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Fig. 13.19 Determination of
the curvature R in a neck
The neck curvature radius is to be defined at each instant in all above models with
exception of Mirone’s model. For this, the points, representing external contour of
specimen, are approximated by polynomial r(z) (Fig. 13.19). The curvature radius
in the point of the minimum section (min(r)) is calculated by the formula:
R =
1 +
dr
z
2
3
2
d
2 r (z)
dz 2
z at min(r )
.
(13.15)
The model’s nodes are shown as points in Fig. 13.19, solid lines represent the
polynomial approximation; the black circle demonstrates contour curvature.
Figure 13.20 illustrates time histories of plastic strains in different points in a neck
section (colored lines) and compares them with plastic strain estimation, calculated
by formula (13.6). It is clear that plastic strain determined using current neck radius
is in good agreement with the real mean value of plastic strain in a neck. The nonuniformity of plastic strain occurs at early stages of deformation for short 5 mm length
specimen. The difference between maximum and minimum strain is significant.