Tensors matrix form

Tensors can easily be written in matrix form. For example, a vector aj can be represented with a column matrix ... 

In this section, we develop some useful relationships involving the determinants and inverses of projected tensors. Let S ap be the Riemannian representation of an arbitrary symmetric covariant tensor with a Cartesian representation S v We may write the Riemannian representation in block matrix form, using the indices a,b to denote blocks in which a or p mns over the soft coordinates and i,j to represent hard coordinates, as... 

The extra terms in the bottom row are a result of nonvanishing unit-vector derivatives. The tensor products of unit vectors (e.g., ezer) are called unit dyads. In matrix form, where the unit vectors (unit dyads) are implied but usually not shown, the velocity-gradient tensor is written as... 

It should be noted that the velocity-gradient tensor is not symmetric. The matrix form of the velocity-gradient tensor for other coordinate systems is stated in Section A.8. 

The principal strain rates are eigenvalues of the strain-rate tensor (matrix). As described more fully in Section A.21, the direction cosines that describe the orientation of the principal strain rates are the eigenvectors associated with the eigenvalues. In solving practical fluids problems, there is rarely a need to determine the principal strain rates or their orientations. Rather, these notions are used theoretically with the Stokes postulates to form general relationships between the strain-rate and stress tensors. It is perhaps worth noting that in solid mechanics, the principal stresses and strains have practical utility in understanding the behavior of materials and structures. 

Write the strain-rate tensor (in matrix form) for these two circumstances. 

D is called the diffusivity tensor and acts as an object that connects one vector to another (e.g., the flux vector with the gradient vector). This connection can be written in matrix form as in Eq. 4.57. The diffusivity tensor D is symmetric (i.e., Dtj = Dji) for any underlying material symmetry. 

The reduced matrix elements of a double-tensor operator (formed as a scalar product of tensor operators of rank k and 1) become... 

These simplifications reduce the size of the elasticity tensors from [9 x 9] to [6 x 6], with 36 elastic coefficients. The shorthand notation normally used for the elasticity tensors are now introduced, namely, that the subscripts become 1 11 2 22 3 33 4 23, 32 5 31, 13 and 6 12, 21. With this change, the elastic stiffness tensor may be written in matrix form as ... 

Molecular polarizability, a, is a measure of the ability of an external electric field, E, to induce a dipole moment, = aE, in the molecule. As such, it can be viewed as contributing to a model for induced dipole (dispersive) interactions in molecules. Because the polarizability is a tensor (matrix) quantity, there is the question of how to represent this in a scalar form. One approach is to use the average of the diagonal components of the polarizability matrix, (a x + otyy + Since the polarizability increases with size (and... 

Thus, when acted on by an electric (or indeed any) external field, naturally isotropic bodies become endowed with anisotropic properties like those possessed by naturally anisotropic ones, such as crystals. The electric permittivity tensor e(Ep) describing this induced anisotropy can be represented in matrix form, as follows, in a Cartesian system of co-ordinate axes X, y, z ... 

We can give a simple, but important physical interpretation of the expressions for sensitivities, (9.55), based on the reciprocity principle. Note that, according to definition (see Chapter 8), the Green s tensors (r r") and Gh (r r"), are the electric and magnetic fields at the receiver point, r, due to a unit electric dipole source at the point r" of the conductivity perturbation. Let us introduce a Cartesian system of coordinates x,y,z, and rewrite these tensors in matrix form ... 

To proceed further we need to develop our representation of the mode specific atomic displacement, presently given as a vector, u. We shall find it most convenient to express this as a tensor in matrix form, B. Where the unit vectors along the Cartesian axes are e, Cy, e. ... 

The Tensor Suface Harmonic functions may be expressed in the following matrix form ... 

The use of the expansions (10) and (11) reduces the number of parameters needed to describe nonlinear MO phenomena (in comparison to the general formula (8). We can show this by comparing equation (8) with equations (9) and (10). Formula (10) can naturally be represented in a matrix form, just as equation (8), where the third-rank tensor has the following form in Voigt s notation ... 

When written in matrix form these equations relate the properties to the crystallographic directions. For ceramics and other crystals the piezoelectric constants are anisotropic. For this reason, they are expressed in tensor form. The directional properties are defined by the use of subscripts. For example, d i is the piezoelectric strain coefficient where the stress or strain direction is along the 1 axis and the dielectric displacement or electric field direction is along the 3 axis (i.e., the electrodes are perpendicular to the 3 axis). The notation can be understood by looking at Figure 31.19. 

In matrix form, the second-rank deformation gradient tensor can be expressed by... 

In order to determine the EO tensor for crystals belonging to the tetragonal 4 class, we will begin with the EO tensor for the triclinic 1 crystal group in matrix form [27] ... 

Book content is otganized in seven chapters and one Appendix. Chapter 1 is devoted to the fnndamental principles of piezoelectricity and its application including related histoiy of phenomenon discoveiy. A brief description of crystallography and tensor analysis needed for the piezoelectricity forms the content of Chap. 2. Covariant and contravariant formulation of tensor analysis is omitted in the new edition with respect to the old one. Chapter 3 is focused on the definition and basic properties of linear elastic properties of solids. Necessary thermodynamic description of piezoelectricity, definition of coupled field material coefficients and linear constitutive equations are discussed in Chap. 4. Piezoelectricity and its properties, tensor coefficients and their difierent possibilities, ferroelectricity, ferroics and physical models of it are given in Chap. 5. Chapter 6. is substantially enlarged in this new edition and it is focused especially on non-linear phenomena in electroelasticity. Chapter 7. has been also enlarged due to mary new materials and their properties which appeared since the last book edition in 1980. This chapter includes lot of helpful tables with the material data for the most today s applied materials. Finally, Appendix contains material tensor tables for the electromechanical coefficients listed in matrix form for reader s easy use and convenience. 

Let h x) = hiCi be a vector-valued function, e.g., a displacement u x). The gradient of h x) is given as a second order tensor, and the components are written in a matrix form as... 

The tensor form of the constitutive equation in (3.1) provides a concise and effective statement of Hooke s law for use in theoretical developments. For purposes of measurement and calculation, however, it is often more convenient to adopt a matrix form of the constitutive equation. Such a form is... 

Depending on the circumstances, it makes sense to apply either tensor or matrix calculus. Occasionally it may be useful to switch the representation. Typically, the results of a derivation requiring tensors are written in the more accessible matrix form. While operations involving scalars and vectors are applicable for both, the more general case is subjected to restrictions ... 

The discrete atomic nanotube structure replaced the effective hollow cylinder having the same length and outer diameter as a discrete nanotube with effective Young s nanotube modulus determined from atomic structure [7], The stress transfer between fiber and matrix in RVE was determined using modified shear-lag model [8], For example, in the region laxial component of stress tensor (dimensionless form) for fiber matrix has the following form... 

The previous equation can be put in a matrix form as well, introducing the so-called dielectric tensor (e), namely D = eF. Now, combining linearly the dependence of the electric displacement on F with a law cross-coupling D and the applied stress, [Pg.340]

The shearing flows are commonly analyzed using a standard Cartesian coordinate system % = %i, X2,X where Xi is directed along the flow and X2 along the velocity gradient. In this coordinate system, the velocity vector v is v = y t)x2,0,0, and the tensors of strain rate e(f) and vorticity m(t) for homogeneous shearing flows have the matrix forms ... 

Collecting all multipole moments of center A into a vector q" (A), those of B into q (B), and arranging the elements of the interaction tensor in matrix form T(R), the multipole expansion can also be formulated in matrix notation as... 

Approach to restoring of stresses SD in the three-dimensional event requires for each pixel determinations of matrix with six independent elements. Type of matrixes depends on chosen coordinate systems. It is arised a question, how to present such result for operator that he shall be able to value stresses and their SD. One of the possible ways is a calculation and a presenting in the form of image of SD of stresses tensor invariants. For three-dimensional SDS relative increase of time of spreading of US waves, polarized in directions of main axises of stresses tensor ... 

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