Creep compliance time dependence

The creep (/) at time / depends on the compliance function J(t), which is a characteristic of the polymer at a given temperature, and on the initial stress time scale has to be employed in J (i.e., ( -0 the time over which that stress was applied. Furthermore, while (0 for any load is given by the product AO17, the stress of concern is the incremental added stress or... 

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. 

K(l) is the function defining the time dependence of the creep. The constant ac is a critical stress characteristic of the material, and at stresses greater than (r< the creep compliance increases rapidly with stress.. ... 

Figure 8.12 Creep compliance (inverse of modulus) as a function of log (time). The rate of transition from the unrelaxed compliance (higher modulus) to the relaxed compliance (lower limiting modulus) depends on the parameter m.

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

Integral representations for the time dependent compliance and modulus may be written down similarly as above (9). The creep compliance function is given by... 

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... 

There is strong interest to analytically describe the fzme-dependence of polymer creep in order to extrapolate the deformation behaviour into otherwise inaccessible time-ranges. Several empirical and thermo-dynamical models have been proposed, such as the Andrade or Findley Potential equation [47,48] or the classical linear and non-linear visco-elastic theories ([36,37,49-51]). In the linear viscoelastic range Findley [48] and Schapery [49] successfully represent the (primary) creep compliance D(t) by a potential equation ... 

The results of creep experiments are usually expressed in the quantity creep compliance, the time-dependent quotient of strain and stress. 

TABLE 13.18 The basic elastic constants gd and ech, the highest filament values of the modulus, and the strength, average values of the creep compliance, j[t) (ratio of time dependent creep and local stress), at 20 °C and the interchain bond for a variety of organic polymeric fibres (after Northolt et al., 2005)... 

As an example of the concentration dependence of viscoelastic properties in Fig. 16.11 the shear creep compliance of poly(vinyl acetate) is plotted vs. time for solutions of poly(vinyl acetate) in diethyl phthalate with indicated volume fractions of polymer, reduced to 40 °C with the aid of the time temperature superposition principle (Oyanagi and Ferry, 1966). From this figure it becomes clear that the curves are parallel. We may conclude that the various may be shifted over the time axis to one curve, e.g. to the curve for pure polymer. In general it appears that viscoelastic properties measured at various concentrations may be reduced to one single curve at one concentration with the aid of a time-concentration superposition principle, which resembles the time-temperature superposition principle (see, e.g. Ferry, General references, 1980, Chap. 17). The Doolittle equation reads for this reduction ... 

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. 

In the preceding sections, we have looked at the various types of relaxation processes that occur in polymers, focusing predominantly on properties like stress relaxation and creep compliance in amorphous polymers. We have also seen that there is an equivalence between time (or frequency) and temperature behavior. In fact this relationship can be expressed formally in terms of a superposition principle. In the next few paragraphs we will consider this in more detail. First, keep in mind that there are a number of relaxation processes in polymers whose temperature dependence we should explore. These include ... 

Rheological properties of mayonnaise have been studied using different rheological techniques steady shear rate-shear stress, time dependent shear rate-shear stress, stress growth and decay at a constant shear rate, dynamic viscoelastic behavior, and creep-compliance viscoelastic behavior. More studies have been devoted to the study of rheological properties of mayonnaise than of salad dressings, probably because the former is a more stable emulsion and exhibits complex viscous and viscoelastic rheological behavior. 

Strictly speaking, there are no static viscoelastic properties as viscoelastic properties are always time-dependent. However, creep and stress relaxation experiments can be considered quasi-static experiments from which the creep compliance and the modulus can be obtained (4). Such tests are commonly applied in uniaxial conditions for simphcity. The usual time range of quasi-static transient measurements is limited to times not less than 10 s. The reasons for this is that in actual experiments it takes a short period of time to apply the force or the deformation to the sample, and a transitory dynamic response overlaps the idealized creep or relaxation experiment. There is no limitation on the maximum time, but usually it is restricted to a maximum of 10" s. In fact, this range of times is complementary, in the corresponding frequency scale, to that of dynamic experiments. Accordingly, to compare these two complementary techniques, procedures of interconversion of data (time frequency or its inverse) are needed. Some of these procedures are discussed in Chapters 6 and 9. 

The / = 0 intercept of the long-time creep compliance is a measure of the stored elastic energy in flow, and is called the steady state compliance J q-The time-dependent strain of a viscoelastic solid in creep is sketched as the bottom curve in Fig. 7.24. The long-time creep compliance of any solid is simply a time-independent compliance /eq that is the reciprocal of its equilibrium modulus Geq. 

Two steady states are recognized for the long-time creep compliance of materials. Either the sample is a solid and the compliance becomes time independent or the sample is a liquid and the compliance becomes linear in -time. Once steady state has been achieved in creep, the stress can be removed (a = 0) and the elastic recoil, called creep recovery, can be measured. Recovery strain is defined as 7r(0 s 7(0) — 7(0 for t > 0, where t is defined to be zero at the start of recovery. The recoverable compliance is defined as the ratio of the time-dependent recovery strain 7r(0 and the initially applied stress a, where both 7r and t are now defined to be zero at the start of recovery ... 

For a solid, the viscosity is infinite, and = sq all deformation in creep is subsequently recovered in creep recovery, with precisely the same time dependence, as shown in the lower curves in Fig. 7.25. In contrast, only the elastic part of the compliance of a liquid is recovered, as shown in the upper curves of Fig. 7.25 ... 

In Figure 5.8d an intermediate behavior, called viscoelastic, is depicted such a relation is often called a creep curve, and the time-dependent value of the strain over the stress applied is called creep compliance. On application of the stress, the material at first deforms elastically, i.e., instantaneously, but then it starts to deform with time. After some time the material thus exhibits flow for some materials, the strain can even linearly increase with time (as depicted). When the stress is released, the material instantaneously loses some of it deformation (which is called elastic recovery), and then the deformation decreases ever slower (delayed elasticity), until a constant value is obtained. Part of the deformation is thus permanent and viscous. The material has some memory of its original shape but tends to forget more of it as time passes. 

Examples include the time-dependent modulus from stress-strain experiments, and the constant-force creep compliance.2,11... 

The four commonly used techniques to extract information on the viscoelastic behavior of suspensions are creep-compliance measurements, stress-relaxation measurement, shear-wave velocity measurements, and sinusoidal oscillatory testing (25-27). In general, transient measurements are aimed at two types of measurements, namely, stress relaxation, which is to measure the time dependence of the shear stress for a constant small strain, and creep measurement, which is to measure the time dependence of the strain for a constant stress. 

Measurements of creep and elastic recovery also provide sensitive, direct means of detecting yield stress, either by simultaneous fit of time dependent strain, y(x), at a constant stress, to the compliance equation ... 

Creep tests are made mostly in tension, but creep experiments can also be done in shear, torsion, flexure, or compression. Creep data provide important information for selecting a polymer that must sustain dead loads for long periods. The parameter of interest to the engineer is compliance (J), which is a time-dependent reciprocal of modulus. It is the ratio of the time-dependent strain to the applied constant stress [J(t) = e(t)/Oo]. Figure 13.3 shows creep curves for a typical polymeric material. 

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