The Ramberg–Osgood equation was created to describe the non linear relationship between
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
and
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
—that is, the
stress–strain curve
In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and ...
—in materials near their
yield points. It is especially applicable to metals that ''harden'' with plastic deformation (see
work hardening
In materials science, work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context.
This strengt ...
), showing a ''smooth'' elastic-plastic transition. As it is a
phenomenological model
A phenomenological model is a scientific model that describes the empirical relationship of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. In other words, a phenomenological ...
, 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
[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]
/ref>
:
here
: is strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
,
: is stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
,
: is Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
, 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
:::
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
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one qua ...
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 , when .
Accordingly, (see Figure 1):
: ''elastic strain at yield'' =
: ''plastic strain at yield'' = = ''yield offset''
Commonly 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 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.
See also
* Viscoplasticity#Johnson–Cook flow stress model
References
{{DEFAULTSORT:Ramberg-Osgood relationship
Mechanics
Materials science