Creep functions

By adopting a logarithmic creep function j(t)=j110log(t), Eq. Ill yields the following equation for the lifetime of a fibre... 

Figure 3.11 Qualitative creep function of a Voigt polymer solid...

At long times, the deformation rate becomes constant, and the creep function approaches a straight line ... 

An analytic expression for the reduced creep function, obtained from Eq. (3.11) with Eq. (3.6) for the linear array, is shown in Table 4 Eq. (T4). Je is the elastic equilibrium compliance and y (1/2, w) is the incomplete gamma function of order 1/2, extensively tabulated and available in most computer subroutine libraries. The last term accounts for contribution of plastic flow, where present. Reduced creep functions (recoverable and including flow contributions) are plotted in Fig. 4 as a function of Z. The... 

To express creep, use of the reciprocal modulus, the compliance, D = HE, is to be preferred the creep function then reads ... 

With all these models, the simple ones as well as the spectra, it has to be supposed that stress and strain are, at any time, proportional, so that the relaxation function E(t) and the creep function D(t) are independent of the levels of deformation and stress, respectively. When this is the case, we have linear viscoelastic behaviour. Then the so-called superposition principle holds, as formulated by Boltzmann. This describes the effect of changes in external conditions of a viscoelastic system at different points in time. Such a change may be the application of a stress or also an imposed deformation. 

A well-known simplified equation for the tensile creep function is Nutting s empirical formula (1921) ... 

A second approximation of the tensile creep function is the Voigt-Kelvin model... 

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. 

The time-temperature equivalence principle makes it possible to predict the viscoelastic properties of an amorphous polymer at one temperature from measurements made at other temperatures. The major effect of a temperature increase is to increase the rates of the various modes of retarded conformational elastic response, that is, to reduce the retarding viscosity values in the spring-dashpot model. This appears as a shift of the creep function along the log t scale to shorter times. A secondary effect of increasing temperature is to increase the elastic moduli slightly because an equilibrium conformational modulus tends to be proportional to the absolute temperature (13). 

Another method of handing this would be by inversion of the Prony series, remembering that the product of the Laplace transforms of the modulus and the creep functions equals l7. 

The experimental creep function is often analyzed as a nnm-ber of Kelvin elements in series (Figure 40.32), each having the property of a spring and dashpot in parallel (Genevaux, 1989 Martensson, 1992 Mohager and Toratti, 1993 Hanhijarvi, 1999 Passard and Perre, 2005). In the case of uniaxial load, this leads to... 

It is apparent that in the case of orthorhombic S3fmmetries 9 independent creep functions are necessary to characterise fully the creep behaviour of... 

It should be noted that in the case of fibre symmetry (in which there are only 5 independent creep functions) tensile creep measurements on specimens cut at only three angles to the fibre axis (say 0°, 45° and 90°) will give three independent combinations of the five, say S22it [2S23(t)+S44(t)] and SssU). Whilst if lateral contraction measurements are also carried out during creep then all 5 functions, including the... 

For orthorhombic symmetry on the other hand, tensile creep and lateral compliance measurements on specimens cut from oriented sheet will yield only 6 of the 9 required creep functions those not accessible by this method being Suit), Sssit) and S66(t)- The two shear compliances 555(1) and Seeit) can be obtained by torsional creep experiments, but these need to be carefully designed and involve complex experimental procedures. The only possibility for measurement of Su(t) on sheet appears to be by compressive creep techniques, however, one would expect substantial experimental difficulties largely associated with strain measurement and specimen geometry. There appears to be no reported evaluation of the full characterisation of creep for the case of orthorhombic synunetry. 

Substitution of the simple special form of the creep function of eq. (5.2) gives... 

J(A.,T, ) has known the meaning of the creep function at constant temperature as a function of the effective time A.. All these results may be transferred immediately to the problem of viscoelastic behavior under the influence of aging at constant temperature [9,15]. The temperature history has to be replaced by the degree of aging. A, of the sample and the time-temperature shift function, a(T,T ), by the time-age shift function, b (A,A ). The degree of aging, also called the age of the sample, defines the time elapsed from the last quench from the equilibrium state down to the aging temperature so far A, can then be expressed ... 

Since one, universal creep function can hardly fit all the experimental data, many functions were suggested to provide an empirical description of creep phenomena. As seen above, functions for logarithmic, transient and linear creep have been suggested in various forms and combinations. 

Figure 4.14 Simulation of creep function through combining Kelvin units in series (schematic)...

Similarly, for the strain in terms of the stress history a creep function C was defined as... 

For an isotropic material, the creep functions B t) and C(r) are determined using the same procedure. For an axisymmetric triaxial stress state, let cti be the axial stress, CT3(< o) (with >2 = 0 ) the confining pressure, i the measured axial strain and 3 the measured lateral strain. Then setting... 

An alternative to the preceding experiment is, after steady shear flow of a viscoelastic liquid under stress creep recovery is simply given by the creep function. The strain at 2 is 7(t2) = + tV Jo) the strain during the re-... 

If the slope m of the creep function J(t) on a doubly logarithmic plot is quite small (precluding the presence of viscous flow, of course), it is related to the loss tangent J /J by the equation3 ... 

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