Elastic rotator

Figure 3.19. Variation of the energy transfer into the surface in scattering of NO from Ag(l 11) as a function of Ee = f ccsO,-. Solid lines and solid points are for rotationally elastic scattering. /, = Jj = 0.5 and the open points are for non-state-resolved scattering experiments (and therefore also contains a contribution from rotationally inelastic scattering). From Ref. [181].

Wahr, J. (1981). Body Tides on an Elliptical, Rotating Elastic and Oceanless Earth. Geophys. J. Roy. Astron. Soc., 64, 677—703. 

Wolgemuth, C.W., Powers, T.R., and Goldstein, R.E. (1998) Twirling and whirling viscous dynamics and rotating elastic filaments. Phys. Rev. Lett., 84, 1623-1226. 

Manghi, M., Schlagberger X., and Nelz, R. (2006) Propulsion with a rotating elastic nanorod. Phys. Rev. Lett., 96, 068101-1-068101-4. 

In (a), an ion and a gas atom approach each other with a total kinetic energy of KE, + KEj. After collision (b), the atom and ion follow new trajectories. If the sum of KE, + KEj is equal to KE3 + KE4, the collision is elastic. In an inelastic collision (b), the sums of kinetic energies are not equal, and the difference appears as an excess of internal energy in the ion and gas molecule. If the collision gas is atomic, there can be no rotational and no vibrational energy in the atom, but there is a possibility of electronic excitation. Since most collision gases are helium or argon, almost all of the excess of internal energy appears in the ion. 

Machine components ate commonly subjected to loads, and hence stresses, which vary over time. The response of materials to such loading is usually examined by a fatigue test. The cylinder, loaded elastically to a level below that for plastic deformation, is rotated. Thus the axial stress at all locations on the surface alternates between a maximum tensile value and a maximum compressive value. The cylinder is rotated until fracture occurs, or until a large number of cycles is attained, eg, lO. The test is then repeated at a different maximum stress level. The results ate presented as a plot of maximum stress, C, versus number of cycles to fracture. For many steels, there is a maximum stress level below which fracture does not occur called the... 

Dual-beUows assembhes, ie, universal-type expansion joints, are particularly vulnerable to squirm, and can experience elastic squirm at one-fourth the pressure of an individual bellows. When large amounts of offset are encountered, as is often the design basis, a pinwheel effect occurs because of unbalanced pressure forces. This effect tends to rotate the center-spool pipe which may lead to bellows mpture. Eor this reason the center spool should always be stabilized by hinges or tie-rod lugs to prevent such rotation. 

The study of flow and elasticity dates to antiquity. Practical rheology existed for centuries before Hooke and Newton proposed the basic laws of elastic response and simple viscous flow, respectively, in the seventeenth century. Further advances in understanding came in the mid-nineteenth century with models for viscous flow in round tubes. The introduction of the first practical rotational viscometer by Couette in 1890 (1,2) was another milestone. 

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. 

The Weissenberg Rheogoniometer (49) is a complex dynamic viscometer that can measure elastic behavior as well as viscosity. It was the first rheometer designed to measure both shear and normal stresses and can be used for complete characteri2ation of viscoelastic materials. Its capabiUties include measurement of steady-state rotational shear within a viscosity range of 10 — mPa-s at shear rates of, of normal forces (elastic... 

Elasticity is another manifestation of non-Newtonian behavior. Elastic Hquids resist stress and deform reversibly provided that the strain is not too large. The elastic modulus is the ratio of the stress to the strain. Elasticity can be characterized usiag transient measurements such as recoil when a spinning bob stops rotating, or by steady-state measurements such as normal stress ia rotating plates. 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

Alternately, a benign shock compression model has been invoked to explain the polarizations. An elastic dipole-rotation model was first proposed... 

The consequences of this approximation are well known. While E s is good enough for calculating bulk moduli it will fail for deformations of the crystal that do not preserve symmetry. So it cannot be used to calculate, for example, shear elastic constants or phonons. The reason is simple. changes little if you rotate one atomic sphere... 

---