Ramberg–Osgood relationship

The Ramberg–Osgood equation was created to describe the non linear 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

${\displaystyle \varepsilon ={\frac {\sigma }{E}}+K\left({\frac {\sigma }{E}}\right)^{n}}$

here

${\displaystyle \varepsilon }$ is strain,

${\displaystyle \sigma }$ is stress,

${\displaystyle E}$ is Young's modulus, and

${\displaystyle K}$ and ${\displaystyle n}$ are constants that depend on the material being considered. In this form K and n are not the same as the constants commonly seen in the Hollomon equation.

The equation is essentially assuming the elastic strain portion of the stress-strain curve, ${\displaystyle \varepsilon _{e}}$, can be modeled with a line, while the plastic portion, ${\displaystyle \varepsilon _{p}}$, can be modeling with a power law. The elastic and plastic components are summed to find the total strain.

${\displaystyle \varepsilon =\varepsilon _{e}+\varepsilon _{p}}$

The first term on the right side, ${\displaystyle {\sigma }/{E}\,}$, is equal to the elastic part of the strain, while the second term, ${\displaystyle \ K({\sigma }/{E})^{n}}$, accounts for the plastic part, the parameters ${\displaystyle K}$ and ${\displaystyle n}$ describing the hardening behavior of the material. Introducing the yield strength of the material, ${\displaystyle \sigma _{0}}$, and defining a new parameter, ${\displaystyle \alpha }$, related to ${\displaystyle K}$ as ${\displaystyle \alpha =K({\sigma _{0}}/{E})^{n-1}\,}$, it is convenient to rewrite the term on the extreme right side as follows:

${\displaystyle \ K\left({\frac {\sigma }{E}}\right)^{n}=\alpha {\frac {\sigma }{E}}\left({\frac {\sigma }{\sigma _{0}}}\right)^{n-1}}$

Replacing in the first expression, the Ramberg–Osgood equation can be written as

${\displaystyle \varepsilon ={\frac {\sigma }{E}}+\alpha {\frac {\sigma }{E}}\left({\frac {\sigma }{\sigma _{0}}}\right)^{n-1}}$