The Generalised Hookes Law

The most general linear relationship between stress and strain is obtained by assuming that each of the tensor components of stress is linearly related to all the tensor components of strain and vice versa. Thus 

In tensor notation, we relate the second-rank tensor ay to the second-rank strain tensor Sij by fourth-rank tensors Ciju and Siju- Thus, 

The fourth-rank tensors Syke and Cyki contain the compliance and stiffness constants respectively, with i, j, k, I taking values 1,2 and 3. 

It is customary to adopt an abbreviated nomenclature in which the generalised Hooke s law relates the six independent components of stress to the six independent components of the engineering strains. 

A typical relationship between stress and strain is now written as 

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