Strain Energy in Terms of Principal Stresses:

Here, we have derived expressions for strain energy assuming that at a time only one stress component is acting. For example, we have taken εx as equal to σx/E which is true only if σx alone is acting. Otherwise, we know that,

ε_x =σ_x/E -vσ_y/E -vσ_y/E,

and the use of such expressions will result in more complex expressions for strain energy and strain energy density. Hence, in the case of a general stress field, we may be able to get simpler expressions if we reduce the number of terms to be considered. A general stress field with six components of stresses namely σx, σy, σz, τ_xy, τ_yz and τ_zx may be expressed in terms of equivalent principal stress components σ₁, σ₂ and σ₃. In terms of these three components, let us now derive expressions for total strain energy.