Time-dependent relaxation modulus

Viscoelastic characteristics of polymers may be measured by either static or dynamic mechanical tests. The most common static methods are by measurement of creep, the time-dependent deformation of a polymer sample under constant load, or stress relaxation, the time-dependent load required to maintain a polymer sample at a constant extent of deformation. The results of such tests are expressed as the time-dependent parameters, creep compliance J t) (instantaneous strain/stress) and stress relaxation modulus Git) (instantaneous stress/strain) respectively. The more important of these, from the point of view of adhesive joints, is creep compliance (see also Pressure-sensitive adhesives - adhesion properties). Typical curves of creep and creep recovery for an uncross-Unked rubber (approximated by a three-parameter model) and a cross-linked rubber (approximated by a Voigt element) are shown in Fig. 2. 

As in the case of the relaxation modulus, the dependence on temperature is quite weak, because within the experimentally accessible temperature range, the variation of p T is small. According to the Bueche-Rouse theory, over some range of times or frequencies away from the terminal zone, where the two or three longest relaxation times can be neglected, the relaxation modulus can be approximated by Eq. 6.17 ... 

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

The time-dependent rheological behavior of liquids and solids in general is described by the classical framework of linear viscoelasticity [10,54], The stress tensor t may be expressed in terms of the relaxation modulus G(t) and the strain history ... 

Using this factorizability of response into a time-dependent and a strain-dependent function. Landel et ai. (61,62) have proposed a theory that would express tensile stress relaxation in the nonlinear regime as the product of a time-dependent modulus and a function of the strain ... 

Here is the element viscosity, is the element modulus, and Ay is the relaxation time. If this model material is placed in a constant stress environment for a fixed time and then the stress is removed, then the time-dependent strain will have the recovery characteristics shown in Fig. 3.11. 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

Comparisons of the theory with experiment can not be presently made due to the lack of data on well characterized molecular IPN. Indications about its validity can, however, be deduced by examining its consistency at extreme cases of material behavior. The agreement at the one-component limit, for example, provided that the rubber is not very weak (iji not very small), has been successfully demonstrated by Ferry and coworkers [ ]. A useful result is obtained at the version of the theory applicable to the fluid state (i.e., at the limit of zero crosslinking). From the last two terms of Equation 13, the following relationship can be derived for the plateau [ ] and time dependent relaxation modulus of miscible polymer blends ... 

In polymers the time dependence of an modulus plays a more important role than in metals. If polymers are loaded with a constant stress they undergo a deformation e, which increases with time. This process is named creep. Conversely, if a test specimen is elongated to a certain amount and kept under tension, the initial stress s decreases with time. This decay is called stress relaxation. 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

These differences in the mechanical behavior are not reflected, within the experimental error, in the temperature dependence of the mechanical properties. As shown by the examples of Figures 1 and 3, the relaxation modulus and creep compliance data showed very little scatter and could be shifted smoothly into superposition along the logarithmic time axis. The amounts of shift, log Or, required to effect superposition are plotted against the temperature, T, in Figure 5 for the relaxation data, and in... 

The results of measurements of the dependencies G (w,t) for three circular frequencies w0 = 27tf0, wi= 4rcwo, and w2 = 16jtwo are shown in Fig. 3.1. The lack of coincidence in the shapes of die time dependencies of the dynamic modulus components for different frequencies is obvious. This phenomenon is especially true for G", because the position of the maximum differs substantially along the time axis. In the most general sense, this reflects the contributions of the main relaxation mechanisms of the material to its measured viscoelastic properties. 

It is well established that between Tg and about Tg + 50 K, the relaxation kinetics obeys the WLF law (Williams et al., 1955). If Pr is a property depending on the macromolecular mobility (relaxation modulus, complex modulus, viscosity, diffusion rate, etc.), the time-temperature equivalence principle may be formulated as... 

Stress relaxation is the time-dependent change in stress after an instantaneous and constant deformation and constant temperature. As the shape of the specimen does not change during stress relaxation, this is a pure relaxation phenomenon in the sense defined at the beginning of this section. It is common use to call the time dependent ratio of tensile stress to strain the relaxation modulus, E, and to present the results of the experiments in the form of E as a function of time. This quantity should be distinguished, however, from the tensile modulus E as determined in elastic deformations, because stress relaxation does not occur upon deformation of an ideal rubber. 

The data are not usually reported as a stress/time plot, but as a modulus/time plot. This time-dependent modulus, called the relaxation modulus, is simply the time-dependent stress divided by the (constant) strain (Equation 13-71) ... 

We will first consider the parameters we are trying to model. Let us start with stress relaxation, where it is usual to describe properties in terms of a relaxation modulus, defined in Table 13-5 for tensile [ (r)] and shear [G(r)] experiments. The parameter used to describe the equivalent creep experiments are the tensile creep compliance [D(r)] and shear creep compliance [7(0]. It is important to realize that the modulus and the compliance are inversely related to one another for linear, tune-independent behavior, but this relationship no longer holds if the parameters depend on time. 

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