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

$$\epsilon ={\displaystyle \frac{\sigma }{E}}+\alpha {\displaystyle \frac{\sigma }{E}}({\displaystyle \frac{\sigma }{{\sigma }_0}}{)}^{n-1}$$

where $\alpha$ is the offset and $n$ is the material constant controls hardening. Noting that $\epsilon ={\epsilon }_e+{\epsilon }_p={\displaystyle \frac{\sigma }{E}}+{\epsilon }_p$, hence

$${\displaystyle \frac{\sigma }{E}}+{\epsilon }_p={\displaystyle \frac{\sigma }{E}}+\alpha {\displaystyle \frac{\sigma }{E}}({\displaystyle \frac{\sigma }{{\sigma }_0}}{)}^{n-1}$$

so

$${\epsilon }_p=\alpha {\displaystyle \frac{\sigma }{E}}({\displaystyle \frac{\sigma }{{\sigma }_0}}{)}^{n-1}.$$

At the yield stress, viz., $\sigma ={\sigma }_0$, then

$${\epsilon }_p=\alpha {\epsilon }_e.$$

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