Tensors Symmetrical

In this book, we confine ourselves only to the special case of fluids mixture (4.128) which is linear in vector and tensor variables.We denote it as the chemically reacting mixture of fluids with linear transport properties or simply the linear fluid mixture [56, 57, 64, 65]. Then (see Appendix A.2) the scalar, vector and tensor isotropic functions (4.129) linear in vectors and tensors (symmetrical or skew-symmetrical) have the forms ... 

At this point, it is useful to review the above discussion of second-order tensors, especially in relationship to the ideas put forward on strain and thermal expansion. A second-order tensor T.p can be defined as a physical quantity with nine components that transform, such that T. j=a.iPjiTi.i or perhaps more simply, as an operator that relates two vectors. There are two special types of second-order tensors symmetric, in which T.j= Tjp and antisymmetric, in which 7V=—Tj. for i =j. There are some properties of these tensors that are important to this text and these are given here without proof... 

We separate the symmetric part and the antisymmetric part of the tensor ... 

In the special case when a = 0, any symmetric 2-D tensor field can be represented as... 

Braun and Hauck [3] discovered that the irrotational and solenoidal components of a 2-D vector field can be imaged separately using the transverse and longitudinal measurements, respectively. This result has a clear analogy in a 2-D tensor field. We can distinguish three types of measurements which determine potentials of the symmetric tensor field separately ... 

Dipolar D / 10 -10 Through space spin-spin interaction, axially symmetric traceless tensor... 

A second-oi der tensor whose components satisfy Ty - Tji is called symmetric and has six distinct components. If Ty - -Ty then the tensor is said to be antisymmetric. 

Any tensor may be represented as the sum of a symmetric part and an antisymmetric part... 

It can hence be seen that the magnitude of a symmetric tensor S is related to its second invariant as... 

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

In the final column of the character table are given the assignments to symmetry species and These are the components of the symmetric polarizability tensor... 

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. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

It may first be noted that the referential symmetric Piola-Kirchhoff stress tensor S and the spatial Cauchy stress tensor s are related by (A.39). Again with the back stress in mind, it will be assumed in this section that the set of internal state variables is comprised of a single second-order tensor whose referential and spatial forms are related by a similar equation, i.e., by... 

The symmetric and antisymmetric parts of I are the tensors of stretching d and spin w, respectively. 

Since the deformation tensor F is nonsingular, it may be decomposed uniquely into a proper orthogonal tensor R and a positive-definite symmetric tensor U by the polar decomposition theorem... 

The symmetric stress tensor S was first used by Piola and Kirchhoff. In component form... 

Because the scalar inner product of a symmetric and an antisymmetric tensor vanishes, from (A.l 1)... 

In this section, well-known properties of second-order positive-definite symmetric tensors and functions involving them will be cited without proof. The principal values and principal vectors (m = 1, 2, 3) of a symmetric second-order tensor A are given by... 

A scalar-valued function/(/4) of one symmetric second-order tensor A is said to be symmetric if... 

A scalar-valued function f(A, B) of two symmetric second-order tensors A and B is said to be isotropic if... 

Obviously, the number of free indices no longer denotes the order of the tensor. Also, the range on the indices no longer denotes the number of spatial dimensions, if the stress and strain tensors are symmetric (they are if no body couples act on an element), then... 

A fourth-order tensor can be written as a 9x9 array in analogy to Equation (A.51) but, by use of contracted notation, is sometimes drastically simplified to a 6 x 6 symmetric array. 

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