Non associated vs associated flow rule

[bondmatt]

I have a couple of items related to plasticity that I am not clear on.

Non associated vs associated flow rule. Non associated uses plastic potential function. Associated assumes a yield criterion in place of the plastic potential function. Is this a good description of the difference? Is there a more practical explanation?

Plastic vs elastic strain. If a tensile specimen is loaded to failure (ductile metal) is the elastic strain a constant once plastic deformation occurs? If a strain increment in the plastic region is considered would the strain increment be a plastic strain increment? Every description I come across shows loading, unloading, and reloading. In these examples if a strain increment in the plastic region is considered there is an elastic component that comes from the loading and unloading. It seems like the textbooks try to say that any increment in the plastic region has elastic and plastic components. Am I correct in assuming these elastic components are the result of loading and unloading? If the total strain is considered I am under the impression that strain has elastic and plastic strain components where the elastic strain is the strain at the point of yielding.

Thank you,
- Matt Bondy

 

[Studiot]

I am going to try to answer your query indirectly since I am guessing that you have just started what is known as 'limit state analysis' which is the formal application of plastic analysis to structural and materials engineering.

I think your difficulty arises because you are muddling up the actual experimental test curves for a specimen and the formal models we use for analysis (which are rather simpler).

So with the aid of the attached sketches here goes.

Sketch A
Shows a typical ductile test. Four points and four regions are clearly identified.
The limit of proportionality - region 1
The elastic limit - region 2
The yield point - region 3
The failure point - region 4

This graph is too complicated to use in practice so we simplify or idealise it to

Sketch B
The elastic-plastic model. Here we have a linear elastic region, followed by a linear plastic region. The plastic region can support some load in a strain recoverable manner, and may be the model which give rise to the comment in your textbook.
This model is still to difficult to use in practice so we further idealise it to

Sketch C
The elastic-perfectlyplastic model. We assume that the material is elastic up to the yield point and perefectly plastic thereafter. By perfectly plastic we mean that it can support no further load. So any load that appears beyond this is redistributed to other parts of the body or structure.
The limit criterion is that failure cannot occur until all the stressed material has become plastic by this load redistribution.

Sketch D

Here I have shown this principle applied to the tensile and compressive stress induced in a beam by bending.

Elastic only analysis is shown at 1.
Here the stresses increase with distance from the neutral axis to a maximum at the edges.
The integral of the stresses matches the loading.
This is known as a triangular stress block.

If however the increase in stress takes the stress beyond the yield point we get the situation in 2.
The triangle is truncated a shown, but the section has not failed, even though some of the material has yielded.
The integral of the stresses still matches the loads as the extra stress is redistributed to the material closer to the axis.

This is the fundamental principle of plastic analysis.

The redistribution continues until the situation at 3 occurs.

At this point the section forms what is known as a plasic hinge and fails.

At this point the integral of the stresses still equals the loads, but the effect is failure.
The integral of the stresses cannot increase beyond this.

So if you like, the theory says that the stress at any point cannot exceed the yield stress, but that strength may be 'borrowed' from another point in the body where the stress has not reached yield.

hope this helps

 

[bondmatt]

Thank you for the response. I should have added a figure, I was not very clear in what I was asking. In figure I have attached (plasticity.jpg) is the labeling of the strain correct?

I have less confidence in the second figure. Textbooks always show unloading and reloading when discussing plastic strain increments. What if there is no unloading and just steady loading to failure?

 

[Studiot]

Give me a clue.

Where are you coming from?
Did you understand what I said?
Textbooks probably try to get several things onto one diagram. I doubt they say you have to unload.
I expect they want to demonstrate 'permanent set'.