Creep compliance Method

The quasielastic method as developed by Schapery [26] is used in the development of the viscoelastic residual stress model. The use of the quasielastic method is motivated by the fact that the relaxation moduli are required in the viscoelastic analysis of residual stresses, whereas the experimental characterization of composite materials is usually in terms of the creep compliances. An excellent account of the development of the quasielastic method is given in [27]. The underlying restriction in the application of the quasielastic method is that the compliance response of the material shows little curvature when plotted versus log time [28]. Harper [27] shows excellent agreement between the quasielastic method and direct inversion for AS4/3510-6 graphite/epoxy composite. For most graphite/thermoset systems, the restrictions imposed by the quasielastic method are satisfied. 

Laws and McLaughlin30 discuss viscoelastic creep compliances of composite materials using another approach to the problem of the elastic properties of heterogeneous materials - the self-consistent method. 

The master curves obtained from specimens cast from tetrahydro-furan solution at 2 and 4% strain, respectively, are slightly different. These differences, however, are probably within the experimental error. An idea of the reproducibility can be obtained from Figure 4, which shows the master curves of the creep compliances obtained on specimens cut from two sheets of Kraton 102 cast from benzene solution. Although the method of preparation appeared to be identical, there are noticeable differences between the two curves. Even larger differences exist between these curves and the master curve obtained from the relaxation data after conversion to creep. Again, there were no apparent differences in the method of preparation of the sheets from which the specimens for the relaxation and creep tests were cut. 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

As mentioned above, it is very difficult, for experimental reasons, to measure the relaxation modulus or the creep compliance at times below 1 s. In this time scale region, dynamic mechanical viscoelastic functions are widely employed (5,6). However, in these methods the measured forces and displacements are not simply related to the stress and strain in the samples. Moreover, in the case of dynamic experiments, inertial effects are frequently important, and this fact must be taken into account in the theoretical methods developed to calculate complex viscoelastic functions from experimental results. 

Once a relaxation (retardation) spectrum is obtained from a relaxation (creep compliance) viscoelastic function, any other function can be obtained. Alternatively, approximate methods have been developed to calculate viscoelastic functions from one another (10). By taking into account... 

The proposed method of data treatment has two advantages (1) It allows assessment of the status of blend miscibility In the melt, and (11) It permits computation of any linear viscoelastic function from a single frequency scan. Once the numerical values of Equation 20 or Equation 21 parameters are established Che relaxation spectrum as well as all linear viscoelastic functions of the material are known. Since there Is a direct relation between the relaxation and Che retardation time spectra, one can compute from Hq(o)) the stress growth function, creep compliance, complex dynamic compliances, etc. 

The viscoelastic behavior is evaluated by means of two types of methods static tests and dynamic tests. In the first calegtuy a step change of stress or strain is applied and the stress or strain response is recorded as a function of time. Stress relaxation, creep compliance, and creep recovery are static methods. The dynamic tests involve the imposition of an oscillatory strain or stress. Every technique is described in the following sections. 

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. 

Sel Seltzer, R., Mai, Y.-W. Depth sensing indentation of linear viscoelastic-plastic solids A simple method to determine creep compliance. Eng. Fract. Mech. 75 (2008) 4852-4862. 

Averaged experimental creep compliance data are represented by dots in Figure 12.12 as a function of log time. The tests show that even after 14 months the creep continued and deformations kept increasing. After tests of 10000 h the compliance of the materials examined exceeds the instantaneous elastic compliance by a factor varying from 3.8 (for pure PP) to 3.2 (for the 80% PP + 20% PLC blend). This shows that the inelastic strain of these materials manifests itself substantially methods of accelerated testing and prediction have to be subjected to very detailed scrutiny. 

Use the shear creep data in Figure 4.4, together with the method of time-temperature superposition, to estimate the shear creep compliance for linear polyethylene at 20°C and a creep time 10 s. Ust the assumptions that you make in this long extrapx>lation of the creep data. 

The US practice uses the creep compliance and indirect tensile strength test, according to AASHTO T 322 (2011), as a suitable performance test for low-temperature cracking in hot mix asphalt mix design. The same procedure was selected as the material characterisation test method for the prediction of low-temperature cracking of flexible pavements in the AASFITO (2008) Mechanistic-Empirical Pavement Design Guide. 

The test method determines the creep compliance, D(f), the tensile strength, S , and Poisson s ratio, v, of hot mix asphalts. The procedure is applied to test specimens having a maximum aggregate size of 38 mm or less. 

The superposition approach can be used to produce a constitutive equation which expresses the creep compliance (Ca) of the adhesive in terms of a reference creep compliance (O and shift factors for stress (a ), temperature (at) and resin content (Cv) such that Ca = Cr X a X Ut X Oy X f". The method has been used by Dharmarajan et al. (30) to characterise the creep behaviour of epoxy, polyester and acrylic mortars in the form of prism specimens under 3 point loading. From relatively short-term tests, strain v. time curves such... 

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. 

The problem of the reduction of the J (t) curves in the short- time region is still present. Dependent on T and t the variation of the J (t) values at short creep times is up to 15-20% in the interval regarded. The only quantity that can explain this strong time and temperature dependence of the creep compliance is the free volume. Other reduction methods [13,30], for example pT/p T , lead in this temperature interval to a variation of about 1%. 

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. 

Finally, a numerical method has been outlined by Benbow for calculating the components of the dynamic compliance from the creep compliance, based on a Fourier analysis of the latter. 

The methods previously discussed in this chapter can be used to determine the differential equations, solutions and parameters for a number of mechanical models using a variety of combinations of springs and damper elements. Table 5.1 is a tabulation of the differential equation, parameter inequalities, creep compliances and relaxation moduli for frequently discussed basic models. Note that the equations are given in terms of the pj and qj coefficients of the appropriate differential equation in standard format. The reader is encouraged to verify the validity of the equations given and is also referred to Flugge (1974) for a more complete tabulation. 

The relaxation time spectrum can be calculated exactly from the measured stress relaxation modulus using Fourier or Laplace transform methods, and similar eonsiderations apply to the retardation time spectrum and the creep compliance. It is more convenient to consider these transformations at a later stage, when the final representation of linear viscoelasticity, that of the complex modulus and complex compliance, has been discussed. 

Therefore, with the exception of the Giesekus model, the parameters for all of these constitutive equations can be deduced from the relaxation time spectrum of the material which can be obtained from the small strain linear viscoelasticity measurements alone. There are various numerical methods in the literature which allow the determination of this spectrum from measured viscoelastic master curves, such as dynamic modulus, relaxation modulus, and creep compliance. 

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