Elastic constants and polymer symmetry

The mechanical properties of an anisotropic elastic solid where the stresses are linearly related to the strains are defined by the generalized Hooke s law where each component of stress can relate to all six independent components of strain, and equivalently each component of strain can relate to all six independent components of stress. In the former case we have, for example 

An Introduction to the Mechanical Properties of Solid Polymers I. M. Ward and J. Sweeney 2004 John Wiley Sons, Ltd ISBN 0471 49625 1 (HB) 0471 49626 X (PB) 

In practical applications of oriented polymers we are concerned with films or fibres, which reduces the number of independent elastic constants to nine or six, respectively. It is also convenient at this stage to limit the discussion to the compliance constants because these can be related directly to the readily measured engineering elastic constants, such as Young s moduli, shear moduli and Poisson s ratios. 

The compliance matrix for these axes involves nine independent elastic constants  

Note that in each of these expressions the denominator indicates the direction x, y, z (or 1,2,3) in which the tensile stress is applied. 

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