Maxwell model structural relaxation

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

Figure 12.10 shows the mechanical response as a funetion of time of two structures formed by combining a spring and a dashpot in series (Maxwell model) and in parallel (Voigt model). Creep is slow deformation of a viseoelastic material under constant stress (o), while relaxation is the time response of the stress after imposing a constant deformation (e). Thus, a simple mathematieal expression accounts for the relation between structure (combination of elements) and a property (creep or relaxation). Evidently, more complex responses ean be obtained by eombining several elements in series and in parallel. 

The material properties (moduli and relaxation times) are then calculated from knowledge of network structure and free volume. Strains imparted by processing and reaction are determined. These inputs are then applied to a viscoelastic bi-Maxwell model, whereby stress in the polymer is determined. Time is then incremented and the procedure repeated until the cure profile is complete. 

This figure shows that at low frequencies the system can be described by the normal Maxwell model with a rise in G with a slope 2 and in G with a slope 1. However, the second increases of G and G" do not fit into the Maxwell model. Although the second plateau value of G is not reached at the available frequencies, it was possible to fit the data with a Burger model, which consists of two Maxwell elements with different shear moduli and structural relaxation times, as follows ... 

For polyacrylamide there are two rheological effects which can be explained in terms of its random coil structure. Firstly, it was discussed above that polyacrylamide is much more sensitive than xanthan to solution salinity and hardness. This is explained by the fact that the salinity causes the molecular chain to collapse, which results in a much smaller molecule and hence in a lower viscosity solution. The second effect which can be explained in terms of the polyacrylamide random coil structure is the viscoelastic behaviour of this polymer. This is shown both in the dynamic oscillatory measurements and in the flow through the stepped capillaries (Chauveteau, 1981). When simple models of random chains are constructed, such as the Rouse model (Rouse, 1953 Bird et al, 1987), the internal structure of these bead and spring models gives rise to a spectrum of relaxation times, Analysis of this situation shows that these relaxation times define response times for the molecule, as indicated in the simple Maxwell model for a viscoelastic fluid discussed above. Thus, because of the internal structure of a flexible coil molecule, one would expect to observe some viscoelastic behaviour. This phenomenon is discussed in much more detail by Bird et al (1987b), in which a range of possible molecular models are discussed and the significance of these to the constitutive relationship between stress and deformation rate and deformation history is elaborated. 

Our goal was to measure the viscoelastic properties of the human brain under practical conditions. Therefore, we used the tactile resonant sensor with the stress-strain function that simulated manual palpation. In this study, the stiffness was 2.837 0.709 (N), Young s elastic modulus was E = 5.08 1.31, and the shear modulus was G = 1.94 0.49 for a depth of 3.0 mm. Poisson s ratio (u) was calculated as 0.31-0.62 using the equation E = 2G (1h- u). These values were approximately equal to those previously reported for the viscoelasticity properties of the brain in vivo [1-7]. The results of indentation fitted the Maxwell model as expressed by the equation G = Ge - Gi exp (-t/x), where Ge is the instantaneous modulus in shear, Gi is the relaxation in the shear modulus, t is time, and x is the relaxation time. Thus, G = 1.94-1- 3.3 exp(-t/0.5) under the assumption that Ge = 1.94, Gi = 3.3, t = h/1.5, and x = 0.5. The results obtained in this study by an indentation method, reflected those of a previous model [9-12]. However, this measurement method evaluated brain viscoelasticity via multiple structural layers including the skin, subcutaneous tissues, muscle fascia, and dura. Moreover, some assumptions had to be made to approximate the expression for elasticity. 

The Maxwell model can successfully describe monoexponential stress relaxation properties. This holds for the regime of small deformations, shear rates, or shear stresses. In the region of elevated mechanical forces, severe departures from simple Newtonian and Hookean occur. In viscoelastic surfactant solutions, it is often observed that the shear viscosity decreases markedly with increasing amount of shear. This t3q)ical behavior is called shear thinning or pseudo plastic. The non-Newtonian behavior of such solutions is of great practical interest, and it is intimately connected with orientation processes or structural changes that occur during flow. 

The structure is essentially that of a Maxwell fluid if Equation 9.23 is multiplied by Y, but with a relaxation time equal to that decreases with increasing stress.) Values of s are typically of order 0.02 or less, and at this level the shear viscosity and normal stress differences are insensitive to e. The parameter f arises in the network model as a slip coefficient that reflects the motion of the network relative to the continuum. 

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