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 σt 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.