Strain Energy due to Shear Stress:

A small element of dimensions dx, dy and dz is subjected to a shear stress component τ_xy and undergoing shear strain τ_xy

As shown here, we may calculate the work done as product of force dF and average displacement dδ/2.

Displacement, dδ =ϒ_xy dy

Force, dF =τ_xy dx dy

Shear strain, ϒ_xy =τ_xy/G

∴ dδ = (τ_xy/G)dy

∴          Work done or energy stored in the element,

dU = 1/2 τ_{xy dx dz.} τ_xy/G.dy                                                                                           . . . (18)

dU = ½ τ²_xy /G.dv

∴          Total Strain Energy

                                                                                                            . . . (19)

Strain energy density at any point,

u = τ²_xy /2G                                                                                                                 . . . (20)

Expressions similar to Eqs. (18), (19) and (20) can be developed for other shear stress components τ_yz and τ_zx also.