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Mixed hardening Amstrong-Frederick and Ludwik J2 plasticity model VUMAT implementation

bitamendi's picture

Hi, I’m trying to implement a mixed hardening J2 plasticity model. The idea is to use the Ludwik law to represent the isotropic hardening and the Amstrong-Frederick law for the kinematic hardening, both combine in a J2 classic von Mises model.


I need some advice for the return mapping algorithm.


Once that I have check that the elastic trial state is not plastically admissible I have to solve a three equation system, where the first two are a tensor equations and the third one is the J2 yield function equation.


How I have to do to solve the system? Is possible to use an New-Raph algorithm?


If I use a New-Raph algorithm, after the first guess of null plastic multiplier, the idea is to find the derivative of the yield function with respect to the plastic multiplier to find the new plastic multiplier guess. And follow this iterative scheme until be under the tolerance, or not?

I have tried to find the derivative, but it is very tedious, and as I’m not sure if this is the correct way to solve the problem…


Thanks a lot in advance,



   Use secant method instead of NR to avoid differentiation. But NR has quadriatic convergence when compared with secant method.




For stress integration scheme I suggest you to look at the following article from int journal of plasticity

volume 26 (2010) 288- 309

A non - associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming.



bitamendi's picture

HI Rohith,


Thanks, I will try first with the secant, and if the convergence is too slow I will try with the paper.


Thanks again

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