Modulus and Compliance

The nature of modulus and compliance depend on the nature of the deformation. The three most important elementary modes of deformation and the moduli (and compliances) derived from them are given in Fig. 13.1 and Table 13.1 compression, tensile and shear moduli, K, E and G, respectively and the corresponding compliances, k, S and /, respectively... 

This is because although 0 = (10), in general, cr(10) oQ (it will usually be less). In principle, the quantities we have defined, E(t), Dit), Gif), and J(i), provide a complete description of tensile and shear properties in creep and stress relaxation (and equivalent functions can be used to describe dynamic mechanical behavior). Obviously, we could fit individual sets of data to mathematical functions of various types, but what we would really like to do is develop a universal model that not only provides a good description of individual creep, stress relaxation and DMA experiments, but also allows us to relate modulus and compliance functions. It would also be nice to be able formulate this model in terms of parameters that could be related to molecular relaxation processes, to provide a link to molecular theories. 

Develop expressions for the complex modulus and compliance for a Maxwell body... 

Equations (7.26) are called dispersion relations and analogous equations can be derived for /, and J2 (see problem 7.7) and for more general models. They can also be derived for the real and imaginary parts of the dieleetrie constant (see section 9.2.4). The limiting values of G, and /, at low frequencies are called the relaxed modulus and compliance, G, and J, and the limiting values at high frequencies are called the unrelaxed modulus and compliance, G and / ... 

Compliance, elastic n. Symbol S. An elastic constant, which is the ratio of a strain or strain component to a stress or stress component. For a perfectly elastic material it is the reciprocal of the elastic modulus. For a viscoelastic material the modulus and compliance are not reciprocally related due to their different time dependencies. Sepe MP (1998) Dynamic mechanical analysis. Plastics Design Library, Norwich, New York. 

A fundamental quantity relating the basic viscoelastic functions (i.e., storage, loss modulus and compliance, shear viscosity) is the monomeric friction coefficient, which is a measure of the frictional resistence per monomer unit encountered by a moving chain segment. This co-... 

Hence, one can relate the stresses to any applied state of strain. Furthermore, the strains can be determined as functions of the applied stresses. Note that, in general, the theory of elasticity does not demand the development of compliance functions, which relate strains to stresses, as they are the inverse of the moduli. In the case of viscoelasticity, this is not so, and both modulus and compliance functions are developed in the next section. 

FIG. 1 -8. Vectorial resolution of components of complex modulus and compliance in sinusoidal shear deformations. 

The various moduli and compliances which have been introduced for infinitesimal deformations are summarized in Table l-I, including the glasslike modulus and compliance which will be explained in Chapter 2. All moduli have the dimensions of stress (units usually dynes/cm or alternatively Pa) all compliances have the dimensions of reciprocal stress (usually cm /dyne or Pa" )- Certain quantities in any row of the table can be interrelated by equations of the form of 55 and 58. The symbols follow rather closely the recommendations of a Committee of the Society of Rheology. ... 

The loss modulus G" and compliance are often called the imaginary parts of their complex counterparts by analogy with the language of dielectrics and optics in which such notation is conventional. However, G" and J" are of course real quantities which are the coefficients of the imaginary terms in equations 24 and 26. The complex notation is particularly convenient for interconverting modulus and compliance as expressed by equations 27 to 30. 

The introduction of chemical cross-links into an uncross-linked polymer converts it from a viscoelastic liquid to a viscoelastic solid in the sense of the definitions of Chapter 1 and the classification of Chapter 2 the viscosity becomes infinite and the material acquires an equilibrium modulus and compliance, so the properties in the plateau and terminal zones change profoundly. However, the properties in the transition zone may change very little. The effects of cross-linking are discussed in this chapter, as well as the effects in the plateau and terminal zones of incorporating fillers (finely divided particles, usually of high modulus) or other combinations of more than one phase. 

Four important mechanical properties — modulus and compliance, elastic recovery, vibration damping and energy loss, flow and creep - wiU be dealt with in this section. 

Variation of the relaxation modulus and creep compliance of a Maxwell model on linear-linear (left) and log-log (right) scales are shown in Figs. 7.15-7.16. Notice the rapid decay of the modulus as the time approaches the selected relaxation time and the flow at long times due to the fluid nature of the Maxwell model. The behavior of the modulus and compliance for a simple Maxwell element is similar to that for many polymers in the glassy and transition region. 

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