Viscoelastic behaviour, linear viscoelasticity

The simplest theoretical model proposed to predict the strain response to a complex stress history is the Boltzmann Superposition Principle. Basically this principle proposes that for a linear viscoelastic material, the strain response to a complex loading history is simply the algebraic sum of the strains due to each step in load. Implied in this principle is the idea that the behaviour of a plastic is a function of its entire loading history. There are two situations to consider. 

For a linear viscoelastic material in which the strain recovery may be regarded as the reversal of creep then the material behaviour may be represented by Fig. 2.49. Thus the time-dependent residual strain, Sr(t), may be expressed as... 

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. 

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

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... 

The constant Tr is called the Trouton ratio10 and has a value of 3 in this experiment with an incompressible fluid in the linear viscoelastic limit. The elongational behaviour of fluids is probably the most significant of the non-shear parameters, because many complex fluids in practical applications are forced to extend and deform. Studying this parameter is an area of great interest for theoreticians and experimentalists. 

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... 

This section is primarily concerned with the behaviour of simple homo-polymers. The development of viscoelastic theory was intimately linked with the study of polymeric species. This area of activity has led the way in the development of rheological models and experimental design and so is a very important area for the proto-rheologist to understand. So far in this chapter we have taken the approach of developing phase diagrams from a rheological perspective in order to understand linear viscoelastic... 

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... 

The surface characteristics of these species are determined by the particulates and stress transfer across the membrane will tend to be low, reducing internal circulation within the drop. The structure of the interface surrounding the drop plays a significant role in determining the characteristics of the droplet behaviour. We can begin our consideration of emulsion systems by looking at the role of this layer in determining linear viscoelastic properties. This was undertaken by... 

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. 

Model of the linear viscoelastic behaviour of a liquid in which (q ) + P)cT= Dy... 

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