Hi,

Never,

Joao Fonseca has explained:

Necking occurs in metals tested in tension when the rate of work hardening is lower than the flow stress. This corresponds to the point of highest load in the load displacement curve, well before failure and any cracks have appeared in the material.

To understand why this happens, we need to understand 2 main ideas. The first one is that metals usually strengthen when they are deformed. This is called hard-working. The second is that when metals deform plastically, the change in volume is negligible. This means that the cross-sectional area decreases as the sample elongates. Since the stress on the sample is force divided by area, the stress on the sample increases as the area decreases. Here’s the clincher: as long as the rate of change in stress due to a change in area is lower that the rate of increase in strength because of hard-working, a metal sample will elongate uniformly and will not neck.

The point at which this happens is notoriously difficult to define in terms of strain but it can be more easily derived in terms of stress. This necking criterion, which states that necking will start when the rate of hardening equals the true flow stress, is called Considère’s criterion.

And for better understanding, please refer to below paper:

Uniaxial True Stress-Strain after Necking

Bests,

The answer is no.

For necking, the condition is dP=0. This is called Considere's condition. During plastic deformation, that is, well before, during and after necking till fracture, the volume change is zero; thatt is, dV=0. From, these two conditions, we arrive at the relationship that during necking, the rate of change of stree with respect to strain (of course, all true values) is numerically equal to the stress.

Most of the answers provided here are applicable for describing the onset of necking. However, the original poster has clearly mentioned that he wants to know what happens during necking. It is indeed true that dV=0 is valid only for the onset of necking and not when the specimen is undergoing necking.

The straightforward, single line answer to your question, when it is 'undergoing' necking, is YES. In most materials, the volume of the material in the neck increases as the applied load is increased. This is because the necked section experiences a tri-axial state of stress during necking, which triggers void formation as a precursor to failure (Note that in most materials, necking precedes failure). The formation of voids increases the volume of the material locally and the condition, dV = 0, is no longer valid.

Detailed explanation:

The formation of a neck in the specimen completely changes the mechanics of the test. The specimen is no longer under a uniaxial state of stress, even though the applied load is still along the same axis. To visualize this, consider the example of a circumferentially notched tensile specimen. On loading, the notched section, owing to its smaller cross section area, experiences higher stress and hence will have a tendency to undergo more lateral contraction than the rest of the specimen. However, the rest of the specimen will try to constrain deformation in the lateral direction. Therefore unnotched ligament experiences a tri-axial state of stress.

The example is analogous to a specimen undergoing necking. Typically, the tri-axial stress acts on local flaws in the necked section and opens them up. This leads to volume expansion during necking. Hope this helps.

Hello!

The answer is YES, definitely! Even though in theoretical models it is usually neglected. And the more strain in the neck region is, the more volume changes. I have the test results of measuring dencity of samples cutted from the neck of mild steel specimen, showing that the dencity at the neck region where local strain is 40% is 7.2g/sm2, while the dencity of original steel is7.8g/sm2. So, at high level of strain the volume change can be significant.

I am very interested in the effects of necking on mechanical property tests. Therefore, your question is very interesting to me.

I agree with all above answers, although I would perhaps hedge my bets and simply say that volume "often" increases during necking for the reasons given above.

Sometimes necking could occur without voids such as when shear dominates. The only other point I would make is that when we model material behaviour we have to make many assumptions and the assumption dV=0 is a very useful assumption and in many cases can be made and still allow good reliable predictions to be made. Indeed the assumption of dV=0 can really help to simplify models.

Hi Michael,

You do have a point regarding shear flow dominating up to the point of failure. In such a condition, the specimen would neck to a point and fail. Although that is pretty rare, it does happen in some fcc metals.

However, in most cases, either voids form or the cavitation instability sets in leading to catastrophic crack propagation. In either of these conditions, the volume of the specimen would change. The assumption dV = 0 is an approximate one and should ideally not be used for modelling finite strains

Rajorshi

I am so forgetful. Of course the other reason for the dV=0 assumption being only valid up to the onset of necking is that there is another assumption which we should not forget:

The "homogeneous strain assumption".

These two assumptions (i) dV=0 and (ii) homogeneous strain. Are usually used together and of course the "homogeneous strain" assumption does not allow for any localisation of strain.

It is so important in science/engineering to know all of the assumptions that have been made, by you and by others and how they might affect your conclusions.

We also usually assume a smooth surface finish but you can easily see how a coarsely turned piece may produce different results, from a turned and polished sample, due to localized notching.

From a fundamental perspective: for every dislocation (of void etc) you generate the 'activation volume' increases as the hole it creates in the material is now part of the volume of the system. Therefore the system never actually has conservation of volume. This becomes important if you are looking at activation energies as most methods use a canonical approach with defined volume that influences the accuracy.

On a macro scale approach, you need to also consider it in terms of activation volume. Before necking the activation volume is the entire gauge section. As the specimen necks the activation volume reduces to the necking section. Think about it in terms of the neck acting as a new smaller gauge section within the specimen, which then makes a smaller gauge section, and smaller again as necking progresses. This essentially means that the activation volume changes every time a new smaller gauge section is created and therefore volume is not conserved on this scale either (as there will be voids and dislocations adding to the volume on a micro level too).

You don't lose matter, you gain holes and therefore activation volume. Once you start necking you essentially have a rapidly diminishing gauge volume.