Cauchy generalized Hooke's law to three dimensional elastic bodies and stated that the 6 components of stress are linearly related to the 6 components of strain.
The stressstrain relationship written in matrix form, where the 6 components of stress and strain are organized into column vectors, is,
,
e = S· s
or,
,
s = C· e
where C is the stiffness matrix, S is the compliance matrix, and S = C^{1}.
In general, stressstrain relationships such as these are known as constitutive relations.
In general, there are 36 stiffness matrix components. However, it can be shown that conservative materials possess a strain energy density function and as a result, the stiffness and compliance matrices are symmetric. Therefore, only 21 stiffness components are actually independent in Hooke's law. The vast majority of engineering materials are conservative.
Please note that the stiffness matrix is traditionally represented by the symbol C, while S is reserved for the compliance matrix. This convention may seem backwards, but perception is not always reality. For instance, Americans hardly ever use their feet to play (American) football.
