Linear viscoelasticity time-temperature equivalence

The major features of linear viscoelastic behavior that will be reviewed here are the superposition principle and time-temperature equivalence. Where they are valid, both make it possible to calculate the mechanical response of a material under a wide range of conditions from a limited store of experimental information. 

The linear viscoelastic properties of all samples were characterized by dynamic shear measurements in the parallel-plate geometry. Experimental details have been previously published [9]. Using time-temperature equivalence, master curves for the storage and loss moduli were obtained. Fig. 1 shows the master curves at 140°C for the relaxation spectra and Table 3 gives the values of zero-shear viscosities, steady-state compliances and weight-average relaxation times at the same temperature. 

For the tensile strength of a rubber to follow the time-temperature equivalence principle of linear viscoelasticity it is necessary that the extension at break also follow it. This is most easily verified by use of Equation (23), i.e., with the simplifying assumption of strain-time factorization. In an experiment conducted at fixed rate of strain, i = constant, the stress at any temperature and strain may be shown to be (200) ... 

In spite of these complications, the viscoelastic response of an amorphous polymer to small stresses turns out to be a relatively simple subject because of two helpful features (1) the behavior is linear in the stress, which permits the application of the powerful superposition principle and (2) the behavior often follows a time-temperature equivalence principle, which permits the rapid viscoelastic response at high temperatures and the slow response at low temperatures to be condensed in a single master curve. 

In the following sections we discuss the two superposition principles that are important in the theory of viscoelasticity. The first is the Boltzmann superposition principle, which is concerned with linear viscoelasticity, and the second is time-temperature superposition, which deals with the time-temperature equivalence. 

Fortunately for linear amorphous polymers, modulus is a function of time and temperature only (not of load history). Modulus-time and modulus-temperature curves for these polymers have identieal shapes they show the same regions of viscoelastic behavior, and in each region the modulus values vary only within an order of magnitude. Thus, it is reasonable to assume from such similarity in behavior that time and temperature have an equivalent effect on modulus. Such indeed has been found to be the case. Viscoelastic properties of linear amorphous polymers show time-temperature equivalence. This constitutes the basis for the time-temperature superposition principle. The equivalence of time and temperature permits the extrapolation of short-term test data to several decades of time by carrying out experiments at different temperatures. 

Experimental Studies of Linear Viscoelastic Behaviour as a Function of Frequency and Temperature Time-Temperature Equivalence... 

In addition to the Boltzmann superposition principle, the second consequence of linear viscoelasticity is the time-temperature equivalence, which will be described in greater detail later on. This equivalence implies that functions such as a=/(s), but also moduli, behave at constant temperature and various exten-sional rates similarly to analogues that are measured at constant extensional rates and various temperatures. Time- and temperature-dependent variables such as the tensile and shear moduli (E, G) and the tensile and shear compliance (D, J) can be transformed from E =f(t) into E =f(T) and vice versa, in the limit of small deformations and homogeneous, isotropic, and amorphous samples. These principles are indeed not valid when the sample is anisotropic or is largely strained. 

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