In 1910, Basquin found a relation between the superficial stress and the number of cycles until the crack initiation. He made experimental analysis on some specimens and he also made a linear regression about the results. He noticed that when logarithmic scales were used, the relation between the two variables was linear. Another important conclusion was that when the plastic limit was overcome the relation disappeared.
Later, Coffin and Manson developed a relation between the plastic strain and the number of cycles until the crack initiation, obtaining a relation very similar to the one described by Basquin.
On their own, both methods are able to calculate the life until crack initiation for cases with elastic strain or plastic strain. However, they cannot be used to do these calculations when the strain has both components of the same magnitude order.
Total strain is the addition of elastic and plastic. Therefore, by adding both components it is possible to know the number of cycles until crack initiation for any value of the strain. Consequently, the local strain method is described by the expression written in the picture, that is the addition of the Basquin and Coffin and Manson expressions.