Viscoelastic rod

Calculate the longitudinal strain of a viscoelastic rod of a material that behaves like (a) a Maxwell model, (b) a Maxwell solid in shear but an elastic solid in bulk and (c) a viscoelastic solid standard in shear but an elastic solid in bulk. The material is constrained in such a way that the lateral dimensions cannot vary when it is under uniform forces of compression at both ends of the rod. 

The analysis of the stresses and strains in beams and thin rods is a subject of great interest with many practical applications in the study of the strength of materials. The geometry associated with problems of this type determines the specific type of solution. There are cases where small strains are accompanied by large displacements, flexion and torsion in relatively simple structures being the most relevant examples. Problems of this type were solved for the elastic case by Saint Venant in the nineteenth century. The flexion of viscoelastic beams and the torsion of viscoelastic rods are studied in this chapter. 

The methods utilized to measure the viscoelastic functions are often close to the stress patterns occurring in certain conditions of use of polymeric materials. Consequently, information of technological importance can be obtained from knowledge of these functions. Even the so-called ultimate properties imply molecular mechanisms that are closely related to those involved in viscoelastic behavior. Chapters 16 and 17 deal with the stress-strain multiaxial problems in viscoelasticity. Application of the boundary problems for engineering apphcations is made on the basis of the integral and differential constitutive stress-strain relationships. Several problems of the classical theory of elasticity are revisited as viscoelastic problems. Two special cases that are of special interest from the experimental point of view are studied viscoelastic beams in flexion and viscoelastic rods in torsion. 

Derive, from first principles, the equations giving G and tan 8 from the frequency and attenuation of oscillations of a torsion pendulum, when the specimen is a linearly viscoelastic rod of circular cross-section and A c 1 (see Section 4.2.3) ... 

Hunter, S. C. (1967) The transient temperature distribution in a semi-infinite viscoelastic rod, subject to longitudinal oscillations. Int. J. Eng. Sci. 5, 119-143 Hunter, S.C. (1968) The motion of a rigid sphere embedded in an adhering elastic or viscoelastic medium . Proceedings of the Edinburgh Mathematical Society 16 (Series II), Part I, pp. 55-69 Hunter, S.C. (1983) Mechanics of Continuous Media, 2nd edition (Wiley, New Ycrk)... 

Penetration—Indentation. Penetration and indentation tests have long been used to characterize viscoelastic materials such as asphalt, mbber, plastics, and coatings. The basic test consists of pressing an indentor of prescribed geometry against the test surface. Most instmments have an indenting tip, eg, cone, needle, or hemisphere, attached to a short rod that is held vertically. The load is controlled at some constant value, and the time of indentation is specified the size or depth of the indentation is measured. Instmments have been built which allow loads as low as 10 N with penetration depths less than mm. The entire experiment is carried out in the vacuum chamber of a scanning electron microscope with which the penetration is monitored (248). 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid and a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study(18). Two common and easily observable examples of viscoelastic behaviour in a liquid are ... 

Another well-known phenomenon is the Weissenberg effect, which occurs when a long vertical rod is rotated in a viscoelastic liquid. Again, the shearing generates a tension along the streamlines, which are circles centred on the axis of the rod. The only way in which the liquid can respond is to flow inwards and it therefore climbs up the rod until the hydrostatic head balances the force due to the normal stresses. 

A characteristic of viscoelastic behaviour is the tendency for flow to occur at right angles to the applied force. An extreme example of this behaviour is illustrated in Figure 9.9. When a rotating rod is lowered into a Newtonian liquid, the liquid is set into rotation and tends to move outwards, leaving a depression around the rod. When the rotating rod is lowered into a viscoelastic liquid, the liquid may actually climb up the rod. The rotation of the rod causes the liquid to be sheared circularly and, because of its elastic nature, it acts like a stretched rubber band, tending to squeeze liquid in towards the centre of the vessel and, therefore, up the rod. 

The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics variational formulations computational mechanics statics, kinematics and dynamics of rigid and elastic bodies vibrations of solids and structures dynamical systems and chaos the theories of elasticity, plasticity and viscoelasticity composite materials rods, beams, shells and membranes structural control and stability soils, rocks and geomechanics fracture tribology experimental mechanics biomechanics and machine design. 

Viscoelastic fluids have elastic properties in addition to their viscous properties. When under shear, such fluids exhibit a normal stress in addition to a shear stress. For example, if a vertical rod is partly immersed and rotated in a non-viscoelastic liquid the rod s rotation will create a centrifugal force that drives liquid outwards toward the container walls, as shown in Figure 6.16(a). If, on the other hand, the liquid is viscoelastic then as the liquid is sheared about the rod s axis of rotation, a stress normal to the plane of rotation is created which tends to draw fluid in towards the centre. At some rotational speed, the normal force will exceed the centrifugal force and liquid is drawn towards and up along the rod see Figure 6.16(b). This is called the Weissenberg effect. Viscoelastic fluids flow when stress is applied, but some of their deformation is recovered when the stress is removed [381]. 

The elastic-melt extruder makes use of the Weissenberg or rod climbing effect—which is observed when an elastic fluid is sheared or rotated inside a container by a rod. Because of viscoelasticity, the fluid climbs the rod. 

As a consequence of their tendency to form lamellar phases or rod shaped micelles at low concentration, cationic surfactants are frequently employed as the primary surfactants to thicken high salt formulas [72,73]. The viscoelastic nature of certain cationic surfactant solutions has been employed in a novel way to allow for a solution of sodium hypochlorite not to be easily diluted and therefore to remain at a higher concentration for the purpose of oxidizing clogs of human hair which form in drains [73 ]. Low concentrations of cetyl trimethyl ammonium chloride in combination with two hydro tropes form viscoelastic solutions with the values of viscosity and Tau/Go shown in Table 6.6. 

For a viscoelastic material both K and G are complex quantities. When the material sample has finite dimensions other modes of wave propagation may occur as a result of multiple scattering from the material boundaries. Mode conversion from longitudinal wave to shear wave, and vice versa, occurs on reflection at a solid boundary. For material samples in the form of thin rods or plates the modes of wave propagation are extenional waves (with speed determined by the Young s modulus, or the plate modulus), and flexural (bending) waves [8]. 

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