Uniaxial Return Mapping Algorithm

August 21, 2001

Michael H. Scott

PEER, University of California, Berkeley

This document outlines the return mapping algorithm for a rate-independent uniaxial material model with combined isotropic and kinematic hardening. The algorithm and its derivation are given in Simo and Hughes [@Simo:1998].

The material parameters are the elastic modulus, \(E\), yield stress, \(\sigma_y\), isotropic hardening modulus, \(H_{iso}\), and kinematic hardening modulus, \(H_{kin}\). Path dependence is tracked by the plastic strain, \(\varepsilon^p\), internal hardening variable, \(\alpha\), and back stress, \(\kappa\).

Model Description

The model assumes an elastic stress-strain relationship with elastic modulus \(E\). The onset of plastic flow occurs upon yielding, after which the elastoplastic tangent is given by \(\frac{E(H_{iso}+H_{kin})}{E+H_{iso}+H_{kin}}\), as shown in figure [fig:StressStrain].

Isotropic hardening can be thought of as an “expansion” of the elastic region. The internal hardening variable, \(\alpha\), tracks the growth of the elastic region. Kinematic hardening corresponds to a “translation” of the elastic region. The back stress, \(\kappa\), is the center of the elastic region. When there is no kinematic hardening, \(\kappa\) is zero. These two hardening rules are shown in figure [fig:HardeningBehavior], and they can be combined to give mixed hardening behavior.

Continuum Equations

1. Elastic stress-strain relationship \[\sigma = E \left( \varepsilon-\varepsilon^p \right)\]

2. Flow rule \[\dot{\varepsilon}^p = \gamma \mbox{sign}\left( \sigma-\kappa \right)\]

3. Isotropic and kinematic hardening laws $$

   \[\begin{aligned} \dot{\alpha} &= \gamma \ \dot{\kappa} &= \gamma H_{kin} \mbox{sign}\left( \sigma-\kappa \right)\end{aligned}\]

   $$

4. Yield condition \[f(\sigma, \kappa, \alpha) = \left| \sigma-\kappa \right| - \left( \sigma_y + H_{iso}\alpha \right) \leq 0\]

5. Kuhn-Tucker complementary conditions $$

   \[\begin{aligned} \gamma &\geq 0 \ f(\sigma, \kappa, \alpha) &\leq 0 \ \gamma f(\sigma, \kappa, \alpha) &= 0\end{aligned}\]

   $$

6. Consistency condition \[\gamma \dot{f}(\sigma, \kappa, \alpha) = 0 \:\:\:\:\: \mbox{(if $f(\sigma, \kappa, \alpha)$ = 0)}\]

Return Mapping Algorithm

1. Committed state at time \(t_n\) \[\left\{ \varepsilon_n^p, \alpha_n, \kappa_n \right\}\]

2. Given trial strain at time \(t_{n+1}\), \[\varepsilon_{n+1} = \varepsilon_n + \Delta\varepsilon_n,\]

   determine the corresponding stress, \(\sigma_{n+1}\), and tangent, \(D_{n+1}\); proceed to step 3.

3. Compute trial stress and test for plastic loading $$

   \[\begin{aligned} \sigma_{n+1}^{trial} &= E \left( \varepsilon_{n+1} - \varepsilon_n^p \right) \ \xi_{n+1}^{trial} &= \sigma_{n+1}^{trial} - \kappa_n \ f_{n+1}^{trial} &= \left| \xi_{n+1}^{trial} \right| - \left( \sigma_y + H_{iso}\alpha_n \right)\end{aligned}\]

   $$

   If \(f_{n+1} \leq 0\), this is an elastic step; set \(\sigma_{n+1} = \sigma_{n+1}^{trial}, D_{n+1} = E\), and exit. Else, this is a plastic step; proceed to step 4.

4. Return mapping $$

   \[\begin{aligned} \Delta\gamma &= \frac{f_{n+1}^{trial}}{E+H_{iso}+H_{kin}} \ \sigma_{n+1} &= \sigma_{n+1}^{trial} - \Delta\gamma E \mbox{sign}(\xi_{n+1}^{trial}) \ \varepsilon_{n+1}^p &= \varepsilon_n^p + \Delta\gamma \mbox{sign}(\xi_{n+1}^{trial}) \ \kappa_{n+1} &= \kappa_n + \Delta\gamma H_{kin} \mbox{sign}(\xi_{n+1}^{trial}) \ \alpha_{n+1} &= \alpha_n + \Delta\gamma \ D_{n+1} &= \frac{E(H_{iso}+H_{kin})}{E+H_{iso}+H_{kin}}\end{aligned}\]

   $$

99 J.C. Simo and T.J.R. Hughes, Computational Inelasticity. Springer-Verlag, 1998.