Modulus time behavior

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)... 

In this general approach to viscoelasticity, appropriate models are constructed for the interpretation of the stress-strain-time behavior of a polymer. Then, values of Young s modulus G of the elastic elements and the viscosities i] of the viscous elements are used to characterize and predict the general behavior of the material. 

Accordingly, the loss compliance function presents a maximum in the frequency domain at lower frequency than the loss relaxation modulus. This behavior is illustrated in Figure 8.18, where the complex relaxation modulus, the complex creep compliance function, and the loss tan 8 for a viscoelastic system with a single relaxation time are plotted. Similar arguments applied to a minimum in tan 8 lead to the inequalities... 

The 10-second modulus at Tg is read directly from the master curve. Now, however, the master curve can be shifted to exhibit the behavior of the polymer at some other temperature. Applying this horizontal shift, with the slight additional vertical correction, if significant, allows one to "predict" the 10-second modulus, at this new temperature from the shifted curve. This procedure is repeated until the entire modulus-time curve is generated (Figure 4-7). 

Fortunately for linear amorphous polymers, modulus is a function of time and temperature only (not of load history). Modulus-time and modulus-temperature curves for these polymers have identieal shapes they show the same regions of viscoelastic behavior, and in each region the modulus values vary only within an order of magnitude. Thus, it is reasonable to assume from such similarity in behavior that time and temperature have an equivalent effect on modulus. Such indeed has been found to be the case. Viscoelastic properties of linear amorphous polymers show time-temperature equivalence. This constitutes the basis for the time-temperature superposition principle. The equivalence of time and temperature permits the extrapolation of short-term test data to several decades of time by carrying out experiments at different temperatures. 

These equations cannot be used indiscriminately. Each case must be considered on its merits, with account being taken of the plastic behavior in time imder load, mode of deformation, static and/or dynamic loads, service temperature, fabrication method, environment, and others. The traditional engineering equations are derived using the relationship that stress equals modulus time s strain, where the modulus... 

Maitra et al. determined the hardness of PVA with 0.6 wt% oxidized NDs using nanoindentation and the Oliver-Pharr method addition of nanofiller enhanced the hardness 80% of the neat polymer [48]. Hardness and modulus of FG/epoxy composites increased steadily with the incorporation of up to 1.5 wt% nanofiller. An increased amount of agglomerates was obtained at a loading of 2 wt% amino FGs as observed by the dramatic drop in the modulus this behavior also affected the nanocomposite hardness, as depicted in Figure 10.20 [113], The microhardness of amino FGs/PI nanocomposites showed a dependence on nanofiller content, although the dependence diminishes at loadings > 1 wt%, where the effect starts to saturate [115]. Covalently bonded amino NDs/epoxy composites showed a 200 times higher hardness compared to the neat... 

An adhesive is often subjected to a rupture test, in which the stress response of the material is measured in order to determine the utility of the adhesive. In such a test, one or a combination of several different modes of deformation—shear, extension, compression, torsion, or flexure—can be important. While one of these modes may resemble the application of interest more closely than the other modes, the knowledge obtained regarding material behavior from the different tests is similar in some cases, the information is the same. In other words, the information gathered in one experiment can often be predicted from the results of the other experiments. Although this is a gross simplification, one can, for purposes of illustration, cite the behavior of linearly elastic solids and purely viscous Newtonian liquids. While the former material is characterized by its elastic modulus, the behavior of the latter is determined by the (shear) viscosity. In the case of incompressible Hookean solids, the modulus of elasticity is three times the shear modulus. (See also Chapter 2 by Krieger.)... 

Coherent states and diverse semiclassical approximations to molecular wavepackets are essentially dependent on the relative phases between the wave components. Due to the need to keep this chapter to a reasonable size, we can mention here only a sample of original works (e.g., [202-205]) and some summaries [206-208]. In these, the reader will come across the Maslov index [209], which we pause to mention here, since it links up in a natural way to the modulus-phase relations described in Section III and with the phase-fiacing method in Section IV. The Maslov index relates to the phase acquired when the semiclassical wave function haverses a zero (or a singularity, if there be one) and it (and, particularly, its sign) is the consequence of the analytic behavior of the wave function in the complex time plane. 

At very short times the modulus is on the order of 10" ° N m comparable to ordinary window glass at room temperature. In fact, the mechanical behavior displayed in this region is called the glassy state, regardless of the chemical composition of the specimen. Inorganic and polymeric glasses... 

This result shows that the highest modes of response have the shortest relaxation times and influence the initial response of the sample. Conversely, the longest relaxation time is ti, which we can identify with the terminal behavior of the sample. For example, in Fig. 3.9 the final collapse of the modulus at long times occurs at Ti. An example will show how we can use this idea. 

The coefficient Tj is termed the modulus of rigidity. The viscosities of thixotropic fluids fall with time when subjected to a constant rate of strain, but recover upon standing. This behavior is associated with the reversible breakdown of stmctures within the fluid which are gradually reestabflshed upon cessation of shear. The smooth sprea ding of paint following the intense shear of a bmsh or spray is an example of thixotropic behavior. When viscosity rises with time at constant rate of strain, the fluid is termed rheopectic. This behavior is much less common but is found in some clay suspensions, gypsum suspensions, and certain sols. 

Master curves can be used to predict creep resistance, embrittlement, and other property changes over time at a given temperature, or the time it takes for the modulus or some other parameter to reach a critical value. For example, a mbber hose may burst or crack if its modulus exceeds a certain level, or an elastomeric mount may fail if creep is excessive. The time it takes to reach the critical value at a given temperature can be deduced from the master curve. Frequency-based master curves can be used to predict impact behavior or the damping abiUty of materials being considered for sound or vibration deadening. The theory, constmction, and use of master curves have been discussed (145,242,271,277,278,299,300). 

It is important to differentiate between brittie and plastic deformations within materials. With brittie materials, the behavior is predominantiy elastic until the yield point is reached, at which breakage occurs. When fracture occurs as a result of a time-dependent strain, the material behaves in an inelastic manner. Most materials tend to be inelastic. Figure 1 shows a typical stress—strain diagram. The section A—B is the elastic region where the material obeys Hooke s law, and the slope of the line is Young s modulus. C is the yield point, where plastic deformation begins. The difference in strain between the yield point C and the ultimate yield point D gives a measure of the brittieness of the material, ie, the less difference in strain, the more brittie the material. 

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). 

---