Basic extensional response

Transforming this deviatoric shear-rate expression, eq. (10.7), into a uniaxial extensional rate expression gives 

With these preparatory developments we consider the condition for the impending necking instability in extensional deformation for a rigid plastic idealization of the material, for which 

In a constant-elongation-rate experiment where d( = 0 the impending necking instability occurs when the load L reaches a maximum, i.e., L = 0. Then the condition for necking becomes 

For a rate-insensitive solid m is very large, giving for the condition of the instability 

This is the well-known Considere (1885) criterion for impending necking for a rigid plastic, rate-insensitive material. It is expressed graphically with a simple construction in Fig. 10.1 showing the intersection of the stress-strain curve with the curve of the decreasing slope of that curve, where the point of intersection is at u, gives the uniform extension strain. In Fig. 10.1, for strains e u,dr/d t e) 

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