##### Identifying Necking in Metals and Plastics

When characterizing materials such as Metals and Plastics in LS-DYNA, most constitutive models provide a yield criteria that accounts for a 3D state of stress which reduces to a uniaxial yield stress in 1D. This allows us to directly input the true stress-strain curve from a one-dimensional state of stress testing such as in uniaxial testing which then can be used for any 3D problems. One important consideration in the usage of such one-dimensional testing data is the phenomenon called necking which occurs in both Metals and Plastics. It is well known that post-necking stress state is no longer uni-axial but is a 3D state of stress. This limits us to use uniaxial information only upto necking and use some sort of iterative process to characterize the post-necking behavior. The first task therefore is to identify the necking point on a uni-axial true stress-strain curve. This is done by plotting the true stress-strain curve against its derivative. The intersection point of these curves is the necking point. When this curve overlay is done using the raw test data, which is sampled at a high frequency, the derivative may be extremely noisy and this may provide multiple intersection points. A sample of such noisy overlay is shown here:

To overcome this, it is usually recommended to digitize the raw curve using anywhere from 10-100 equi-distant points which provides a very smooth derivative of the true-stress-strain curve ensuring a unique point of intersection (necking point). A digitized curve using the raw test data is shown here.

• Nihar says:

Dear Suri,

This is a very valuable piece of information. Thanks.
To know more about this, I would like to know the theory behind the consideration of derivative of True Stress-Strain values.

Thanks & Regards,
Nihar

• Suri Bala says:

Dear Nihar,

Thanks for the comments. The theory is simple and can be expressed as shown below. For more elaborative derivations, you can refer to the Advanced Class Notes by Paul Dubois.

Necking (atleast for metals) is known to coincide with the ultimate stress value. On a engineering stress-strain curve, mathematically, it is when its derivative is zero. However, we need to highlight the necking point on a true-stress strain curve. The derivative of true stress-strain is non-zero at necking which can be then derived to show that the non-zero value is the true-stress that coincides with the initiation of the necking. The derivation is performed by first expressing the true-stress and true-strain in terms of the nominal stress and nominal strains.

Optionally, you and find the nominal strain (x-intercept) at which the necking occurs by taking the derivative of the engineering stress-strain curve. This can then be converted to the true-strain and then the necking true-stress.

• Gopi says:

Hi,

I have some doubts on this.

I have a engineering stress-strain curve (generated at high temperature) which shows gradual reduction in stress almost immediately after yield strength and continues to deform quite a bit that way till fracture. Now I need to convert this engineering stress-strain curve to true stress-strain curve. Now, my doubts are :

1. If I can’t convert the eng stress-strain curve to true stress-strain curve beyond the maximum stress point on engineering curve, then I really do not have any true curve in the plastic regime. How can this happen ?. Is it possible that point of necking is not at the maximum stress ?. Can necking start at some other point other than the maximum stress ?. If so, what is the way to determine the point at which necking is initiated ?.

2. If my understanding is correct true stress-strain curves obtained by converting eng stress-strain curve cannot come down at all. But the plots you have put above shows true stress dropping towards the end. Is this correct ?

Regards
Gopi

• Suri Bala says:

Gopi,

It is common for certain materials to underego necking early on but they stabilize later due to hardening effects. Some plastics are a good example. This can be validated with any test images of the test.
If the drop is due to any visible necking, then the same method can be employed as mentioned in the post.

Post fracture and until rupture, the true-stress-strain curve will show a drop. Usually, the true-stress-strain curve is terminated at the initiation of the drop (at fracture point) and is represented with a damage curve as in MAT_089.

Hope this helps.