Viscoelastic behaviour, linear creep

Note 5 Creep is sometimes described in terms of non-linear viscoelastic behaviour, leading, for example, to evaluation of recoverable shear and steady-state recoverable shear compliance. The definitions of such terms are outside the scope of this document. 

The mechanical response of polypropylene foam was studied over a wide range of strain rates and the linear and non-linear viscoelastic behaviour was analysed. The material was tested in creep and dynamic mechanical experiments and a correlation between strain rate effects and viscoelastic properties of the foam was obtained using viscoelasticity theory and separating strain and time effects. A scheme for the prediction of the stress-strain curve at any strain rate was developed in which a strain rate-dependent scaling factor was introduced. An energy absorption diagram was constructed. 14 refs. 

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. 

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.

When plastics are unloaded, the creep strain is recoverable. This contrasts with metals, where creep strains are permanent. The Voigt linear viscoelastic model predicts that creep strains are 100% recoverable. The fractional recovered strain is defined as 1 — e/cmax, where e is the strain during recovery and Cmax is the strain at the end of the creep period. It exceeds 0.8 when the recovery time is equal to the creep time. Figure 7.9 shows that recovery is quicker for low Cmax and short creep times, i.e. when the creep approaches linear viscoelastic behaviour. 

Just as linear viscoelastic behaviour with full recovery of strain is an idealisation of the behaviour of some real polymers under suitable conditions, so ideal yield behaviour may be imagined to conform to the following for stresses and strains below the yield point the material has time-indepen-dent linear elastic behaviour with a very low compliance and with full recovery of strain on removal of stress at a certain stress level, called the yield stress, the strain increases without further increase in the stress if the material has been strained beyond the yield stress there is no recovery of strain. This ideal behaviour is illustrated in fig. 8.1 and the differences between ideal viscoelastic creep and ideal yield behaviour are shown in table 8.1. 

The variation of the isochronous modulus at 100 s with the magnitude of the creep strain at 100 s for strains in the region OT-10% in samples cut at various angles to the fibre axis is shown in Fig. 3. The data were obtained using the isochronous stress-strain procedure, previously referred to, on LDPE drawn at 20 C so as to produce fibre symmetry with a draw ratio of 4-2. In this figure horizontal straight lines would indicate linear viscoelastic behaviour. The strain at which significant deviation firom... 

The variation of the 100 second tensile creep modulus with 100 second tensile strain is presented in Fig. 16. The behaviour of a specimen cut from an isotropic sheet (which had been subjected to the same thermal cycle as the drawn sheet) is included for comparison. It is apparent that all specimens exhibited non-linear viscoelastic behaviour, but there is little anisotropy of non-linearity. Furthermore the degree of non-linearity exhibited by the specimens from the drawn sheet is similar to that of a specimen from the isotropic sheet. At any chosen creep strain the anisotropy of modulus for the drawn sheet is relatively low. 

A phenomenological model has been proposed for the non-linear viscoelastic behaviour of thermorheologjcally complex polymer glasses prior to and including yield. The approach was based upon stress additivity. A linear viscoelastic material will exhibit stress-strain additivity. The molecular processes modelled were resolved into two parallel processes, each with a characteristic relaxation time spectrum. The model described the yield behaviour and creep experiments at increasing stress. " ... 

An alternative experimental procedure to creep and stress relaxation is to subject the specimen to an alternating strain and simrdtaneously measure the stress. For linear viscoelastic behaviour, when equilibrium is reached, the stress and strain will both vary sinusoidally, but the strain lags behind the stress. Thus we write... 

Fig. 4.2. Isochronals taken at r, (see Fig. 4.1) after the initiation of the creep experiment y(r.) and at r , v(fb)- diagram illustrates the transition from linear to non-linear viscoelastic behaviour. Note that this is not the y-o plot which would be obtained in a conventional stress strain test the data are taken from creep experiments at different stresses.

It is convenient to introduce the discussion of linear viscoelastic behaviour with the onedimensional situation of creep under a fixed load. For an elastic solid, the following is observed at the two levels of stress cto and 2cto (Figure 5.2(a)). 

In the diagram illustrating creep under constant load recovery curves are also displayed. We will presently show that the recovery behaviour is basically similar to the creep behaviour if we neglect the quantity C3, the Newtonian flow. This is a direct consequence of linear viscoelastic behaviour. 

Consider again the general linear differential equation, which represents linear viscoelastic behaviour. From the present discussion, it follows that to obtain even an approximate description of both stress relaxation and creep, at least the first two terms on each side of Equation (5.9) must be retained, that is the simplest equation will be of the form... 

Two principal approaches have been used to model the yield behaviour of polymers. The first approach addresses the temperature and strain-rate dependence of the yield stress in terms of the Eyring equation for thermally activated processes [39]. This approach has been applied to many amorphous and crystalline polymers (see Section 12.5.1) and links have been established with molecular relaxation processes determined by dynamic mechanical and dielectric measurements and with non-linear viscoelastic behaviour determined by creep and stress relaxation. The Eyring approach assumes that the yield process is velocity controlled, i.e. the yield process relates to existing thermally activated processes that are accelerated by the application of the yield stress to the point where the rate of plastic deformation reaches the applied macroscopic strain rate. This approach has... 

For a linear viscoelastic material in which the strain recovery may be regarded as the reversal of creep then the material behaviour may be represented by Fig. 2.49. Thus the time-dependent residual strain, Sr(t), may be expressed as... 

There is strong interest to analytically describe the fzme-dependence of polymer creep in order to extrapolate the deformation behaviour into otherwise inaccessible time-ranges. Several empirical and thermo-dynamical models have been proposed, such as the Andrade or Findley Potential equation [47,48] or the classical linear and non-linear visco-elastic theories ([36,37,49-51]). In the linear viscoelastic range Findley [48] and Schapery [49] successfully represent the (primary) creep compliance D(t) by a potential equation ... 

Boltzmann extended the idea of linearity in viscoelastic behaviour to take account of the time dependence He assumed that, in a creep experiment ... 

In chapter 7 the phenomenon of creep in a viscoelastic solid is considered. For an ideal linear viscoelastic medium the deformation under a constant stress eventually becomes constant provided that in equation (7.4) is zero. If the load is removed at any time, the ideal material recovers fully. For many polymers these conditions are approximately satisfied for low stresses, but the curves (b) and (c) in fig. 6.2 indicate a very different type of behaviour that may be observed for some polymers under suitable conditions. For stresses above a certain level, the polymer yields. After yielding the polymer either fractures or retains a permanent deformation on removal of the stress. 

These observations suggest that, for real materials, there need not be a very clear-cut distinction between yield and creep. In fact polymers present a complete spectrum of behaviours between the two ideal types. Fortunately, however, many polymers under conditions of temperature and strain-rate that are within the ranges important for applications do have behaviours close to one of these ideals. It is therefore useful both from a practical and from a theoretical point of view to try to understand these approximately ideal behaviours before attempting to study the more complicated behaviours exhibited by other materials. In chapter 7 this approach is used in discussing creep and linear viscoelasticity and in the present chapter it is used in discussing yield. 

There appears to be no rigorous theoretical scheme for describing anisotropy of creep behaviour in these materials. However, simple extensions of linear viscoelastic theory are presented and shown to be useful though not completely rigorous. Further development is clearly desirable. 

Inspection of this equation shows that it models reasonably well, on a very superficial level, a stress-strain curve of the type shown in Fig. 1(b), curve (4). In other words it raises the question as to whether the deviations from linear stress-strain relationships observed in constant strain-rate tests might not be merely resulting from the intrinsic time-dependence of the linear viscoelasticity, which can be more clearly studied in creep or stress-relaxation and not due to some new process starting at high stresses. It does not take long to show that at the strain-levels of 3-5% experienced at yield, the response of most polymers is highly non-linear (r(t)/ is a function of strain-rate S as well as t, and so eqn. (14) cannot adequately describe the behaviour. However, it is also clear that at... 

This short discussion of non-linear viscoelasticity has been included in order to show that most of the features associated with yield in constant strain-rate tests are directly related to aspects of the creep behaviour. It is not yet clear what significance, if any, can be attributed to a measurement of a yield stress from a load-maximum (especially if the latter is complicated by the occurrence of a necking instability), or to an arbitrary proof strain, although the flow stress (Tf, at which the long-term creep... 

Under static loading conditions where either the stress or strain is keeping constant polymer materials (especially thermoplastics) show non-linear viscoelastic deformation behaviour to appear as retardation (creep) or relaxation. Long-term investigations to analyse creep or relaxation can be accomplished at flexural, indentation, or uniaxial tensile or compression loading as a function of time and loading level as well as environmental conditions such as temperature, media etc. (see [13Gre], p. 171 - 183). 

The theory of linear viscoelasticity is phenomenological there is no attempt to discover the time and frequencty response of the solid in an altogether a priori fashion. The aim is to predict behaviour under certain circumstances, having observed it under others for example, to correlate creep, stress relaxation, and (fynamic properties so that if one of these has been determined then all the others are known. This is closety related to electrical network theory, both in aim and, as will soon be apparent, in method. 

Creep, stress relaxation and deformation under constant strain rate can be described assuming a viscoelastic response. Application of a constant strain can give rise to yield in thermoplastic materials. At yield the viscoelastic behaviour is non-linear, though the transition to non-linear is likely to occur prior to yield. 

The transition from linear to non-hnear viscoelastic behaviour and the transition threshold has been investigated with time and temperature. Thermal and mechanical tests were applied followed by isothermal creep tests at temperature steps and with differing stress levels. Isochronal creep curves were constructed to reveal the non-linearity threshold at different times and temperatures. The study was important for design and performance of elastomeric components. ... 

In this chapter we describe the common forms of viscoelastic behaviour and discuss the phenomena in terms of the deformation characteristics of elastic solids and viscous fluids. The discussion is confined to linear viscoelasticity, for which the Boltzmann superposition principle enables the response to multistep loading processes to be determined from simpler creep and relaxation experiments. Phenomenological mechanical models are considered and used to derive retardation and relaxation spectra, which describe the time-scale of the response to an applied deformation. Finally we show that in alternating strain experiments the presence of the viscous component leads to a phase difference between stress and strain. 

The models discussed here, which are phenomenological and have no direct relation with chemical composition or molecular structure, in principle enable the response to a complicated loading pattern to be deduced from a single creep (or stress-relaxation) plot extending over a long time interval. Interpretation depends on the assumption in linear viscoelasticity that the total deformation can be considered as the sum of independent elastic (Hookean) and viscous (Newtonian) components. In essence, the simple behaviour is modelled by a set of either integral or differential equations, which are then applicable in other situations. 

Figure 4.6 The creep behaviour of a linear viscoelastic solid...

In Chapter 4 we introduced linear viscoelasticity. In this scheme, observed creep or stress relaxation behaviour can be viewed as the defining characteristic of the material. The definition of the creep compliance function J t), which is given as the ratio of creep strain e t) to the constant stress o, may be recalled as... 

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