Viscoelastic cylinders

Study the forced oscillations in a straight circular viscoelastic cylinder (see Fig. P17.7.1). 

The radial and hoop stresses are plotted versus position in Fig. 9.3, where it is easily seen that the boundary condition of a (r = a) = -p0 is met on the inner boundary and the limit case of uniform stresses at long time for incompressible behavior is also apparent. The elastic solution is included, which overlays the viscoelastic response at t=0. While the internal pressure applied is compressive leading to compressive radial stresses at aU positions and all times, the hoop stress at the inner surface is tensile due to the expansion of the cylinder. While the hoop stress remains tensile for all time for an elastic cylinder, this tensile stress relaxes in the viscoelastic cylinder, ultimately becoming compressive. 

Fig. 9.3 Variation of the radial stress and hoop stress with position in the viscoelastic reinforced cylinder loaded with a step input of internal pressure. Parameters used are K /Go=3, t=1000, where the viscoelastic cylinder has an elastic bulk modulus and is a single Maxwell element in shear modulus. Response parameterized with time from the initial application of load at t=0 to asymptotic response at long times.

Edelstein, W. S. (1969 b) Ablation and thermal effects in a viscoelastic cylinder. Acta Mech. 8, 174-182... 

Harvey, R.B. (1975) On the deformation of a viscoelastic cylinder, rolling without slipping. Q. J. Mech. Appl. Math. 28, 1-24... 

Morland, L.W. (1968) Rolling contact between dissimilar viscoelastic cylinders. Q. i. ppl. Math. 25, 363-376... 

The radiation and temperature dependent mechanical properties of viscoelastic materials (modulus and loss) are of great interest throughout the plastics, polymer, and rubber from initial design to routine production. There are a number of laboratory research instruments are available to determine these properties. All these hardness tests conducted on polymeric materials involve the penetration of the sample under consideration by loaded spheres or other geometric shapes [1]. Most of these tests are to some extent arbitrary because the penetration of an indenter into viscoelastic material increases with time. For example, standard durometer test (the "Shore A") is widely used to measure the static "hardness" or resistance to indentation. However, it does not measure basic material properties, and its results depend on the specimen geometry (it is difficult to make available the identity of the initial position of the devices on cylinder or spherical surfaces while measuring) and test conditions, and some arbitrary time must be selected to compare different materials. 

Normal Stress (Weissenberg Effect). Many viscoelastic fluids flow in a direction normal (perpendicular) to the direction of shear stress in steady-state shear (21,90). Examples of the effect include flour dough climbing up a beater, polymer solutions climbing up the inner cylinder in a concentric cylinder viscometer, and paints forcing apart the cone and plate of a cone—plate viscometer. The normal stress effect has been put to practical use in certain screwless extmders designed in a cone—plate or plate—plate configuration, where the polymer enters at the periphery and exits at the axis. 

Piston Cylinder (Extrusion). Pressure-driven piston cylinder capillary viscometers, ie, extmsion rheometers (Fig. 25), are used primarily to measure the melt viscosity of polymers and other viscous materials (21,47,49,50). A reservoir is connected to a capillary tube, and molten polymer or another material is extmded through the capillary by means of a piston to which a constant force is appHed. Viscosity can be determined from the volumetric flow rate and the pressure drop along the capillary. The basic method and test conditions for a number of thermoplastics are described in ASTM D1238. Melt viscoelasticity can influence the results (160). 

Dyna.mic Viscometer. A dynamic viscometer is a special type of rotational viscometer used for characterising viscoelastic fluids. It measures elastic as weU as viscous behavior by determining the response to both steady-state and oscillatory shear. The geometry may be cone—plate, parallel plates, or concentric cylinders parallel plates have several advantages, as noted above. 

Throughout this chapter the viscoelastic behaviour of plastics has been described and it has been shown that deformations are dependent on such factors as the time under load and the temperature. Therefore, when structural components are to be designed using plastics, it must be remembered that the classical equations which are available for the design of springs, beams, plates, cylinders, etc., have all been derived under the assumptions that... 

Kumar, S., and Homsy, G. M., Chaotic advection in creeping flow of viscoelastic fluids between slowly modulated eccentric cylinders. Phys. Fluids 8,1774-1787 (19%). 

The simplest model that can show the most important aspects of viscoelastic behaviour is the Maxwell fluid. A mechanical model of the Maxwell fluid is a viscous element (a piston sliding in a cylinder of oil) in series with an elastic element (a spring). The total extension of this mechanical model is the sum of the extensions of the two elements and the rate of extension is the sum of the two rates of extension. It is assumed that the same form of combination can be applied to the shearing of the Maxwell fluid. 

Oscillatory shearing is used to characterize viscoelastic fluids using coaxial cylinders or cone and plate instruments. 

When a polymer is extruded through an orifice such as a capillary die, a phenomenon called die swell is often observed. In this case, as the polymer exits the cylindrical die, the diameter of the extrudate increases to a diameter larger than the diameter of the capillary die, as shown in Fig. 3.9. That is, it increases in diameter as a function of the time after the polymer exits the die. Newtonian materials or pure power law materials would not exhibit this strong of a time-dependent response. Instead they may exhibit an instantaneous small increase in diameter, but no substantial time-dependent effect will be observed. The time-dependent die swell is an example of the polymer s viscoelastic response. From a simplified viewpoint the undisturbed polymer molecules are forced to change shape as they move from the large area of the upstream piston cylinder into the capillary. For short times in the capillary, the molecules remember their previous molecular shape and structure and try to return to that structure after they exit the die. If the time is substantially longer than the relaxation time of the polymer, then the molecules assume a new configuration in the capillary and there will be less die swell. 

Rolling friction is often found to be proportional to the velocity, but more complex relationships may be observed, depending on the combination of the bodies. For a soft, viscoelastic sphere on a hard substrate, Brilliantov et al. [464] predicted a linear dependence of rolling friction on speed. For a hard cylinder on a viscous surface, a much more complex behavior was found [465,466], At lower speeds, the rolling friction increases with speed to reach a maximum value and then decreases at higher speeds. The reason is an effective stiffening of the substrate at higher speeds. 

With the above information, it becomes possible to combine viscous characteristics with elastic characteristics to describe the viscoelasticity of polymeric materials.86-90 The two simplest ways of combining these features are shown in Figure 2.49, where a spring having a modulus G models the elastic response. The viscous response is modelled by what is called a dashpot. It consists of a piston moving in a cylinder containing a viscous fluid of viscosity r. If a downward force is applied to the cylinder, more fluid flows into it, whereas an upward force causes some of the fluid to flow out. The flow is retarded because of the high viscosity and this element thus models the retarded movement and flow of polymer chains. 

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

Figure 3 presents the time evolution of tangential stresses at the surface and the core of the cylinder according to the viscoelastic and elastic models. The stress reverse can be explained as follows when the body dries, the drier surface attempts to shrink but is restrained by the wet core. The surface is stressed in tension and the core in compression and inelastic strain occurs. Later, under a surface with reduced shrinkage, the core dries and attempts to shrink causing the stress state to reverse [4],... 

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