Instability in Tensile Deformation - Solution Procedure

General Concept: The strength of strain hardening materials continually increases with increasing deformation. At necking the load carrying capacity starts to decrease ( load carrying ability decreases, strength which is a stress continues to increase). The load carrying capacity decreases because the effect decreasing cross-section area overcomes the effect of increasing material strength due to strain hardening

To identify the onset of necking the load, P, is used. When necking starts the load starts to decrease. Engineering stress is defined in terms of the specimen initial cross-section area, seng = P / Ao. So, a decrease in P results in a decrease in seng since Ao is constant. The onset of necking can be associated with the start of decreasing stress on the engineering stress-engineering strain plot.

Since there is a relation between engineering strain and true strain, the true strain at necking can be found if the engineering strain at necking is known.

Model of Instability in Tensile Deformation

The model development is based on describing the load carrying behavior at the maximum load state.

The load P is supported over cross section area A by stress s

P = s A
Since the question posed is to find the strain at necking, the area A in the force description is put in terms of tensile true strain e
e = ln ( l / lo )
for constant volume, V
volume at any time V = initial volume Vo
l (Pi/4) D² = lo ( Pi/4) D²o
Do² / D² = l / lo
e = 2 ln ( Do / D )
ln ( Do / D ) = e / 2
D = Do exp[ -(e/2) ]
A = (Pi/4) D2 = (Pi/4) D²o exp( -e )
P = s (Pi/4) D²o exp( -e )

The maximum load point has zero slope and the description of this is

dP / de = 0
Using the expression for P above and checking the second derivative the maximum load condition is
dP / de = (Pi/4) D²o { ds / de [ exp( -e ) ] - s exp( -e ) } = 0
ds / de = s
This general result can be applied to any material.

With the specific material behavior s = K en

ds / de = s
n K e(n-1) = K en
The result is
e = n
For the particular material behavior used localized deformation in tension starts when the strain is numerically equal to the strain hardening exponent.

The end result will differ depending on the material behavior description, but the model development procedure will be the same.

So? If tensile deformation is imposed during part production, the amount of deformation that can be produced before localized deformation starts is determined by the material strain hardening behavior. For this particular case, the strain imposed should be less than the value of n. This means that we can predict whether localized deformation will occur in a particular process.

This is useful information for tool and process design. Or we can use the result to help in selecting the material to use if the part and tooling are fairly firmly fixed.

While the end result is different, the same kind of process modeling can be applied to two and three dimensional tensile deformation.

Answers to the questions related to tensile deformation instability are HERE

Back to Tensile Instability Exercise

© 2001 by Barney E. Klamecki. All rights reserved