Relaxation, stress

Stress relaxation is rarely measured in plastics, creep testing being preferred. There are a number of standardised test methods for rubber as detailed by Brown [24]. 

Typical relaxation modulus data for several different materials. 

Creep experiments are particularly useful for the long time end of the relaxation spectra. As illustrated in Hgure 3.3.2, in a creep experiment the stress is increased instantly from 0 to tq and the strain is recorded versus time. 

Creep experiment, (a) Stress is increased from 0 to ro at t = 0. (b) Strain is recorded versus time, (c) Data are usually plotted as creep compliance. 

If we apply to the creep experiment the single relaxation time model, the simple Maxwell model, eq. 3.2.9, or eq. 3.2.16, we obtain 

Nevertheless, for the more general model it is not simple to calculate the compliance. Since r is the independent variable, we must invert the integral in eq. 3.2.7. 

In a stress relaxation test, a polymer test specimen is deformed by a fixed amount, eo, and the stress required to hold that amount of deformation is recorded over time. This test is very cumbersome to perform, so the design engineer and the material scientist have tended to ignore it. In fact, several years ago, the standard relaxation test ASTM D2991 was dropped by ASTM. Rheologists and scientists, however, have been consistently using the stress relaxation test to interpret the viscoelastic behavior of polymers. 

While the stress relaxation of a filled plastic as a function of temperature can be characterized in terms of a WLF-type function (Section 12.1.1.2 Williams et a/., 1955), a significant effect of strain has been noted by Narkis and Nicolais (1971) for copolymers of styrene and acrylonitrile filled with glass beads and examined at temperatures above the. As mentioned above. 

In other words, just as was shown by Andrews et al. (1948) for unfilled polymers, the ratio of two relaxation moduli for a given system at a common time but at different strains was found to be essentially independent of time, at least within a certain range of time and strain. Thus stress relaxation curves for a filled polymer at different strains 8 could be shifted along the modulus axis to give a master curve at a reference strain (Table 12.1). The corresponding modulus function becomes 

If a sample is subjected to a fixed strain the stress rises immediately and then falls with time to a final constant value, assuming that there is no flow, as shown in fig. 7.4. 

4 Stress-relaxation. The upper graph shows the applied strain as a function of time and the lower graph the resulting stress. 

A plot of log G t) against log t is similar to an inverted form of log J i) against logt. A characteristic relaxation time x analogous to x can be defined. 

Other factors that affect the stress relaxation behavior of polymer fibers include, but are not limited to molecular weight, molecular orientation, molecular polarity, crystalhnity, and moisture or other additives. In general, the stress relaxation of polymer fibers decreases with increases in molecular weight, molecular orientation, molecular polarity, crystallinity, and glass transition temperature. However, the introduction of moisture or other small molecules into polymer fibers can facilitate faster stress relaxation since these small molecules can improve the molecular mobility of polymer chains. 

The term r /E, the relaxation time, is the reciprocal of the rate at which stress decays. The linear viscoelastic region, of course, corresponds to E(t) being independent of Go. 

The alternative step-function experiment is stress-relaxation. A constant shear strain, say y, is applied at r = 0 and the stress r(t) required to maintain y, constant is observed. T(r) is found to decrease with lime, as shown in Fig. 4.5(a) (4.N.2). Suppose the specimen to be allowed to recover and a larger strain yj is then applied. The time dependence of the stress is shown in Fig. 4.5(b). At low strains (as in creep) it is observed that the isochronals are linear. 

The stresses in the two experiments at the same time t are proportional to the imposed strains. This fact leads to the definition of the stress-relaxation modulus at time t 

This is another manifestation of linear viscoelasticity it is observed in all polymers at strains below 0.5 x 10 .  

It is a simple matter to determine the strain range in which a specimen exhibits linear stress-relaxation behaviour it suffices to determine several isochronals, such as those shown in Fig. 4.5(c). The transition from the linear behaviour of Fig. 4.5(c) to non-linear is illustrated in Fig. 4.6. 

G (t) is determined most easily with a thin-walled tube (see Fig. 4.2). At r = 0 the tube is rotated through an angle 0 and the time dependence of the torque r(r) which keeps 0 constant is determined. Now the shear stress acting on a section (from Fig. 4.2) is, from eqn (4.5), 

Git) is measured 1 determining r(0 for fixed 6. Conversefy, in a design calculation, the time dependence of the torque in a tube twisted through an angle 6 may be obtained if Git) is known. 

7 Stress relaxation modulus observed in tension E(t) of polyisobutylene at different temperatures in the region of the glass-rubber relaxation (Tg -atW). At -83 at short time. E(t) approaches asymptotically the modulus of the glass at -40 C at long time. E(t) approaches asymptotically the modulus of the rubber. The relaxation is centred in the region of -66. Note the immense reduction in (t) of over 3 decades in a temperature rise of 43 C this behaviour is typical of amorphous polymers at the glass-rubber relaxation. 

When a viscoelastic material is subjected to a constant strain, the stress initially induced within it decays in a time-dependent manner. This behavior is called stress relaxation. The viscoelastic stress relaxation behavior is typical of many TPs. The material specimen is a system to which a strain-versus-time profile is applied as input and from which a stress-versus-time profile is obtained as an output. Initially the material is subjected to a constant strain that is maintained for a long period of time. An immediate initial stress gradually approaches zero as time passes. The material responds with an immediate initial stress that decreases with time. When the applied strain is removed, the material responds with an immediate decrease in stress that may result in a change from tensile to compressive stress. The residual stress then gradually approaches zero. 

The stress-relaxation behavior of a material is normally determined in either the tensile or the flexural mode. In these experiments, a material specimen is rapidly elongated or compressed to produce a specified strain level and the load exerted by the specimen on the test apparatus is measured as a function of time. Specimens of certain plastics may fail during tensile or flexural stress-relaxation experiments. 

Viscoelastic stress-relaxation data are usually presented in one of two ways. In the first, the stress manifested as a function of time. Families of such curves may be presented at each temperature of interest. Each curve representing the stress-relaxation behavior of the material at a different level of 

There are two further related sets of tests that can be used to give information on the mechanical properties of viscoelastic polymers, namely creep and stress relaxation. In a creep test, a constant load is applied to the specimen and the elongation is measured as a function of time. In a stress relaxation test, the specimen is strained quickly to a fixed amount and the stress needed to maintain this strain is also measured as a function of time. 

In creep tests, the parameter of interest is the creep compliance, y, defined as the ratio of the creep strain to the applied stress, i.e. 

Immediately the load is applied, the specimen elongates corresponding to an instantaneous elastic modulus. This is followed by a relatively fast rate of creep, which gradually decreases to a smaller constant creep rate. Typically this region of constant creep in thermoplastics essentially corresponds to viscous flow. In terms of the spring and dashpot model, the retardation is dominated by the viscous liquid in the dashpot. As before, 

Since the stress is constant, it follows that so also is the creep rate, The creep compliance at time I, /, can be considered to consist of three terms, an instantaneous compliance, /n, a term covering a variety of retardation processes, tp[t), and a viscous term, t/t. These are related by  

The last term represents irrecoverable flow which occurs in these polymers such that there is a permanent deformation which remains in the specimen after the load is removed. 

Having discussed the microscopic dynamical properties of a system of Rouse-chains, we now inquire about the resulting mechanical behavior and consider as an example the shear stress relaxation modulus, G t). G t) can be determined with the aid of the fluctuation-dissipation theorem, utilizing Eq. (6.7) 

To see the background of this equation, consider a unit area normal to the z-axiSj as indicated in Fig. 6.5. Stress on this plane is produced by all springs which cross it. The term 

the extensions along x and 2 of the spring I on the chain k are denoted Xk i and zjc i and those of the spring V on the chain k correspondingly. The sum includes all chains contained in i . As the extensions of springs in different chains are uncorrelated, we may furthermore write 

Chain dynamics may be represented as a superposition of independent Rouse-modes. The displacements of mode m, polarized in 2 -direction, are given by 

Equivalently, the extension associated with mode m, polarized along ar, is given by 

FIGURE 13-74 Schematic diagrams of (A) purely elastic response and (B) purely viscous response. 

The data are not usually reported as a stress/time plot, but as a modulus/time plot. This time-dependent modulus, called the relaxation modulus, is simply the time-dependent stress divided by the (constant) strain (Equation 13-71)  

FIGURE 13-76 Stress relaxation—a constant strain experiment. 

FIGURE 13-77 Stress relaxation of poIy(methyl methacrylate) [redrawn from the data of J. R. McLoughlin and A, V. Tobolsky. J. Colloid Sci., 7, 555(1952)]. 

The only reason for time is so that everything doesn t happen at once (Albert Einstein) 

This is tnie as well for PUs subjected to stress relaxation . And it applies here too. As is widely known, PUs are not perfectly elastic. Thus, when deformation is held constant, the induced stress relaxes gradually. 

Values of percentage stress relaxation, 10 minutes after straining to 300% elon- 

I = 110) following 300% elongation at nominal strain-rate 3.1x10 s . Time t is expressed in hours [61] 

Polymer PU structure tr300% MPa Stress relaxation % F IH % F 2H % Residual elongation % 

In order to express the bilinear form (k r) as a sum of three independent coordinate variables, let us first find the eigenvalues of the matrix k. For a shear deformation (9.169), they are given by 

Upon moving onto these coordinates, the average over a Gaussian chain distribution can be explicitly carried out by performing Gaussian integrals. 

In order to do this, we first use an identity using the identity 

In a similar way, the three eigenvalues of the strain tensor (9.116) in the case of elongational deformation are given by xj = X and xl = xl = l/X. Hence we have 

The number of active chains is given by the same formula as above, but the A s must be replaced. The elongational stress takes the form 

Properties of unreinforced plastics are strongly dependent on temperature and time. This is also true, to a lesser degree, for RPs, particularly RTSs, 

In structural design, it is important to distinguish between various modes in the product. The behavior of any material in tension, for example, is different from its behavior in shear, as with plastics, metals, concrete, etc. For viscoelastic materials such as plastics, the history of deformation also has an effect on the response of the material, since viscoelastic materials have time- and temperature-dependent material properties. 

Recognize that there are many stresses that cannot be accurately analyzed in plastics, metals, aluminum, etc. Thus one relies on properties that correlate with performance requirements. Where the product has critical performance requirements, such as ensuring safety to people, production prototypes will have to be exposed to the requirements it is to meet in service. 

The designer starts by one visualizing a certain family of material, makes approximate calculations to see if the contemplated idea is practical to meet requirements that includes cost, and, if the answer is favorable, proceeds to collect detailed data on a range of materials that may be 

In structural applications for plastics, which generally include those in which the product has to resist substantial static and/or dynamic loads, it may appear that one of the problem design areas for many plastics is their low modulus of elasticity. The moduli of unfilled plastics are usually under 7x10 MPa (1 x 10 psi) as compared to materials such as metals and ceramics where the range is usually 7 to 28 x 10 MPa (10 to 40 x 10 psi). However with reinforced plastics (RPs) the high moduli of metals are reached and even surpassed as summarized in Fig. 2.3. 

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Stress relaxation time, obtained from rheograms based on viscometric flows, is used to define a dimensionless parameter called the Deborah number , which quantifies the elastic character of a fluid... 

Bernstein, B., Kearslcy, E.A. and Zapaa, L., 1963. A study of stress relaxation with finite strain. Trans. Soc. Rheol. 7, 391-410. 

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. 

The purpose of these comparisons is simply to point out how complete the parallel is between the Rouse molecular model and the mechanical models we discussed earlier. While the summations in the stress relaxation and creep expressions were included to give better agreement with experiment, the summations in the Rouse theory arise naturally from a consideration of different modes of vibration. It should be noted that all of these modes are overtones of the same fundamental and do not arise from considering different relaxation processes. As we have noted before, different types of encumbrance have different effects on the displacement of the molecules. The mechanical models correct for this in a way the simple Rouse model does not. Allowing for more than one value of f, along the lines of Example 3.7, is one of the ways the Rouse theory has been modified to generate two sets of Tp values. The results of this development are comparable to summing multiple effects in the mechanical models. In all cases the more elaborate expressions describe experimental results better. 

Stress relaxation studies were conducted on samples of nylon yarn at a constant strain of 2% and the following results were obtained ... 

The isothermal curves of mechanical properties in Chap. 3 are actually master curves constructed on the basis of the principles described here. Note that the manipulations are formally similar to the superpositioning of isotherms for crystallization in Fig. 4.8b, except that the objective here is to connect rather than superimpose the segments. Figure 4.17 shows a set of stress relaxation moduli measured on polystyrene of molecular weight 1.83 X 10 . These moduli were measured over a relatively narrow range of readily accessible times and over the range of temperatures shown in Fig. 4.17. We shall leave as an assignment the construction of a master curve from these data (Problem 10). 

Figure 4.17 Experimental stress relaxation moduli of polystyrene measured over about two orders of magnitude in time at the temperatures indicated. [Reprinted with permission from H. Fujita and K. Ninomiya, J. Polym. Sci. 24 233 (1957).]...

Another aspect of plasticity is the time dependent progressive deformation under constant load, known as creep. This process occurs when a fiber is loaded above the yield value and continues over several logarithmic decades of time. The extension under fixed load, or creep, is analogous to the relaxation of stress under fixed extension. Stress relaxation is the process whereby the stress that is generated as a result of a deformation is dissipated as a function of time. Both of these time dependent processes are reflections of plastic flow resulting from various molecular motions in the fiber. As a direct consequence of creep and stress relaxation, the shape of a stress—strain curve is in many cases strongly dependent on the rate of deformation, as is illustrated in Figure 6. 

Weathering. Articles fabricated from FEP are unaffected by weather, and thek resistance to extreme heat, cold, and uv kradiation suits them for apphcations in radar and other electronic components. For example, after 15 years of solar exposure in Florida, the tensile strength (73) and light transmission (96%) of a 25-p.m thick film was unchanged and the film remained crystal clear. Elongation increased slightly for the first 5 to 7 years of outdoor exposure, probably as a result of stress relaxation. Beyond 10 years, a small decrease was observed. 

Snap-Fit and Press-FitJoints. Snap-fit joints offer the advantage that the strength of the joint does not diminish with time because of creep. Press-fit joints are simple and inexpensive, but lose hoi ding power. Creep and stress relaxation reduce the effective interference, as do temperature variations, particularly with materials with different thermal expansions. 

Creep, creep mpture, and stress relaxation tests are multiple-point tests requiring long periods of time (1000 h min) to generate useflil data these are standard tests for determining more fundamental polymer properties (202,203). Data for these tests are generated under several... 

Fig. 38. Stress—relaxation curve for a lightly vulcanized rubber (242). To convert Pa to dyn/cm, multiply by 10.

The stress—relaxation process is governed by a number of different molecular motions. To resolve them, the thermally stimulated creep (TSCr) method was developed, which consists of the following steps. (/) The specimen is subjected to a given stress at a temperature T for a time /, both chosen to allow complete orientation of the mobile units that one wishes to consider. (2) The temperature is then lowered to Tq T, where any molecular motion is completely hindered then the stress is removed. (3) The specimen is subsequendy heated at a controlled rate. The mobile units reorient according to the available relaxation modes. The strain, its time derivative, and the temperature are recorded versus time. By mnning a series of experiments at different orientation temperatures and plotting the time derivative of the strain rate observed on heating versus the temperature, various relaxational processes are revealed as peaks (243). 

Tensile Testing. The most widely used instmment for measuring the viscoelastic properties of soHds is the tensile tester or stress—strain instmment, which extends a sample at constant rate and records the stress. Creep and stress—relaxation can also be measured. Numerous commercial instmments of various sizes and capacities are available. They vary greatiy in terms of automation, from manually operated to completely computer controlled. Some have temperature chambers, which allow measurements over a range of temperatures. Manufacturers include Instron, MTS, Tinius Olsen, Apphed Test Systems, Thwing-Albert, Shimadzu, GRC Instmments, SATEC Systems, Inc., and Monsanto. 

A typical stress—strain curve generated by a tensile tester is shown in Eigure 41. Creep and stress—relaxation results are essentially the same as those described above. Regarding stress—strain diagrams and from the standpoint of measuring viscoelastic properties, the early part of the curve, ie, the region... 

Another resonant frequency instmment is the TA Instmments dynamic mechanical analy2er (DMA). A bar-like specimen is clamped between two pivoted arms and sinusoidally oscillated at its resonant frequency with an ampHtude selected by the operator. An amount of energy equal to that dissipated by the specimen is added on each cycle to maintain a constant ampHtude. The flexural modulus, E is calculated from the resonant frequency, and the makeup energy represents a damping function, which can be related to the loss modulus, E". A newer version of this instmment, the TA Instmments 983 DMA, can also make measurements at fixed frequencies as weU as creep and stress—relaxation measurements. 

The Imass Dynastat (283) is a mechanical spectrometer noted for its rapid response, stable electronics, and exact control over long periods of time. It is capable of making both transient experiments (creep and stress relaxation) and dynamic frequency sweeps with specimen geometries that include tension-compression, three-point flexure, and sandwich shear. The frequency range is 0.01—100 H2 (0.1—200 H2 optional), the temperature range is —150 to 250°C (extendable to 380°C), and the modulus range is 10" —10 Pa. 

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