RambergOsgood

Ramberg-Osgood Steel Model

Syntax

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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

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

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1 2 3	`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`

example one

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1 2 3	`material RambergOsgood 1 100.0 8.0 1 10.0 materialTest1D 1 1E-2 20 40 40 40 40 40 40 exit`

example two

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1 2 3	`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`

example three