Viscoelastic behaviour, linear 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. 

Another approach we can use to describe the stress relaxation behaviour and all the linear viscoelastic responses is to calculate the relaxation spectrum H. Ideally we would like to model or measure the microstructure in the dispersion and include the role of Brownian diffusion in the loss of structural order. The intermediate scattering... 

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 ... 

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. 

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 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. 

Ilie anomalous behaviour in the linear viscoelasticity has been explained by the tube model.Figure 7.26 shows schematically how the stress relaxation takes place in star polymers. In the crude theory, it is assumed that the centre of the star is fixed during the viscoelastic relaxation time and that the relaxation takes place only by the contour length fluctuation, i.e., by the process that the polymer retracts its arm down the tube and evacuates from the deformed tube as shown in Fig. 7.26. 

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. ... 

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. 

When an instantaneous strain is applied to an ideal elastic solid a frnite and constant stress will be recorded. For a linear viscoelastic solid subjected to a nominally instantaneous strain the initial stress will be proportional to the applied strain and will decrease with time (Figure 4.4), at a rate characterized by the relaxation time r. This behaviour is called stress relaxation. For amorphous linear polymers at high temperatures the stress may eventually decay to zero. In the following discussion we shall ignore transient behaviour. 

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. 

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... 

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... 

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... 

In Chapter 5, we introduced linear viscoelasticity. In this scheme, the observed creep or stress relaxation behaviour can be viewed as the defining characteristic of the material. The creep compliance function - the ratio of creep strain e t) to the constant stress a - is a function of time only and is denoted as J t). Similarly and necessarily, the stress relaxation modulus, the ratio of stress to the constant strain, is the function G(r). Any system in which these two conditions do not apply is non-linear. Then, the many useful and elegant properties associated with the linear theory, notably the Boltzmann superposition principle, no longer apply and theories to predict stress or strain are approximations that must be supported by experiment. 

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... 

The flow properties of molten polyethylene (PE) are of primary practical concern in forming, moulding, or extrusion processes. Deviating considerably from ideal Newtonian flow, melts of PE show non-ideal viscoelastic behaviour their shear rate versus shear stress plot is non-linear. Very closely related to this non-linear behaviour of pseudoelastic materials is their elastic nature, in that some recovery may be observed when an applied stress is relaxed. Molecularly, chain disentanglement in the course of increasing stress can explain these phenomena. 

It is clear that any theoretical explanations of the above phenomena should be able to account for the dependence of the stress and strain upon time. Ideally it should be possible to predict, for example, the stress relaxation behaviour from knowledge of the creep curve. In practice, with real polymers this is somewhat difficult to do but the situation is often simplified by assuming that the polymer behaves as a linear viscoelastic material. It can be assumed that the deformation of the polymer is divided into an elastic component and a viscous component and that the deformation of the polymer can be described by a combination of Hooke s... 

The most surprising result is that such simple non-linear relaxation behaviour can give rise to such complex behaviour of the stress with time. In Figure 6.3(b) there is a peak termed a stress overshoot . This illustrates that materials following very simple rules can show very complex behaviour. The sample modelled here, it could be argued, can show both thixotropic and anti-thixotropic behaviour. One of the most frequently made non-linear viscoelastic measurements is the thixotropic loop. This involves increasing the shear rate linearly with time to a given... 

Abstract The discussion of relaxation and diffusion of macromolecules in very concentrated solutions and melts of polymers showed that the basic equations of macromolecular dynamics reflect the linear behaviour of a macromolecule among the other macromolecules, so that one can proceed further. Considering the non-linear effects of viscoelasticity, one have to take into account the local anisotropy of mobility of every particle of the chains, introduced in the basic dynamic equations of a macromolecule in Chapter 3, and induced anisotropy of the surrounding, which will be introduced in this chapter. In the spirit of mesoscopic theory we assume that the anisotropy is connected with the averaged orientation of segments of macromolecules, so that the equation of dynamics of the macromolecule retains its form. Eventually, the non-linear relaxation equations for two sets of internal variables are formulated. The first set of variables describes the form of the macromolecular coil - the conformational variables, the second one describes the internal stresses connected mainly with the orientation of segments. 

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). 

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