Viscoelastic behaviour, linear Maxwell model

It is likely that most biomaterials possess non-linear elastic properties. However, in the absence of detailed measurements of the relevant properties it is not necessary to resort to complicated non-linear theories of viscoelasticity. A simple dashpot-and-spring Maxwell model of viscoelasticity will provide a good basis to consider the main features of the behaviour of the soft-solid walls of most biomaterials in the flow field of a typical bioprocess equipment. 

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. 

The WLF equation can now be obtained in a simple manner. The model representations of linear viscoelastic behaviour all show that the relaxation times are given by expressions of the form r = rjlE (see the Maxwell model in Section 4.2.3 above), where t] is the viscosity of a dashpot and E the modulus of a spring. 

An important and sometimes overlooked feature of all linear viscoelastic liquids that follow a Maxwell response is that they exhibit anti-thixo-tropic behaviour. That is if a constant shear rate is applied to a material that behaves as a Maxwell model the viscosity increases with time up to a constant value. We have seen in the previous examples that as the shear rate is applied the stress progressively increases to a maximum value. The approach we should adopt is to use the Boltzmann Superposition Principle. Initially we apply a continuous shear rate until a steady state... 

This model is known as the standard linear solid and is usually attributed to Zener [4]. It provides an approximate representation to the observed behaviour of polymers in their main viscoelastic range. As has been discussed, it predicts an exponential response only. To describe the observed viscoelastic behaviour quantitatively would require the inclusion of many terms in the linear differential Equation (5.9). These more complicated equations are equivalent to either a large number of Maxwell elements in parallel or a large number of Voigt elements in series (Figures 5.13(a) and (b)). 

Fig. 5.7 Mechanical models used to represent the viscoelastic behaviour of polymers, (a) Maxwell model, (b) Voigt model, (c) Standard linear solid.

Materials, and, in particular, polymers, that show viscoelastic behaviour, can be modelled by a combination of perfectly elastic Hookean springs and Newtonian viscous dashpots. For many polymers the behaviour at temperatures above when strains are small 1 per cent) is approximately represented by the so-called standard linear substance. This consists of a dashpot and spring in series (called a Maxwell element) and this combination in parallel with a second spring of different elastic modulus. In deformation, the strain of the Maxwell element and of the spring will be the same, say e. At a time t let Oi be the stress in the Maxwell element and 02 that in the spring. Then, if Ei is the Young s modulus of the Maxwell element and E2 that of the spring ... 

At very short times this simplifies to elastic behaviour. Then at t = r, the stress is 1/e of its value at steady state, where it is cr= 77/, i.e. purely steady-state viscous behaviour. The start-up of real viscoelastic liquids may need to be modelled using a number of Maxwell elements. However, for most realistic experiments using this kind of test, the response quickly enters the non-linear region since the strain is continually increasing. 

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