Shear response function

Note here that the A.[ s are stretches and not relaxation times as used previously. Also note that for polymer melts and solutions, the shear response function (shear modulus) is several orders of magnitude smaller than is the bulk modulus. Therefore, these materials can be treated as incompressible. For the linear case, the incompressibility assnmption has 2 oo or v 0.5. Here, I3 = 1 and the constitntive law for the incompressible material becomes... 

Polymer rheology can respond nonllnearly to shear rates, as shown in Fig. 3.4. As discussed above, a Newtonian material has a linear relationship between shear stress and shear rate, and the slope of the response Is the shear viscosity. Many polymers at very low shear rates approach a Newtonian response. As the shear rate is increased most commercial polymers have a decrease in the rate of stress increase. That is, the extension of the shear stress function tends to have a lower local slope as the shear rate is increased. This Is an example of a pseudoplastic material, also known as a shear-thinning material. Pseudoplastic materials show a decrease in shear viscosity as the shear rate increases. Dilatant materials Increase in shear viscosity as the shear rate increases. Finally, a Bingham plastic requires an initial shear stress, to, before it will flow, and then it reacts to shear rate in the same manner as a Newtonian polymer. It thus appears as an elastic material until it begins to flow and then responds like a viscous fluid. All of these viscous responses may be observed when dealing with commercial and experimental polymers. 

Figure 15 Schematic of the method employed to calculate friction coefficient. The corrugated surfaces are immobile, and a shear flow is generated in the confined fluid using SLLOD equations of motion. The difference in momentum between the fluid and the surfaces results in a frictional force, which is the response function.

However, in the solution process, the same two opposing forces reach a different balance. In Figure 83, the shear response of polymer is again shown as a function of the catalyst activation temperature, but in this case the polymers were made in the solution process. Because the LCB level of polymers made in the solution process is much lower, it is the breadth that now dominates the response to shear stress. The data of Figure 83 show that the response to shear stress decreases with rising activation temperature. Again, the comparison is made with polymers produced to have the same three MI values shown in Figure 82. Three parallel lines are produced that decline with activation temperature. 

Liao P, Carter EA (2010) Ab initio density functional theory-tU predictions of the shear response of iron oxides. Acta Mater 58 5912-5925... 

Figure 7.20(a) gives the simulation results of the plastic response of this particular structure in a tensile-flow experiment at a given constant strain rate at 0 K and 300 K. The response is given as a deviatoric shear resistance (stress) shear resistance r at 0 K, plotted as a function of the total deviatoric shear strain y, where, in a formal application of a Tresca connection between tensile and shear response, a in shear is taken as half of the tensile deviatoric plastic resistance and y is twice the total uniaxial deviatoric strain (McClintock and Argon 1966). The initial quenched-in level of tp for this alloy is... 

Holliday-Ankeny, C.J., et al. The function of shear-responsive and side-dependent microRNA-486-5p in aortic valve endothelium. Cardiovascular Pathology 22(3), e50... 

Here, p is the magnitude of pressure fluctuation, c is the speed of sound,/is the frequency of acoustic wave, k = Inf/c is the wave number, z is the axial direction, p is the fluid viscosity, p is the density, R is the radius of tube, v is the kinematic viscosity, and / is the Bessel function. The frequency response function H(f) from Eq. 27 is used to determine the frequency response of the shear stress sensor as... 

This represents the solid lines in Fig. 1, which shows data for three concentrations (500, 250 and 100 wppm) each of the fresh and shear degraded solutions. The parameters that govern the viscosity function in (11) also strongly influence other predicted response functions such as normal stress behavior, elongational viscosity, transient response, etc. 

Figure 6. Interfacially confined hexadecane measured by SFM shear modulation spectroscopy. The shear response is measured simultaneously with the normal force deflection of the cantilever as function of the cantilever-siicon sample distance. The difference in the bending onset of the two curves defines the interfacially confined boundary layer thickness.

The high shear rate tail of the universal response function in Figure 5 follows a power law over three decades in shear rate. The observed exponent is about -2/3 with a best fit value of x==-0.69(3). As long as the system can be taken close enough to the glass transition that one sees a substantial drop in viscosity, this exponent should be observable. If one can only observe part of the curve near the beginning of the drop in p, the apparent exponent will be smaller. This may have been the situation in Ref. 36 where crystallization occurred before the relaxation rate had slowed substantially relative to the bulk value. 

Using the same assumptions of the example solved in the Laplace domain (step input in pressure, elastic bulk modulus and Maxwell behavior in shear) with Eq. 9.58, the solution of the integral equation (Eq. 9.58) will yield the same results. (See homework problem 9.4). Since polymers are such that many Maxwell or Kelvin elements are needed to represent actual behavior, this example shown here is simplistic. However, such simple solutions can show trends in behavior and may give insight to the differences between thermosets and thermoplastics. The next section discusses briefly use of broadband material response functions for more physically realistic... 

Our concern is to find out how the characteristic material parameters rjo and Jg are included in the various response functions. To begin with, consider a perfectly viscous system in a dynamic-mechanical experiment. Here the dynamic shear compliance is given by... 

With the aid of ap we can express response functions at any temperature in terms of the respective response function at Tq. Explicitly, for the dynamical shear modulus, the following relation holds... 

Finally, if the system is subjected to a time varying stress, the steady state response function for the shear strain is given by ... 

Rheometric Scientific markets several devices designed for characterizing viscoelastic fluids. These instmments measure the response of a Hquid to sinusoidal oscillatory motion to determine dynamic viscosity as well as storage and loss moduH. The Rheometric Scientific line includes a fluids spectrometer (RFS-II), a dynamic spectrometer (RDS-7700 series II), and a mechanical spectrometer (RMS-800). The fluids spectrometer is designed for fairly low viscosity materials. The dynamic spectrometer can be used to test soHds, melts, and Hquids at frequencies from 10 to 500 rad/s and as a function of strain ampHtude and temperature. It is a stripped down version of the extremely versatile mechanical spectrometer, which is both a dynamic viscometer and a dynamic mechanical testing device. The RMS-800 can carry out measurements under rotational shear, oscillatory shear, torsional motion, and tension compression, as well as normal stress measurements. Step strain, creep, and creep recovery modes are also available. It is used on a wide range of materials, including adhesives, pastes, mbber, and plastics. 

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