Tensor transformation laws

Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. 

C.2 Tensor Transformation Laws 1167 Then, if A is a second order tensor, or a dyad, the divergence of A is ... 

Stress and strain are second-order tensors while stiffness and compliance are fonrth-order tensors [14]. Hence these entities are mled by the tensor transformation laws that establish the relationships between the components... 

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. 

The third-rank tensors, as occur in the expression for the anharmonic temperature factor (chapter 2), the restrictions may be derived by use of the transformation law ... 

As discussed in Appendix A, symmetric tensors have properties that are important to the subsequent derivation of conservation laws. As illustrated in Fig. 2.9, there is always some orientation for the differential element in which all the shear strain rates vanish, leaving only dilatational strain rates. This behavior follows from the transformation laws... 

Many materials properties are anisotropic they vary with direction in the material. When anisotropic materials properties are characterized, the values used to represent the properties must be specified with respect to particular coordinate axes. If the material remains fixed and the properties are specified with respect to some new set of coordinate axes, the properties themselves must remain invariant. The way in which the properties are described will change, but the properties themselves (i.e., the material behavior) will not. The components of tensor quantities transform in specified ways with changes in coordinate axes such transformation laws distinguish tensors from matrices [6]. 

A further point of interest is the transformation law for vector (tensor) operators w.r.t. the laboratory system. For the electric dipole moment one may show by the same arguments as used for the case of SRMs41 that... 

Note that each component of p is related to all three components of q. Thus, each component of the tensor is associated with a pair of axes. For example, X32 gives the component of p parallel to aa when q is parallel to aa. In general, the number of indices assigned to a tensor component is equal to the rank of the tensor. Tensors of all ranks, like vectors, are defined by their transformation laws. For our purposes, we need not consider these. 

This transformation law is quite simple, and on it relies the main advantages of using spherical tensors in problems involving rotations. The Wigner matrices defined by Eq. (B.2) provide a set complete and orthogonal in the space of Euler angles, thereby making it possible to use them as a suitable expansion basis set. 

A key property of a tensor is the transformation law of its components. This law expresses the way in which the tensor components in one coordinate system are related to its components in another coordinate system. The precise... 

The last expression gives the potential matrix in the standard representation. The transformation law (89) gives precisely the Poincare transformation of the electromagnetic field strengths E and B, which can be combined into a tensor field on Minkowski space. 

The components of a Cartesian pseudotensor satisfy the same transformation law as do those of a true Cartesian tensor of the same rank, except when the transformation alters the right- or left-handedness of the axes (an improper rotation), as for the case of reflection in a plane (inversion). In this case the two transformation laws differ by an algebraic sign. 

These relations show that by starting from the tensor product of j (rank 1 tensor or vector operator) with itself, we can construct a scalar quantity (Bqq), a vector quantity pt = 0, 1), and a quadrupole B, /a = 0, 1, 2, not shown here). It is important to understand that we use the terms scalar, vector, and so on, with respect to spherical [SO(3)] transformation laws. This is equivalent to saying that the must... 

As is well known, explicit expressions for the polarization fields can be given such that equation (3) has the requisite properties these expressions, involving multipole series or line integrals, are by no means tmique. That this must be so can be seen from at least two levels of theory. We noted earlier that the notion of an electric field (or a magnetic field) is not invariant with respect to Lorentz transformations under such transformations however V should be an invariant scalar and this implies a definite transformation law for (P(x), M(x) that mixes them, and mirrors that for E(x), B(x) classically both pairs can be shown to be components of skew-symmetric second-rank tensors [7]. Although... 

As these functions are clearly determined in each coordinate system we describe them as components of geometrical (or physical) objects. If these components are given in a coordinate system, they are determined in all other coordinate systems by a simple linear transformation law. Because of the special linear shape of these transformation laws this geometrical object is called a tensor, more precisely, the fundamental tensor of the Riemannian space. 

Stresses transform from one coordinate-axis system to another according to well-defined transformation laws that utilize direction cosines of the angles of rotation between the final and initial coordinate-axis systems. Matrixes that obey such transformation laws are referred to as tensors (McClintock and Argon 1966). There are three sets of stress relations that are scalar and invariant in coordinate-axis transformations. The first such stress invariant of particular interest is the mean normal stress o- , defined as. 

As with stresses, transformations of strain elements from one coordinate-axis system into another obtained by rotations of axes obey the same transformation laws as those of stresses, utilizing the same direction cosines and making the symmetrical strain matrix also a tensor (McClintock and Argon 1966). 

We emphasize that we will here deal, in general, with double tensor fields, which are defined with respect to 3 and surface S(t) embedded in 3. Thus, we will deal with the quantities which obey the transformation law for a tensor under the following groups of transformations ... 

The PM tensor that couples the axial magnetic field vector H with the polar strain tensor Uy, is a third rank axial tensor As any tensor components are defined by their transformation laws, we consider the form of the piezomagnetic tensors in nanomaterials quantitatively. To find the nonzero components of third rank tensors we use the system of linear equations generated by the transformation law for the axial third rank tensor describing PM (m) ... 

Therefore it becomes necessary to use the transformation laws which relate the tensors in one coordinate system to another in a rotated coordinate system. Briefly, the relationship between the stresses in the principal material and global coordinates [14] is given by... 

We can show that a second-rank tensor transforms like the product x, ag if we use the transformation law (2.16)... 

Having established the transformation law Eq. (1), first rank tensor quantities (i.e. vectors) may be defined as those properties which transform according to Eq. (1). Higher order tensors can be defined in terms of different transformation laws, and they arise in a general sense when one vector or ten-... 

Although the simplest direction-dependent property is a vector, most physical properties of liquid crystals are higher order tensors. All tensor properties can be categorized by their transformation properties under rotation of the coordinate frame, and the transformation law for a third rank tensor such as the piezoelectric tensor Papyi ... 

Thus, in order to describe completely the state of stress at a point in a continuum, we must specify the stress tensor T. A key property of a tensor is the tranrformation law of its components. This law expresses the way in which the tensor components in one coordinate system are related to its components in another coordinate system. The precise form of this transformation law is a consequence of the physical or geometric meaning of the tensor. 

This transformation law characterizes rank-2 tensor quantities, with respect to the group U(m) of transformations in the m-dimensional vector space the product functions are said to span an m -... 

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