What's the broken diagonal line during stress-strain curves?

[Femme_physics]

So in my mechanics of materials class we were taught about stress-strain curves. I asked a couple of times for the meaning of the broken diagonal line on the graph but no one seemed to give me a logical answer so I decided to ask here.

 

[Studiot]

Good Morning FP,

The diagram you have posted is the stress/strain curve for a ductile material.

If you start at the origin (unstrained material) and the specimen bar out the stress in the bar increases from zero through point 1 to point 2 etc.

From zero to point 2 the graph is a straight line and the slope is called Young’s modulus.

As far as point 2, if we let go, the bar returns to its original length.Reducing the applied force, but not letting go completely reduces the extension and therefore the strain.
We can repeat the stretching and relaxation as often as we like and the bar always returns to its unstressed length.
This is called the hookean (after Hooke) or linear elastic region.

If we now stretch the bar beyond point 2
You will notice that the graph curves over after 2 has been reached.

The bar is still elastic in that if we let go it will shorten, but this time not to its original length. It will remain a bit longer than before.The extra length is called the ‘permanent set’

If we pull the bar right out as far as point 4 and then let go, the bar will adopt a new unloaded length 0.2% longer than the original.
The stress at which this happens is called the 0.2% proof stress (0.1% is perhaps more usual to quote).

The interesting thing about all this is that if we now pull the bar out again it the stress strain graph will now follow the dashed curve you are asking about, which has the same slope as the original linear region, so Young's Modulus is the same in this new region.
The linear elastic region will be longer than before.

Note that for a brittle material the graph is different.

Go well.

 

[Femme_physics]

Great reply! 4 questions though...

1) So I assume that 0.2% (or 0.1%) has some sort of special significance that it gets its own name (0.2% proof stress)...why, why 0.2% and not 0.3% or 0.4%?

2) Why is the 0.2% represented in the epsilon axis (strength applied) and not on the sigma axis (how the material is stretched)? Seems it should be opposite if anything.

3) Why is that dashed line representing the 0.2% necessarily diagonal?

The interesting thing about all this is that if we now pull the bar out again it the stress strain graph will now follow the dashed curve you are asking about, which has the same slope as the original linear region, so Young's Modulus is the same in this new region.
The linear elastic region will be longer than before.

4) The fact that the linear elastic region will be longer means that it is harder to stretch (takes more energy investment), right?

 

[Dadface]

Hello
1.My guess is that 0.1 percent or whatever value is quoted is an arbitary choice but convenient in that it makes it easy to compare the elastic /plastic behaviour of different metals
2.It refers to the permanent extension caused by overloading this corresponding to the quoted percentage stress.
3.The material still retains elasticity with any initial applied stretching forces increasing the interatomic separations.The interatomic forces remain pretty much the same as they were previously so the lines are parallel.When the metal goes into the plastic region atomic planes start to slip.
4.The work done in the linear regions to reach a certain extension is the same in both cases

 

[Studiot]

Strictly 0.2% proof stress is the stress which causes 0.2% permanent set. That is the intercept on the strain (epsilon) axis is 0.2%.

Note the dashed line is not really 'diagonal' it slopes parallel to the original elastic straight line, but displaced along the strain axis by the permanent set.

It is lost in the mists of time why we call it proof stress not proof strain.
The term comes from the very practical testing of sample of material. A series of samples are stretched (beyond any working stress) until 0.2% set is observed and the stress recorded as the 0.2% proof stress, thus proving the material for normal working loads.

As I said before 0.1% is more usual - You may never reach 0.4% as the sample might fail or other effects may intervene, before then.

In energy terms, the strain energy input is recovered on unloading up to the limit of proportionality (the point where the graph starts to deviate from a straight line).

Loading beyond this point uses some of the energy to create the permanent set.
At this point the material is still well capable of supporting the load, it's just that the response is no longer linear, ie stress is no longer proportional to strain. Then the mathematics becomes more hairy.
After a permanent set, if you unload and reload, the elastic slope is the same so energy per unit strain is the same so the energy to get to a given elastic strain point in the graph is the same. However the linear elastic line now extends further so the energy to get to this point will obviously be greater.

Hope this helps

 

[Femme_physics]

Ah...that clears it actually...both your replies :)

Thanks.

 

[stpmmaths]

Sorry for bumping old thread. I just want to ask how can we find the 0.2% permanent set? Is it 0.002 strain? Or 0.002 multiply by sth, and what's that sth?

Thanks.

 

[stpmmaths]

Any help?

 

[Studiot]

If we pull the bar right out as far as point 4 and then let go, the bar will adopt a new unloaded length 0.2% longer than the original.
The stress at which this happens is called the 0.2% proof stress

So yes, .2% = .002 strain units or 2 millistrain.

 

[stpmmaths]

So yes, .2% = .002 strain units or 2 millistrain.

Thanks :D