Split-radix FFT algorithm

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 useful for metals that harden with plastic deformation (see strain hardening), showing a smooth elastic-plastic transition.

In its original form, the equation for strain (deformation) is

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

where

$\epsilon$ is strain,

$\sigma$ is stress,

$E$ is Young's modulus, and

$K$ and $n$ are constants that depend on the material being considered.

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

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

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

$\epsilon ={\frac {\sigma }{E}}+\alpha {\frac {\sigma _{0}}{E}}\left({\frac {\sigma }{\sigma _{0}}}\right)^{n}$

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 $\alpha \,$ and $n\,$. 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 $\alpha$ and $n$, the plastic strain remains negligible compared to the elastic strain. On the other hand, for stress levels higher than $\sigma _{0}$, plastic strain becomes progressively larger than elastic strain.

The value $\alpha {\frac {\sigma _{0}}{E}}$ can be seen as a yield offset, as shown in figure 1. This comes from the fact that $\epsilon =(1+\alpha ){{\sigma _{0}}/{E}}\,$, when $\sigma =\sigma _{0}\,$.

Accordingly (see Figure 1):

elastic strain at yield = ${{\sigma _{0}}/{E}}\,$

plastic strain at yield = $\alpha ({\sigma _{0}}/E)\,$ = yield offset

Commonly used values for $n\,$ are ~5 or greater, although more precise values are usually obtained by fitting of tensile (or compressive) experimental data. Values for $\alpha \,$ 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:

$\alpha {\frac {\sigma _{0}}{E}}=0,002$

Figure 1: Generic representation of the Stress-Strain curve by means of the Ramberg–Osgood equation. Strain corresponding to the yield point is the sum of the elastic and plastic components.