Elastic continuum

The free vibration of an elastic continuum is harmonic in time, so Whitney chose a harmonic solution... 

Consider first the case where the scattering projection of the Hamiltonian, PHMP, can induce only elastic scattering (referred to as an uncoupled or elastic continuum). In this situation the partial wave in Eq. (37) reduces to... 

In the case that an isolated vibronic resonance interacts with the uncoupled (or elastic ) continuum of Eq. (48), Eq. (43) simplifies to... 

Polyatomic molecules provide a still richer environment for studying phase control, where coupling between different dissociation channels can occur. Indeed, one of the original motivations for studying coherent control was to develop a means for bond-selective chemistry [25]. The first example of bond-selective two-pathway interference is the dissociation of dimethyl-sulfide to yield either H or CH3 fragments [74]. The peak in Fig. 11 is indicative of a resonance embedded in an elastic continuum (case 4). 

The Debye s elastic continuum model for solids [1-9] gives for cph ... 

Still, the strain enthalpy is of particular importance. An elastic continuum model for this size mismatch enthalpy shows that, within the limitations of the model, this enthalpy contribution correlates with the square of the volume difference [41,42], The model furthermore predicts what is often observed experimentally for a given size difference it is easier to put a smaller atom in a larger host than vice versa. Both the excess enthalpy of mixing and the solubility limits are often asymmetric with regard to composition. This elastic contribution to the enthalpy of mixing scales with the two-parameter sub-regular solution model described in Chapter 3 (see eq. 3.74) ... 

An elastic continuum model, which takes into account the energy of bending, the dislocation energy, and the surface energy, was used as a first approximation to describe the mechanical properties of multilayer cage structures (94). A first-order phase transition from an evenly curved (quasi-spherical) structure into a... 

As early as 1829, the observation of grain boundaries was reported. But it was more than one hundred years later that the structure of dislocations in crystals was understood. Early ideas on strain-figures that move in elastic bodies date back to the turn of this century. Although the mathematical theory of dislocations in an elastic continuum was summarized by [V. Volterra (1907)], it did not really influence the theory of crystal plasticity. X-ray intensity measurements [C.G. Darwin (1914)] with single crystals indicated their mosaic structure (j.e., subgrain boundaries) formed by dislocation arrays. Prandtl, Masing, and Polanyi, and in particular [U. Dehlinger (1929)] came close to the modern concept of line imperfections, which can move in a crystal lattice and induce plastic deformation. 

Next, let us compile some quantitative relations which concern the stress field and the energy of dislocations. Using elastic continuum theory and disregarding the dislocation core, the elastic energy, diS, of a screw dislocation per unit length for isotropic crystals is found to be... 

DEBYE THEORY OF SPECIFIC HEAT. The specific heal of solids is attributed to the excitation of thermal vibrations of the lattice, whose spectrum is taken to be similar to that of an elastic continuum, except that it is cut off at a maximum frequency in such a way that the total number of vibrational modes is equal to the total number of degrees of freedom of the lattice. 

Fluidization of solid particles with these two widely different classes of fluids—liquids and gases—leads to vastly different phenomena of solids behavior, as shown in Fig. 3. For L/S fluidization, as liquid velocity increases beyond the incipient fluidization point, the solids bed continues to expand as if it were an elastic continuum stretching under the dynamic forces of augmented flow, until, near the terminal velocity of the particles, the solid particles can be noted to be suspended sparsely. Throughout this process of liquid-velocity increase, the solid particles are dispersed quite uniformly, fully exhibiting their discrete behavior, essentially independent of one another. Therefore, L/S fluidization was named particulate. ... 

Fig. 11. Total energy of an exciton in an anisotropic elastic continuum for different Pt-Pt distances Rm). The energy is calculated for Mg[Pt(CN)4] 7 H20 from Eq. (6). a(ai ct ) represents a localization parameter which describes a free exciton (FE) with a = 0 and a localized exciton (self-trapped exciton STE) with a = 1. The exciton binding energy EB is normalized to zero for different R-values...

Note that for the asymptotic equations of Eqns. (2) and (3) to be valid, r

characteristic length, and is normally the crack length or the remaining ligament, whichever is the smaller, of a fracture specimen. Also, the above asymptotic equations are not valid for an orthotropic elastic continuum, such as a ceramic fiber/ceramic matrix composite. While the static crack tip state for an orthotropic elastic continuum has been derived, to the author s knowledge, no dynamic counterpart is available to date. Nevertheless, the above crack tip state should be applicable to particulate/whisker-filled ceramic matrix composites which macroscopically behave like an isotropic homogeneous continuum. 

In our discussion of elastic constants we have imagined uniform distortions of the crystal. An elastic medium can sustain vibrations, which at any instant consist of nonuniform distortions. The normal modes of an elastic continuum arc sound waves (longitudinal and transverse) propagating in the medium, and these normal modes will also exist in the crystal. Indeed, viewing the crystal as an elastic... 

The distribulion of fi cquencics for one branch of the vibration spectrum, described in the Debye approximation by treating the crystal as an elastic continuum. The Debye frequency is... 

Generally, for any dimension therefore, if a crack of length I already exists in an infinite elastic continuum, subject to uniform tensile stress a perpendicular to the length of the crack, then for the onset of brittle fracture, Griffith equates (the differentials of) the elastic energy E with the surface energy E ... 

Doublets of folded longitudinal acoustic (LA) phonons due to the superlattice periodicity [143] can also be seen in the Raman spectra of the SLs (indicated by arrows in Fig. 21.2). The positions of the doubled peaks agree well with the first doublet frequencies calculated within the elastic continuum model [144]. The observation of the LA phonon folding suggests that these superlattices possess the requisite structural quality for acoustic Bragg mirrors and cavities to be used for potential coherent phonon generation applications [145-147]. 

Therefore, when Jenike developed his methods to mathematically model the flow of bulk solids, he concluded that a bulk solid must be modeled as a plastic, and not a visco-elastic, continuum of solid particles (1). This approach included the postulation of a flow-no-flow criterion that states the bulk solid would flow from a bin when the stresses applied to the bulk solid exceed the strength of the bulk solid. The terms stress and strength are further discussed in this section on cohesive strength tests below. The flow properties test methods discussed are used to obtain the equipment parameters required to provide consistent, reliable flow. 

Theoretically 7 was first defined as the variation of the eigenfrequencies of a solid as the volume changed. Equation (6.26) can be derived from this definition, using the model of an elastic continuum. The value of 7 is not predicted, but operationally it can be measured using Equation (6.26). For many solids Cy is not equal to 3k, as in Equation (6.25), because the highest frequencies are not excited. This causes no error, since x is not increased by such frequencies either. 

We begin by examining what continuum mechanics might tell us about the structure and energetics of point defects. In this context, the point defect is seen as an elastic disturbance in the otherwise unperturbed elastic continuum. The properties of this disturbance can be rather easily evaluated by treating the medium within the setting of isotropic linear elasticity. Once we have determined the fields of the point defect we may in turn evaluate its energy and thereby the thermodynamic likelihood of its existence. 

Fig. 7.11. Collection of force dipoles used to rniinic the presence of a point defect as idealized within an elastic continuum.

According to the elastic continuum theory of liquid crystals which was introduced in Chapter 1, the three kinds of deformations can be described by three elastic constants, An(splay), / (twist) and / (bend). In the case of small molecular mass liquid crystals, the three constants are mainly determined by the chemical composition of the liquid crystalline molecules. Among them, K22 is the smallest while the other two are approximately close. All three elastic constants are of the order of 10 12 N. The elastic constants of some important liquid crystals are listed in Table 6.1. Each kind of liquid crystals is a mixture of R5-pentyl and R6-hexyl homologues in the ratio of 40 60. The data are obtained at the temperature of T = Tc — 10 °C where Tc is the clear temperature. 

Theoretical Estimates The use of the Debye model (Figure 3.2), which assumes that a solid behaves as a three-dimensional elastic continuum with a frequency distribution/(j ) = allows accurate prediction of the temperature dependence of the vibrational heat capacity C / of solids at low temperatures Cy oc r ), as well as at high temperatures (Cy = Wks). One may also use the same model with confidence to evaluate the temperature dependence of the surface heat capacity due to vibrations of atoms in the surface. 

Let us consider a surface of area (J at the termination of the bulk lattice in which the atoms have the same properties as in the three-dimensional elastic continuum. 

In general, for an elastic continuum of (x dimensions, the frequency distribution IS f(p) p. If N surface atoms still have 3N vibrational modes (they are allowed to have out-of-plane vibrations), we have... 

---