Tensor components

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

Denote hy W = (w, w ), w horizontal and vertical displacements of the mid-surface points, respectively, and write down the formulae for strain and integrated stress tensor components y(lL), aij W) ... 

We shall consider an equilibrium problem with a constitutive law corresponding to a creep, in particular, the strain and integrated stress tensor components (IT ), ay(lT ) will depend on = (lT, w ), where (lT, w ) are connected with (IT, w) by (3.1). In this case, the equilibrium equations will be nonlocal with respect to t. 

Consider an inclined crack with the nonpenetration condition of the form (3.173), (3.176). Let % = (IL, w) be the displacement vector of the midsurface points. Introduce the strain and stress tensor components Sij =... 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

Table 6-1. C2(l molecular poinl group. The electronic stales of the flat T6 molecule are classified according lo the lwo-1 old screw axis (C2). inversion (/). and glide plane reflection (o ) symmetry operations. The A and lt excited slates transform like translations Oi along the molecular axes and are optically allowed. The Ag and Bg stales arc isoniorphous with the polarizability tensor components (u), being therefore one-photon forbidden and Iwo-pholon allowed.

Because an applied field in the y direction Ev can induce a dipole M with a component in the x direction Mx as well as the component in the y direction My, it is necessary that we specify the components of the polarizability tensor by two subscripts (Fig. 3). If the bond A—B of a diatomic molecule stretches during a vibrational mode, Mx and Mv will vary and therefore the corresponding polarizability tensor components will vary. 

To obtain for 71 and jk compact dispersion formulas similar as Eq. (79) for 7, these hyperpolarizability components must be written as sums of tensor components which are irreducibel with respect to the permu-tational symmetry of the operator indices and frequency arguments ... 

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

The Fe hyperfine tensor components were determined by Mossbauer spectroscopy in the case of the rubredoxin from Clostridium... 

The asymmetry parameter can be obtained from the corresponding results for the tensor components in the x- and y-directions,... 

Hyperfine-coupling tensor components in Tesla for the intrinsic spin of iron, Spe = 3/2... 

Interestingly, the contributions from the gradient of the electromagnetic field across the interface, Tfg xx and Tfg zzz, which scale with the mismatch in the optical dielectric constants of the media forming the interface [37], only appear in the susceptibility tensor components nd xl zzz- Therefore, these contributions may be rejected with a... 

TABLE 1 Absolute Magnitude of the Nonvanishing Susceptibility Tensor Component at Several Air-Liquid and Liquid-Liquid Interfaces [40]. All units are in esu but Transformation into SI Units is Obtained Using the Relationship 1 esu = 3.72 x 10 m ... 

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

The five second-moment spherical tensor components can also be calculated and are defined as the quadrupolar polarization terms. They can be seen as the ELF basin equivalents to the atomic quadrupole moments introduced by Popelier [32] in the case of an AIM analysis ... 

Determination of g-tensor components from resolved 327-670 GHz EPR spectra allows differentiation between carotenoid radical cations and other C-H jt-radicals which possess different symmetry. The principal components of the g-tensor for Car"1 differ from those of other photosynthetic RC primary donor radical cations, which are practically identical within experimental error (Table 9.2) (Robinson et al. 1985, Kispert et al. 1987, Burghaus et al. 1991, Klette et al. 1993, Bratt et al. 1997) and exhibit large differences between gxx and gyy values. 

The g-tensor principal values of radical cations were shown to be sensitive to the presence or absence of dimer- and multimer-stacked structures (Petrenko et al. 2005). If face-to-face dimer structures occur (see Scheme 9.7), then a large change occurs in the gyy component compared to the monomer structure. DFT calculations confirm this behavior and permitted an interpretation of the EPR measurements of the principal g-tensor components of the chlorophyll dimers with stacked structures like the P 00 special dimer pair cation radical and the P700 special dimer pair triplet radical in photosystem I. Thus dimers that occur for radical cations can be deduced by monitoring the gyy component. 

A major advantage of studying pure compounds is that single crystals can be used, and hence e.s.r. parameters, which are generally anisotropic, can be accurately extracted. Furthermore, if the crystal structure is known, and if, as is frequently the case, the paramagnetic centres retain the orientation of the parent species, the directions of the g- and electron-nuclear hyperfine tensor components can be identified relative to the radical frame. 

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