Net Strain Energy Density:
Net strain energy density in an element subjected to a general stress field can be obtained by adding (algebraically) the energy density due to all the stress components. That is
u = ½ [σ1ε1 + σ2ε2 + σ3ε3] = ½ σT ε
but ε = Cσ
∴ u =1/2 σT Cσ
Eq. may, if desired, be expanded as,
u = 1/2E (σ12 +σ22 + σ32 -2v(σ1σ2 +σ2σ3 +σ3σ1))