15.1 LOCALIZED NECKING

Many sheet-forming operations involve biaxial stretching in the plane of the sheet. Failures occur by the formation of a sharp local neck. Localized necking should not be confused with diffuse necking, which precedes failure in round tensile specimens. Diffuse necking of sheet specimens involves contraction in both the lateral and width directions. In sheet specimens, local necking occurs after diffuse necking. During local necking the specimen thins without further width contraction. Figure 15.1 illustrates local necking. At first the specimen elongates uniformly. At maximum load, a diffuse neck forms by contraction of both the width and thickness when ? ₁ =n (Figure 15.1(a)). Finally a local neck develops (Figure 15.1(b)).

Figure 15.1: Diffuse neck (a) and localized neck (b) coordinate system used in analysis (c).

In a tension test the strain in the width direction cannot localize easily, but the condition is reached where a sharp local neck can form at some characteristic angle ? to the tensile axis. Typically the width of the neck is roughly equal to the thickness so very little elongation occurs after local necking. The strain parallel to the neck, d ? _2? , must be zero, but

For an isotropic material under uniaxial tension in the 1-direction,

Substituting into equation 15.1, cos ² ? ?(sin ² ?)/2=0, or

If the metal is anisotropic,

In this case,

The cross-sectional area of the neck is A ^? = ?. Because ? is constant, d A ?