Applying displacement control loading using Lagrange multipliers in the material non-linear finite element method

Question

Hi I am trying to implement a simple plasticity based finite element code. I am not clear how to set up displacement control applied through Lagrange multipliers. In case of a linear problem, I did the following to apply displcament control and it worked. If the energy balance was linear then the following equations would be obtained:

However, incase of non-linear problems, the K is K(u) and is non-linear. So the energy balance equation and its derivative will be like as follows

It feels wrong. How do I evaluate this? How are the matrix form of equations modified in the non-linear case? So how do the matrix equations get modified? I used the https://codecogs.com/latex/eqneditor.php to add images typed in latex. Let me know if its not clear.

Answer 1

This isn't the way plasticity was modeled when I did FEA for a living.

Plasticity is modeled as a non-linear stress-strain material relationship, not with displacements and Lagrange multipliers.

Non-linear problems are typically expressed in terms of linearized increments:

K(u)*du = F(t)
u(t+dt) = u(t) + du
F(t+dt) = F(t) + dF

1. Start with an initial guess for displacement.
2. Apply a load increment.
3. Solve iteratively for the displacement increment until it converges.
4. Update the displacement and load vectors,
5. Go to step 2 and repeat.