Using force equilibrium ie internal force caused by stress

Question

Using force equilibrium; i.e., internal force caused by stress is equal to external force.

Problem

A force F is applied at the tip of the uniform bar shown in the figure. The displacement of the bar is given as u(X) = λX where λ is the principal stretch. The initial length and the cross-sectional area of the bar are, respectively, A₀ and L₀.

The elastic modulus of the material is E. Calculate the tip displacement by solving for the principal stretch using the total Lagrangian formulation with the St. Venant-Kirchhoff material model.

Assume the following numerical values: E = 700 MPa, A₀ = 1.0 X 10^-4 m², L₀ = 1.0 m, and F = 10 kN. Compare the tip displacement with that of the linear elastic model when (a) E = 700 MPa and (b) E = 70 GPa.