Transient creep compliance

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. 

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 transient creep compliance, D (i i), can be expressed in the following exponential form ... 

In creep measurements of polyacrylonitrile gels [78,82], the shear creep compliance J(t) behawd as JAt/tof/U + (t/to)"l where is the steady state compliance, the time constant to could be of the order of a minute, and n 0.75. This implies G co) (ko)" for coto 1 and S(q, t) for t to. We thus expect emergence of the power law (6.39) or more complicated transient decays in many cases. 

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. 

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)... 

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. 

Transient Response Creep. The creep behavior of the polsrmeric fluid in the nonlinear viscoelastic regime has some different features from what were found with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation... 

Oscillatory studies are useful for materials with short relaxation times, comparable to the period of oscillation. For long relaxation times, however, transient methods are used. The creep test subjects the material to constant stress, and follows the strain as a function of time. Creep compliance J(t) is the ratio of strain to stress, and is a function of elapsed time from the instant of application of the stress ... 

Methods exist which, in principle, permit all of the transient and dynamic moduli to be calculated from the results of any one of the analyses described above. For example, from the stress relaxation test it should be possible to calculate creep compliance and dynamic moduli. In practice, however, these calculations are not so simple. They require accurate stress relaxation data over a wide range of time, including times close to zero. The calculation procedures involve convolution techniques which are not very successful if the data are inaccurate or incomplete. 

Similarly, the creep compliance can be separated into an instantaneous component and a transient component such that. 

We will now consider another type of transient response. Let a shear stress a be applied at the viscoelastic solution at = 0. In a general case, a time-dependent shear strain is developed that can be measured to get the creep comphance J(t). The creep compliance has the dimensions of a reciprocal modulus, and it is therefore an increasing function of time. From theories of viscoelasticity, it is possible to calcrflate the creep compliance from the relaxation modrflus and inverse ... 

The creep function J(t) is the transient strain per unit stress in a step-stress experiment. The resolution at short times is also limited from instrument response and sensitivity. J(t) at short times may also be derived from the high frequency complex compliance data. 

---