Ramberg–Osgood relationship
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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
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—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) isRamberg, 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> :\varepsilon = \frac + K \left(\frac \right)^ here : \varepsilon 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 ...
, : \sigma 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 ...
, : E 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 : K and 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, \varepsilon_e, can be modeled with a line, while the plastic portion, \varepsilon_p, can be modeling with a power law. The elastic and plastic components are summed to find the total strain. \varepsilon = \varepsilon_e + \varepsilon_p The first term on the right side, /\,, is equal to the elastic part of the strain, while the second term, \ K(/)^, 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 (/)^\,, it is convenient to rewrite the term on the extreme right side as follows: :::\ K \left(\frac \right)^n = \alpha \frac \left(\frac \right)^ Replacing in the first expression, the Ramberg–Osgood equation can be written as ::: \varepsilon = \frac + \alpha \frac \left(\frac \right)^


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 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 \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 can be seen as a ''yield offset'', as shown in figure 1. This comes from the fact that \varepsilon = (1+\alpha)\,, when \sigma = \sigma_0\,. Accordingly, (see Figure 1): : ''elastic strain at yield'' = \, : ''plastic strain at yield'' = \alpha(/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 = 0.002


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 K is the strength coefficient (Pa) and n is the strain hardening coefficient (no units). \varepsilon = \frac + \left(\frac\right)^ Alternatively, if the yield stress, \sigma_y, is assumed to be at the 0.2% offset strain, the following relationship can be derived. Note that n is again as defined in the original Ramberg-Osgood equation and is the inverse of the Hollomon's strain hardening coefficient. \varepsilon = \frac + 0.002\left(\frac\right)^


See also

* Viscoplasticity#Johnson–Cook flow stress model


References

{{DEFAULTSORT:Ramberg-Osgood relationship Mechanics Materials science