# Considère's construction: stable necking and cold drawing

Considère's construction offers a simple method for determining whether a polymer will form a stable neck and cold draw. For a sample under uniaxial tension, there is an observed drop in nominal stress at the onset of necking. At this point:

\[{{d\sigma } \over {d\lambda }} = 0\]

Where σ is the nominal stress and λ is the extension ratio, which is defined as the current length divided by the initial length and is therefore related to the nominal strain; an extension ratio of 1 corresponds to 0% strain. By conservation of volume it can be shown that:

$$\sigma = {{{\sigma _t}} \over \lambda }$$

Therefore

$${{d\sigma } \over {d\lambda }} = {1 \over \lambda }{{d{\sigma _t}} \over {d\lambda }} - {{{\sigma _t}} \over \lambda^2 }$$

In these expressions σ_{t} is the true stress, which is equal to the applied
force divided by the instantaneous area. Since \({{d\sigma } \over {d\lambda }} = 0\)
at the onset of necking, it follows that at the onset of necking the condition below is satisfied.

$${{d{\sigma _t}} \over {d\lambda }} = {{{\sigma _t}} \over \lambda }$$

From the plot above, of true stress versus extension ratio, it is possible to determine whether necking will occur. A tangent to the curve constructed from λ = 0, touches the curve at the point where necking starts. This is Considère's construction.

As in the plot below, it is sometimes possible to draw a second tangent to the curve from λ = 0, which touches the curve at a higher extension ratio. In the region between the points where the tangents touch the curve, the polymer exhibits stable cold drawing; the point where the second tangent touches the curve corresponds to the limiting strain in the neck. In conclusion three types of behaviour may be recognised:

- If \({{d{\sigma _t}} \over {d\lambda }} > {{{\sigma _t}} \over \lambda }\) at all times, no necking occurs; the material extends uniformly until failure.
- If \({{d{\sigma _t}} \over {d\lambda }} > {{{\sigma _t}} \over \lambda }\) at one point on the curve, then the material will neck and fail.
- If \({{d{\sigma _t}} \over {d\lambda }} > {{{\sigma _t}} \over \lambda }\) at two points, then necking will be followed by cold drawing.