Tensor characterized

The dipole polarizability tensor characterizes the lowest-order dipole moment induced by a unifonu field. The a tensor is syimnetric and has no more than six independent components, less if tire molecule has some synnnetry. The scalar or mean dipole polarizability... 

An electrostatic quadrupole moment is a second-rank tensor characterized by three components in its principal-axis system. Since the trace of the quadrupole moment tensor is equal to zero, and atomic nuclei have an axis of symmetry, there is only one independent principal value, the nuclear quadrupole moment, Q. This quadrupole moment interacts with the electrostatic field-gradient tensor arising from the charge distribution around the nucleus. This tensor is also traceless but it is not necessarily cylindrically symmetrical. It therefore needs in general to be characterized by two independent components. The three principal values of the field-gradient tensor are represented by the symbols qxx, qyy and qzz with the convention ... 

A symmetric tensor ft is called a rotational tensor. It depends both on the shape and size of the particle and on the choice of the origin. The rotational tensor characterizes the drag under rotation of the body and has the diagonal form with entries fii, Q2, 3 in the principal axes (the positions of the principal axes of the rotational and translational tensors in space are different). For axisymmetric bodies, one of the major axes (for instance, the first) is parallel to the symmetry axis, and in this case = O3. For a spherical particle, we have fii = fl2 = flj. 

The required increases in pressure are mechanically induced by local coil expansion and give rise to stresses, tensions, in the macromolecular chains of the cluster. The tensor characterizing the stresses present locally in the gel matrix must have nonzero components that generate (i.e., compensate) the pressure increase both in the liquid and solid parts of the cluster. 

The harmonic functions are examples of the tensor operators of section 3.4. Their matrix elements, for a given J level, are thus proportional to the CG coefficients (31) with J =J. Bethe (1929) had already found some algebraic expressions for some special cases for which k = 2 and 4. Stevens (1952) extended the analysis to k = 6 and showed how to determine the proportionality constants. Since the matrix elements of other tensors characterized by the same k and q must, by the Wigner-Eckart theorem, be proportional to the same set of CG coefficients, Stevens... 

In this respect, a theory that takes into account the deformation of one droplet (Doi and Ohta 1991) can be applied to describe the shear and normal stress transients. According to this model, blend morphology is characterized by a scalar (referring to a specific interfacial area) and a tensor (characterizing interface anisotropy). These parameters may be expressed in two equations—one describing the stresses of the interfacial structures and the other for the evaluation of the scalar and interface tensor. For immiscible blends with Newtonian or weakly viscoelastic fluids and an increase in shear, the droplets deform into fibrils while maintaining their initial diameter, d. In comparison, in a highly elastic matrix where droplet shape is... 

In the new coordinate system, one essential feature that was not visible in the principal coordinate system became revealed. Namely, an arbitrary stressed state can be viewed as the sum of two tensors the spherical tensor characterizing the equilateral extension or compression and the shear stress deviator tensor. Either of these tensors (or both of than) can be zero. 

An arbitrary symmetric strain tensor has been thus subdivided into two additive components a spherical hydrostatic expansion (or contraction) tensor and a deviator tensor, characterizing shear deformations without any change in volume. 

In contrast to conductive material with the ability to accommodate electric flow fields, dielectric matter, as well as vacuum, may exhibit electrostatic fields. Although the physical condition of the examined dielectric domain is not limited to solid state, it may be described in analogy with deformable structures as a continuum. In comparison to the mechanical fields, the tensors characterizing the electrostatic fields will be one order lower. A comprehensive description of electrical engineering is given by Paul [140], while electromagnetic fields are detailed by Fischer [74], Lehner [120], and Reitz et al. [153]. 

Just as the stress tensor characterizes that state of stress at any point through its ability to describe the force acting on any plane, the deformation gradient describes the state of deformation and rotation at any point through the relation above. However, unlike the stress tensor, which depends only on the current state, the deformation gradient depends on both the current and a past state of deformation. 

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