7® Elastic shear compliance

According to a thorough calculation by Janeschitz-Kriegl (General references, 1983, Appendix A3) the elastic shear compliance is equal to... 

According to Eq. (15.80) recoverable shear strain is proportional to the elastic shear compliance, J°. The elastic shear compliance depends on the molecular-weight distribution in the following way... 

The first normal stress coefficient at low shear rates, vPio, and the elastic shear compliance, /g, are related by Eq. (15.78) which reads... 

The ratio Sq, which is defined to be equal to 1 /i /rj1, is equal to the elastic shear compliance /° for shear rates approaching zero. Both Th and 77 are shear rate dependent, but it is previously not clear whether it increases or decreases with increasing shear rate. In Fig. 16.20 both 77/r]0 and Sq //( are plotted vs. qi0 on double logarithmic scales for solutions and melts of poly (a-methyl styrene) as well (Sakai et al., 1972). First, it appears that the... 

Polymers owe much of their attractiveness to their ease of processing. In many important teclmiques, such as injection moulding, fibre spinning and film fonnation, polymers are processed in the melt, so that their flow behaviour is of paramount importance. Because of the viscoelastic properties of polymers, their flow behaviour is much more complex than that of Newtonian liquids for which the viscosity is the only essential parameter. In polymer melts, the recoverable shear compliance, which relates to the elastic forces, is used in addition to the viscosity in the description of flow [48]. 

The molecules of a liquid start to move relative to one another and the shear compliance increases rapidly after the shearing force has been applied for only c. 10 6 s. On the other hand, hard solids, such as diamond, sodium chloride crystals and materials at a low enough temperature to be in the glassy state, show only the above rapid elastic deformation, even after the shearing stress has been applied for a considerable time. 

The compliance is defined as the reciprocal of the modulus of elasticity (unit Pa 7 Hence, shear compliance is defined as / = 1/G, while tensile compliance is defined by D = /E. 

In the special case of branched polyethylene, the elastic modulus at 45° to the fiber axis is exceptionally small this means that shear compliance along the fiber axis is very high (16). Such deformation involves reversible shearing displacement of adjacent fibrils. Since a similarly high shear compliance does not occur with linear polyethylene and isotactic polypropylene, the difference may be attributable to the substantial difference in draw ratio (4.5 in branched, 20 in linear polyethylene and propylene) which results in proportionately shorter microfibrils and fibrils in the former. The shorter the microfibril, the shorter the fibril and the smaller the surface-to-cross section ratio and hence the smaller the resistance to shear displacement. 

If a system is to possess elasticity, it must possess a physical mechanism for storing energy. In flow, droplet distortion causes an increase in surface free energy (aAA) which is released on cessation of flow, manifesting as, for example, recoverable shear compliance. The magnitude of the (dimensionless) droplet distortion is proportional to the Weber number, We, which is the ratio of the deforming stress (n y) to the restoring Laplace stress (4(j/dd). Thus, the drop Weber number based on drop radius is defined as ... 

At low shear rates, polymeric liquid properties are characterized by two constitutive parameters zero shear rate viscosity t]o and recoverable shear compliance Jq, which indicates fluid elasticity. At higher shear strain rates, rheological behavior is measured with a viscometer. Extensional strain viscosity, associated with extensional flow, occurs with film extrusion. 

For an isotropic polymer there are two independent elastic constants, and the two alternative schemes predict a value for the isotropic shear compliance 544 and the isotropic shear stiffness C44 respectively. 

The rheology of blends of linear and branched PLA architectures has also been comprehensively investigated [42, 44]. For linear architectures, the Cox-Merz rule relating complex viscosity to shear viscosity is valid for a large range of shear rates and frequencies. The branched architecture deviates from the Cox-Merz equality and blends show intermediate behavior. Both the zero shear viscosity and the elasticity (as measured by the recoverable shear compliance) increase with increasing branched content. For the linear chain, the compliance is independent of temperature, but this behavior is apparently lost for the branched and blended materials. These authors use the Carreau-Ya-suda model. Equation 10.29, to describe the viscosity shear rate dependence of both linear and branched PLAs and their blends ... 

In order to do this, it is best to deal with another useful parameter of elasticity, namely, the steady-state shear compliance which gives a measure of the stored energy or the elastic recovery of the system. When represented mathematically as = (xn — X2i)/2T 2r the plot of Jg vs. y would give the correct trend based on a molecular interpretation. 

The shear compliance 66 relates to shear in the 12 plane and is related to the compliance constants in and. Si2, such that 66 = 2(in - 512). This relationship expresses the fact that these specimens are isotropic in a plane perpendicular to the symmetry axis, i.e. that the elastic behaviour in this plane is specified by only two elastic constants as for an isotropic material. It will be seen that this property is very important in determining the elastic constants for fibres. 

There are three independent shear moduli Gi = 1/ 44, G2 = I/S55 and G3 = l/see corresponding to shear in the 23, 13 and 12 planes respectively. For a sheet of general dimensions, torsion experiments where the sheet is twisted about the 1, 2 or 3 axis will involve a combination of shear compliances. This will be discussed in greater detail later, when methods of obtaining the elastic constants are described. 

Alternative to the shear compliances, J t) and J (o ), one can also use for the description of the properties under shear the shear moduli, G t) and As we shall find, this can drastically change the values of the relaxation times. Let us first consider a single Debye-process, now in combination with a superposed perfectly elastic part, and calculate the associated dynamic modulus. We have... 

For a Newtonian low molar mass liquid, knowledge of the viscosity is fully sufficient for the calculation of flow patterns. Is this also true for polymeric liquids The answer is no under all possible circumstances. Simple situations are encountered for example in dynamical tests within the limit of low frequencies or for slow steady state shears and even in these cases, one has to include one more material parameter in the description. This is the recoverable shear compliance , usually denoted and it specifies the amount of recoil observed in a creep recovery experiment subsequent to the unloading. Jg relates to the elastic and anelastic parts in the deformation and has to be accounted for in all calculations. Experiments show that, at first, for M < Me, Jg increases linearly with the molecular weight and then reaches a constant value which essentially agrees with the plateau value of the shear compliance. 

In a polymer melt, the viscous properties of Newtonian liquids combine with elastic forces. The latter ones contribute a real part to the dynamic shear compliance, to be identified with Jg... 

Shear compliance (steady-static elastic compliance). Source Ref. 24. 

It is apparent from Equations 8.42 and 8.43 that four material elastic properties (compliance or stiffness) are needed to characterize the in-plane behavior of a linear elastic orthotropic lamina. It is convenient to define these material properties in terms of measured engineering constants (Young s moduli, El > d Ej, shear modulus Glt, and Poisson s ratios u,lt and (Xtl). The longitudinal Young s... 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

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