Evaluation from modulus

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. 

Figure 4 shows stress-strain curves measured at an extension rate of 94% per minute on the TIPA elastomer at 30°, —30°, and —40°C. With a decrease in temperature from 30° to -40°C, the ultimate elongation increases from 170% to 600%. The modulus Ecr(l), evaluated from a one-minute stress-strain isochrone, obtained from plots like shown in Figure 1, increases from 1.29 MPa at 30°C to only 1.95 MPa at —40°C. This small increase in the modulus and the large increase in the engineering stress and elongation at fracture results from viscoelastic processes. 

To evaluate Pj. g from modulus measurements, a value of A has to be assumed and used together with the experimental value of Mg/AM. Accordingly, Figures 4 and 5 show pj- g plotted versus Pr.c affine (A=l) and phantom (A = 1 - 2/t) chain behaviour using the results in Figure 1. One condition that should be obeyed in the plots is that Pr e Pr c Pr e includes pre-gel and post-gel reaction. 

For small damping, the storage modulus M) may be evaluated from co through... 

Note 1 For a Voigt-Kelvin solid of negligible mass, the absolute modulus can be evaluated from the ratio of the flexural force (/o) and the amplitude of the flexural deflection (y) with... 

If the elastic modulus is obtained from the slope of the elastic stress-strain curve, then we can evaluate the first term on the right-hand side in Equation (8.3) from experimental data elastic stress-strain curves. The second term on the right-hand side in Equation (8.3) can be evaluated from the product of the strain rate, which is set in a constant strain-rate experiment, and the viscosity. As we discussed in Chapter 3, the viscosity of a macromolecule is related to the shape factor v, therefore we can evaluate the second term on the right-hand side of Equation (8.3) from the product of the shape factor and the strain rate. 

Several comprehensive reviews on the BDS measurement technique and its application have been published recently [3,4,95,98], and the details of experimental tools, sample holders for solids, powders, thin films, and liquids were described there. Note that in the frequency range 10 6-3 x 1010 Hz the complex dielectric permittivity e (co) can be also evaluated from time-domain measurements of the dielectric relaxation function (t) which is related to ( ) by (14). In the frequency range 10-6-105 Hz the experimental approach is simple and less time-consuming than measurement in the frequency domain [3,99-102], However, the evaluation of complex dielectric permittivity in the frequency domain requires the Fourier transform. The details of this technique and different approaches including electrical modulus M oo) = 1/ ( ) measurements in the low-frequency range were presented recently in a very detailed review [3]. Here we will concentrate more on the time-domain measurements in the high-frequency range 105—3 x 1010, usually called time-domain reflectometry (TDR) methods. These will still be called TDS methods. 

Small Debond Energy. For SDE, when cr< crs, the unloading modulus E depends on r0, but is independent of T, and Cl. However, the permanent strain e0 depends on T, and Cl, as well as r0. These differing dependencies of E and e0 on constituent properties have the following two implications. (1) To simulate the stress-strain curve, both e0 and E are required. Consequently, r0, T, and Cl must be known. (2) The use of unloading and reloading to evaluate the constituent properties has the convenience that the hysteresis is dependent only on tq. Consequently, precise determination of r0 is possible. Moreover, with t0 known from the hysteresis, both T,- and Cl can be evaluated from the permanent strain. The principal SDE results are as follows. 

The elasticity modulus is evaluated from the measured film tension and local film thickness as follows... 

Hinr. You could use Equation (P4,23-i), which would include dp and an unknown proportionality constant which you could evaluate from the data. For very small values of the Thiele modulus we know t = 1 and for very large values of the Thiele modulus we know that -n = 3/ = VCd. ] P4-24c Spherical reactor) Because it is readily available from coal, methanol has been investigated as an alternative raw material for producing valuable olefins... 

Taking the square modulus of this result to evaluate the overall rate from Eq. (5.6) again introduces a factor of K, which can be evaluated from Eq. (6.5) subject to replacement of by (which is independent of coherence length), the subscripts in the latter case denoting the fact that both laser photons are absorbed by the same molecule. The final result for the rate is then given by ... 

In the same work, the damping behavior of those three component LIPNs was evaluated from the integrals of the linear loss shear modulus vs. temperature (loss area, LA) and linear tan 8 vs. temperature (tan 8 area, TA) curves measured by... 

The results are shown in Figure 4, where Mc/Mc° is plotted versus Pp The molar mass between junction points of the perfect network(Mc°) is calculable from the molar mass and structure of the reactants (1, ) and was evaluated from the measured modulus using Equation 6 with A=l. pr Q is the extent of intramolecular reaction at gelation, given by the... 

The mechanical characteristics of SPU - tensile strength, o, (the maximum stress calculated on the initial cross-section area) and the relative critieal deformation, 8, (relative deformation at the stress o ) have been evaluated for the strain rate, v = 0.28 s. The mechanical modulus, E , was evaluated from the initial, linear part of the strain diagrams. 

Based on damping theory (Kim and Sperling 1997 Sophiea et al. 1994a), the damping behavior of a polymer can be evaluated from its dynamic mechanical behavior by expressing it as the area under the tan 8 versus temperature curve (Keskkula et al. 1971) or the area under the linear loss modulus versus temperature... 

Integration constants A2 and A3 can be evaluated from the thermal boundary conditions. The most general thermal boundary conditions are obtained by prescribing the local heat flux with the Biot modulus. The thermal boundary condition at the screw surface can thus be written as ... 

A number of tests are possible to evaluate the behavior of soils under dynamic loads such as wave or earthquake loads. Dynamic tests generally are strength tests with the sample subjected to some sort of cyclic loading. Tests can be performed to evaluate variations of strength, modulus, and damping, with variations in rate and magnitude of cyclic stresses or strains. Small strain parameters for earthquake loading cases can be evaluated from resonant column tests. 

An expression for elastic modulus evaluation from force-distance data can be derived from (7) taking into account condition (6) ... 

---