Configuration tensor

If we assume the configuration tensor W = (RR as an independent variable, the Gibbs equation is... 

Quasispin for complex electronic configurations tensors Tk and at II = 1, to the CFP with two detached electrons... 

The theories of internal variables are applied in rheology, dielectric, and magnetic relaxation where the structure of the macromolecules plays a relevant role (Beris and Edwards, 1994). In the theories of internal variables, it is usual to propose purely relaxational equations for the internal variables and associate the additional variables with some structure of underlying molecules. If we assume the configuration tensor W = (RR) where R is the end-to-end vector of the macromolecules as an independent variable, the Gibbs equation is... 

Here, C denotes the configuration tensor and D describes the rate-of-strain tensor of the material continuum. The dimensionless anisotropy factor, cc, characterizes the anisotropic character of the particle mobility. It is easy to show that a attains values between zero and one. The limiting case a = 0 corresponds to the isotropic motion and ultimately leads to an upper converted Maxwell material. In order to derive a deformation-dependent constitutive equation, a Hookean law connecting the tensor of external stresses S, the configuration tensor C, and the shear modulus G was suggested ... 

The configuration tensor c can be considered as describing the stretching and orientation of the polymer chain. It is seen that one must first solve Eq. (3.26) for the configuration tensor c in order to calculate stresses using Eq. (3.25). Because the stress tensor a is coupled, via Eq. (3.25), with the configuration tensor c. 

The end-to-end vectors of the subchains have distributions in their length and orientation. Equation [10] clearly indicates that the deviatoric part (measurable part) of the stress tensor due to the entropy elasticity of the polymer chains, hereafter referred to as the polymeric stress, reflects the orientational anisotropy of the subchains specified by the configuration tensor S(n,t). Consequently, the polymeric stress relaxes, even though the material keeps its distorted (e.g., sheared) shape, when the orientational anisotropy induced by tbe applied strain relaxes tbrougb tbe tbermal motion of tbe cbains. (In tbis relaxed state, S(n,t) is equal to 1/3 and tbe subcbain tension is transmitted isoUopically in aU tUrecrions to balance tbe isotropic pressure.) Thus, tbe relaxation time of the polymeric stress is identical to the orientational relaxation time of the polymer chains. 

In what follows the Kirchhoff-Love model of the shell is used. We identify the mid-surface with the domain in R . However, the curvatures of the shell are assumed to be small but nonzero. For such a configuration, following (Vol mir, 1972), we introduce the components of the strain tensor for the mid-surface,... 

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. 

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. 

For some purposes, it is convenient to express the constitutive equations for an inelastic material relative to the unrotated spatial configuration, i.e., one which has been stretched by the right stretch tensor U from the reference configuration, but not rotated by the rotation tensor R. The referential constitutive equations of Section 5.4.2 may be translated into unrotated terms, using the relationships given in the Appendix. 

While r is a spatial vector with components relative to the current configuration, F and its inverse are dual tensors, with one index relative to the current configuration and one index relative to the reference configuration. 

The triple product of three noncolinear line elements in the reference configuration provides a material element of volume dV. Another well-known theorem in tensor analysis provides a relation with the corresponding element of volume dv in the current spatial configuration... 

Clearly, /, d, and w are spatial tensors with components relative to the current configuration. Since the trace of an antisymmetric tensor vanishes, from (A.9)... 

Consequently, E has components relative to the reference configuration, and is a referential strain tensor. A complementary strain tensor may be defined from the inverse deformation gradient F ... 

Consequently, e has components relative to the current configuration and is a spatial strain tensor. If (A.18) is premultiplied by F and postmultiplied by F, it is seen from (A. 17) that... 

The components of strain ej- relative to the unrotated spatial configuration are shifted to components of strain relative to the reference configuration by the stretch U, or to components of strain Cy relative to the current spatial configuration by the rotation R. The tensors E, e, and e all are measures of the same irrotational part of the deformation, but with components relative to different configurations. 

The spatial Cauchy stress tensor s is defined at time by f = sn, where t(x, t, n) is a contact force vector acting on an element of area da = n da with unit normal i and magnitude da in the current configuration. The element of area... 

If a complementary stress tensor S is defined in terms of the vector T acting on the area dA in the reference configuration by 7 = SN, then, from these equations,... 

This induced dipole moment is independent of any dipole moment the molecule may possess in its equilibrium configuration. The molecular polarizability, a, has the properties of a tensor because both M and E are vectors. 

Actually transversality in all the k variables already follows from transversality in any one of the k variables because of the symmetric character of the tensor alll...Un(k1, , kn). Again due to the freedom of gauge transformations an n photon configuration is not described by a unique amplitude but rather by an equivalence class of tensors. We define the notion of equivalence for these tensors, as follows a tensor rfUl. ..Bn( i, , kn) will be said to be equivalent to zero ... 

The correlation can readily be extended to the higher photon configurations. We shall choose as the representative of the n photon configuration the tensor ktt) defined on the positive... 

The metric term Eq. (2.8) is important for all cases in which the manifold M has non-zero curvature and is thus nonlinear, e.g. in the cases of Time-Dependent Hartree-Fock (TDHF) and Time-Dependent Multi-Configurational Self-Consistent Field (TDMCSCF) c culations. In such situations the metric tensor varies from point to point and has a nontrivial effect on the time evolution. It plays the role of a time-dependent force (somewhat like the location-dependent gravitational force which arises in general relativity from the curvature of space-time). In the case of flat i.e. linear manifolds, as are found in Time-Dependent Configuration Interaction (TDCI) calculations, the metric is constant and does not have a significant effect on the dynamics. 

We restrict ourselves to the local valence part of the EFG tensor to illustrate the principle. Since the EFG operator is spin-free, there are no off-diagonal elements M M and an inspection of Table 5.6 reveals that there are also no off-diagonal components between different configurations I A J- Hence ... 

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