Modulus absolute

Fluids. The previous methods were designed for soHd specimens, although some can be used for fluids if a soHd support or carrier is used. The fluid must be highly viscoelastic for data to register, and absolute modulus values are difficult to determine because of the presence of the support. 

Note 2 The absolute modulus is related to the storage modulus and the loss modulus by the relationship... 

Note 3 The absolute compliance is the reciprocal of the absolute modulus. 

Note 3 From Definition 5.14, the absolute modulus in uniaxial deformation... 

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... 

Note 3 The flexural modulus has been given the same symbol as the absolute modulus in uniaxial deformation as it becomes equal to that quantity in the limit of zero amplitudes of applied force and deformation. Under real experimental conditions it is often used as an approximation to i . ... 

Quantitative tests of the absolute modulus value from kinetic theory [Eq. (7.2)] are quite scarce independent determination of the network strand concentration is generally the difficulty. Schaefgen and Flory (278) prepared model... 

When the stress is decomposed into two components the ratio of the in-phase stress to the strain amplitude (j/a, maximum strain) is called the storage modulus. This quantity is labeled G (co) in a shear deformation experiment. The ratio of the out-of-phase stress to the strain amplitude is the loss modulus G"(co). Alternatively, if the strain vector is resolved into its components, the ratio of the in-phase strain to the stress amplitude t is the storage compliance J (m), and the ratio of ihe out-of-phase strain to the stress amplitude is the loss compliance J"(wi). G (co) and J ((x>) are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. Tlie loss parameters G" w) and y"(to) on the other hand reflect the nonrecoverable use of applied mechanical energy to cause flow in the specimen. At a specified frequency and temperature, the dynamic response of a polymer can be summarized by any one of the following pairs of parameters G (x>) and G" (x>), J (vd) and or Ta/yb (the absolute modulus G ) and... 

Since the absolute modulus square of the resonance function varies with time, it is of interest to see the result of applying the continuity equation, or more precisely of the continuity equation integrated along the internal region of the interaction. 

The late 1970s saw Polymer Laboratories develop their DMTA using dual cantilever bending, which works well for most small. samples from -150 C to the onset of melt. Shear, tensile, torsion, and simple compression options followed, as did the complementary Di-clectric Thermal Analyser (DETA), and computers were used from 1982 to both control and analyze the data. Seiko Instruments copied this and tried to patent it. and others such as Netzsch, Perkin-Elmer, and TA Instruments looked very closely at this before launching their own. For comparative data and fast thermal scans they all can give good data, but for absolute modulus numbers most systems need to consider the frame compliance, sample end corrections, and relative dimensions, and hence only a limited range of sample dimensions can be used for accurate measurement of modulus in a particular mode of deformation. 

It is evident from the above description that G (oj) and are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. The loss parameters G" oj) and on the other hand, reflect the nonrecoverable use of applied mechanical energy to cause viscous flow in the material. At a specified frequency and temperature, the dynamic response of a polymer in shear deformation can be summarized by any one of the following pairs of parameters G ( w) and G"( w), J ioj) and or absolute modulus G and tan 6. 

To determine such a modulus, the measurement of the deformation is not made at the macroscopic level but rather at the molecular one using Raman spectroscopy and/or X-ray diffractometry. These experimental methods give access to the absolute modulus from the modifications induced by the applied stress in the Bragg refiection-—and therefore in the position of atoms—, one can indeed evaluate Young s modulus in the direction of stress—and even in the perpendicular direction—thanks to the Hooke law. The values of moduli obtained in this manner are remarkably high. Diamond, which is exclusively constituted of carbon-carbon bonds, has a tensile modulus equal to llbOGPa in the [110] direction for a cross-section of 0.049 nm. In comparison, polyethylene chains which also consist of C-C bonds substituted by hydrogen atoms and whose cross-section is 0.180 nm, should exhibit a tensile modulus of about 310 GPa [i.e 1160 (0.049/0.180)]. This value corresponds almost ideally to the absolute modulus of polyethylene fibers determined at the molecular level by X-ray diffractometry. In contrast, the tensile modulus obtained from a macroscopic measurement of the deformation represents... 

This expression shows that the smaller the cross-section of the chain, the higher Fthl it applies to any chain with local transoidal conformation irrespective of its nature. For instance, the theoretical modulus obtained from this expression gives a value of 180 GPa for fibers of poly(p-phenyleneterephtalamide), which is hardly different from the absolute modulus measured at the molecular level (200 GPa) and even from the tensile modulus (132 GPa) determined from a macroscopic sample. For polyethylene chains, the calculated value of 340 GPa is in agreement with the absolute modulus value of 325 GPa. 

For the data presented in Figure 1-4, it can be noted that measured permittivity is low at high frequencies and often becomes very high at low frequencies (due to interfacial polarization). The modulus representation shows very small absolute modulus values at low frequencies (and the low frequency semicircle is invisible in Figure 2-6B) where the impedance notation is particularly powerful but will resolve high-frequency dielectric responses well. Figure 2-6 illustrates this example for the - R,ntI dl circuit. 

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