A formulation of the criterion for multiaxial fatigue in terms of complex number as proposed by Macha

This criterion by Macha has been repeatedly verified, analyzed and special cases of this criterion have been taken into account, reducing the complex stress to its uniaxial equivalent. Since both normal and shear stresses are vectors, we encounter the mathematical problem of adding these vectors, and the question arises how to consider the obtained equivalent stress, because two perpendicular vectors are added with weighting factors. Therefore, in this work it is proposed to adopt a system of complex numbers. Normal stress is defined here as a real part and shear stress as an imaginary part. As a result, on the basis of the defined complex number and based on pendulum bending and double-sided torsion, after transformations, the expression for equivalent stress proves to be identical to those previously defined on the basis of the concept of Macha.