Isochronous stress-strain curves

Strain and constant time can give respectively isometric stress-log time curves and isochronous stress-strain curves Figure 9.10). Whilst not providing any new information, such alternative presentations of the data may be preferred for certain purposes. 

ISOCHRONOUS STRESS - STRAIN CURVE CREEP MODULUS - TIME CURVE... 

Example of isochronous stress-strain curves for PCs resulting from stress relaxation. 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

The isochronous stress-strain curves for the creep of PP bead foams (254) were analysed to determine the effective cell gas pressure po and initial yield stress do as a function of time under load (Figure 11). po falls below atmospheric pressure after 100 second, and majority of the cell air is lost between 100 and 10,000 s. Air loss is more rapid than in extruded PP foams, because of the small bead size and the open channels at the bead boundaries, do reduces rapidly at short yield times <1 second, due to proximity of the glass transition, and continues to fall at long times. 

An isochronal stress-strain curve is established by applying deformation to a sample at various strain rates and cross-plotting the results at a fixed time. In this fashion the effects of strain can be separated from the effects of time. 

The isochronal stress-strain curve can be found by analysis of the stress-growth function cr(t y) as given by the Boltzmann superposition relationship of equation (2-46). For a generalized Maxwell model, the result is the familiar... 

The quantity in the brackets is a function of time only, so will be a constant for the isochronal stress-strain experiment. Thus the isochronal stress-strain curve for a linear material will be straight. 

It is evident from these expressions that the isochronal stress-strain response depends on the type of experiment. For example, at t t = 1, the isochronal stress-strain curves will be cr = 0.632 Gy, cr = 0.5 Gy and cr = 0.368 Gy respectively. 

Creep data is usually obtained for a number of different stresses, as creep modulus will only be independent of stress over limited ranges. It may also be important to obtain data as a function of temperature. Commonly, isochronous stress-strain curves are derived from the creep curves at different stress levels as a useful way of displaying the information. 

The isochronous stress strain curve, which is a cartesian plot of stress against creep strain at a specific lime after the application of the load (Fig. 22b). 

There is linear viscoelastic behaviour in the stress region where the isochronous stress-strain curve is linear (to within 5%). The creep compliance /( ), defined by Eq. (7.4), is independent of stress. However, above this stress region (stresses >1 MPa for the data in Fig. 7.7 for a time of 1 year) there is non-linear viscoelastic behaviour and the creep compliance becomes stress dependent... 

Figure 7.7 Isochronous stress-strain curve at a time of I year constructed from the creep data in Figure 7.6. The broken line represents linear viscoelastic behaviour.

Figure 7.8 shows a possible cross section for the beam. The 2 mm thick section was chosen so the injection moulding cycle time is short, while the I beam is efficient in bending (Chapter 13). From the linear portion of the isochronous stress-strain curve, the linear viscoelastic compliance is 7(1 year) = 3.3 x 10 m N . Substituting this and the deflection limit in... 

For constant stress applications, the isochronous stress-strain curve can be used with standard equations by choosing the appropriate effective... 

The use of this modulus based on the maximum stress in the part should provide a conservative estimate of the time and temperature dependent deflection of the part. When the isochronous stress-strain curve is highly nonlinear or the part geometry is complex, finite-element structural analysis techniques can be used. Then, the complete nonlinear, isochronous stress-strain curve can be used in a nonlinear finite-element analysis or a linear effective modulus can be used in a linear analysis. 

The value of the simplified technique for producing isochronous stress-strain curves for non-linear isotropic materials by successive loading and unloading of a single sample have been amply demonstrated over many years and fully described elsewhere.These techniques become even more valuable in studies of anisotropy, where samples may be difficult to obtain in large numbers and where the scope of the problem is much larger. A considerable proportion of work on oriented materials reported in the literature is essentially confined to this measurement and does not include studies of time dependence of behaviour. Detailed work has been carried out validating this procedure for oriented materials by comparison of the isochronous stress-strain data with isochronous sections from families of creep curves. ... 

In studying time dependence, Le. creep behaviour, it is necessary to carry out tests at several stress levels in each direction of interest in the oriented material. I his can involve a prohibitive amount of experimental work and, in practice, little is generally lost by reducing the tests to, say, one creep curve and one isochronous stress-strain curve in each direction. The problem then becomes one of selection of the absolute value of stress for each of the creep curves and is most severe when non-linearity and its anisotropy are well developed. The choice of stress levels is arbitrary but interesting special cases are (a) equal stress levels at all angles and (b) equal strain ranges at all angles. 

In the case of (a), since there can be substantial variations in both compliance and strength (creep rupture) with angle, this may result in creep in some directions involving extremely low strains, and therefore presenting severe measurement problems, whilst in other directions very rapid large creep or rupture may occur thus limiting the information available. It has therefore been found preferable to employ (b) and to choose stress levels in different directions so as to produce equal strains after a specified creep time in all directions. Furthermore if correlation of creep behaviour with deformation mechanisms is sought it may well be desirable to compare the polymer response when the different mechanisms produce similar strains. Selection of appropriate stress-levels is achieved by use of the isochronous stress-strain curves. 

Fig. 18. Isochronal stress-strain curves for oriented PET tested in tension at right angles to the initial draw (O - ) constant strain-rate data (A, A), creep data (after...

Figure 12.6 Isochronous stress-strain curves at t = 0 and t = 120 min for creep at 20°C (After [36].)...

For ease of reference, the creep data are usually replotted in one or more different ways, as Ulustrated in Figures 8.14(a) and (b). Isochronous stress-strain curves (Figure 8.14(a)), which are discussed in Chapter 4, are included in most discussions of creep characteristics. From isochronous curves of this type, the engineer can determine the secant modulus of the pofymer at any given strain or applied stress and time under load. This creep modulus E(creep compliance at the appropriate stress and time creep data is in the form of isometric curves, as shown in Figure 8.14(b), which are helpful in designing plastic components to a... 

This is an example of strain-limited design. We apply the pseudo-elastic design method, specifying the duration of loading as S hours (which is 18000 seconds). In order to determine the modulus, we need the isochronous stress-strain curve for 18000 s. Substituting in the equation,... 

A conventional creep curve as exhibited by most materials is illustrated in Fig. 2.25 although many engineers present the data using log axes to produce a graph of the form shown in Fig. 2.26. Data from families of strain-time curves at various values of constant stress are used to produce isochronous stress-strain curves (Fig. 2.27). These are obtained by cross-plotting stresses and strains at various times from the commencement of loading. The results of creep tests can also be used to derive constant strain, or isometric, curves of stress versus time, also as illustrated in Fig. 2.27. 

For simple material comparisons, determine the stress to produce 1% strain in 1000 h. Select several loads to produce strains in the approximate range of 1% strain and plot the 1000-h isochronous stress-strain curve from which the stress to produce 1% strain may be determined by interpolation. 

Figure 1-12. Average Isochronous Stress-Strain Curves. (Reprinted by permission, ASME.)...

Isochronal stress-strain curves contructed from all of the available experimental results. 

A further complication of creep is that it is nonlinear in strain, just as the stress-strain relationship is nonlinear for plastics. Since plastics are typically subjected to large deformations during their life, it is unfortunately essential to characterize this phenomenon. This phenomenon is best seen in the classic isochronous stress-strain curve which plots stress-strain relationships at several times, decades apart. These curves are invaluable for design and product performance evaluation. 

Where it is necessary to compare several different materials, basic creep curves alone are not completely satisfactory. This is particularly so where the stress levels used are not the same for each material. If the stress endurance time relevant to a particular application can be agreed, a much simpler comparison of materials for a specific application can be made by means of isochronous stress-strain curves. 

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