Fractional approach to dynamic magnetic power losses in low-carbon steel under static mechanical stress and alternating magnetic field

This study extends an analytical fractional approach for describing magnetic losses to thick, low-carbon steel specimens, focusing on the influence of uniaxial mechanical stress on magnetic loss variations. The main objective is to develop a simple, yet accurate analytical model for describing magnetic losses in a low-carbon steel under static mechanical stress. The model is based on a fractional-order approach that captures the material’s dissipative behavior across a wide range of frequencies and stress levels. The high conductivity and thickness of the tested specimens amplify the classical loss contribution, making the methods using a fixed fractional derivative order n inadequate. To address this, the order n is made dependent on the maximal value of the average flux density ( B a max ), reducing the relative Euclidean distance (RED(%)) between simulation predictions and experimental results to 5.4 % under stress-free conditions. Under applied stress, a fitting procedure linking n to both maximum flux density ( B a max ) and stress σ achieves a RED(%) of 3.16 % across 225 data points, albeit with excessive degrees of freedom. Simplifying the definition of n to a three-parameter polynomial maintains reasonable accuracy (RED(%) = 6.7 %) while improving efficiency. Deviations are observed at B a max = 0.5 T and 1.7 T. They are attributed to specific low-field material behaviors or experimental inconsistencies. Finally, using a sigmoid-type function with the same number of parameters achieves an improved RED(%) of 6.4 % while ensuring physical coherence.