When Does the Nonlinear Stress-Strain Equation Apply?

[tandoorichicken]

My statics text says the following:

The relationship between axial stress and strain can be represented by the equation
$$\sigma = E\epsilon$$

"At higher levels of stress, the following nonlinear equation may be a better fit to describe the correlation between axial stress and strain:
$$\sigma = E e^{\epsilon-1}$$
"

Where $\sigma$ is force per unit area, $\epsilon$ is axial strain and E is Young's modulus.

Out of curiosity, at what level of stress does the second equation begin to better represent the situation than the first?

 

[Cyrus]

More than likely in the non-linear elastic region.

 

[Gamma]

First equation is valid below the material's elastic limit. Below the elastic limit, when the stress is removed the material comes back to it's original length. If you apply sufficiently large stress, then you can pass the elastic limit. Meaning, that upon the removal of the stress the object does not return to its original length.

 

[Pyrrhus]

I haven't seen that equation before, but for non-linear behaviour the equations most used are Ramberg-Osgood, especially for materials that have a gradual transition between the elastic linear region to the plasticity. Some examples are aluminium-alloys. Mostly high-strength alloys in aircrafts.

 

[FredGarvin]

More than likely in the non-linear elastic region.

You mean the non-linear plastic region.