Hi Marco, this topic has been addressed before. Please refer to

https://www.researchgate.net/post/How_can_i_calculate_the_equivalent_plastic_strain

Regards,

Nicolas

If you want to implement the intergration of plasticity equations yourself, you need to implement a return mapping algorithm in UMAT.

A good description of these algorithms is given here:

http://csm.mech.utah.edu/content/wp-content/uploads/2011/03/GoBagRadialReturn.pdf

A sample implementation of a return mapping algorithm as an UMAT for Abaqus is given in the book "Introduction to Computational Plasticity" by Dunne and Petrinic

https://global.oup.com/academic/product/introduction-to-computational-plasticity-9780198568261?cc=at&lang=en&

(you can download the code if you buy the book)

How I did it is described here:

https://vulcanhammer.net/2017/03/21/improved-methods-for-forward-and-inverse-solution-of-the-wave-equation-for-piles/

Anton Ishmurzin:

I was looking at how to do it mathematically rather than using UMAT in Abaqus.

from "Introduction to Computational Plasticity" by Dunne and Petrinic", Section 2.3.1. "The normality hypothesis of plasticity"

"The direction of the plastic strain increment (or equivalently, plastic strain rate) is given by \partial f / \partial \sigma (derivative of the yield function by stress tensor - A.I.), while the magnitude is determined, for the plastic strain rate, by lambda dot. This is called plastic multiplier (...)."

This book contains examples on how to find the plastic multiplier analytically, for example for the case of uniaxial loading with linear isotropic hardening.

I recommend you to read the Chapter "2. Continuum plasticity" of the aforementioned book, I think you will find your answers there.