Strain Stress decay

Stress-relaxation measurements, where stress decay is measured as a function of time at a constant strain, have also been used extensively to predict the long-term behavior of styrene-based plastics (9,12). These tests have also been adapted to measurements in aggressive environments (13). Stress-relaxation measurements are further used to obtain modulus data over a wide temperature range (14). 

One example of this occurs with stress relaxation. If a polymer is deformed to a fixed strain at constant temperature the force required to maintain that strain will decay with time owing to viscous slippage of the molecules. One measure of this rate of decay or stress relaxation is the relaxation time 0, i.e. the time taken for the material to relax to 1/e of its stress on initial application of strain. 

Many designs incorporate the phenomenon of stress-relaxation. For example, in many products, when plastics are assembled they are placed into a permanently deflected condition, as for instance press fits, bolted assemblies, and some plastic springs. In time, with the strain kept constant the stress level will decrease, from the same internal molecular movement that produces creep. This gradual decay in stress at a constant strain (stress-relaxation) becomes important in applications such as preloaded bolts and springs where there is concern for retaining the load. The amount of relaxation can be measured by applying a fixed strain to a sample and then measuring the load with time. 

We can get a first approximation of the physical nature of a material from its response time. For a Maxwell element, the relaxation time is the time required for the stress in a stress-strain experiment to decay to 1/e or 0.37 of its initial value. A material with a low relaxation time flows easily so it shows relatively rapid stress decay. Thus, whether a viscoelastic material behaves as a solid or fluid is indicated by its response time and the experimental timescale or observation time. This observation was first made by Marcus Reiner who defined the ratio of the material response time to the experimental timescale as the Deborah Number, Dn-Presumably the name was derived by Reiner from the Biblical quote in Judges 5, Song of Deborah, where it says The mountains flowed before the Lord. ... 

For shear strains greater than approximately 2 the ratio cr(r)/> 0 for a concentrated polystyrene solution was reduced at all observable times. For the large strains, relaxation proceeded more rapidly at short times, but at longer times the residua] stress decayed with about the same time dependence as that in the linear viscoelastic region. 

Natural rubber exhibits unique physical and chemical properties. Rubbers stress-strain behavior exhibits the Mullins effect and the Payne effect. It strain crystallizes. Under repeated tensile strain, many filler reinforced rubbers exhibit a reduction in stress after the initial extension, and this is the so-called Mullins Effect which is technically understood as stress decay or relaxation. The phenomenon is named after the British rubber scientist Leonard Mullins, working at MBL Group in Leyland, and can be applied for many purposes as an instantaneous and irreversible softening of the stress-strain curve that occurs whenever the load increases beyond... 

The model represents a liquid (able to have irreversible deformations) with some additional reversible (elastic) deformations. If put under a constant strain, the stresses gradually relax. When a material is put under a constant stress, the strain has two components as per the Maxwell Model. First, an elastic component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied. The Maxwell model predicts that stress decays exponentially with time, which is accurate for most polymers. It is important to note limitations of such a model, as it is unable to predict creep in materials based on a simple dashpot and spring connected in series. The Maxwell model for creep or constant-stress conditions postulates that strain will increase linearly with time. However, polymers for the most part show the strain rate to be decreasing with time [23-26],... 

Because all tissues are viscoelastic this means that their mechanical properties are time dependent and their behavior is characterized both by properties of elastic solids and those of viscous liquids. The classic method to characterize a viscoelastic material is to observe the decay of the stress required to hold a sample at a fixed strain (stress relaxation) or by the increasing strain required to hold a sample at a fixed stress (creep) as diagrammed in Figure 7.1 and explained further in Figure 7.2. Viscoelastic materials undergo processes that both store (elastic) and dissipate (viscous)... 

Stress relaxation is an alternative procedure. Here an instantaneous, fixed deformation is imposed on a sample, and the stress decay is followed with time. A very useful modification of these two basic techniques involves the use of a periodically varying stress or deformation instead of a constant load or strain. The dynamic responses of the body are measured under such conditions. 

Most real bodies are viscoelastic and obey laws (349) and (353) only under certain conditions. Hence, the concept of the stress decay time or the relaxation time x is introduced to characterize the stress-strain state of real bodies. For absolutely elastic bodies, x —> 0, whereas, for ideally viscous bodies, x > oo. Real viscous, anomalous viscous, and viscoelastic media are described in the interval 0 < x < oo. 

Transient shear flows involve examining the shear stress and viscosity response to a time-dependent shear. The stress build up at the start of steady flow (<7+) and at the cessation of steady flow (a ) and the stress decay (ff(0) after a dynamic instantaneous impulse of deformation strain (y) can be used to characterize transient rheological behaviour. 

Another example of a transient measurement is given in Figure 3.73. Figure 3.73(a) shows the start up of an imposed strain rate and the subsequent measurement of the stress build-up ((t" " (0) over time. Figure 3.73(b) shows the relaxation of an applied strain rate and the subsequent measurement of the stress decay (stress build-up for a polymer melt is shown in Figure 3.74. 

Figure 3.73. A transient rheological test showing (a) the start up of an imposed strain rate and the subsequent measurement of the stress build-up, and (b) the cessation of an imposed shear rate and the subsequent stress decay.

Host outstanding properties of these products were high resilience and good resistance to stress decay. Resilience Is Illustrated In Figure 1 which shows the stress-strain relationship of a poly[(plvalolactone-b-lsoprene-b-plvalolactone)-g-plvalo-lactone] fiber as It was stretched 300% and then allowed to relax. The shaded area Is the work lost as the fiber was loaded and then unloaded. This area amounts to 13% of the total, which shows that work recovered was 87%. Such high resilience compares very favorably with that of chemically-cured natural rubber. 

In stress relaxation experiments, the specimen is rapidly (ideally, instantaneously) extended a given amount, and the stress required to maintain this constant strain is measured as a function of time (Figure 13.4). The stress that is required to maintain the strain constant decays with time. When this stress is divided by the constant strain, the resultant ratio is the relaxation modulus (Ef(t,T), which is a function of both time and temperature. Figure 13.5 shows the stress relaxation curves for PMMA at... 

FIGURE 3.27 The time over which stress decay was observed for four NR elastomers. Initially viscoelasticity governs the relaxation time at higher strains crystallization commences (Choi and Roland, 1997). 

After a constant shear strain is created in a previously relaxed material, the resulting shear stress decays with ensuing time to zero for a viscoelastic liquid and to a finite equilibrium value for a viscoelastic solid. The shear stress relaxation modulus (Pa or dynes/cm )... 

If the standard linear solid (SLS) is unloaded from a constant stress, the spring (modulus ,) closes immediately and the elastic strain is removed. The anelastic strain then decays to zero as the second spring closes the dashpot, i.e., there is complete recovery. Under the action of a constant strain, the SLS model will also show stress relaxation but, in this case, the time constant, Tf =rf /(E +E2). In applying a constant stress to the SLS model, the strain can be considered to lag behind the stress, both on loading and unloading. This lag concept is also very important in considering the effect of a dynamic stress or strain. 

Methods to study crystallization of deformed elastomers include x-ray diffraction [207,208,263-265], optical birefringence [266,267], infrared or Raman spectroscopy, electron microscopy [268], dilatometry [269, 270], NMR [271], and mechanical measurements [193,262,272]. Strain-induced crystallization is manifested in the latter by both greater hysteresis (Fig. 23) and a longer time for stress decay (Fig. 24). However, the shape of the stress-strain curve during extension does not obviously reveal the onset of crystallization [207,208,262]. 

Tobolsky and his coworkers made extensive efforts to characterize the stress relaxation characteristics of elastomers, notably polyisobutylene. The stress would decay over time to zero at a rate dependent on temperature and molecular weight (Fig. 2). They expressed the relaxation through a series of exponentials or a spectrum of relaxation times. Consider the shear stress decay a t) following an shear imposed strain yo- This may be used to define a shear relaxation modulus G i) through... 

Upon instantaneous application of a constant strain, the stress will gradually relax down over time. This process is called the stress-relaxation experiment. From the condition de/df = 0, the Maxwell model solves the stress decaying with an exponential function, as given by... 

The effect of strain level on the relaxation behaviour has also been investigated and is shown in Figure 10.8. It was observed that the increase in strain level from 20 to 200% does not have any effect on the rate of relaxation of the unfilled rubber and the composites. The effect of ageing on the stress decay was also investigated and the rate of stress relaxation was found to decrease after ageing and is shown in Figure 10.9. 

Figure 10.9 Effect of strain level for NR filled with 40 phr Ti02- Effect of ageing on the stress decay T40.

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