Relationship Between Viscoelastic Moduli Compliances

For isotropic elastic materials, there are only two independent elastic constants and relations exist between various constants as given in Table 2.1 such as, 

For an isotropic viscoelastic material only two time dependent properties are independent and it is clear from the correspondence principle that similar relationships to Eqs. 9.15 hold for the Laplace transformed moduli such that. 

Using relations 9.11 will convert the Eqs. 9.16 to relations between the Laplace transform of the of relaxation moduli, creep compliances. For example. 

Similar integral equations can be developed for each relationship given in Eqs. 9.16 or Table 2.1. 

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Tackifying resins enhance the adhesion of non-polar elastomers by improving wettability, increasing polarity and altering the viscoelastic properties. Dahlquist [31 ] established the first evidence of the modification of the viscoelastic properties of an elastomer by adding resins, and demonstrated that the performance of pressure-sensitive adhesives was related to the creep compliance. Later, Aubrey and Sherriff [32] demonstrated that a relationship between peel strength and viscoelasticity in natural rubber-low molecular resins blends existed. Class and Chu [33] used the dynamic mechanical measurements to demonstrate that compatible resins with an elastomer produced a decrease in the elastic modulus at room temperature and an increase in the tan <5 peak (which indicated the glass transition temperature of the resin-elastomer blend). Resins which are incompatible with an elastomer caused an increase in the elastic modulus at room temperature and showed two distinct maxima in the tan <5 curve. 

Chapters 5 and 6 discuss how the mechanical characteristics of a material (solid, liquid, or viscoelastic) can be defined by comparing the mean relaxation time and the time scale of both creep and relaxation experiments, in which the transient creep compliance function and the transient relaxation modulus for viscoelastic materials can be determined. These chapters explain how the Boltzmann superposition principle can be applied to predict the evolution of either the deformation or the stress for continuous and discontinuous mechanical histories in linear viscoelasticity. Mathematical relationships between transient compliance functions and transient relaxation moduli are obtained, and interrelations between viscoelastic functions in the time and frequency domains are given. 

A polymer material will show viscoelastic properties if the ambient temperature is above its glass transition temperature (Tg). The relationship between compliance (inverse of elastic modulus) and time for a polymer undergoing creep was illustrated in Figure 6.2 (Chapter 6). At low temperatures, the polymer exists as a glass. As the temperature is raised, the Tg is exceeded, the compliance increases, and the polymer takes on leathery and rubbery properties. The value of Tg is influenced by the presence of plasticizers—compounds of low MW thaf reduce the elastic modulus. For dough, water is the most relevant plasticizer. 

At intermediate times it will be seen that, in creep, the compliance passes from /u to /r with time constant r . In stress relaxation the modulus passes from G to Gr with time constant r. Thus, at very short and very long times the stress and strain are Hookean, but at intermediate times when the time t is of the order of the relaxation times this k not true and it in this region that we see viscoelastic effects. The relationship between theory (Figure 4.15) and experiment (Figures 4.4 and 4.7, for e mple) will be explored later the reader may well however compare these figmres now and see in outline how theory is in broad agreement with eqieriment... 

Note that the relationships between the viscoelastic compliance and the relaxation modulus are given by... 

Transient Response Creep. The creep behavior of the polsrmeric fluid in the nonlinear viscoelastic regime has some different features from what were found with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation... 

The differential equation governing the relationship between stress and strain for a given mechanical model is quite valuable, but needs to be solved in order to determine the model response to specific loading conditions. Fundamental viscoelastic properties such as the creep compliance or relaxation modulus can be found by solution of the differential equation to the appropriate loading. For example, the creep compliance can be determined using the conditions for a creep test of constant stress, as shown in Fig. 5.2. 

The linear viscoelastic properties G(t)md J t) are closely related. Both the stress-relaxation modulus and the creep compliance are manifestations of the same dynamic processes at the molecular level in the liquid at equilibrium, and they are closely related. It is not the simple reciprocal relationship G t) = 1/J t) that applies to Newtonian liquids and Hookean solids. They are related through an integral equation obtained by means of the Boltzmann superposition principle [1], a link between such linear response functions. An example of such a relationship is given below. 

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