Stress-relaxation modulus

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. 

Object in this section is to review how rheological knowledge combined with laboratory data can be used to predict stresses developed in plastics undergoing strains at different rates and at different temperatures. The procedure of using laboratory experimental data for the prediction of mechanical behavior under a prescribed use condition involves two principles that are familiar to rheologists one is Boltzmann s superposition principle which enables one to utilize basic experimental data such as a stress relaxation modulus in predicting stresses under any strain history the other is the principle of reduced variables which by a temperature-log time shift allows the time scale of such a prediction to be extended substantially beyond the limits of the time scale of the original experiment. 

Stress Relaxation Modulus vs Age for Times of 0.01, 0.1 andiO Seconds. P 429... 

A8. The Helmholtz elastic free energy relation of the composite network contains a separate term for each of the two networks as in eq. 5. However, the precise mathematical form of the strain dependence is not critical at small deformations. Although all the assumptions seem to be reasonably fulfilled, a simpler method, which would require fewer assumptions, would obviously be desirable. A simpler method can be used if we just want to compare the equilibrium contribution from chain engangling in the cross-linked polymer to the stress-relaxation modulus of the uncross-linked polymer. The new method is described in Part 3. 

A new stress-relaxation two-network method is used for a more direct measurement of the equilibrium elastic contribution of chain entangling in highly cross-linked 1,2-polybutadiene. The new method shows clearly, without the need of any theory, that the equilibrium contribution is equal to the non-equilibrium stress-relaxation modulus of the uncross-linked polymer immediately prior to cross-linking. The new method also directly confirms six of the eight assumptions required for the original two-network method. 

Equation (3) is plotted with two different time scales in Figures 1 and 2 for values somewhat typical of an elastomer. All the initial deformation takes place in the spring at a later time the dashpot starts to relax and allows the spring to contract. Most of the relaxation takes place within one decade of time on both sides of the relaxation time, but this is shown clearly only in Figure 2. On the logarithmic time scale, the stress-relaxation curve has a maximum slope at the time / = T and the stress ratio cr/cr is 0.3679 ore. The stress relaxation may also be given in terms of a stress-relaxation modulus Er(t) ... 

The temperature-time superposition principle is illustrated in Figure 8 by a hypothetical polymer with a TK value of 0°C for the case of stress relaxation. First, experimental stress relaxation curves are obtained at a series of temperatures over as great a time period as is convenient, say from 1 min to 10 min (1 week) in (he example in Figure 8. In making the master curve from the experimental data, the stress relaxation modulus ,(0 must first be multiplied by a small temperature correction factor/(r). Above Tg this correction factor is where Ttrt is the chosen reference... 

If the Boltzmann superposition principle holds, the creep strain is directly proportional to the stress at any given time, f Similarly, the stress at any given lime is directly proportional to the strain in stress relaxation. That is. the creep compliance and the stress relaxation modulus arc independent of the stress and slrai . respectively. This is generally true for small stresses or strains, but the principle is not exact. If large loads are applied in creep experiments or large strains in stress relaxation, as can occur in practical structural applications, nonlinear effects come into play. One result is that the response (0 l,r relaxation times can also change, and so can ar... 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

For glassy and crystalline polymers there are few data on the variation of stress relaxation with amplitude of deformation. However, the data do verily what one would expect on the basis of the response of elastomers. Although the stress-relaxation modulus at a given time may be independent of strain at small strains, at higher initial fixed strains the stress or the stress-relaxation modulus decreases faster than expected, and the lloltz-nuinn superposition principle no longer holds. 

Figure 22 Stress-relaxation modulus as a function of Crystallinity at temperatures above I f Numbers on the curves are rough values of the degree of Crystallinity.

The temperature dependence of the compliance and the stress relaxation modulus of crystalline polymers well above Tf is greater than that of cross-linked polymers, but in the glass-to-rubber transition region the temperature dependence is less than for an amorphous polymer. A factor in this large temperature dependence at T >> TK is the decrease in the degree of Crystallinity with temperature. Other factors arc the reciystallization of strained crystallites ipto unstrained ones and the rotation of crystallites to relieve the applied stress (38). All of these effects occur more rapidly as the temperature is raised. 

Analogous results have been found for stress relaxation. In fibers, orientation increases the stress relaxation modulus compared to the unoriented polymer (69,247,248,250). Orientation also appears in some cases to decrease the rate, as well as the absolute value, at which the stress relaxes, especially at long times. However, in other cases, the stress relaxes more rapidly in the direction parallel to the chain orientation despite the increase in modulus (247.248,250). It appears that orientation can in some cases increase the ease with which one chain can slip by another. This could result from elimination of some chain entanglements or from more than normal free volume due to the quench-cooling of oriented polymers. 

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

The stress relaxation properties of a high molecular weight polybutadiene with a narrow molecular weight distribution are shown in Figure 1. The behavior is shown in terms of the apparent rubber elasticity stress relaxation modulus for three differrent extension ratios and the experiment is carried on until rupture in all three cases. A very wide rubber plateau extending over nearly 6 decades in time is observed for the smallest extension ratio. However, the plateau is observed to become narrower with increasing extension... 

PVF PVP Q R Polyvinyl formal Polyvinyl fluoride Polyvinyl pyrrolidone Charge (electric) Heat flow Universal gas constant Stress relaxation modulus Weight change per day Resistance... 

The shear stress relaxation modulus of the fluid, G(t), is a monotonically decreasing function of time, with G(oo) = 0. If the fluid initially at rest is given a small shear deformation y0 at t=0, the shear stress at later times becomes simply ... 

Finally, tensile deformations provide the same information as shear deformation as long as the incompressibility assumption is not violated. In this case, the tensile stress relaxation modulus E(t) is directly related to the shear modulus E(t) = 3G(f), and all other relationships follow accordingly. 

Completely rigid models appear to provide rather peculiar short time response. The stress relaxation modulus for rigid dumbbells is (102) ... 

For our present purposes, the network theories suffer from an additional defect. They supply no information on the form of the memory function. The memory function must be obtained for each system by rheological experiments, and there is no way at present to predict how it should vary with the molecular structure of the polymer. For example, M(t) can be obtained from the stress relaxation modulus G(t) ... 

For shear strains of sufficiently small amplitude, the response is linear and the shear stress is governed by the stress relaxation modulus ... 

The application of stress relaxation is shown in Figure 3. The relaxation modulus (G) is determined after a step strain as a function of time. A step strain is applied to the sample causing a stress. The modulus is measured as the stress relaxes. The stress relaxation modulus shows how molecular weight affects the relaxation process as a function of time as depicted in Figure 4. 

Shows how the stress relaxation modulus is related to molecular weight. 

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