Steels_ Metallurgy and Applications, Third Edition

22 Steels: Metallurgy and Applications

thinner and narrower. The strain ratio r is defined, therefore, as the ratio of

the true strain, ew, in the width direction to the true strain, st, in the thickness

direction at a particular moment during the test, usually when the engineering

strain is either 15 or 20%. Thus:

r ~ Ew/6t

In practice, it is usual to assume that the sample retains its original volume during

the test which means that the sum of the true width, thickness and length strains

would be equal to zero. Only the width wf, therefore, needs to be measured if

the length-engineering strain is fixed at either 15 or 20%, corresponding to true

length strains of either 0.1398 or 0.1823. The final formula then becomes, for

20% engineering strain:

In wf/wo

(ln wo/w f - 0.1823)

The strain ratio usually has different values for different directions in the plane
of any sheet. A mean r value, rm or L may, therefore, be defined by measuring
a value in the rolling, transverse and diagonal directions and calculating a mean
giving the diagonal value double weight as follows:

rm = (r0 + 2r45 + r9o)/4

where the subscripts refer to the angle of the test direction to the rolling direction.
The variation in r value around the rolling plane is also important and a Ar
value is defined as:

Ar = (r0 - 2r45 + r9o)/4

The main structural feature influencing the r value is the crystallographic texture,
but the relationship between r value and crystallographic texture is a complicated
one. It is useful to note here, however, that a high proportion of grains with (111)
planes parallel to the surface, which is the ~, fibre texture component mentioned
in the previous section, leads to high r values. A high proportion of grains
with (100) planes parallel to the surface, which is one component of the ct fibre
texture, tends to lead to a low r value. Various empirical relationships between
the strength of texture components and r value have been established. One of the
most simple relationships is illustrated in Figure 1.18, which shows a substantially
linear relationship between the rm value and the ratio of the strengths of the (111)
and (100) texture components.
The rm value is important because it affects the distribution of strain in a
pressing by influencing the thinning tendency and gives a measure of the deep
drawability of the steel. This is illustrated in Figure 1.19, which gives a plot
of rm value versus limiting drawing ratio for a number of steels drawn to form
cylindrical cups. There is a linear correlation between the rm value and the
limiting drawing ratio of the steel. This arises partly because the flange being
drawn in to form the wall of a cup exhibits less thickening for a high rm value
steel than for a low rm value steel, as illustrated in Figure 1.20. The rm value also
influences the strain distribution during stretching, as illustrated in Figure 1.21,