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

here
is strain,
is stress,
is Young's modulus, and
and 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, , can be modeled with a line, while the plastic portion, , can be modeling with a power law. The elastic and plastic components are summed to find the total strain. 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:

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

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 where is the strength coefficient (Pa) and is the strain hardening coefficient (no units). Alternatively, if the yield stress, , is assumed to be at the 0.2% offset strain, the following relationship can be derived. Note that is again as defined in the original Ramberg Osgood equation and is the inverse of the Hollomon's strain hardening coefficient.