Voigt notation

Equation (4) expresses G as a function of temperature and state of applied stress (pressure) (o. Pa), (/(a) is given by the force field for the set of lattice constants a, Vt is the unit cell volume at temperature T, and Oj and are the components of the stress and strain tensors, respectively (in Voigt notation). The equilibrium crystal structure at a specified temperature and stress is determined by minimizing G(r, a) with respect to die lattice parameters, atomic positions, and shell positions, and yields simultaneously the crystal structure and polarization of minimum free energy. 

Because of the asumed transverse isotropy it follows that = 1/2 (Cu - Cn). The terms Qj are the elastic stiffnesses expressed in the contracted (Voigt) notation. 

The Voigt notation may be extended to a symmetric 7(4) tensor when Tljk becomes Tpq. 

These symmetries in eq. (3) reduce the number of independent tensor components for a triclinic crystal from eighty-one to twenty-one, which in Voigt notation form a symmetric... 

Example 15.4-3 Both the piezoelectric effect and the Pockels effect involve coupling between a vector and a symmetric 7(2). The structure of K is therefore similar in the two cases, the only difference being that the 6 x 3 matrix [rqi is the transpose of the 3x6 matrix [diq where i 1, 2, 3 denote the vector components and q= l,. .., 6 denote the components of the symmetric 7(2) in the usual (Voigt) notation. Determine the structure of the piezoelectric tensor for a crystal of C3v symmetry. 

Taking into account the symmetry expected for the spin-coated and evaporated films and fitting the experimental theoretical expressions [75] to hie experimental data (SHG yield vs rotation angle 0) for the three different polarizations, the values for the three non-zero independent components of the tensor (xsi , Ti5 , using Voigt notation 1 o 11, 2 o 22, 3 o 33, 4 o 23, 5 o 31,... 

These corrections to the elastic constants were calculated using the above equation and compared with those determined numerically. The numerical approach was to apply small stresses (Sj Voigt notation) of + 0.2 GPa and —0.2 GPa with the elastic compliances related to the resulting strains, e, to e6, by... 

Equation (4.40) is a good approximation, independent of the shape of the solid. 4.3.3 VOIGT NOTATION... 

Right-handed quartz, crystal class 32 (example from Sections 4.4.1 and 4.4.2). The tensor d has the form (Voigt notation) ... 

By using the Voigt notation, we represent equation (4.89) by a 10x10 symmetric matrix. Figure 4.17 shows this matrix and its inverse. The energy change per unit volume of a crystal subjected to o, E and AT in a reversible manner is equal to the sum of the strain energy (4.62), the electric polarization... 

Since, in our consideration, both the stress and the strain tensors are symmetrical about the diagonal, there are only six independent stress and strain elements, making many of the elastic compliances equal to each other. This makes it possible to simplify the generalized Hooke s law so that it involves at most 36 elastic compliances or constants, but requires the introduction of a shorthand notation both for stress and for strain for a unique representation that is referred to as the Voigt notation that we state as follows, for stresses and strains ... 

On the basis of the eontracted Voigt notation of stresses and strains, the generalized Hooke s law ean be written out in the form of a set of linear relations, which for the strain-stress relationships are... 

Table 4.2 Surface and bulk PE, PM and ME tensors in Voigt notation... 

The equation of state 8F12 /8q3 = 0 in Voigt notation has the form ... 

In this Eq. (1.48), the Ax and Au in Fig. 1.25 are replaced by dx and du or in terms of their partials as dx and du, thus indicating the instantaneous change of u with respect to x. Recall that ei = en = e = e x, depending on the type of notation. Nomenclatures may vary in accordance with the original research applications. For instance, the Voigt notation is useftil in calculations involving constitutive models, such as the generalized Hooke s Law, as well as for finite element analysis. 

Here Sx are the elastic strain components in Voigt notation, d x are the piezoelectric coefficients and M-,jx are the electrostrictive coefficients. 

Calculations of the full stress tensor is a method ideally suited to the derivation of elastic constants, since it contains up to six independent pieces of information that otherwise would require extensive calculations of total energy. The c- and c. 2 elastic constants can be found from the stress-strain relation with the application of an ei-strain. (The Voigt notation is used, see e.g. (Nye, 1957), i.e. 11- -1, 22- 2, 33- 3, 23 4, 13 5, 12- -6 thus... 

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),... 

We can also invert the elasticity matrix in the Voigt notation instead Sap) =... 

If we consider an arbitrary 5deld surface fipij) = 0 and prescribe a plastic strain rate (in the Voigt notation) e P equation (3.41) holds if the projection of o onto e P becomes maximal. This is sketched in figure 3.27. To achieve a maximum power of plastic energy dissipation... 

OC strain vector (Voigt notation, TYli sbp direction... 

Ta stress vector (Voigt notation, Vi direction of movement of a dis-... 

Voigt notation, possible values for the tensor components, pos-... 

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