Viscoelastic behaviour, linear stress-strain

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... 

The terms are arranged into sections dealing with basic definitions of stress and strain, deformations used experimentally, stresses observed experimentally, quantities relating stress and deformation, linear viscoelastic behaviour, and oscillatory deformations and stresses used experimentally for solids. The terms which have been selected are those met in the conventional mechanical characterization of polymeric materials. 

Note 1 In linear viscoelastic behaviour, stress and strain are assumed to be small so that the squares and higher powers of crand f may be negleeted. 

Note 4 For linear viscoelastic behaviour, a sinusoidal stress (o) results from the sinusoidal strain with... 

Note 5 For linear viscoelastic behaviour interpreted in terms of complex stress and strain (see notes 2 and 3)... 

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. 

The viscoelastic behaviour of rubbers is not linear stress is not proportional to strain, particularly at high strains. The non-linearity is more pronounced in tension or compression than in shear. The result in practice is that dynamic stiffness and moduli are strain dependent and the hysteresis loop will not be a perfect ellipse. If the strain in the test piece is not uniform, it is necessary to apply a shape factor in the same manner as for static tests. This is usually the case in compression and even in shear there may be bending in addition to pure shear. Relationships for shear, compression and tension taking these factors into account have been given by Payne3 and Davey and Payne4 but, because the relationships between dynamic stiffness and the basic moduli may be complex and only approximate, it may be preferable for many engineering applications to work in stiffness, particularly if products are tested. 

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. 

Dynamic mechanical analysis involves the determination of the dynamic properties of polymers and their mixtures, usually by applying a mechanical sinusoidal stress For linear viscoelastic behaviour the strain will alternate sinusoidally but will be out of phase with the stress. The phase lag results from the time necessary for molecular rearrangements and this is associated with the relaxation phenomena. The energy loss per cycle, or damping in the system, can be measured from the loss tangent defined as ... 

Though a simple Maxwell model in the form of equations (1) and (2) is powerful to describe the linear viscoelastic behaviour of polymer melts, it can do nothing more than what it is made for, that is to describe mechanical deformations involving only infinitesimal deformations or small perturbations of molecules towards their equilibrium state. But, as soon as finite deformations are concerned, which are typically those encountered in processing operations on pol rmers, these equations fail. For example, the steady state shear and elongational viscosities remain constant throughout the entire rate of strain range, normal stresses are not predicted. 

In this section we introduce the matter of equivalent mechanieal circuits on an elementary level. First we restrict ourselves to linear viscoelastic behaviour. Second, to show the basic elements, idealized cases will be emphasized (mainly strain retardation and stress relaxation), ignoring for the time being the problem of how to carry out such experiments. As a rule, however, we keep in mind that stress-wise the monolayer is always at equilibrium, and strain has to adjust to it. In this section only dilational rheology will be considered, but this is not a real restriction because for shear the formalism is the same mutatis mutandis. 

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.

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. 

Ideal yielding behaviour is approached by many glassy polymers well below their glass-transition temperatures, but even for these polymers the stress-strain curve is not completely linear even below the yield stress and the compliance is relatively high, so that the deformation before yielding is not negligible. Further departures from ideality involve a strain-rate and temperature dependence of the yield stress. These two features of behaviour are, of course, characteristic of viscoelastic behaviour. 

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... 

It has long been recognised that the mechanical properties of polymers are time-dependent. The behaviour at very small strains (less than 0 5%) can be described by the theory of linear viscoelasticity. Conventionally the stress a at time t is related to the strain e at all previous instants by the equation... 

The second model (Figure 4.15d) describes the complicated viscoelastic behaviour of bitumen. Upon application of stress, the model immediately presents elastic deformation and continues to deform at a non-linear rate. Thus, for a given temperature, if a constant stress (oi) is applied, the strain (e) after time (t) could be calculated using the Burgers model by the following equation ... 

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. 

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. " ... 

Crosslinked NR nanocomposites were prepared with montmorillonite. Morphology was characterized using transmission electron microscopy (TEM), wide-angle X-ray scattering (WAXS), and dynamic mechanical analysis (DMA). X-ray scattering patterns revealed clay intercalation and TEM showed dispersion with partial delamination. The loss modulus peak broadened with clay content, while Tg remain constant. Montmorillonite reinforced the rubber. The DMA exhibited non-linear behaviour typified as a Payne effect (see Section 20.11) that increased with clay content and was more pronounced for this type of nanocomposite. Viscoelastic behaviour was observed under large strains via recovery and stress relaxation. ... 

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. 

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. 

A simple possible formulation of linear viscoelastic behaviour combines these equations, making the assumption that the shear stresses related to strain and strain rate are additive ... 

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... 

The viscoelasticity can be categorized as either linear or nonlinear, but only the linear viscoelasticity can be described theoretically with uncomplicated mathematics. The fundamental viscoelastic parameters of a linear viscoelastic system do not depend on the magnitude of the stress or strain. Therefore, the linear viscoelastic regime is always used for studying the mechanical properties of viscoelastic blended materials. One of the accepted techniques for investigating the viscoelastic behaviours of natural rubber blended materials is the... 

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.

For small stresses, we can use the approximation sinha w a in equation (8.3) so that the strain rate is proportional to the applied stress. In this case, the behaviour is linear and viscous. As stresses are small, the deformation is not plastic, but elastic, for there is a restoring force corresponding to the spring element in figure 8.7(a), whereas equation (8.3) describes the dashpot element of the Kelvin model. The behaviour is thus linear viscoelastic. At larger stresses, deviations from linearity occur, although the behaviour is still viscoelastic. 

---