Viscoelastic yield behaviour

The above observations enable a qualitative relation, based upon the role of crack tip blunting, to be postulated between the fracture behaviour and the viscoelastic yield behaviour of the materials1 8- 44,45). The basis for this relationship is the idea that crack-tip blunting decreases the local crack tip stress concentration and thus higher applied loads are required to cause failure. 

For the investigation of the time and the temperature dependence of the fibre strength it is necessary to have a theoretical description of the viscoelastic tensile behaviour of polymer fibres. Baltussen has shown that the yielding phenomenon, the viscoelastic and the plastic creep of a polymer fibre, can be described by the Eyring reduced time (ERT) model [10]. The shear deformation of a domain brings about a mutual displacement of adjacent chains, the... 

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. 

Just as real materials may have behaviour close to ideal yielding behaviour but with some features similar to those of viscoelastic materials, materials that exhibit behaviour close to the ideal linear viscoelastic may exhibit features similar to those of yield, particularly for high stresses and long times of application. If the term e i) in equation (7.4) is not zero, a linear viscoelastic material will not reach a limiting strain on application of a fixed stress at long times the strain will simply increase linearly with time. The material may also depart from linearity at high or even moderate stresses, so that higher stresses produce disproportionately more strain. These are features characteristic of 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. " ... 

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... 

As we have seen, the strain rate dependence does suggest that yield behaviour often indicates the presence of two thermally activated processes, as discussed above. In some cases, notably polyethylene, a double yield point is observed. Ward and co-workers [64], Seguala and Darras [65] and Gupta and Rose [66] concur that these two deformation processes are essentially interlamellar shear and intra lamellar shear (or c-slip). They are akin to the dynamic mechanical relaxation processes identified in Chapter 10.7.1 for the specially oriented PE sheets, and Seguala and Darras have related them to the a and o 2 transitions reported by Takayanagi [67]. This establishes a direct link between yield and viscoelastic behaviour. 

On this approach, yield can be considered to be nucleation controlled as distinct from the viscoelastic approach that can be considered to be velocity controlled. It implies a direct link between the shear modulus and the shear yield stress. Brown [77] proposed that there is good empirical evidence for this supposition, which had been suggested by other workers previously [25,78,79], and it is an essential ingredient of nucleation controlled yield behaviour as developed formally by Bowden and Raha [42], Argon [43] and others. 

Since the publication of the second edition in 1983, the subject has advanced considerably in many respects, especially with regard to non-linear viscoelasticity, yield and fracture. We have altered some chapters very little, notably those dealing with viscoelastic behaviour and the earlier research on anisotropic mechanical behaviour and rubber elasticity, only adding sections to deal with the latest developments. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

Much less is known about the settling of particles in fluids exhibiting a yield stress. Barnes (39) suggests that this is partly due to the fact that considerable confusion exists in the literature as to whether or not the fluids used in the experiments do have a true yield stress 39. Irrespective of this uncertainty, which usually arises from the inappropriateness of the rheological techniques used for their characterisation, many industrially important materials, notably particulate suspensions, have rheological properties closely approximating to viscoelastic behaviour. 

It is easy to understand that these solutions must exhibit viscoelastic properties. Under shear flow the vesicles have to pass each other and, hence, they have to be deformed. On deformation, the distance of the lamellae is changed against the electrostatic forces between them and the lamellae leave their natural curvature. The macroscopic consequence is an elastic restoring force. If a small shear stress below the yield stress ery is applied, the vesicles cannot pass each other at all. The solution is only deformed elastically and behaves like Bingham s solid. This rheological behaviour is shown in Figure 3.35. which clearly reveals the yield stress value, beyond which the sample shows a quite low viscosity. 

In Fig. 13.71 the ratio of yield stress and temperature of polycarbonate is plotted vs. log rate of deformation at various temperatures. The lines appear to be parallel and their mutual distance can be described by a WLF equation. This behaviour shows that the yielding process is a viscoelastic phenomenon. 

The sedimentation results obtained with the structured suspensions, are consistent with the predictions from rheological investigations. In the absence of any bentonite clay, the pesticidal suspension exhibits Newtonian behaviour with unmeasurable yield value, modulus or residual viscosity. In this case the particles are free to settle individually under gravity forming a dilatant sediment or clay. On the other hand, at bentonite concentrations above the gel point (> 30 g dm the non-Newtonian behaviour of the suspensions and in particular their viscoelastic behaviour results from the formation of a "three-dimensional" network, which elastically supports the total mass. After 21 weeks standing in 100 ml measuring cylinders, no separation was observed when the bentonite concentration was >37.5 g dm corresponding to a modulus > 60 Nm. Clearly the modulus value required to support the mass of the suspension depends on the density difference between particle and medium. 

The viscoelastic nature of polymers generally determines rate and temperature dependence of their mechanical properties. At low strain levels, i.e. in a linear regime, this dependence is well described by intrinsic material properties defined within constitutive viscoelastic laws [1]. At high strains, in presence of failure processes, such as yielding or fracture, it is more difficult to establish a constitutive behaviour as well as to define material properties able to intrinsically characterise the failure process and its possible viscoelastic features. 

Some more recently published studies (e.g. Hennicke and Tomsu, 1970 Tomsu, 1974) stress the viscoelastic behaviour of refractories at high temperatures on the basis of extensive experiments. According to these authors, slow deformations take place even under very low loads without any perceptible yield value. The viscoelastic behaviour has the result that the deformation is partially reversible but not immediately after the stress has been removed. Such properties can be expected especially with materials whose glassy phase has a viscosity corresponding to the transformation range, or with materials polycrystalline in character, free from flux. 

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