Viscoelastic relaxations

In the previous sections we have seen that the compliance and relaxation viscoelastic functions can be expressed in terms of the retardation and relaxation spectra, respectively. However, the spectra cannot be determined beforehand they can only be calculated from viscoelastic functions. For example, N s) and N s) in Eqs. (9.5) and (9.11) can be obtained by using the expressions... 

Because the relaxation spectra are similar for transient and dynamic relaxation viscoelastic functions, H t) can also be obtained from the storage relaxation modulus. The plot of the kernel of the integral of Eq. (9.8), x /(l + (o x ), versus logcax is a sigmoidal curve that intercepts the ordinate axis at 0.5 and reaches the value of 1 in the limit cox oo (see Fig. 9.5). The kernel can be approximated by the step function... 

Maxwell model A mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus E) in series with a dashpot of coefficient of viscosity (ri). It is an isostress model (with stress 8), the strain (e) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as stress relaxation and creep with Newtonian flow analysis. Also called Maxwell fluid model. See stress relaxation viscoelasticity. Maxwell-Wagner efifect See dielectric, Maxwell-Wagner effect. 

The various approaches to model polymer dissolution have accounted for stress relaxation, viscoelasticity of the polymer, disentanglement of polymer... 

Figure 3.44 Thermomechanical cycle in the compression direction for a spedmen of T25C25 (the deformation history starts with the Poisson effect due to the first tension programming at temperatures above Tg, followed by cooling down to room temperature. Then the second programming starts step 1 compression to 25% additional strain, step 2 —> hold the strain for 30 minutes, and step 3 —> unloading. After unloading, relaxation (viscoelastic rehound) occurs and after 24 hours it is stahiUzed, completing the second programming. Step 4 free shape recovery). Source [59] Reproduced with permission from the American Society of Civil Engineers...

Polymers are viscoelastic materials that exhibit characteristics of both solids and liquids. Their mechanical response is both time and temperature dependent, with what is often described as time-dependent elastic properties. As described in Section 4.2.3, viscoelastic materials will undergo creep and stress relaxation. Viscoelasticity is discussed in more detail in Chapter 5. 

In Section 28.2.2.2, we demonstrated that E-wave DT is determined by both chamber stiffness and chamber relaxation/viscoelasticity. Chamber stiffness can be measured both invasively, by AP/AV, and noninvasively, via the PDF parameter k. Chamber relaxation/viscoelasticity can be determined nonin-vasively from the PDF parameter c, but until recently, there was no established invasive, pressure-based analog of c. In current practice, the primary pressure-based index of relaxation is the time constant of isovolumic relaxation (x), but t is only a weak correlate of c [7]. Indeed, matched patients with indistinguishable X values may have distinguishable c values [79], and this maybe due to the temporal difference between when x (before MVO) and c (after MVO) are measured. Thus, the search for a pressure-based (hemodynamic) analog of c is likely to be informative. 

Shmuylovich L and Kovacs SJ. E-wave deceleration time may not provide an accurate determination ofLV chamber stifihess if LV relaxation/viscoelasticity is unknown. AmerfcaM Journal of Physiology-Heart and Circulatory Physiology 2007 292 H2712-20. 

The issues of the correlation of adliesion and of viscoelastic relaxation with friction are currently being investigated using AFM and LFM. Although friction does not correlate with the adliesion energy between two... 

Direct determination of relaxation time through viscoelastic studies (all mechanical properties involve this important parameter). 

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

The Maxwell and Voigt models of the last two sections have been investigated in all sorts of combinations. For our purposes, it is sufficient that they provide us with a way of thinking about relaxation and creep experiments. Probably one of the reasons that the various combinations of springs and dash-pots have been so popular as a way of representing viscoelastic phenomena is the fact that simple and direct comparison is possible between mechanical and electrical networks, as shown in Table 3.3. In this parallel, the compliance of a spring is equivalent to the capacitance of a condenser and the viscosity of a dashpot is equivalent to the resistance of a resistor. The analogy is complete... 

The relaxation and creep experiments that were described in the preceding sections are known as transient experiments. They begin, run their course, and end. A different experimental approach, called a dynamic experiment, involves stresses and strains that vary periodically. Our concern will be with sinusoidal oscillations of frequency v in cycles per second (Hz) or co in radians per second. Remember that there are 2ir radians in a full cycle, so co = 2nv. The reciprocal of CO gives the period of the oscillation and defines the time scale of the experiment. In connection with the relaxation and creep experiments, we observed that the maximum viscoelastic effect was observed when the time scale of the experiment is close to r. At a fixed temperature and for a specific sample, r or the spectrum of r values is fixed. If it does not correspond to the time scale of a transient experiment, we will lose a considerable amount of information about the viscoelastic response of the system. In a dynamic experiment it may... 

Whether a viscoelastic material behaves as a viscous Hquid or an elastic soHd depends on the relation between the time scale of the experiment and the time required for the system to respond to stress or deformation. Although the concept of a single relaxation time is generally inappHcable to real materials, a mean characteristic time can be defined as the time required for a stress to decay to 1/ of its elastic response to a step change in strain. The... 

Figure 36 is representative of creep and recovery curves for viscoelastic fluids. Such a curve is obtained when a stress is placed on the specimen and the deformation is monitored as a function of time. During the experiment the stress is removed, and the specimen, if it can, is free to recover. The slope of the linear portion of the creep curve gives the shear rate, and the viscosity is the appHed stress divided by the slope. A steep slope indicates a low viscosity, and a gradual slope a high viscosity. The recovery part of Figure 36 shows that the specimen was viscoelastic because relaxation took place and some of the strain was recovered. A purely viscous material would not have shown any recovery, as shown in Figure 16b. 

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... 

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