Understanding Stress Strain Graphs: Exploring Yield Point and Material Behavior

[prasannapakkiam]

Just a question. The internet is mixed with this. But in a Stress Strain Graph; after the YIELD POINT, does the material experience more strain for a lesser stress? I.e. does it slightly curve down before going up?

 

[AlephZero]

It depends on the material.

It also depends how you define "stress" and "strain" for large deformations - Green or logarithmic strain, Piola-Kirchoff or Cauchy stress, etc. See http://www.shodor.org/~jingersoll/weave/tutorial/node3.html

 

[billiards]

lesser stress? do you mean if you reverse the loading so you are actually reducing the stress, or are you talking (as i imagine,) about a lesser increase in (differential) stress per % increase in strain?

If you're talking about the second, after the yield point, E (the young's modulus) goes down, at least for all the materials that I have studied (i.e. rocks). In other words, the material experiences the same amount of strain for a lesser increase in differential stress (I say same amount because it is common for these experiments to be conducted at constant strain rates).

 

[prasannapakkiam]

Okay. But why is it that the curve goes down? I mean in a microscopic view, what is the reason?

 

[billiards]

I'll talk about rocks only, although you might find that these concepts cross over to other materials. Initially the compression is due to compaction, cracks in the rock which are perpendicular to the max principal stress close up. Once these start closing up the curve goes up because the material is gretting stiffer. Then the rock will deform along a straight line, this is hookean deformation a bit like a spring. Then dilatancy will begin to dominate, cracks open up parallel to the maximum principal stress axis, the rock volume actually expands, this expansion is accommodated perpendicular to max stress. Along the max stress axis the rock will shorten, it becomes less stiff and under goes more strain per unit differential stress. Eventually the thing will reach max stress, if the rock is leading to shear failure the curve will roll over as the cracks align to form a fault plane. Then you get a sudden stress drop, this is associated with failure rather like an earthquake.

 

[Astronuc]

Okay. But why is it that the curve goes down? I mean in a microscopic view, what is the reason?

Atoms move in the atomic/crystal lattice, and that causes some permanent or plastic deformation.

In the straight portion of the stress-strain curve, there is a linear relationship (Hooke's law) between stress and strain, and the slope (proportionality constant) is the elastic modulus (also called Young's modulus). Also placing a material in tension imposes a mechanical energy into the material - and there is another parameter called strain energy density which is related to the mechanical energy.

http://em-ntserver.unl.edu/Mechanics-Pages/Marina-Gandelsman/strain.html

When a material begins to yield, atoms have started to move within the lattice. Now it is complicated in a polycrystalline material because some grains will permanently deform before others, because stresses are not uniformly distributed, because grains have different sizes and orientations, not to mention composition. There is even grain boundary slippage.

The differences in orientation also mean that some grains experience mostly tension while others may experience more shear, and with localized shear, some grains may actually experience compression.

The tensile test measures a bulk (average) material behavior and one must keep that in mind when applying the results to simulations of materials on microscopic or nanoscopic (atomistic) level.

 

[rdt2]

Just a question. The internet is mixed with this. But in a Stress Strain Graph; after the YIELD POINT, does the material experience more strain for a lesser stress? I.e. does it slightly curve down before going up?

It does for ferrous metals. I think the simplest explanation is that it's easier to keep dislocations moving (at the 'lower yield stress') than it is to start them moving (at the 'upper yield stress'). Like the difference between static and dynamic friction coefficients. See:

http://www.mssmat.ecp.fr/IMG/pdf/385_bel.pdf

for details.