### Table of Contents

# Non-proportional stress-strain ratcheting

##### by Dr. René Marek

## Problem description

Consider the rod shown subjected to cyclic non-proportional loading. Compute elastic-plastic response using directional-distortional hardening.

## Material properties

$E=2.1\times10^5\text{ MPa},$ $\nu=0.3.$ Feigenbaum-Dafalias directional-distortional hardening model, type $\alpha$ with constant $c.$

$k_0\text{ [MPa]}$ | $150$ |
---|---|

$\kappa_1\text{ [MPa]}$ | $10~000$ |

$\kappa_2\text{ [1/MPa]}$ | $0.008$ |

$a_1\text{ [MPa]}$ | $50~000$ |

$a_2\text{ [1/MPa]}$ | $0.01$ |

$c\text{ [1/MPa]}$ | $0.008$ |

## Support

Clamped at $x=0, y=0.$ Sliding at $y=0.$ Statically determinate.

## Loading

$u_y|_{y=20\text{ mm}}=(+20\ \mu\text{m}, -4\ \mu\text{m}),$ 2.5 cycles. Preload to $u_y|y_{y=20\text{ mm}}=18\ \mu\text{m}$ and hold. $\sigma_{xy}=\pm150\text{ MPa},$ 2.5 cycles.

## Solution

Analytical solution of the proportional part and precise numerical solution of the second non-proportional part with $\mathtt{NSUB}=1000$ and $\mathtt{NINT}=5000$ are shown below.

$\varepsilon_{11}\times10^3$ | $\sigma_{12}$ | $k$ | $\alpha_{11}$ | $\alpha_{12}$ | $\sigma_{11}$ | $\varepsilon_{12}\times10^3$ | $\varepsilon_p\times10^3$ |
---|---|---|---|---|---|---|---|

$0$ | $0$ | $150$ | $0$ | $0$ | $0$ | $0$ | $0$ |

$+50$ | $0$ | $145.807$ | $70.480$ | $0$ | $367.834$ | $0$ | $3.2484$ |

$-20$ | $0$ | $142.436$ | $-65.377$ | $0$ | $-335.642$ | $0$ | $6.8985$ |

$+50$ | $0$ | $139.447$ | $67.179$ | $0$ | $339.292$ | $0$ | $10.6846$ |

$-20$ | $0$ | $136.976$ | $-67.052$ | $0$ | $-334.450$ | $0$ | $14.4763$ |

$+50$ | $0$ | $134.904$ | $67.356$ | $0$ | $332.374$ | $0$ | $18.3009$ |

$+45$ | $0$ | $134.904$ | $67.356$ | $0$ | $227.374$ | $0$ | $18.3009$ |

$+45$ | $+150$ | $134.13$ | $39.95$ | $4.07$ | $142.24$ | $1.545$ | $19.783$ |

$+45$ | $-150$ | $133.23$ | $18.53$ | $-4.24$ | $61.17$ | $-1.280$ | $22.061$ |

$+45$ | $+150$ | $132.43$ | $8.16$ | $4.31$ | $26.52$ | $1.573$ | $24.369$ |

$+45$ | $-150$ | $131.70$ | $3.54$ | $-4.34$ | $11.44$ | $-1.293$ | $29.700$ |

$+45$ | $+150$ | $131.04$ | $1.52$ | $4.37$ | $4.90$ | $1.582$ | $29.050$ |

Loading path and subsequent yield surfaces on the $\sigma$–$\tau$ diagram are shown below.

Loading path and subsequent yield surfaces on the $\varepsilon$–$\gamma$ diagram are shown below.

Stress-strain characteristics with visible relaxation of axial tension during shear is shown below.