Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Crystallinity in Polymers Considère's construction: stable necking and cold drawing

# 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.

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

True stress plotted against extension ratio, showing Considère's construction for the onset of necking

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:

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

True stress plotted against extension ratio, showing the double tangent construction for stable cold drawing