Shear stress relaxation time

Poisson coefficient effective concentration of cross-links Density electron density Relaxation time shear stress Volume fraction... 

Figure 10 Relaxation of shear stress with time upon cessation of steady flow. (O) Resin A, ( ) Resin B, (O) Resin C, (A) Resin D, (A) Resin E. (Refer to Table 2 for symbol code.) Source Ref. 56.

Rheological parameters, such as relaxation time, shear modulus, and stored elastic energy, are determined from the extrudate swell and stress-strain data as previously described. Representative examples of the variation of these parameters with blend ratios for both blends are shown in Figs. 16-18. Figure 16 shows that relaxation time for both preblends without heating and... 

Figures 11.15 and 11.16 show the relaxation of shear stress O12 and first normal stress difference Ni, respectfully, for Titan at 340 ° C after cessation of steady flow with shear rate y = 6 s T Here, experimental data are shown by dots and fitting curve by dashed lines. The values ofboth O12 and during relaxation drop abruptly and reach zero at the time of around 4 s. The simulated relaxation curve for O12 has an excellent agreement with the experimental data, but this is not the case for Ni when t > 4 s. This disagreement is seemingly attributed to the experimental error because the final value of Ni during relaxation should reach zero.

To approve this concept experimentally, single overlap shear specimens were subjected to stress relaxation at room temperature after imposing an initial strain of tan y = 1. During relaxation, time-dependent stress values were determined for 0.5,2,8, and 16 h (O Fig. 34.20). 

Of the adjustable parameters in the Eyring viscosity equation, kj is the most important. In Sec. 2.4 we discussed the desirability of having some sort of natural rate compared to which rates of shear could be described as large or small. This natural standard is provided by kj. The parameter kj entered our theory as the factor which described the frequency with which molecules passed from one equilibrium position to another in a flowing liquid. At this point we will find it more convenient to talk in terms of the period of this vibration rather than its frequency. We shall use r to symbolize this period and define it as the reciprocal of kj. In addition, we shall refer to this characteristic period as the relaxation time for the polymer. As its name implies, r measures the time over which the system relieves the applied stress by the relative slippage of the molecules past one another. In summary. 

Suppose we divide the flow segments into classes according to relaxation times and index the various states by the subscript i. Thus the relaxation time and the component of shear stress borne by the segments in class i are and Fj, respectively. The applied shear force is related to the Fj s through... 

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

This is the fundamental differential equation for a shear stress relaxation experiment. The solution to this differential equation is an equation which gives a as a function of time in accord with experiment. 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Not only are the creep compliance and the stress relaxation shear modulus related but in turn the shear modulus is related to the tensile modulus which itself is related to the stress relaxation time 0. It is therefore in theory possible to predict creep-temperature relationships from WLF data although in practice these are still best determined by experiment. 

The transition strongly affects the molecular mobility, which leads to large changes in rheology. For a direct observation of the relaxation pattern, one may, for instance, impose a small step shear strain y0 on samples near LST while measuring the shear stress response T12(t) as a function of time. The result is the shear stress relaxation function G(t) = T12(t)/ < >, also called relaxation modulus. Since the concept of a relaxation modulus applies to liquids as well as to solids, it is well suited for describing the LST. 

The longest relaxation time. t,. corresponds to p = 1. The important characteristics of the polymer are its steady-state viscosity > at zero rate of shear, molecular weight A/, and its density p at temperature 7" R is the gas constant, and N is the number of statistical segments in the polymer chain. For vinyl polymers N contains about 10 to 20 monomer units. This equation holds only for the longer relaxation times (i.e., in the terminal zone). In this region the stress-relaxation curve is now given by a sum of exponential terms just as in equation (10), but the number of terms in the sum and the relationship between the T S of each term is specified completely. Thus... 

As for the derivation of Eqs. 122,123 and 124 only the transitions 1—>2 have been counted, these equations do not describe recovery processes, where the transitions 2 —>1 are important as well. These approximations have been made for convenience s sake, but neither imply a limitation for the model, nor are they essential to the results of the calculations. Equation 124 is the well-known formula for the relaxation time of an Eyring process. In Fig. 65 the relaxation time for this plastic shear transition has been plotted versus the stress for two temperature values. It can be observed from this figure that in the limit of low temperatures, the relaxation time changes very abruptly at the shear yield stress Ty = U0/Q.. Below this stress the relaxation time is very long, which corresponds with an approximation of elastic behaviour. 

The transition from ideal elastic to plastic behaviour is described by the change in relaxation time as shown by the stress relaxation in Fig. 66. The immediate or plastic decrease of the stress after an initial stress cr0 is described by a relaxation time equal to zero, whereas a pure elastic response corresponds with an infinite relaxation time. The relaxation time becomes suddenly very short as the shear stress increases to a value equal to ry. Thus, in an experiment at a constant stress rate, all transitions occur almost immediately at the shear yield stress. This critical behaviour closely resembles the ideal plastic behaviour. This can be expected for a polymer well below the glass transition temperature where the mobility of the chains is low. At a high temperature the transition is a... 

If the applied shear stress varies during the experiment, e.g. in a tensile test at a constant strain rate, the relaxation time of the activated transitions changes during the test. This is analogous to the concept of a reduced time, which has been introduced to model the acceleration of the relaxation processes due to the deformation. It is proposed that the reduced time is related to the transition rate of an Eyring process [58]. The differential Eq. 123 for the transition rate is rewritten as... 

Consequently, A is called the relaxation time it is the time taken for the shear stress to fall to lie times the initial value. 

Most pigmented systems are considered viscoelastic. At low shear rates and slow deformation, these systems are largely viscous. As the rate of deformation or shear rate increases, however, the viscous response cannot keep up, and the elasticity of the material increases. There is a certain amount of emphasis on viscoelastic behavior in connection with pigment dispersion as well as ink transportation and transformation processes in high-speed printing machines (see below). Under periodic strain, a viscoelastic material will behave as an elastic solid if the time scale of the experiment approaches the time required for the system to respond, i.e., the relaxation time. Elastic response can be visualized as a failure of the material to flow quickly enough to keep up with extremely short and fast stress/strain periods. 

In order to proceed with the evaluation of the time-dependent Poisson ratio v(0, both sets of relaxation behaviour are required. Now from Chapter 2 we know the Poisson ratio is the ratio of the contractile to the tensile strain and that for an incompressible fluid the Poisson ratio v = 0.5. Suppose we were able to apply a step deformation as we did for a shear stress relaxation experiment. The derivation then follows the same course as that to Equation (4.69) ... 

Figure 6.3 Plot of a simple non-linear viscoelastic response for (a) the stress relaxation as a function of the applied strain, (b) stress as a function of time at a shear strain y = 1 and (c) viscosity as a function of shear stress. (r (0) = 33Pas, rj(co) = 3 Pas, a = 1, P = 0.1, m = 0.35 and t = Is). Continued overleaf...

It has been reported (4-6) that elastomers undergo very longterm relaxation processes in stress relaxation and creep experiments. The long time behavior of shear modulus can be represented by (18)... 

Figure 1 The shear stress relaxation function, C(t), obtained from a molecular dynamics simulation of500 SRP spheres at a reduced temperature of 1.0 and effective volume fraction of 0.45. Note that n = 144 and 1152 (from Equation (1)) cases are superimposable with the analytic function of Equation (4) ( Algebraic on the figure) for short times, t (or nt here)...

Figure 5 Time evolution of the shear stress relaxation function, C t), for the 36 18 potential at 4> = 0.2 and T = 0.3. The waiting times are = 3 and 162 for the two curves...

Under constant shear conditions, Eq. (5.68) predicts that the stress will exponentially disappear, so that after a time t = r]/G, the stress will have decayed to l/e of its original value. This time is referred to as the relaxation time, trei, for the material, so that Eq. (5.68) becomes... 

When a material is subjected to a tensile or compressive stress, Eqs. (5.63) through (5.74) should be developed with the shear modulus, G, replaced by the elastic modulus, E, the viscosity, rj, replaced by a quantity known as Trouton s coefficient of viscous traction, k, and shear stress, r, replaced by the tensile or compressive stress, a. It can be shown that for incompressible materials, k = 3r], because the flow under tensile or compressive stress occurs in the direction of stress as well as in the two other directions perpendicular to the axis of stress. Recall from Section 5.1.1.3 that for incompressible solids, E = 3G therefore the relaxation or retardation times are k/E. 

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