RambergOsgood

Ramberg-Osgood Steel Model

Syntax

material RambergOsgood (1) (2) (3) [4] [5] [6]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, initial yield stress
# [4] double, offset alpha, default: 1.0
# [5] double, n, default: 4.0
# [6] double, density, default: 0.0

History Variable Layout

location	value
initialize_history(0)	load_sign
initialize_history(1)	reverse_strain
initialize_history(2)	reverse_stress
initialize_history(3)	previous_reverse_strain
initialize_history(4)	previous_reverse_stress

Remarks

1. Local iterations are required to obtain the stress value.

Theory

The Ramberg-Osgood relationship is defined as

\[ \varepsilon=\dfrac{\sigma}{E}+\alpha\dfrac{\sigma}{E}(\dfrac{\sigma}{\sigma_0})^{n-1} \]

where \(\alpha\) is the offset and \(n\) is the material constant controls hardening. Noting that \(\varepsilon=\varepsilon_e+\varepsilon_p=\dfrac{\sigma}{E}+\varepsilon_p\), hence

\[ \dfrac{\sigma}{E}+\varepsilon_p=\dfrac{\sigma}{E}+\alpha\dfrac{\sigma}{E}(\dfrac{\sigma}{\sigma_0})^{n-1} \]

so

\[ \varepsilon_p=\alpha\dfrac{\sigma}{E}(\dfrac{\sigma}{\sigma_0})^{n-1}. \]

At the yield stress, viz., \(\sigma=\sigma_0\), then

\[ \varepsilon_p=\alpha\varepsilon_e. \]

So the offset \(\alpha\) indicates the magnitude of plastic strain at yield stress.

The cyclic response uses the difference between current reverse stress and previous reverse stress as "yield stress".

Examples

material RambergOsgood 1 100.0 8.0 1 10.0
materialTest1D 1 1E-2 20 20 30 20 30 20 30 20 30 20 30 20
exit

material RambergOsgood 1 100.0 8.0 1 10.0
materialTest1D 1 1E-2 20 40 40 40 40 40 40
exit

material RambergOsgood 1 100.0 8.0 1 10.0
materialTest1D 1 1E-2 20 20 30 15 20 40 15 25 15 20 30
exit