Tensor scalar

Now we come to the main theorem of this Appendix concerning the representation of scalar, vector, and tensor linear isotropic functions of vectors and tensors (scalars as independent variables play the role of parameters). 

Damour, T. and Nordtvedt, K., General relativity as a cosmological attractor of tensor-scalar theories, Phys. Rev. Lett, 70, 2217, 1993 Tensor-scalar cosmological models and their relaxation toward general relativity, Phys. Rev. D, 48, 3436, 1993. 

Note 2 According to the definition of pth order tensor, scalars are zero-order tensors and vectors are first-order tensors. 

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... 

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

Equations (8.20) are not sufficiently specific for practical purposes, so it is important to consider special cases leading to simpler relations. When the pore orientations are isotropically distributed, the second order tensors k, 3 and y are isotropic and are therefore scalar multiples of the unit tensor. Thus equation (8.20) simplifies to... 

The electric moments are examples of tensor properties the charge is a rank 0 tensor (which i the same as a scalar quantity) the dipole is a rank 1 tensor (which is the same as a vectoi with three components along the x, y and z axes) the quadrupole is a rank 2 tensor witl nine components, which can be represented as a 3 x 3 matrix. In general, a tensor of ran] n has 3" components. 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

APPENDIX - SUMMARY OF VECTOR AND TENSOR ANALYSIS The scalar (dot) product of two vectors is a number found as... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

The following three scalars remain independent of the choice of coordinate system in which the components of T are defined and hence are caUed the invariants of tensor T ... 

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... 

Of course, k may be taken to be comprised of a number of such tensors, and it is not difficult to extend the theory to include a number of indifferent scalars and vectors, if desired. 

It is possible to assume other transformation properties for k. For example, for some purposes it may be more desirable to attribute strainlike properties obeying a transformation law like (A. 19), in which case the equations of this section will take a somewhat different form. Of course, k may be taken to be comprised of a number of such tensors, and it is not difficult to extend the theory to include a number of indifferent scalars and vectors, if desired. 

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

The scalar spherical component A and the tensor deviator A of A are defined 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... 

Analogous results are available for scalar-valued functions of more than two tensor variables, see, e.g., [20]. 

The stress-intensity factors are quite different from stress concentration factors. For the same circular hole, the stress concentration factor is 3 under uniaxial tension, 2 under biaxiai tension, and 4 under pure shear. Thus, the stress concentration factor, which is a single scalar parameter, cannot characterize the stress state, a second-order tensor. However, the stress-intensity factor exists in all stress components, so is a useful concept in stress-type fracture processes. For example. 

Matrix and tensor notation is useful when dealing with systems of equations. Matrix theory is a straightforward set of operations for linear algebra and is covered in Section A.I. Tensor notation, treated in Section A.2, is a classification scheme in which the complexity ranges upward from scalars (zero-order tensors) and vectors (first-order tensors) through second-order tensors and beyond. 

A scalar is a tensor of order zero and has 3° = 1 component. Because it has only magnitude and not direction, no transformation relations are needed. Examples of scalars include speed (but not velocity), work, and energy. 

Einstein coefficient b, in (5) for viscosity 2.5 by a value dependent on the ratio between the lengths of the axes of ellipsoids. However, for the flows of different geometry (for example, uniaxial extension) the situation is rather complicated. Due to different orientation of ellipsoids upon shear and other geometrical schemes of flow, the correspondence between the viscosity changed at shear and behavior of dispersions at stressed states of other types is completely lost. Indeed, due to anisotropy of dispersion properties of anisodiametrical particles, the viscosity ceases to be a scalar property of the material and must be treated as a tensor quantity. 

In order to reproduce the temperature variation of the lattice constants, the anisotropy of the lattice expansion has to be taken into account. For this purpose, the tensor of thermal expansion ot is introduced instead of the scalar a , and the tensor of deformation due to the HS <- LS transition is employed instead of the dilation (Fh — Fl)/Fl. Each lattice vector x T) can now be... 

Angular restraints are another important source of structural information. Several empirical relationships between scalar couplings and dihedral angles have been found during the last decades. The most important one is certainly the Karplus relation for -couplings. Another, relaxation-based angular restraint is the so-called CCR between two dipolar vectors or between a dipolar vector and a CSA tensor. 

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