Viscoelasticity moduli

Figure 10.8 Schematic representation of hydrolysis and gel formation in renneted milk H = hydrolysis of K-casein V = changes in the viscosity of renneted milk (second stage of coagulation), G = changes in the viscoelastic modulus (gel formation).

Gels are viscoelastic bodies, the rheological properties of which can be described by two parameters, the storage modulus (G, which is a measure of its elasticity) and the loss modulus (G", which is a measure of its viscous nature). The combined viscoelastic modulus (G ) is a measure of the overall resistance of a gel to deformation. These moduli are often highly dependent on the time-scale of deformation. Another important parameter of a food gel is its yield stress. 

In the first part of this chapter we studied the radial vibrations of a solid or hollow sphere. This problem was considered an extension to the dynamic situation of the quasi-static problem of the response of a viscoelastic sphere under a step input in pressure. Let us consider now the simple case of a transverse harmonic excitation in which separation of variables can be used to solve the motion equation. Let us assume a slab of a viscoelastic material between two parallel rigid plates separated by a distance h, in which a sinusoidal motion is imposed on the lower plate. In this case we deal with a transverse wave, and the viscoelastic modulus to be used is, of course, the shear modulus. As shown in Figure 16.7, let us consider a Cartesian coordinate system associated with the material, with its X2 axis perpendicular to the shearing plane, its xx axis parallel to the direction of the shearing displacement, and its origin in the center of the lower plate. Under steady-state conditions, each part of the viscoelastic slab will undergo an oscillatory motion with a displacement i(x2, t) in the direction of the Xx axis whose amplitude depends on the distance from the origin X2-... 

The corresponding viscoelastic equations can easily be found by using differential operators for the viscoelastic modulus or alternatively the integral representation. For this purpose the following main conclusions of the preceding analysis should be considered ... 

The Timoshenko equation holds only for elastic films on elastic substrates. Polymer layers, however, are viscoelastic, in particular near their glass transition temperatures. The experimental raw data can be deconvoluted to obtain stress measurements, even if the viscoelastic effects are important. This is because we use the thin-film equation, where the viscoelastic modulus of the polymer is not needed for the stress computation. For... 

Screen-Printable Polyimide. Bending beam results for a screen-printable polyimide used for chip protection are presented in Figure 8. The EPO-TEK 600 polyimide paste was applied to a quartz beam, then the beam was spun at 5000 rpm to achieve high uniformity. Results are presented as inverse radius of curvature, 1/R, because the 50 m coating thickness on the 84 im substrate violates the thin film criterion of Equation 2. We have no knowledge of the polyimide s viscoelastic modulus, which is needed in order to convert 1/R into interfacial stress using Equation 1. 

The viscoelasticity is a complex number determined by the dilatational elasticity and viscosity [19, 94, 95]. The viscoelasticity modulus (or surface dilatational modulus) incorporates a real and imaginary constituent, elasticity and viscosity, respectively. 

In panels (b) and (c), we also note that Gp and Gptes (evaluated with 9 = 0) exhibit the Rouse-like power-law increases beyond the level of the high-co entanglement plateau (3.6 x 10 Pa cf. Equation 3.68) expected for the 50 wt% PI/PtBS blend at 30°C. The viscoelastic modulus at co well below the... 

The quasi-equilibrium adsorption layers (the formation time of60,000-70,000 sec) were subjected to compressive/tensile deformation sinusoidally in the field of linear viscoelasticity. The dependences of the complex viscoelastic modulus of adsorption layers (E), as well as its elastic (real part. 

The dependences of complex viscoelastic modulus (E) of mixed adsorption layers, as well as its elastic (real part, and viscous (imaginary part, E,p) components as functions of MR concentration at the smallest 0.007 i s and the biggest 0.62 rad/s frequency values of the applied deformation are presented in Figs. 1 and 2 of Section 5.8. In Fig.3 of Section 5.8, the dependences of phase angle on MR concentration at the same conditions are shown. 

The Viscoelastic Material Functions. In linear viscoelasticity, the moduli discussed for the elastic case can be recast as time- or fi equency-dependent functions. The same is true for the compliance functions that are discussed here. For simplicity, consider the shear modulus G which becomes G(t) or G (a>) in the case of the viscoelastic material. An important point here is that the viscoelastic modulus functions all exhibit time (frequency) dependence. Hence, one will have functions for K(t) and E(t) [or, eg, G t) and v i)] and these are required in the case of a three-dimensional strain or stress field. 

Here Gif) is the linear viscoelastic modulus and /i(y), the damping function in shear is introduced. For small y, the linear response is recovered. As y increases, we see another relaxation in G(y, t) at short times that corresponds to the relaxation of the contour length. Figures 45 and 46 show the expected relaxation... 

Evolution with Time and Parallel to the Viscoelastic Modulus G(t). . 58... 

With the uptake of biomass, energy is stored whereby this seems to happen periodically. All in all, this energy should depend on the mean cell density. The density should thus characterise the mean-field properties of a cell culture analogously to a complex viscoelastic modulus . 

Many technologies and natural phenomena involve processes of fast expansion or compression of fluid interfaces covered with surfactant adsorption layers. The dynamic system properties depend on the mechanisms and rate of equilibrium restoration after a deformation. At small magnitudes of deformation the mechanical relaxation of an interface can be described by the complex dilational viscoelastic modulus [1,2]. For sinusoidal deformations it is deflned as the ratio of complex amplitudes of interfacial tension variation and the relative surface area variation f (I ty) = dy /din A being a function of frequency. This modulus may include... 

An alternative approach to develop the TTSP expression (Eq. 7.25) is to consider an expression of the viscoelastic modulus as a Prony series as given by Eq. 5.21b or Eq. 6.25 with the temperature dependence now included on the basis of the theories of Rouse and Zimm. 

At small strains y 0, so G(y, t) must reduce to the linear viscoelastic modulus G(/), and h y) must approach unity for small y. Note also in Figure 4.4.1 that the normal stress modulus N /y equals the shear stress modulus xn/y this implies that melt I obeys the Lodge-Meissner relationship, eq 4.2.8. Note that this relation follows directly from eqs. 4.4.4 and 4.4.5. 

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