Approximation of the stress-strain curve

To ease the calculation of plastic deformations, stress-strain curves are frequently approximated using simple equations. One commonly used example is the Ramberg-Osgood law 

If plastic deformations are large, as it is the case in metal working processes like deep drawing or forging, the elastic part of the deformation can be neglected and the law simplifies to 

Using this approximation, the strain i neck at which necking of the specimen sets in can be estimated. 

Necking occurs when hardening of the material is not sufficient anymore to compensate for the reduction of the cross-sectional area so that the force transferred by the cross section decreases. At this moment, the external force does not increase further dUneck/d neck = 0. Using the true stress at, we can write F = atS. If we form the differential dU/di, the onset of necking is characterised by  

If we neglect elastic strains, the volume V = S L oi the specimen has to remain constant during plastic deformation (and thus also at the onset of necking)  

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