Compliance functions, creep

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

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

ANALYSIS OF COMPLEX CREEP COMPLIANCE FUNCTIONS AT LOW FREQUENCIES... 

There are a great number of techniques for the experimental determination of viscoelastic functions. The techniques most frequently found in the literature are devoted to measuring the relaxation modulus, the creep compliance function, and the components of the complex modulus in either shear, elongational, or flexural mode (1-4). Although the relaxation modulus and creep compliance functions are defined in the time domain, whereas the complex viscoelastic functions are given in the frequency domain, it is possible, in principle, by using Fourier transform, to pass from the time domain to the frequency domain, or vice versa, as discussed earlier. 

Figure 8.2 Double logarithmic plots of the creep compliance function in the time domain at various temperatures for solutions of polystyrene in tri-m-tolyl phosphate the weight fraction of polymer in the solution is 0.70. The subscript p in Jp t) indicates that the values of this function have been reduced to a common temperature.

Figure 8.5 Double logarithmic plots of the creep compliance function versus t/a-p, where ap = (ri/Tio)(Po7b/p70 and the subindex 0 refers to 100°C. (From Ref. 5.)...

Accordingly, the loss compliance function presents a maximum in the frequency domain at lower frequency than the loss relaxation modulus. This behavior is illustrated in Figure 8.18, where the complex relaxation modulus, the complex creep compliance function, and the loss tan 8 for a viscoelastic system with a single relaxation time are plotted. Similar arguments applied to a minimum in tan 8 lead to the inequalities... 

In the glass-like zone, the values of the creep compliance function seem to be independent of the concentration however, the changes that take place in the values of J t) in the transition zone are larger the lower the concentration. The length of the plateau increases with the concentration, and the plateau and terminal zones merge into a single region at low concentrations. The location of the isotherms on the time scale is shifted to shorter chains as the concentration decreases. 

The shift factors corresponding to different isotherms representing the recovery creep compliance function of a solution of polystyrene in tri-m-tolyl phosphate are given in the table. Find the parameters of the WLF equation. 

In the transient compliance function, J(t), the retardation spectrum L(x) is modulated by the function 1 — exp(—t/x) [see Eq. (9.15)]. Plotting this function against In t/x gives the sigmoidal curve shown in Figure 9.6. We should note that the time of observation ( ) in the first quadrant is greater than the retardation times, and as a result x varies between zero and t. Then the creep compliance function for viscoelastic liquids is approximately given by (1,2)... 

Figure 9.6 Plot of the kernel of the creep compliance function versus log t/x.

The response to the stress input of the Kelvin-Voigt element is schematically represented in Figure 10.5. From Eq. (10.22), the creep compliance function is easily obtained as... 

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. 

C Compliance (A/P) constant for craze growth creep compliance function... 

When the strains or the strain rates are sufficiently small, the creep response is Unear. In this case, when the time-dependent strain is divided by the fixed stress, a unique creep compUance curve results that is, at each time there is only one value for this ratio, which is the compliance—y(t)lao = J t). The unique shear creep compliance function J t) (Pa or cm /dyne, 1 Pa = 0.1 cm /dyne) obtained for an amorphous polymer has the usual contributions... 

The first two terms on the right-hand side of equation [12.6] are viscoelastic terms proposed by Schapery, where e represents uniaxial kinematic (or total) strain at time t, o is the Cauchy stress at time t, is the instantaneous compliance and AD(r[i ) is a transient creep compliance function. The factor g defines stress and temperature effects on the instantaneous elastic compliance and is a measure of state dependent reduction (or increase) in stiffness. Transient compliance factor gi has a similar meaning, operating on the creep compliance component. The factor gj accounts for the influence of loading rate on creep. The function i ) represents a reduced timescale parameter defined by ... 

The creep compliance function AD22(0 is modeled by using a Prony series of the form,... 

The most common technique employed to date has been that of creep in uniaxial tension. It was shown above that with the inclusion of lateral strain measurements this is a powerful technique giving access to up to 6 independent creep compliance functions. This is more than for any other known method. It further has the overwhelming advantage over many methods, such as say torsional or flexural creep, that the stress is sensibly uniform over the working volume of the specimen. This advantage is paramount in studies of materials displaying non-linear behaviour in creep since analysis of the non-uniform stress situation in non-linear systems is not well developed. Attempts to overcome the non-uniform stress situation in torsion, by recourse to, say, torsion of thin walled tubes, lead to severe difSculties in specimen preparation in oriented materials, when anisotropy of behaviour is to be studied. 

Write down the set of differential equations relating strain y, stress a, and time t for the generalized Zener model shown in Figure 4.17. Solve them to show that the creep compliance function Jit) is given by... 

A certain pipe-grade PVC deforms in shear with a creep compliance function at 20°C of the form... 

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