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The Ramberg–Osgood equation was created to describe the nonlinear relationship between stress and strain—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 a ...
—in materials near their yield points. It is especially applicable to metals that ''harden'' with plastic deformation (see
work hardening Work hardening, also known as strain hardening, is the process by which a material's load-bearing capacity (strength) increases during plastic (permanent) deformation. This characteristic is what sets ductile materials apart from brittle materi ...
), 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.


Functional Relationship

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, : \sigma is stress, : E is
Young's modulus Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
, and : K and n are constants that depend on the material being considered. In this form, and 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 modeled 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 relative change in the other quantity proportional to the change raised to a constant exponent: one quantity var ...
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 Parameterizations

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)^


Alternative Models

The Ramberg-Osgood model provides an explicit formula for obtaining strain \varepsilon from stress \sigma, but in general an iterative solve must be performed for the inverse relation from strain to stress. This can be computationally demanding, and is not well suited for applications like
Finite element analysis Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of structural ...
where the inverse mapping from strain to stress is generally required. For this reason, several alternative curves have become common in these contexts. One such example is the curve proposed by \bar(\bar)=b \bar+\frac where \bar=\sigma / \sigma_y, \bar=\varepsilon / \varepsilon_y,\left(\sigma_y, \varepsilon_y\right) is the yield point, b is the strain hardening parameter, and the parameter r influences the shape of the transition curve and takes account of the Bauschinger effect.


See also

* Viscoplasticity#Johnson–Cook flow stress model


References

{{DEFAULTSORT:Ramberg-Osgood relationship Mechanics Materials science