (Section C3 of Chapter 10), equation 3 of Chapter 3 applies approximately and... 

Figure 9.38 Trace of first normal stress difference of a compression-molded 73/27 HBA/HNA copolyester specimen during transient and steady-state shear flow, and dnring the relaxation after cessation of steady-state shear flow. The normal stress before applying a sudden shear flow to the specimen is taken to be zero. (Reprinted from Han and Chang, Journal of Rheology 38 241. Copyright 1994, with permission from the Society of Rheology.)...

The nanocomposite PET-PEN/MMT clay was smd-ied under steady shear, instantaneous stress relaxation, and relaxation after cessation of steady flow [83]. Relaxation times of the slow mode in instantaneous stress relaxation were longer for the systems that have presumably permanent crosslinking networks (PET-PEN) or dynamic networks (PET-PEN-MMT). These results are consistent with those found in relaxation after cessation of flow (Fig. 31.4). Nanoclay addition somehow restricts the slow relaxation (due to polymer-particle interactions). The nanocomposite exhibits lower steady-state viscosity as compared to the polymer-matrix system. This is thought to be caused by polymer-polymer slipping, as revealed by the SEM observations (Fig. 31.5a and b). 

Measurements of normal stress differences during steady shear flow, and of normal stress growth approaching steady-state flow and stress relaxation after cessation of flow, provide additional information about nonlinear viscoelastic properties. The conventional identifications of the normal stresses for simple shear have been shown in Fig. 1-16 their orientations in several examples of experimental geometry are sketched in Fig. 5-5. 

The normal stresses present during the steady-state flow also relax after its cessation, and the course of the primary normal stress difference, (oi j — shear rate which precedes it. However, the normal stress difference relaxes more slowly than the shear stress. 

Most rheological measurements measure quantities associated with simple shear shear viscosity, primary and secondary normal stress differences. There are several test geometries and deformation modes, e.g. parallel-plate simple shear, torsion between parallel plates, torsion between a cone and a plate, rotation between two coaxial cylinders (Couette flow), and axial flow through a capillary (Poiseuille flow). The viscosity can be obtained by simultaneous measurement of the angular velocity of the plate (cylinder, cone) and the torque. The measurements can be carried out at different shear rates under steady-state conditions. A transient experiment is another option from which both y q and ]° can be obtained from creep data (constant stress) or stress relaxation experiment which is often measured after cessation of the steady-state flow (Fig. 6.10). 

System relaxation times have been determined from the relaxation of the stress after abrupt cessation of shear flow. Representative applications of the approach are found in Takahashi, etal. 9), who examined 355-3840 kDa polystyrenes in benzyl- -butylphthalate, at concentrations identified as showing dilute-solution behavior for the steady-state compliance Je and semidilute behavior for the zero shear viscosity. The stress relaxation after shear cessation, identified as the transient viscosity, decreased exponentially with time except at the shortest times studied, leading to an identification of an observed longest relaxation time Xm, whose c and M dependences were determined. 

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