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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: enter image description here

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

enter image description here

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.

Bruce Lee Jun Fan
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1 Answers1

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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.
duffymo
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  • I know this. Iam not interested in this. I want to use lagrange multipliers – Bruce Lee Jun Fan Jul 05 '21 at 05:17
  • Your physics aren’t correct. Dislocation motion in metals is a material effect, not a boundary condition. – duffymo Jul 05 '21 at 15:52
  • Hi my interst is to apply Lagrange multipliers for a non-linear problem. Applying Lagrange multipliers in a non-linear setup. I am not solving dislocations. What I mean by boundary condition is merely a displacement control loading. Is that clear? – Bruce Lee Jun Fan Jul 07 '21 at 06:51
  • Your physics are incorrect. Loading is not material behavior. Metal plasticity is dislocation motion. Is that clear? Perhaps the problem you really have in mind happens to be non-linear in some way but you're using the word "plasticity" in the wrong context. – duffymo Jul 07 '21 at 11:06
  • Yes loading is not nonlinear. Neither loading is a material parameter. I am only trying to solve a non-linear problem. I am not bothered about how the non-linearity is induced. But plasticity could be one way I thought. As I said, I am only interested in arriving at equations for the Lagrange multiplier for any non-linear setup? Have I clarified? – Bruce Lee Jun Fan Jul 08 '21 at 13:40