Steels_ Metallurgy and Applications, Third Edition

Low-carbon strip steels 21

~.- True

stress

Engineering

stress

Strain

Figure 1.17 Stress-strain diagrams

stress continues to increase with increasing strain and represents the actual stress

on the specimen at every moment during the test.

The engineering strain during a test is defined as the change in length divided

by the original length. The concept of true strain has been developed by regarding

any overall strain as being made up of a series of small elements of strain each

related to the length of the specimen at the start of each element. Thus if an

overall engineering strain (L3 - L0)/L0 were to be regarded as made up of three

elements, the true strain e would be given by the sum of the three elements. Thus:

e = (L1 - Lo)/Lo + (L2 - LI)/LI + (L3 - L2)/L2

In the limit as each element becomes very small and the number of elements
becomes very large:

L

and

= lnL/Lo

Thus, the true strain is equal to the natural logarithm of the ratio of the final

length to the original length. With this concept, any material that is extended to

give a particular true strain would need the same numerical value of true strain

but of opposite sign to restore it to its original length. Clearly, the engineering

strains to achieve the same result would not have had the same numerical value.

It is also useful to note that if, for example, any steel is given several separate

strains to make up a complete strain, the total true strain would be equal to the

sum of the separate true strains, whereas the total engineering strain would not

be equal to the sum of the individual engineering strains.

The strain ratio r or Lankford value

The definition of the strain ratio r or Lankford value 3s depends on the observation

that, as a tensile test proceeds, the gauge length of the sample becomes both