Engineering strains and matrix notation

In one dimension Hooke s law defines stiffness (or modulus) as stress divided by strain alternatively, compliance is strain divided by stress. In three dimensions the stress and strain tensors are related through stiffness [c] and compliance [5] tensors (note the confusing nomenclature). As stress and strain are each second-rank symmetric tensors, [c] and [5] are each fourth-rank symmetric tensors each component of strain is linearly related to all nine components of stress, and vice versa, so there are 81 components in the stiffness and compliance tensors, which when written out in full form a 9 X 9 array. 

The fourth-rank tensors syu and cyki define the compliance and stiffness constants, with i,j, k, I taking values 1, 2, 3 in turn. In these equations the use of 1, 2, 3 is s30ionymous with the x, y, z used to define stress and strain components. 

In a fourth-rank symmetric tensor not all 81 terms are independent in general 

It is often more convenient to work in terms of engineering strains rather than use tensor strain components. Such an approach leads to a more compact notation, in which a generalized Hooke s law relates the six independent components of stress to the six independent components of engineering strain  

For stiffness constants the following substitution is used for obtaining p and q in terms of ij, and /  

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