Strain superposition

Firstly, it has been shown that there may be many experimental problems in a direct determination of the experimental fimction. In shear, damping functions obtained from step strain and from step strain rate experiments do not match each other. This poses an important question on the validity of the separability assumption in the short time rai e. Significant departures from this factorization have already been observed in the case of narrow polystyrene fractions by Takahashi et al. [54]. These authors found that time-strain superposition of the linear and nonlinear relaxation moduli was only possible above a cert2un characteristic time. It is interesting to note that this is predicted by the Doi-Edwards theory [10] and according to this theory, this phenomena is attributed to an additional decrease of the modulus connected to a tube contraction process and time-strain separability may hold after this equilibration process has been completed. Other examples of non-separability were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions. 

Time-Strain Superposition. The principle of time-strain superposition is essentially the same as that for time-temperature superposition, thou now there is a strain induced shift [acceleration] in the time scale of the material response (7-9). Again, within the context of the KWW function one can write the time-strain shift function as where Yo and y rqjresent the reference and current strains. Similarly, the vertical strain shifts are by=G y / G y). Because the shear strain in the samples is a ftinction of radial portion r and at large deformations the shear stress is not linear in the deformation, the modulus of the material is a function of r. Hence, for the time-strain superposition we followed an isochronal analysis developed by McKenna and Zapas (10) based on tiie elastic scaling analysis of Penn and Kearsley (11,12). In this case we can write ... 

Figure 3. Tonperature shift ftu tors, log as a function of temperature. Table II. Vertical shift factors for time - temperature and time - strain superposition...

Time-strain superposition was also shown to be possible. However, the resulting master curve was rignificantly dirierent fixxn tii obtained using time-temperarnre superporition. This is problematic if it is assumed tiiat there exists a single underlying re nse function. 

Non-linear viscoelastic mechanical behaviour of a crosslinked sealant was interpreted as due to a Mullins effect. The Mullins effect was observed for a series of sealants under tensile and compression tests. The Mullins effect was partially removed after a mechanical test, when a long relaxation time was allowed, that is the modulus increased over time. Non-linear stress relaxation was observed for pre-strained filler sealants. Time-strain superposition was used to derive a model for the filled sealants. Relaxation over long periods demonstrates that the Mullins effect is caused by non-equilibrium with experimental conditions being faster than return to the initial state. If experiments were conducted over times of the order of a day there may be no Mullins effect. If a filled elastomer were only required to perform its function once per day then each response might be linear viscoelastic. 

P. A. O Connell and G. B. McKenna, Large Deformation Response of Polycarbonate Time-Temperature, Time-Aging Time and Time-Strain Superposition , Polym. Eng. Sci. 37, 1485-1495 (1997). 

The Stress-Rang e Concept. The solution of the problem of the rigid system is based on the linear relationship between stress and strain. This relationship allows the superposition of the effects of many iadividual forces and moments. If the relationship between stress and strain is nonlinear, an elementary problem, such as a siagle-plane two-member system, can be solved but only with considerable difficulty. Most practical piping systems do, ia fact, have stresses that are initially ia the nonlinear range. Using linear analysis ia an apparendy nonlinear problem is justified by the stress-range concept... 

The simplest theoretical model proposed to predict the strain response to a complex stress history is the Boltzmann Superposition Principle. Basically this principle proposes that for a linear viscoelastic material, the strain response to a complex loading history is simply the algebraic sum of the strains due to each step in load. Implied in this principle is the idea that the behaviour of a plastic is a function of its entire loading history. There are two situations to consider. 

Now consider the situation in which the stress, ai, was applied at time, ti, and an additional stress, Boltzmanns Superposition Principle states that the total strain at time, t, is the algebraic sum of the two independent responses. 

The predicted strain variation is shown in Fig. 2.43(b). The constant strain rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle. Of course superposition is not restricted to this simple model. It can be applied to any type of model or directly to the creep curves. The method also lends itself to a graphical solution as follows. If a stress is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54). When a second stress, 0 2 is added then the new creep curve will be obtained by adding the creep due to 02 to the anticipated creep if stress a had remained... 

Fig. 2.44(b) Predicted strain response using Boltzmann s superposition principle... 

A plastic with a time dependent creep modulus as in the previous example is stressed at a linear rate to 40 MN/m in 100 seconds. At this time the stress in reduced to 30 MN/m and kept constant at this level. If the elastic and viscous components of the modulus are 3.5 GN/m and 50 x 10 Ns/m, use Boltzmann s Superposition Principle to calculate the strain after (a) 60 seconds and (b) 130 seconds. 

A plastic which behaves like a Kelvin-Voigt model is subjected to the stress history shown in Fig. 2.87. Use the Boltzmanns Superposition Principle to calculate the strain in the material after (a) 90 seconds (b) 150 seconds. The spring constant is 12 GN/m and the dashpot constant is 360 GNs/m. ... 

The overall distribution of stresses and strains in the local and global directions is shown in Fig. 3.23. If both the normal stress and the bending are applied together then it is necessary to add the effects of each separate condition. That is, direct superposition can be used to determine the overall stresses. 

Object in this section is to review how rheological knowledge combined with laboratory data can be used to predict stresses developed in plastics undergoing strains at different rates and at different temperatures. The procedure of using laboratory experimental data for the prediction of mechanical behavior under a prescribed use condition involves two principles that are familiar to rheologists one is Boltzmann s superposition principle which enables one to utilize basic experimental data such as a stress relaxation modulus in predicting stresses under any strain history the other is the principle of reduced variables which by a temperature-log time shift allows the time scale of such a prediction to be extended substantially beyond the limits of the time scale of the original experiment. 

Indeed, the self-consistent model averages the stresses and strains in either phase of a two-phase material, and it determines them, by solving separate problems, whose superposition yields the final configuration of the model 7). 

Since we are interested in this chapter in analyzing the T- and P-dependences of polymer viscoelasticity, our emphasis is on dielectric relaxation results. We focus on the means to extrapolate data measured at low strain rates and ambient pressures to higher rates and pressures. The usual practice is to invoke the time-temperature superposition principle with a similar approach for extrapolation to elevated pressures [22]. The limitations of conventional t-T superpositioning will be discussed. A newly developed thermodynamic scaling procedure, based on consideration of the intermolecular repulsive potential, is presented. Applications and limitations of this scaling procedure are described. 

The idea is simple consider a polycrystalline material that is subjected to locally varying strain. Then every crystal is probing its local strain by small compression or expansion of the lattice constant. The superposition of all these dilated lattices makes the observable line profiles - and as a function of order their breadth has to increase linearly. According to Kochendorfer the polycrystalline material becomes inhomogeneous or heterogeneous . 

A similar superposition holds for stress-relaxation experiments in which the strain is changed during the course of the experiments. The Bolt/mann superposition principle for stress relaxation is... 

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... 

For glassy and crystalline polymers there are few data on the variation of stress relaxation with amplitude of deformation. However, the data do verily what one would expect on the basis of the response of elastomers. Although the stress-relaxation modulus at a given time may be independent of strain at small strains, at higher initial fixed strains the stress or the stress-relaxation modulus decreases faster than expected, and the lloltz-nuinn superposition principle no longer holds. 

There are many types of deformation and forces that can be applied to material. One of the foundations of viscoelastic theory is the Boltzmann Superposition Principle. This principle is based on the assumption that the effects of a series of applied stresses acting on a sample results in a strain which is related to the sum of the stresses. The same argument applies to the application of a strain. For example we could apply an instantaneous stress to a body and maintain that stress constant. For a viscoelastic material the strain will increase with time. The ratio of the strain to the stress defines the compliance of the body ... 

The ideal stress relaxation experiment is one in which the stress is instantaneously applied. We have seen in Section 4.4.2 the exponential relaxation that characterises the response of a Maxwell model. We can consider this experiment in detail as an example of the application of the Boltzmann Superposition Principle. The practical application of an instantaneous strain is very difficult to achieve. In a laboratory experi-... 

Let us suppose the strain applied at time t0 increases over a time v to a maximum value y. At times less than to — v no strain is applied and at times greater than t0 the strain is constant. This gives the limits to the Boltzmann superposition integral ... 

The application of a linearly ramped strain can provide information on both the sample elasticity and viscosity. The stress will grow in proportion to the applied strain. The ratio of the strain over the applied time gives the shear rate. Applying the Boltzmann Superposition Principle we obtain the following expression ... 

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