Strain Energy due to Shear Stress:

A small element of dimensions dx, dy and dz is subjected to a shear stress component t_xy and undergoing shear strain g_xy is shown in given Figure

                                 Figure

As shown in previous Section, we might calculate the work completed as product of force dF and average displacement dd/2

Displacement, dδ = g _xy  dy

Force, dF = t_xy  dx dy

Shear strain, g_xy = t _xy  /  G

 ∴  dδ = (t _xy /G ) dy

∴          Work done or energy stored in the element,

            dU =    ½ t _xy  dx dz. (t _xy  /G).dy

∴          Total Strain Energy

Strain energy density at any point,