Essentials of Manufacturing

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One-dimensional Hooke's Law
Robert Hooke, who in 1676 stated,

The power (sic.) of any springy body is in the same proportion with the extension.

announced the birth of elasticity. Hooke's statement expressed mathematically is,

where F is the applied force (and not the power, as Hooke mistakenly suggested), u is the deformation of the elastic body subjected to the force F, and k is the spring constant (i.e. the ratio of previous two parameters).

Generalized Hooke's Law (Anisotropic Form)
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 stress-strain relationship written in matrix form, where the 6 components of stress and strain are organized into column vectors, is,

  ,      e = S·s


  ,      s = C·e

where C is the stiffness matrix, S is the compliance matrix, and S = C-1.

In general, stress-strain 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.