Homogeneous strain

After the formation of the network structure has been completed, let the sample be subjected to any type of homogeneous strain (including swelling, to be treated in Chap. XIII) which may be described as an alteration of its dimensions X, Y, and Z by factors and az,... 

Other types of deformation may be handled similarly. Shear, for example, may be treated as a homogeneous strain involving an increase in one coordinate x) while another (z) remains constant, the volume being constant also. Thus, we may let aa = a, oLy = a, and az=. On substitution of these conditions in Eq. (41), the deformation entropy per unit volume becomes... 

Fred H. Pollack, Effects of Homogeneous Strain on the Electronic and Vibrational Levels in Semiconductors... 

F. H. Pollack, Effects of Homogeneous Strain on the Electronic and Vibrational Levels in Semiconductors J. Y Marzin, J. M. Gerard, P. Voisin, and J. A. Brum, Optical Studies of Strained III—V Heterolayers R. People and S. A. Jackson, Structurally Induced States from Strain and Confinement M. Jams, Microscopic Phenomena in Ordered Superlattices... 

With homogeneous strain, the deformation is proportionately identical for each volume element of the body and for the body as a whole. Hence, the principal axes, to which the strain may be referred, remain mutually perpendicular during the deformation. Thus, a unit cube (with its edges parallel to the principal strain directions) in the unstrained body becomes a rectangular parallelepiped, or parallelogram, while a circle becomes an ellipse and a unit sphere becomes a triaxial ellipsoid. Homogeneous strain occurs in crystals subjected to small uniform temperature changes and in crystals subjected to hydrostatic pressure. 

Lumley has solved the equation system for homogeneous shear, and compared the results with homogeneous strain and homogeneous shear experiments. Lumley s model predicts that the time scale T grows without bound, so that homogeneous flows can never attain an equilibrium structure. Champagne et al. (C4) experiments are consistent with Lumley s notion, but Lumley s model does not predict the observed structure very well. Some improvements on Lumley s model based on Eq. (63) are suggested in Section V. 

Kim and Michler have observed the relationship between morphology and strain micromechanisms in cases of both rigid and elastomeric filler growth of voids, by cavitation or debonding [7,31]. Oshyman has reported a transition, at a certain fraction of filler, correlated to the evolution from macroscopic homogeneous strain to micromechanisms such as crazes. It is in fact a transition between independent mode and correlated mode of strain micromechanims [32]. 

The CITE system will be considered as a crystal containing a lattice (or one of the sublattices) of JT ions (structural units). A typical Hamiltonian should describe the JT effect at each of the corresponding centers plus the elastic energy related to the appearance of the homogeneous strain as a result of the structural phase transition. 

In this Hamiltonian k is the wave vector of the phonons, y is the phonon mode branch, go and Vmk are the electron-strain and the electron-phonon interaction constants. It is important to remind that as it was noted for the first time by Kanamori [3], the electron interaction with the homogeneous strain U should be considered separately from the electron-phonon interaction as that type of strain can not be represented by phonons. The introduction of the last ones depends upon the Born-Karman conditions that are changing at the structural phase transition. 

Crinella did not find individual consistency in sham-operated white rats (a rather dull and homogeneous strain—despite Maier s defence) from their study (Thompson et al 1990). However, when all 424 rats were analysed together, including the 348 animals with lesions to one of 50 different brain sites, the correlations across the six categories of tasks were positive, frequently significant, and a general factor contributing to performance across all tasks was found. 

Most real crystals contain imperfections producing local distortions of the lattice, resulting in a non-homogeneous strain field. The effect on position, shape and extension of reciprocal space points, and consequently on PD peak profiles, is usually more complex than that of the domain size. A formal treatment of the strain broadening is beyond the scope of the present book interested readers can refer to the cited literature. In the following a simplified, heuristic approach is proposed. 

First consider the effect of a macroscopically homogeneous strain (or macrostrain), expressed as z = ts.djd. By differentiating Bragg s law (assuming a constant wavelength) ... 

Toxicokinetic and PK research studies are characterized by some uncertainty regarding the process studied and significant variation in the concentration measurements obtained. Variability in PK parameters among homogeneous strains of small laboratory animals has been reported to be between 30% and 50% in some cases (1, 2). In addition to the inherent variability of the biological system, there is the uncertainty associated with the assay and process noise. 

Note that the stiffness coefficients are now evaluated at the strained reference state (as signified by the notation dEtot/ Ria)v) rather than for the state of zero strain considered in our earlier treatment of the harmonic approximation. To make further progress, we specialize the discussion to the case of a crystal subject to a homogeneous strain for which the deformation gradient is a multiple of the identity (i.e. strict volume change given by F = A,I). We now reconsider the stiffness coefficients, but with the aim of evaluating them about the zero strain reference state. For example, we may rewrite the first-order term via Taylor expansion as... 

We are now in a position to reconsider the vibrational frequencies in light of the imposition of a small homogeneous strain. Recall that our strategy was to use the quadratic potential energy to produce equations of motion. Inspection of the potential energy developed above makes it clear that the term that is quadratic in the displacements has acquired additional complexity. In particular, if we derive the equations of motion on the basis of both our earlier quadratic terms and those considered here, then we find that they may be written as... 

In this case, Vmn refers to the distance between the and atoms in the deformed state in which the crystal is subjected to some homogeneous strain. If we assume that the deformed and reference positions are related by r, = FR, then r, i may be rewritten as... 

Here, force is the quantity, F and A represents the unit area. The unit the plant chemist will usually use is the megapascal (MPa). Strain measures the deformation of an elastomer when a stress is applied to it. Here, we will only consider homogeneous strain or equal deformation from all parts of the part resulting from a uniform compound. It can be represented as... 

Figure 6-3a). In the general case of a pure homogeneous strain, the cube is transformed into a rectangular parallelepiped (Figure 6-3b). The dimensions of the parallelepiped are A, /I2 and /L3 in the three principal axes, where the are called the principal extension ratios. Choosing the coordinate axes for the chain to coincide with the principal axes of strain for the sample, then... 

Figure 6-3. A unit cube of elastomer (a) in the unstrained state (b) in the homogeneous strained state (c) under uniaxial extension.

In general, we measure the homogeneous strain of a solid by the relative displacement of two points and P2 separated by the vector r, keeping the coordinate system invariant (Fig. 4.9). The strain displaces the point Piix ) to the point P Xi + < j) and the point P2( i + ) to P 2 Xi H- H- u-). The vector r H- u gives the relative position of the two points of the strained solid. By analogy with equation (4.34) and (4.35), the strain tensor eexpresses the displacement u per unit... 

EXAMPLE. Under the effect of a homogeneous strain , a cube of linear dimension D, characterized by the edges (i = 1, 2,3), is transformed into... 

Pure shear is represented in Fig. 5 and is defined as a homogeneous strain in which one of the principal extensions is zero and the volume is unchanged. If the extension ratio A] = a while At = 1. then is /a. 

For finite strain in isotropic media, only states of homogeneous pure strain will be considered, i.e. states of uniform strain in the medium, with all shear components zero. This is not as restrictive as it might first appear to be, because for small strains a shear strain is exactly equivalent to equal compressive and extensional strains applied at 90° to each other and at 45° to the original axes along which the shear was applied (see problem 6.1). Thus a shear is transformed into a state of homogeneous pure strain simply by a rotation of axes by 45°. A similar transformation can be made for finite strains, but the rotation is then not 45°. All states of homogeneous strain can thus be regarded as pure if suitable axes are chosen. 

The fact that the energy release rate of a small crack surrounded by homogenously strained material scales linearly with the size of the crack, for all multiaxial loading states, has been established from experience (Gough and Muhr, 2005), and can be established mathematically by considering the balance of configurational stresses (Ait-Bachir et al., 2012). 

Hydrogels experience recoverable elasticity if the applied strain is modest (<20% in most instances). Since gel networks are often amorphous and homogeneous, strain recovery is expected whether stresses are isotropic, anisotropic, or bulk, as in the case of solvent swelling. Therefore, mbber elastic theory can be used to determine critical hydrogel parameters, originally developed for vulcanized mbber, and later modified for polymers. Further modifications for gels... 

We first want to recall some basic facts of the theory of elasticity. In the harmonic approximation the lattice energy of a crystal has contributions due to homogeneous strains and to harmonic phonons ... 

For a non-Bravais lattice (e.g., diamond structure) may also have contributions due to product terms of homogeneous strains and optic phonon coordinates (sect. 2.4.1). 

In a perfectly harmonic crystal the elastic constants would be strictly independent of temperature. However, due to the existence of third- and fourth-order anharmonic terms in the crystal potential there is a coupling between the homogeneous strains and the phonon coordinates. This will lead to a background temperature dependence of the elastic constants. It can be described within a quasiharmonic approximation (Ludwig 1967), in which the anharmonic contributions to the crystal potential are implicitly included by assuming a strain dependence of the phonon frequencies which can be characterized by the... 

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