Derive an expression for the force as a function of bar

Question

Calculate the total force required to plastically deform a metallic nanobar of circular cross section taking into account the surface energy Es in the case that the length y of the nanobar increases and its cross section α²(y) homogeneously decreases.

What are the bulk and surface contributions to this force in terms of critical stress for material flow (yield stress) and surface energy density g ?

The work required for plastic elongation of the cylinder on Δy is:

E_y = Πα²σ_yΔy, σ_y = G/30 is the yield stress and the work required to extend the surface is:

α_o²(y) = E_s = γ2ΠαΔγ, γ ≈ 1 J/m² G ≈ 30GPa

a. Derive an expression for the force as a function of bar length y if the length and cross section before deformation are y_o and α_o² respectively. Hint: use volume conservation of the bar and the expression for the force:

dV/dy = dya²(y)/dy = 0

and the expression of force:

F = -d(E_v + E_s)/dy

Determine the length of the bar at which surface contribution to the force is equal to the bulk contribution. What is a equal to in this case?