# 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
```

![example one](https://4006314410-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MQ5rMqEBbzA9NgGETGX%2Fuploads%2Fgit-blob-1973709faad11027919fbbbd5b5ce750da7a4228%2FRambergOsgood.EX1.svg?alt=media)

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

![example two](https://4006314410-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MQ5rMqEBbzA9NgGETGX%2Fuploads%2Fgit-blob-c1a68d144a5e04a7955f9eb796fc192deb2e475b%2FRambergOsgood.EX2.svg?alt=media)

```
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](https://4006314410-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MQ5rMqEBbzA9NgGETGX%2Fuploads%2Fgit-blob-ab0b3b17ce72dfb4621bb11b8dc8a5162abc2b12%2FRambergOsgood.EX3.svg?alt=media)