Extensional modulus

Eor the case of high modulus fibers such as carbon fibers with = 240 GPa (3.5 x 10 psi), in a polymer matrix, such as epoxy resin with = 3.0 GPa (450,000 psi), the extensional modulus is approximately proportional to the fiber volume fraction and the modulus of the fibers ... 

The difference between the bounds defined by the simple models can be large, so that more advanced theories are needed to predict the transverse modulus of unidirectional composites from the constituent properties and fiber volume fractions (1). The Halpia-Tsai equations (50) provide one example of these advanced theories ia which the rule of mixtures expressions for the extensional modulus and Poisson s ratio are complemented by the equation... 

Show that the apparent extensional modulus of an orthotropic material as a function of 0 [the first of Equations (2.97)] can be written as... 

Particulate composites, 26 754-755 ceramics processing, 5 653-654 extensional modulus of, 26 777 fabrication of, 26 766 Particulate emission limits, 13 183 Particulate emissions, reducing, 11 689 Particulate fillers, ll 302t, 303 for rubber, 21 772, 773 Particulate fluidized-bed regime, 11 801 Particulate impurities in gases, 13 464 in high purity gases, 13 466 Particulate matter (PM), 1 798-801 ... 

Moseley extends the derivation to consider the sonic modulus E (Young s extensional modulus) by making the reasonable assumption that the modulus parallel and perpendicular to the molecular segments is proportional to the intramolecular and intermolecular force constants respectively. Then one can write similarly to Eq. (43)... 

The deformation ability of networks strongly swollen with benzene and those slightly swollen in cyclohexane was unexpectedly found to be the same. What is surprising here is the absence of any correlation between the volume increase of model networks on swelling and their deformation under compression or elongation [130], as it would have to foUow from the classic theory of rubber elasticity. This theory does not predict any difference between the extensional modulus and the shear modulus that controls the swelling. Nevertheless, the experimental ratio of Ce(CH)/ Ce(BZ) = 6 is twice as large as the ratio of E(CH)/E(BZ) = 3 (irrespective ofp) [123]. 

Rg. 11.7 A comparison between experimental moduli for low-density polyethylene filaments and values calculated using the pseudo-affine aggregate model E, extensional modulus Ei, transverse modulus and G, torsional modulus (a) experimental, (b) calculated and (c) experimental and calculated. (Reprinted by permission of Kluwer Academic Publishers.)... 

The extensional modulus 33 of the uniaxially oriented system is given by... 

A disadvantage of the technique is that only resonant frequencies are accessible, and these frequencies, which are proportional to the square root of the extensional modulus, vary with temperature. 

At acoustic frequencies, the attenuation goes through a maximum determined by the spectrum of relaxation times in the polymer hence dynamical mechanical analysis can be performed by scanning over a wide frequency range, typically 10 —lO Hz. An example of the technique is sonic DMA of PVC [54] which shows that the shear modulus increase monotonically with frequency, while the longitudinal or extensional modulus displays the transition associated with Tp. The ratio of the loss and storage moduli, or tan delta obtained via DMA can be related to the absorption coefficient through the equation [2] ... 

For a linear and isotropic material the shear modulus G is related to the extensional modulus by ... 

Table 8 gives values of the real part of the extensional modulus (eqs. 3 and 5) and the loss factor. The values for most of the polymers are outside the glass-transition region, while the values for polyurethanes are typical of the glass-transition region. The high values of tan S are of interest for soimd absorption and for vibration damping. 

Table 8. Extensional Modulus and Loss factor at Room Temperature ...

Eiastic Materiai Functions. The material functions that are used here for an isotropic, elastic material are the shear modulus G, the extensional modulus E, the bulk modulus K, and the Poisson s ratio v. However, any two of these provides the full set of information needed to describe such a material, as they are not all independent. For simple deformations, the definitions of these moduli are... 

Birefringence X-ray crystallinity Extensional modulus (GPa) Torsional modulus (GPa)... 

As a result, at the present time, use of one or more of the assumptions provided earlier in the chapter, together with broadband data for shear or extensional modulus, represent the most fruitful approach to the solution of viscoelastic boundary value problems. 

Detailed measurements of the extensional modulus of monofilaments have been made by longitudinal wave-propagation methods, where the relationship of the extensional modulus to molecular orientation and crystallinity has been examined. Early investigations using this technique were made by Hillier and Kolsky [24], Ballou and Smith [25], Nolle [26] and Hamburger [27]. 

The measurement of the extensional modulus was re-examined as a possible method for the measurement of molecular orientation in textile yarns by Charch and Moseley [28], Moseley [29] and Morgan [30], Morgan has developed Hamburger s pulse-propagation method. 

It has been suggested by Morgan [30] and others [28] that the sonic modulus (i.e. the extensional modulus measured at high frequencies by a wave-propagation technique) can be used to obtain a direct measure of molecular orientation in a manner analogous to the derivation of the so-called optical orientation function /o = (1 - sin2 0) from the birefringence. 

---