Hooke's law

Each of the six components of stress at any part of the body is a linear function of the six components of strain at the point.

Thus, following relations between stress and strains hold,

(3.10)

are 36 coefficients which are called elastic constants of the substance.

For isotropic systems, the formulae connecting stress components to the strain components should be independent of the directions. Following simplification results by considering that the directions are equivalent:

(3.11)

Thus the stress strain relationships do not alter when any of the two axes are interchanged.

Furthermore they should not change when the direction of any axis is reversed. However when say the axis is reversed the signs of also get reversed while the signs of the remaining components remain unchanged.

Hence, the constants must vanish, so that,

(3.12)