Consider a part of a tensile test specimen loaded with a tension force

F and momentary cross-section area

A , the momentary yield stress will be σ =

F /

A . Suppose that in that part a small section exists that has been strained additionally with a small strain dε ≪1, with resulting cross-section area

A ′ and stress σ′, see Fig.

19.1 . Assuming the volume to remain constant the cross-section area

A ′ of that part has reduced:

A ′ =

A /(1 + dε) ≈

A .(1 − dε).

Fig. 19.1 Force and stress in a tensile test

The tension force in that section has to be equal to

F , so that for the actual stress in that part σ′.

A ′ =

F = σ.

A , consequently:

$$ \sigma^{\prime} = \sigma .(1 + d\varepsilon ) $$

(19.1)

The question is now if that section has actually become stronger or weaker. If the section has become stronger, meaning that a force larger than

F is required to elongate it more, it will not deform any further but another part of the specimen will start to deform instead; the deformation is stable and no necking will occur. This depends on the change in yield stress, the stress required to deform that part any further. This new yield stress can be expressed as:

$$ \sigma_{\text{yield, \,new}} = \sigma + d\sigma = \sigma + \frac{d\sigma }{d\varepsilon }.d\varepsilon = \sigma .\left( {1 + \frac{1}{\sigma }.\frac{d\sigma }{d\varepsilon }.d\varepsilon } \right) $$

(19.2)

We can now state that no instability will arise, and necking will not occur, if the new yield stress of the section expressed by Eq.

19.2 has become larger than the actual acting stress expressed by Eq.

19.1 , this finally yields:

$$ \frac{1}{\sigma }.\frac{d\sigma }{d\varepsilon } > 1\,,\quad \text {or}:\quad \frac{d\sigma }{d\varepsilon } > \sigma $$

(19.3)

This is the well known Considère condition.