Viscoelastic relaxation modulus

The time dependency of the stress-strain of viscoelastic materials presents a difficulty in testing this class of materials. In order to develop a meaningful test, a time-dependent relaxation modulus Ejft) introduced is defined as 

All thermoplastic polymers start out with a glassy behavior below the glass transition temperature Tg. As T is increased above Tg, they all become more ductile or leathery and later become softer or more rubber-like. A semicrystaUine isotactic pol)nner will retain its strength longer than an amorphous polymer because its crystalline structure permits more elastic deformation while the others deform viscoelactically. Eventually, semicrystalline isotactic polymer as well as the amorphous polymer will turn into a viscous liquid as they eventually melt. A cross-linked polymer softens somewhat at the glass temperature but does not melt because of the strong covalent cross-link bonds. It will decompose before these bonds are broken. 

Sdiematic of the relaxation modulus plotted against temperature for a viscoelastic polymer. 

Ceramics tend to be very brittle because (1) it is more difficult to form dislocations because a dislocation requires two half planes of ions to be inserted and (2) it is difficult for the dislocations to move because of the Coulomb repulsion as large anions must slip past each other. 

Creep is a thermally activated process under which materials will slowly stretch under stress when the temperature exceeds 50% of the melting temperature. Creep mechanisms include dislocation climb, vacancy migration, and grain boimdary slip. Creep is especially important in gas turbine engines and a class of Ni-based superaUoys has been developed to minimize its effects. 

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The temperature Tg represents an asymptotic temperature at which, if nothing else intervened, the flow stress would vanish. It principally serves to prescribe the form of the temperature-dependent decrease of the flow stress in the range T< Tg. At temperatures approaching Tg, as important concurrent diffuse structural relaxations set in and the viscoelastic relaxation modulus undergoes a sharp decline, the flow stress that remains proportional to the relaxation modulus exhibits a fundamental softening making it approach zero at Tg. For typically in the range of 0.05, 0.5, vg = 10 s, and an applied shear strain rate of... 

Stress relaxation for step squeezing of polystyrene at 180°C. (a) Stress versus time for increasing strain steps. Stress increases at short times, 2-10 ms because the plates take a finite time to close. The horizontal stress response signifies transducer overload. The rapid drop for strains e > 1 indicates loss of lubricant, (b) Stress relaxation data plotted as relaxation modulis. Solid line is the linear viscoelastic relaxation modulus calculated from shear dynamic data. Adapted from Soskey and Winter (1985). 

Strain level—maintained constant during viscoelastic relaxation modulus tests ... 

A Standard Model for the viscoelastic behaviour of plastics consists of a spring element in scries with a Voigt model as shown in Fig. 2.86. Derive the governing equation for this model and from this obtain the expression for creep strain. Show that the Unrelaxed Modulus for this model is and the Relaxed Modulus is fi 2/(fi + 2>. 

The relaxation modulus (or any other viscoelastic function) thus obtained is a mean s of characterizing a material. In fact relaxation spectra have been found very useful in understanding molecular motions of plastics. Much of the relation between the molecular structure and the overall behavior of amorphous plastics is now known. 

The linear viscoelastic behavior of liquid and solid materials in general is often defined by the relaxation time spectrum 11(1) [10], which will be abbreviated as spectrum in the following. The transient part of the relaxation modulus as used above is the Laplace transform of the relaxation time spectrum H(l)... 

The time-dependent rheological behavior of liquids and solids in general is described by the classical framework of linear viscoelasticity [10,54], The stress tensor t may be expressed in terms of the relaxation modulus G(t) and the strain history ... 

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,... 

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.

Calculation of the viscoelastic functions proceeds as above where, for example, Eq. (T 7) is the reduced relaxation modulus for the cubic array. The incomplete gamma function of order 5/2 may be obtained in simpler form through a recurrence relation and ... 

The model of a viscoelastic body with one relaxation time used above has one principal disadvantage it does not describe the viscous flow of the reactants before the gel-point at t < t. Thus it is important to use a more general model of a viscoelastic medium to interpret the results obtained. The model must allow for flow and may be constructed by combining viscous and viscoelastic elements the former has viscosity rp and the latter has a relaxation modulus of elasticity Gp and viscosity rp,... 

In this section, the composite system with the properties given by Eq. (58) will be used. Since glassy polymers are not in thermodynamic equilibrium, the change in the nonequilibrium glassy state and its relaxation define the viscoelastic response. The relaxation modulus is given by Eq. (40). 

The Time-Temperature Superposition Principle. For viscoelastic materials, the time-temperature superposition principle states that time and temperature are equivalent to the extent that data at one temperature can be superimposed upon data at another temperature by shifting the curves horizontally along the log time or log frequency axis. This is illustrated in Figure 8. While the relaxation modulus is illustrated (Young s modulus determined in the relaxation mode), any modulus or compliance measure may be substituted. 

Consider two experiments carried out with identical samples of a viscoelastic material. In experiment (a) the sample is subjected to a stress CT for a time t. The resulting strain at f is ei, and the creep compliance measured at that time is D t) = e la. ln experiment (b) a sample is stressed to a level CT2 such that strain i is achieved immediately. The stress is then gradually decreased so that the strain remains at f for time t (i.e., the sample does not creep further). The stress on the material at time t will be a-i, and the corresponding relaxation modulus will be y 2(t) = CT3/C1. In measurements of this type, it can be expected that az> 0 > ct and Y t) (D(r)) , as indicated in Eq. (11-14). G(t) and Y t) are obtained directly only from stress relaxation measurements, while D(t) and J(t) require creep experiments for their direct observation. Tliese various parameters can be related in the linear viscoelastic region described in Section 11.5.2. 

When reptation is used to develop a description of the linear viscoelasticity of polymer melts [5, 6], the same underlying hypothesis ismade, and the same phenomenological parameter Ng appears. Basically, to describe the relaxation after a step strain, for example, each chain is assumed to first reorganise inside its deformed tube, with a Rouse-like dynamics, and then to slowly return to isotropy, relaxing the deformed tube by reptation (see the paper by Montfort et al in this book). Along these lines, the plateau relaxation modulus, the steady state compliance and the zero shear viscosity should be respectively ... 

Although the (Simplex shear modulus is not the most appropriate function to use in all c ses, we wUl describe the linear viscoelastic behaviour in terms of this last function, which is tiie most referred to experimentally furthermore, molecular models are mostly linked to the relaxation modulus, which is the inverse Fourier transform of the complex shear modulus. 

Viscoelastic function in the whole time frequency domain Thus the relaxation modulus may be calculated from a very limited number of... 

The relaxation modulus is the core of most of the viscoelastic descriptions and the above expression can be checked from experimental viscoelastic functions such as the complex shear modulus G (co) for instance. In addition to the molecTilar weight distribution function P(M), one has to know a few additional parameters related to the chemical species the monomeric relaxation time x,... 

One convenient manner of studying viscoelasticity is by stress relaxation where the time-dependent shear stress is studied for step increase in strain. In Figure 1-7, the stress relaxation of a Hookean solid, and a viscoelastic solid and liquid are shown when subjected to a strain instantaneously and held constant. The relaxation modulus can be calculated as ... 

The above equation is a one-dimensional model of linear viscoelastic behavior. It can be also written in terms of the relaxation modulus after noting that ... 

One can express linear viscoelasticity using the relaxation spectrum H X), that is, using the relaxation time X. The relationship between the relaxation modulus and the spectra is ... 

Accordingly, the phenomenological theory of linear viscoelasticity predicts the same frequency dependence for the loss relaxation modulus of solids and liquids in the terminal region. 

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