Compliance, creep

Following the application of a step stress oq at t = 0 as shown in Fig. 4.11, the total strain t) can be measured to yield the creep compliance J t) defined as 

The two viscoelastic response functions J t) and G t) are related to each other as shown in the following  

Setting t = 0 in the equation obtained by substituting Eq. (4.67) into Eq. (4.68), we obtained the relation between the initial relaxation modulus and creep compliance  

It can be shown by applying the Laplace transformation that Eq. (4.73) is equivalent to  

The convolution integral Eq. (4.73) may be solved nmnerically by the method of Hopkins and Hamming. The method is discussed in Appendix 4.A. Shown in Fig. 4.13 is the comparison of the G i) and curves of a nearly monodisperse polystyrene melt with = 4.69 x 10 obtained at 114.5°C (detailed in Chapter 14). The G t) line shape has more features — meaning different processes are better separated or resolved — than the curve, which are smeared by the con- 

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Figure 3.12 shows that, at long times, the creep compliance is directly proportional to time for the polymers shown. For polystyrene (M = 600,000) at 100°C, the following values describe the linear portion of the datat ... 

Not only are the creep compliance and the stress relaxation shear modulus related but in turn the shear modulus is related to the tensile modulus which itself is related to the stress relaxation time 0. It is therefore in theory possible to predict creep-temperature relationships from WLF data although in practice these are still best determined by experiment. 

Johnson [111] addressed the problem of viscoelastic flow by attempting to modify the JKR equation. In his approach, he postulated a creep compliance function... 

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. 

Occasionally in creep analysis it is convenient to use a Creep Compliance instead of the creep modulus. This is simply given by... 

In creep tests, the parameter of interest is the creep compliance, J, defined as the ratio of the creep strain to the applied stress, i.e. 

Since the stress is constant, it follows that so also is the creep rate. The creep compliance at time t, 7, can be considered to consist of three terms, an instantaneous compliance, Jq, a term covering a variety of retardation processes, xj/it), and a viscous term, t/rj. These are related by ... 

Galgali and his colleagues [46] have also shown that the typical rheological response in nanocomposites arises from frictional interactions between the silicate layers and not from the immobilization of confined polymer chains between the silicate layers. They have also shown a dramatic decrease in the creep compliance for the PP-based nanocomposite with 9 wt% MMT. They showed a dramatic three orders of magnitude drop in the zero shear viscosity beyond the apparent yield stress, suggesting that the solid-like behavior in the quiescent state is a result of the percolated structure of the layered silicate. 

Rgure 9 Master curves for creep compliance of polyisoprene of various molecular weights at a reference temperature of - 3()0C ... 

Figure 10 Creep compliance of polyisoprcne at various temperatures. Data are for a fraction with a molecular weight of 1.12 x 10. (Prom Ref. 32.)...

If the Boltzmann superposition principle holds, the creep strain is directly proportional to the stress at any given time, f Similarly, the stress at any given lime is directly proportional to the strain in stress relaxation. That is. the creep compliance and the stress relaxation modulus arc independent of the stress and slrai . respectively. This is generally true for small stresses or strains, but the principle is not exact. If large loads are applied in creep experiments or large strains in stress relaxation, as can occur in practical structural applications, nonlinear effects come into play. One result is that the response (0 l,r relaxation times can also change, and so can ar... 

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 11 Shear creep of polyethylene (density = 0.950) at different loads after 10 min, and as a function of applied stress. Deviation firm the value of 1.0 indicates a dependence of creep compliance on load.

Many other data in the literature show a strong dependence of creep compliance on the applied load, although in some cases the authors did not discuss this aspect of creep. Stress dependence is found with all kinds of plastics. For example, the creep of polyethylene has been studied by... 

Nere- Xf is the weight-average molecular weight of the blend, and Afwl is the weight-average molecular weight of component /. The steady-state creep compliance 7, of the blend is... 

Experiments have been made in which uncross-linkable polymer rubbers have been added to a similar rubber that is subsequently cross-linked (199). As an example, polyisobutylene was added to butyl rubber before it was cross-linked. The polyisobutylene molecules were not attached to the network structure, so they could be extracted by a solvent. As expected, the polyisobutylene greatly increased the creep compliance over that of the pure butyl rubber. 

Figure 21 Creep compliance as a function of degree of crystallinity above Tu. Numbers, on curves are rough values of the degree of crystallinity.

Even in cases where the rigid polymer forms the continuous phase, the elastic modulus is less than that of the pure matrix material. Thus two-phase systems have a greater creep compliance than does the pure rigid phase. Many of these materials craze badly near their yield points. When crazing occurs, the creep rate becomes much greater, and stress relaxes rapidly if the deformation is held constant. 

Plot the creep compliance (cm /dyn) as a function of time using a logarithmic time scale. Would the curve show the upward curvature on a linear time scale ... 

Table 8 The basic elastic constants g and ec, the highest filament values of the modulus ( ) and the strength (q,), together with the average values of the creep compliance (/(f)) at 20 °C (ratio of creep rate and load stress) and the interchain bond for a variety of organic polymer fibres... 

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. ... 

G storage modulus as measured using a cone and plate rheometer G" loss modulus as measured using a cone and plate rheometer J(t) creep compliance... 

Studies have been conducted on creep compliance tests in which paint films were subjected to tensile loads of 4-7 psi (27.2-47.6 x 10 N/m ) and to 6% ozone for 505 h. A typical result for a high-quality emulsion-base paint is shown in Figure 13-1. Creep compliance is reduced by exposure to 6% ozone. If the effect is linearly related to ozone concentration, we might expect the same reduction in creep compliance at 0.1-ppm ozone in 3 x 10 h, or some 30,(XX) yr. Thus, reduction in creep compliance is not viewed as having a serious ozone contribution. 

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