Viscoelastic behaviours

Many mechanical property measurement tests do not deal with samples that change as a function of time. For example, the application of force to a steel sample at room temperature causes an instantaneous extension that does not change with time. If the load is increased the extension immediately increases to a new value, proportional to the load increase. This is Hooke s law and describes the behaviour of materials within their linear elastic range. 

The opposite end of the spectrum to elastic behaviour is viscous behaviour. This is readily evident in liquids where they flow in response to an applied force. Frequently, the action of gravity on the fluid mass is sufficient to cause significant flow (witness the phenomenon of 

One of the major differences between elastic and viscous behaviour is that in the former case a sample will always return to its original size upon removal of the load i.e. the deformation is fully recoverable. Clearly this does not occur in the case of a liquid. The flow in a Newtonian liquid is permanent and the fluid will retain its new shape, even if the load is removed. 

The spring part represents force or energy that is put into the sample that will be fully and instantaneously recoverable upon removal of the load. This can be identified as elastic behaviour. The dashpot part represents the force that is put into the sample that results in permanent deformation and is lost. Neither the sample deformation nor this energy can be recovered. This can be identified as viscous behaviour. 

In practice, the question of whether a sample fuUy recovers will depend on the model that describes how the molecular structure responds to force. If there is a large contribution from a purely dashpot element, then much of the deformation will be permanent. This is the situation in linear polymers with no cross-hnking, LDPE for example. In the absence of 

Loading mode Level Strength (N) Initiala Averageb Finale 

Finally, the rate dependent properties are usually modeled by a three-parameter solid which consists of a spring (m2) and a dashpot (h) in parallel connected to another spring (mf) in series. Viscoelastic properties may also be expressed in terms of the dynamic modulus G. A sinusoidal displacement of the form u = is applied to the specimen 

The anisotropic viscoelastic properties in shear of the meniscus have been determined by subjecting discs of meniscal tissue to sinusoidal torsional loading [35](Table B2.8). The specimens were cut in the three directions of orthotropic symmetry, i.e. circumferential, axial and radial. A definite correlation is seen with the orientation of the fibers and both the magnitude of the dynamic modulus IG I and the phase angle 8. 

The viscoelastic properties of the human intervertebral disc have been modeled [36, 37] using the three-parameter solid. The parameters were obtained by fitting experimentally obtained creep curves to analytical equations using linear regression (Table B2.9). 

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Throughout this chapter the viscoelastic behaviour of plastics has been described and it has been shown that deformations are dependent on such factors as the time under load and the temperature. Therefore, when structural components are to be designed using plastics, it must be remembered that the classical equations which are available for the design of springs, beams, plates, cylinders, etc., have all been derived under the assumptions that... 

Example 2.17 Establish and plot the variation with frequency of the storage and loss moduli for materials which can have their viscoelastic behaviour described by the following models... 

The viscoelastic behaviour of a certain plastic is to be represented by spring and dashpot elements having constants of 2 GN/m and 90 GNs/m respectively. If a stress of 12 MN/m is applied for 100 seconds and then completely removed, compare the values of strain predicted by the Maxwell and Kelvin-Voigt models after (a) 50 seconds (b) 150 seconds. 

During a test on a polymer which is to have its viscoelastic behaviour described by the Kelvin model the following creep data was obtained when a stress of 2 MN/m was applied to it. 

A Standard Model for the viscoelastic behaviour of plastics consists of a spring element in scries with a Voigt model as shown in Fig. 2.86. Derive the governing equation for this model and from this obtain the expression for creep strain. Show that the Unrelaxed Modulus for this model is and the Relaxed Modulus is fi 2/(fi + 2>. 

The grade of polypropylene whose creep curves are given in Fig. 2.5 is to have its viscoelastic behaviour fltted to a Maxwell model for stresses up to 6 MN/m and times up to ICKX) seconds. Determine the two constants for the model and use these to determine the stress in the material after 900 seconds if the material is subjected to a constant strain of 0.4% throughout the 900 seconds. 

The following type of differential equation is encountered in the text, for example, in the analysis of the models for viscoelastic behaviour ... 

Then, for a particulate composite, consisting of a polymeric matrix and an elastic filler, it is possible by the previously described method to evaluate the mechanical and thermal properties, as well as the volume fraction of the mesophase. The mesophase is also expected to exhibit a viscoelastic behaviour. The composite consists, therefore, of three phases, out of which one is elastic and two viscoelastic. 

Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid and a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study(18). Two common and easily observable examples of viscoelastic behaviour in a liquid are ... 

For further information on viscoelastic behaviour, reference should again be made to specialist sources l4-I6). 

Virtual head of centrifugal pumps 332 Viscoelastic behaviour 104,114, 116, 196... 

Some examples where the viscoelastic behaviour of polymer solutions are exploited are their use as thickeners in ... 

Investigations of the rheological properties of disperse systems are very important both from the fundamental and applied points of view (1-5). For example, the non-Newtonian and viscoelastic behaviour of concentrated dispersions may be related to the interaction forces between the dispersed particles (6-9). On the other hand, such studies are of vital practical importance, as, for example, in the assessment and prediction of the longterm physical stability of suspensions (5). 

For a Newtonian fluid, the shear stress is proportional to the shear rate, the constant of proportionality being the coefficient of viscosity. The viscosity is a property of the material and, at a given temperature and pressure, is constant. Non-Newtonian fluids exhibit departures from this type of behaviour. The relationship between the shear stress and the shear rate can be determined using a viscometer as described in Chapter 3. There are three main categories of departure from Newtonian behaviour behaviour that is independent of time but the fluid exhibits an apparent viscosity that varies as the shear rate is changed behaviour in which the apparent viscosity changes with time even if the shear rate is kept constant and a type of behaviour that is intermediate between purely liquid-like and purely solid-like. These are known as time-independent, time-dependent, and viscoelastic behaviour respectively. Many materials display a combination of these types of behaviour. 

The final main category of non-Newtonian behaviour is viscoelasticity. As the name implies, viscoelastic fluids exhibit a combination of ordinary liquid-like (viscous) and solid-like (elastic) behaviour. The most important viscoelastic fluids are molten polymers but other materials containing macromolecules or long flexible particles, such as fibre suspensions, are viscoelastic. An everyday example of purely viscous and viscoelastic behaviour can be seen with different types of soup. When a thin , watery soup is stirred in a bowl and the stirring then stopped, the soup continues to flow round the bowl and gradually comes to rest. This is an example of purely viscous behaviour. In contrast, with certain thick soups, on cessation of stirring the soup rapidly slows down and then recoils slightly. 

The simplest model that can show the most important aspects of viscoelastic behaviour is the Maxwell fluid. A mechanical model of the Maxwell fluid is a viscous element (a piston sliding in a cylinder of oil) in series with an elastic element (a spring). The total extension of this mechanical model is the sum of the extensions of the two elements and the rate of extension is the sum of the two rates of extension. It is assumed that the same form of combination can be applied to the shearing of the Maxwell fluid. 

The function iKt-t ) may be interpreted as a memory function having a form as shown in Figure 3.14. For an elastic solid, iff has the value unity at all times, while for a purely viscous liquid iff has the value unity at thfe current time but zero at all other times. Thus, a solid behaves as if it remembers the whole of its deformation history, while a purely viscous liquid responds only to its instantaneous deformation rate and is uninfluenced by its history. The viscoelastic fluid is intermediate, behaving as if it had a memory that fades exponentially with time. The purely elastic solid and the purely viscous fluid are just extreme cases of viscoelastic behaviour. 

N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behaviour, Springer-Verlag, Berlin, 1989. 

Both of these models show contributions from the viscosity and the elasticity, and so both these models show viscoelastic behaviour. You can visualise a more complex combination of models possessing more complex constitutive equations and thus able to describe more complex rheological profiles. 

The previous sections contain a strong mathematical element, which is inherent in the subject matter. Whilst it is possible to perform and model viscoelastic behaviour without fully applying these ideas they can be... 

There are not a great number of studies on the viscoelastic behaviour of quasi-hard spheres. The studies of Mellema and coworkers13 shown in Figure 5.5 indicate the real and imaginary parts of the viscosity in a high-frequency oscillation experiment. Their data can be normalised to a characteristic time based on the diffusion coefficient given above. 

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