### Table of Contents

# Non-proportional stress 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

$\sigma_{xx}=(+400\text{ MPa}, -300\text{ MPa}),$ 2.5 cycles. Preload to $\sigma_{xx}=300\text{ MPa}$ 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.

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

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

$+400$ | $0$ | $144.067$ | $76.129$ | $0$ | $6.3040$ | $0$ | $4.3992$ |

$-300$ | $0$ | $141.627$ | $-57.350$ | $0$ | $-0.0842$ | $0$ | $7.4541$ |

$+400$ | $0$ | $136.528$ | $78.246$ | $0$ | $9.3069$ | $0$ | $13.5119$ |

$-300$ | $0$ | $134.933$ | $-59.953$ | $0$ | $2.7119$ | $0$ | $16.7736$ |

$+400$ | $0$ | $131.387$ | $79.669$ | $0$ | $13.0175$ | $0$ | $23.7459$ |

$+300$ | $0$ | $131.387$ | $79.669$ | $0$ | $12.5413$ | $0$ | $23.7459$ |

$+300$ | $+150$ | $129.67$ | $60.92$ | $44.58$ | $15.42$ | $2.179$ | $27.84$ |

$+300$ | $-150$ | $127.91$ | $61.13$ | $-45.00$ | $20.48$ | $-1.382$ | $34.28$ |

$+300$ | $+150$ | $126.75$ | $61.27$ | $45.28$ | $25.89$ | $2.283$ | $41.14$ |

$+300$ | $-150$ | $126.02$ | $61.36$ | $-45.46$ | $31.58$ | $-1.465$ | $48.34$ |

$+300$ | $+150$ | $125.58$ | $61.41$ | $45.56$ | $37.45$ | $2.343$ | $55.78$ |

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 softening of the isotropic part $k$ is shown below.