The Storage Compliance

The compliance / (co) is defined as the strain in a sinusoidal deformation in phase with the stress divided by the stress it is a measure of the energy stored and re- 

Storage compliance plotted logarithmically for the eight systems identined as in Fig. 2-1. 

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These were calculated using the relationships given in Section (4.6.1). The storage compliance starts from an initial compliance Je given by... 

Note 2 The absolute compliance is related to the storage compliance (5.16) and the loss compliance (Definition 5.17) by the relationship... 

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... 

Owing to the fact that at the glass-rubber relaxation temperature the storage compliance function is an increasing function of temperature, the inequality... 

Let us now proceed with the development of the expressions for the components of the complex compliance function. According to Eqs. (6.23) and (9.11), the storage compliance function is given by... 

Figure 9.3 Double logarithmic plots of the storage compliance function ( ) and the reciprocal of the storage relaxation modulus (O ) for a 40% solution of polystyrene in tri-m-tolyl phosphate.

The approximate value of L(x) can be derived from the storage compliance function J m) following the method outlined to calculate H(x) from G (o)). Thus by taking into account Eq. (9.18), / ((o) can be written as... 

If, on the other hand, the strain vector is resolved into its components, the ratio of the in-phase strain to stress amplitude (maximum stress) is the storage compliance J (oj) and the ratio of the out-of-phase strain to the stress amplitude is the loss compliance... 

Fig. 7.14 The storage compliance of poly(n-octyl methacrylate) in the glass-transition region plotted against frequency on a logarithmic scale. Left-hand side values obtained at 24 temperatures, ranging from -14.3 °C (bottom curve) to 129.5 °C (top curve) right-hand side the master curve obtained by using appropriate shifts log aj. Note that the horizontal scales differ by a factor of two for the two parts of the figure. (Adapted by permission of Academic Press.)...

The ratio of the stress to the strain is used to define a complex modulus "(iw), the real part of which is the storage modulus and the imaginary part the loss modulus, i.e., E ( )=E+ E. Alternatively, one can define a complex compliance < (iw) as the ratio of the strain to the stress. For this case, real part is the storage compliance and the imaginary part is the negative of the loss compliance (note that E = l/d> ). 

Applying time-temperature equivalence (as discussed in Section 6.2 above) gives the storage compliance as a function of frequency over a very wide range of frequencies, as shown in Figure 6.13. Thus it is now possible to calculate the retardation time spectrum, and compare this with any theoretical models that may be proposed. 

The storage compliance J is the ratio of the strain in the phase with the stress to the stress, whereas the loss compliance J" is the ratio of the strain 90° out phase with the stress to the stress (Fig. 1-8). The storage modulus G and compliance J are so named because they are directly proportional to the average energy storage in a cycle of deformation. The loss modulus G" and compliance J" are directly proportional to the average dissipation or loss of energy as heat in a cycle of deformation. Note that J"jJ = G"/G = tan b. 

FIG, 11-18. Composite curve for the storage compliance of poly( -butyl methacrylate), with ar factors derived from data in the transition zone. 

FIG. II -19. Composite curve for the storage compliance of the mechanism of poly(n-butyl meth-acrylate), attributed to motions of the side chains. 

The idea of plotting one material function versus another has been applied to the viscoelastic properties of polymer. For example, one might plot the loss modulus versus the storage modulus or the loss compliance versus the storage compliance. We note that in this type of... 

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