Tensor matrix notation

The formal conversion of Eqns 3-2 to Eqns 3-3 is straightforward substitution and simplification. The theoretical basis for this transformation as given in the treatment by M. Reiner [3] is in tensor matrix notation, the physical significance of which requires considerable experience to see. 

The generalized Hooke s law in tensor (matrix) notation is given as Cfjj = Ejjkq Ekq. Expand and find the algebraic expansion for a. ... 

Although we are concerned with linear algebra it will be convenient to use a notation slightly different from the usual matrix notation and to adopt the summation convention from tensor analysis. Thus asr will denote the element in the rth row and 5 th column of the matrix a = (a ). The product a fi of a with a matrix ft = (, ) is the matrix (otsrfi s), where summation over s = 1, 2,. .., s is implied. The range of a particular affix (i.e., s = 1,2, s) is given in the table of nomenclature. In this notation the order of any matrix is apparent and the use of partitioned matrices can be represented rather easily. 

The piezoelectric coefficients are third rank tensors, hence the piezoelectric response is anisotropic. A two subscript matrix notation is also widely used. The number of non-zero coefficients is governed by crystal symmetry, as described by Nye [2], In most single crystals, the piezoelectric coefficients are defined in terms of the crystallographic axes in polycrystalline ceramics, by convention the poling axis is referred to as the 3 axis. 

Draw, in matrix notation, the elastic-stiffness tensor for a tetragonal monocrystal in the 422 class. 

Write the elastic-comphance tensor in matrix notation for an orthorhombic monocrystal, showing only the nine independent constants from Table 10.4. 

In sec. 1.6b we have used a similar matrix notation for the stress tensor t imd in sec. I.app.yf we did so for the polarizability tensor a. We assume the system to be at mechanical equilibrium, i.e. the fluids are at rest. This means that the isotropic pressures in the bulk phases must the same everywhere (p = p ). Then... 

When represented in matrix notation the Trace ( ) operation, or Spur, of a tensor is the sum of its diagonal components, its scalar value. The contraction operator implies that the two matrices on either side should be multiplied and the trace of the resulting matrix obtained. This equation, Eq. (2.41), is less troublesome than it first appears, as we shall demonstrate by substituting the results of Eq. (2.38) and (2.40) for the vibration, v= 1. First let us evaluate the a") term. 

Any second-rank tensor, T, can be decomposed into the sum of a symmetric, S, and an antisymmetric, A, tensor, i.e. in matrix notation... 

The quantities G contain contributions from the hyperfine tensor A and from the nuclear Zeeman term bi = B j li where lx, ly and 1 are the direction cosines of the applied magnetic field with respect to a suitably chosen coordinate system of a crystal or molecule. In the general case with non-negligible g-anisotropy the effective direction of the field is u = following the notation of Weil and Anderson [36]. The matrix notation (4.8b) is also employed, e.g. in [26] that also provides the 2nd order corrections reproduced in Chapter 3. 

Table 4.4-1 The relations between ij (tensor notation) and m (matrix notation), jk and n, and kl and n...

For relations between tensor and matrix notation, see Table 4.4-1. The sound velocity in the direction mn in a crystal is given by... 

The symmetry of tensors for strain and stress and the related reduction in the number of coordinates has given rise to the introduction of an abbreviated notation and a representation of these quantities by matrices. This abbreviated matrix notation is almost exclusively in use in the literature on piezoelectricity and in the... 

It was shown earlier that there are only six independent tensor coordinates for strain and stress. The matrix notation allows us to assign them just six different symbols. Following Table 3.1 the assignment Ty = 7 is chosen and... 

This paragraph gives the necessary conventions and abbreviation for the tensor components of elastic, piezoelectric, pyroelectric and dielectric properties and for the thermal expansion coefficient (Tables A. 1 and A.2). Tensor and matrix notation is adopted according to Nye (1957) material tensors tables according to Sirotin and Shaskolskaya (1982). 

Table A.1 Tensor and matrix notation - abbreviation of indeces Tensor notation 11 22 33 23 or 32 13 or 31 12 or 21...

Since the stress and strain tensors are symmetric. the piezoelectric coefficients can be converted from tensor to matrix notation. Table 13 provides the piezoelectric matrices for a-quartz together with several values rfyk and Cjjk. including the corresponding temperature coefficients [259], [261]. 

In this last equation b is any vector and I is a unit tensor, which in matrix notation is given by... 

The reduced number of components enables us to use a simplified matrix notation Voigt notation), rewriting the tensors of second order as column matrices and the tensor of fourth order as a quadratic matrix (<7jj) —> (uq),... 

If all components of a tensor are to be described, this is done by adding parentheses to the tensor written in index notation, for example, (a ), Aij), Cijki). Implicitly, it is assumed that each index runs from 1 to 3. In second-order tensors, the first index denotes the row and the second index denotes the column of the component in the matrix notation. The components of the tensor... 

In this section I aim to describe in some detail continuum theory for nematics, and also to draw attention to some points of current interest, particularly surface conditions and surface terms. At this juncture it does seem premature to discuss new developments concerning smectics, polymers and lyotropics, although a brief discussion of an equilibrium theory for certain smectics seems appropriate, given that it relates to earlier work on this topic. Throughout, to encourage a wider readership, we endeavour to employ vector and matrix notation, avoiding use of Cartesian tensor notation. 

Due to the symmetry of T and E, the number of components of the stiffness and compliance tensors is reduced from a total of 81 to 36 independent ones. Thus, it is possible to represent these fourth-order tensors alternatively in the form of (6 x 6) matrices (which, of course, do not have the transformation properties of a tensor), and to express Hooke s law and inverse Hooke s law in direct matrix notation (engineering notation) as... 

The elastic constants (elastic moduli) A, and ju are called Lame constants (or Lame moduli, units [GPa]). Switching over from matrix notation to tensor notation the elasticity tensor is... 

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