Instantaneous shear modulus

The viscoelastic behavior of concentrated (20% w/w)aqueous polystryene latex dispersions (particle radius 92nm), in the presence of physically adsorbed poly(vinyl alcohol), has been investigated as a function of surface coverage by the polymer using creep measurements. From the creep curves both the instantaneous shear modulus, G0, and residual viscosity, nQ, were calculated. 

Creep Curves Fig. 4 shows the creep curves for the formulations containing 30 and 45 g dm bentonite. Suspensions containing less than 30 g dm bentonite did not give a measurable creep curve. The creep curves shown in Fig. 4 are typical of those obtained with viscoelastic systems. They consist of three regions (a) directly after the application of the stress one observes a rapid elastic deformation resulting in an elastic compliance (instantaneous shear modulus = T/y =... 

Another class of polymeric materials is the three dimensional viscoelastic amorphous sohds. They range from soft gels, which are two component systems, to hard Bakehte. Alfrey divided the materials into six arbitrary categories. What characterizes all of them is that they are solids the shear viscosity is infinite and the static shear modulus is finite and measurable. They are also characterized by a distribution of relaxation times. The instantaneous shear modulus, G(0), is characteristic of a typical liquid for most of the classes, but can achieve high values for Bakelite. 

In order to obtain a general model of the creep and recovery functions we need to use a Kelvin model or a Kelvin kernel and retardation spectrum L. However, there are some additional subtleties that need to be accounted for. One of the features of a Maxwell model is that it possesses a high frequency limit to the shear modulus. This means there is an instantaneous response at all strains. The response of a simple Kelvin model is shown in Equation 4.80 ... 

Gg (instantaneous modulus), Hg (residual viscosity) and G (shear modulus) all showed a rapid increase above 30g dm bentonite. This was attributed to the formation of a gel network structure in the continuous medium and the strength of such a gel increased with increase in bentonite concentration. The results could be qualitatively described in terms of the elastic floe model of Hunter and co-workers. Moreover, the settling characteristics of the structured suspensions were found to be consistent with the predictions from the rheological measurements. This demonstrates the value of rheological studies in predicting the longterm physical stability of suspension concentrates. 

A perfectly elastic solid subjected to a non-destructive shear force will deform almost instantaneously an amount proportional to its shear modulus and then deform no further, strain energy being stored in the bonds of the material. A fluid, on the other hand, continues to deform under the action of a shear stress, the energy imparted to the system being dissipated as flow. 

In order to better understand the shear modulus (and the time dependence of the shear modulus) we shall describe it mathematically. Elastic behavior is the instantaneous response to a stress as shown in Figure 4.11. When a stress is imposed, the material deforms instantaneously. When the stress is removed the material relaxes completely. The amount of deformation (strain) is related to the stress (pressure) by the shear modulus according to ... 

Instantaneously deformed high molar mass polymer melts (long polymer chains in their liquid state) behave at intermediate times as networks with well-defined values of shear modulus, called the plateau modulus Ge, which is independent of molar mass for long-chain polymers. This rubbery plateau is seen for all polymer melts with... 

When a constant stress is imposed (its time derivative cr = 0), this equation describes the ideal Newtonian fluid under steady shear flow. When 77 —> 00, this equation describes the ideal elastic solid. The instantaneous response of the solid to an imposed stress is elastic, and the shear modulus E corresponds to the modulus 00 at high frequency. Consequently, the shear stress will relax down to zero exponentially. Under the condition of y = 0, the exponential function (6.18) can be solved from (6.24), which defines the characteristic relaxation time as... 

Coefficients G and K represent the instantaneous shear and compressibility modulus, respectively. In addition to elastic coefficients, one has to identify, K parameters and. Considering the hypothesis that for 3D isotropic materials leads to... 

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. 

If the bar is simply twisted, there is no volume change ( = 0), only distortion. (Lines scratched along the sides of the bar would be converted to helices by the strain.) This type of deformation is called shear, and the constant of proportionality between the instantaneous elastic shear stress and strain is the shear modulus, G, The shear modulus is related to Young s modulus by... 

The shear relaxation modulus instantaneous value G(0) The equilibrium shear modulus (G = 0 for a fluid)... 

The experiment here is a small rapid shear-strain at time zero - after this the shear stress in a viscoelastic liquid will not vanish instantaneously, but decay as a characteristic function with time. When normalised by the strain to yield the dimensions of modulus, this is G(f). 

If the solid does not shows time-dependent behavior, that is, it deforms instantaneously, one has an ideal elastic body or a Hookean solid. The symbol E for the modulus is used when the applied strain is extension or compression, while the symbol G is used when the modulus is determined using shear strain. The conduct of experiment such that a linear relationship is obtained between stress and strain should be noted. In addition, for an ideal Hookean solid, the deformation is instantaneous. In contrast, all real materials are either viscoplastic or viscoelastic in nature and, in particular, the latter exhibit time-dependent deformations. The rheological behavior of many foods may be described as viscoplastic and the applicable equations are discussed in Chapter 2. 

One convenient manner of studying viscoelasticity is by stress relaxation where the time-dependent shear stress is studied for step increase in strain. In Figure 1-7, the stress relaxation of a Hookean solid, and a viscoelastic solid and liquid are shown when subjected to a strain instantaneously and held constant. The relaxation modulus can be calculated as ... 

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