ABSTRACT
For a small deformation the stress tensor σik is proportional to the strain tensor (Hooke’s law). In a macroscopically isotropic medium (such as, for example, a polycrystalline) this relation takes the form
σik = Kullδik + 2µ
( uik − 1
3 ullδik
) , (5.3)
uik = 1
9K σllδik +
2µ
( σik − 1
3 σllδik
) , (5.4)
which involves two elastic coefficients: themodulus of compression,K, and the modulus of rigidity, µ. Then the free elastic energy per unit volume of a deformed medium reads
F = 1 2 σikuik =
2 K(ull)
2 + µ
( uik − 1
3 ullδik
)2 (5.5)
Second Edition
In the absence of external forces a state of thermodynamic equilibrium with no deformation should correspond to a minimum of the free energy. Therefore, it follows from expression (5.5) that the moduli of compression and rigidity are always non-negative: K ≥ 0, µ ≥ 0.
