# Tutorial: Radial Return

A tutorial on the underlying theory of projecting a stress state back to the plastic yield surface.

You may download the document here, or view just view the graph.

## 2 thoughts on “Tutorial: Radial Return”

1. Phu Nguyen says:

Dear Prof Brannon,

Thanks a lot for sharing this. I have one question though. The presented algorithms look much simpler than the commonly used implicit algorithms which require second derivatives of the yield functions. My question is are they equivalent?

Bests,
Phu

• They are not equivalent. The provided algorithm is an effective and simple choice for high-rate dynamics codes, which are usually based on explicit solvers of the momentum equation. I’ve always been mystified that so much effort has gone into making plasticity algorithms second-order accurate in the case of a constant strain rate. If such a model is put into an implicit-solver host code, the results will still be only first-order accurate unless strain acceleration (i.e., the increment of the strain increment over the step) is included.

Return algorithms are sometimes called backward-Euler methods (because they returns the elastic trial stress at the end of the time step), but that’s not rigorously accurate because a truly implicit method would need to account for changes in elastic moduli as well as rotation of the stress eigenvectors (the eigenvectors of the trial elastic stress are only a first-order accurate approximation of the exact updated stress). For a challenging verification test, see Example 1 in http://www.mech.utah.edu/~brannon/pubs/7-2009BrannonLeelavanichkul-IJF.pdf, which involves plastic loading for which the stress does not move AT ALL in principal stress space, but it does change by having spinning eigenvectors (this problem is somewhat like plastically rolling out a piece of clay while simultaneously imposing superimposed rotation in the opposite direction so that the problem is always a pure stretch for which the stretch directions rotate).

A word of caution about implicit algorithms: don’t just believe the authors when they make assertions about the accuracy and convergence of their model. Always check it yourself against ANALYTICAL solutions like the ones in the article quoted above. There is a rampant tendency for people to check their model against an answer using lots of timesteps, but such an approach only verifies convergence — not accuracy. Plasticity models are notorious for converging to wrong answers (which do not satisfy the stated governing equations). For models that have pressure dependence of strength, the most common source of convergence to the wrong answer is lack of awareness that closest point algorithms are not supposed to go to the closest point in stress space by an ordinary notion of distance. This point is made in the text part of the Simo Hughes Computational Inelasticity book, but their picture misleadingly shows a closest point return. Always check against basic verification tests with analytical solutions! Don’t forget to test frame indifference too.