Contracted notation

Contracted notation is a rearrangement of terms such that the number of indices is reduced although their range increases. For second-order tensors, the number of indices is reduced from 2 to 1 and the range increased from 3 to 9. The stresses and strains, for example, are contracted as in Table A-1. Similarly, the fourth-order tensors for stiffnesses and compliances in Equations (A.42) and (A.43) have 2 instead of 4 free indices with a new range of 9. The number of components remains 81 (3 = 9 ). 

In contracted notation, the stress-strain and strain-stress relations. Equations (A.42) and (A.43), are written as 

Obviously, the number of free indices no longer denotes the order of the tensor. Also, the range on the indices no longer denotes the number of spatial dimensions, if the stress and strain tensors are symmetric (they are if no body couples act on an element), then 

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The generalized Hooke s law relating stresses to strains can be written in contracted notation as... 

Table 2-1 Tensor versus Contracted Notation for Stresses and Strains...

A fourth-order tensor can be written as a 9x9 array in analogy to Equation (A.51) but, by use of contracted notation, is sometimes drastically simplified to a 6 x 6 symmetric array. 

The stress-strain relations in this book are typically expressed in matrix form by use of contracted notation. Both the stresses and strains as well as the stress-strain relations must be transformed. First, the stresses transform for a rotation about the z-axis as in Figure A-1 according to... 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

In the literature relating to second harmonic generation d-coefficients are sometimes used rather than vs when, employing a contracted notation (jk = s (1,2,. . ., 6))... 

The contracted notation for two-electron integrals over the basis functions, (rs tu), is based on the same convention outlined in note (17). 

Contracted notation for second-order susceptibilities. Traditionally the SHG non-linearity of crystals is described by a tensor d which is related to the SHG susceptibility by (29) (Zernike and Midwinter, 1973). 

When there is no doubt as to the formula of the product of the reaction, we may often make use of a contracted notation, viz. 

Since the dielectric tensor is real and symmetric, its inverse is also real and symmetric, and the tensor f must be symmetric in its hrst two indices. The electro-optic tensor elements as dehned in Eqs. (40) and (41) are expressed in the contracted notation ... 

Using this contracted notation the results for the distinct DCI matrix elements are ... 

The determinants involved in the DCI expansion obviously contain lots of orbitals in common (all except i, j, a and b, in fact) and sD it is useful to use a contracted notation which only involves the active orbitals, we write... 

Hooke s law, in a generalized form, using the contracted notation can be written as... 

B.2 Contracted Notation for Stiffness Tensor and Compliance Tensor... 

Since Ci and 8 are symmetric tensors, each of them has 6 independent components, the fourth order stiffness tensor Cyia contains at most 36 independent constants, such that it can be displayed as a 6 x 6 matrix of components using contracted notation, Cto), where m,n — 1,2,3,4,5,6. There is a unique correspondence between of and Cijki- The index m is related to ij, and n is related to Id, as shown in Table B.l. For instance, Cu22 = C12, C1323 = C54- Since C = C the number of independent constants is generally 21. For orthotropic materials, the the number of independent constants further reduces to 9. When the fourth order tensor is transformed to the principal axes, all Qj — 0, except for Cn, C22, C33, C12. 13. 23. 44, 55, and... 

For converting Cijia to engineering moduli, one first converts it to the contracted notation, and takes a matrix inverse to find the etaresponding eompliance matrix, and then use Eq. B.22 to obtain the engineering eonstants. 

Going back to contracted notation, the electro-optic tensor (r, ) for crystals belonging to the tetragonal 4 group can be represented by the well-known matrix [15]... 

When the stresses and strains are symmetric the nnmber of independent constants is rednced to 36. Hooke s law can be written in a contracted notation ... 

A three-dimensional representation of the stresses in contract notation is presented in Fig. 11.5. 

Eor this reason and the confusion that can arise in defining the momentum and strain-displacement equations, one must use care in solving problems with the contracted notation. The value of p in terms of i,j varies notoriously between authors, but the IEEE standard [4] is the most commonly accepted version here p= (i +f) (1 - sgn(i -j) I) -b sgn(j -j) (9 - i -j) for i,j = 1,2,3 here the vertical bars represent an absolute value while sgn(-) is the signum, zero when x is zero and the sign of x otherwise. 

In the natural material coordinate system, the contracted notation whereby. ..,. .., is merely an alternative way to represent... 

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