Ramberg–Osgood relationship explained
The Ramberg–Osgood equation was created to describe the nonlinear relationship between stress and strain—that is, the stress–strain curve—in materials near their yield points. It is especially applicable to metals that harden with plastic deformation (see work hardening), showing a smooth elastic-plastic transition. As it is a phenomenological model, checking the fit of the model with actual experimental data for the particular material of interest is essential.
In its original form, the equation for strain (deformation) is[1]
\varepsilon=
+K\left(
\right)n
here
is
strain,
is
stress,
is
Young's modulus, and
and
are constants that depend on the material being considered. In this form, and are not the same as the constants commonly seen in the
Hollomon equation.
[2] The equation is essentially assuming the elastic strain portion of the stress-strain curve,
, can be modeled with a line, while the plastic portion,
, can be modeled with a power law. The elastic and plastic components are summed to find the total strain.
\varepsilon=\varepsilone+\varepsilonp
The first term on the right side,
, is equal to the elastic part of the strain, while the second term,
, accounts for the plastic part, the parameters
and
describing the
hardening behavior of the material. Introducing the
yield strength of the material,
, and defining a new parameter,
, related to
as
, it is convenient to rewrite the term on the extreme right side as follows:
K\left(
\right)n=\alpha
\right)n-1
Replacing in the first expression, the Ramberg–Osgood equation can be written as
\varepsilon=
+\alpha
\right)n-1
Hardening behavior and yield offset
In the last form of the Ramberg–Osgood model, the hardening behavior of the material depends on the material constants
and
. Due to the
power-law relationship between stress and plastic strain, the Ramberg–Osgood model implies that plastic strain is present even for very low levels of stress. Nevertheless, for low applied stresses and for the commonly used values of the material constants
and
, the plastic strain remains negligible compared to the elastic strain. On the other hand, for stress levels higher than
, plastic strain becomes progressively larger than elastic strain.
The value
can be seen as a
yield offset, as shown in figure 1. This comes from the fact that
\varepsilon=(1+\alpha){{\sigma0}/{E}}
, when
.
Accordingly, (see Figure 1):
elastic strain at yield =
plastic strain at yield =
=
yield offsetCommonly used values for
are ~5 or greater, although more precise values are usually obtained by fitting of tensile (or compressive) experimental data. Values for
can also be found by means of fitting to experimental data, although for some materials, it can be fixed in order to have the
yield offset equal to the accepted value of strain of 0.2%, which means:
Alternative Formulations
Several slightly different alternative formulations of the Ramberg-Osgood equation can be found. As the models are purely empirical, it is often useful to try different models and check which has the best fit with the chosen material.
The Ramberg-Osgood equation can also be expressed using the Hollomon parameters[3] where
is the strength coefficient (Pa) and
is the strain hardening coefficient (no units).
[4] \varepsilon=
+\left(
\right)1/n
Alternatively, if the yield stress,
, is assumed to be at the 0.2% offset strain, the following relationship can be derived.
[5] Note that
is again as defined in the original Ramberg-Osgood equation and is the inverse of the
Hollomon's strain hardening coefficient.
\varepsilon=
+0.002\left(
\right)n
See also
- Viscoplasticity#Johnson–Cook flow stress model
References
- Ramberg, W., & Osgood, W. R. (1943). Description of stress–strain curves by three parameters. Technical Note No. 902, National Advisory Committee For Aeronautics, Washington DC. https://ntrs.nasa.gov/api/citations/19930081614/downloads/19930081614.pdf
- Web site: Mechanical Properties of Materials MechaniCalc. mechanicalc.com. 2020-05-27.
- Hollomon. J. R.. 1945. Tensile Deformation. Transactions of AIME. 162. 268–277.
- Gadamchetty. Geethanjali. Pandey. Abhijeet. Gawture. Majnoo. 2016-01-05. On Practical Implementation of the Ramberg-Osgood Model for FE Simulation. SAE International Journal of Materials and Manufacturing. 9. 1. 200–205. 10.4271/2015-01-9086. 1946-3987.
- Book: Hill, H. N.. Determination of stress-strain relations from "offset" yield strength values. 1944. National Advisory Committee for Aeronautics. 647978489.
