# Distribution of Magnetic Flux Density under Stress and Its Application in Nondestructive Testing
###### Abstract  
Carbon steels are commonly used in railroad, shipment, building, and bridge construction. They provide excellent ductility and toughness when exposed to external stresses. They are able to resist stresses and strains effectively, and guarantee safe operation of the devices through nondestructive testing (NDT). The magnetic metal memory (MMM) can be used as an NDT method to measure the residual stress. The ability of carbon steel to produce a magnetic memory effect under stress is explored here, and enables the magnetic flux density to be analyzed. The relationship between stress and magnetic flux density has not been fully presented until now. The purpose of this paper is to assess the relationship between stress distribution and the magnetic flux density measured by the experiment. For this, an experimental method for examining a carbon steel plate (SA 106), based on the four-point loading test, was used. The effect of stresses resulting from the applied loads on the response of the experimented SA 106 specimen was examined. A three directional tunnel magnetoresistance (TMR) measurement system was used to collect the triaxial magnetic flux density distribution in the SA 106 specimen. In addition, finite element method (FEM) analyses were performed, and provided information on the direction and distribution of the stress over the studied SA 106 specimen. Indeed, a correlation was derived by comparing the stress analysis by FEM and the measured triaxial magnetic flux density. m 2022 25 July 2022 28 July  
the conventional NDT methods [17; 18; 19]. This approach is novel for the nondestructive testing of ferromagnetic devices, and is actually based on the residual magnetization measurement, which occurs in the given material under the action of a stress load or external fields [20; 21]. The MMM testing technique was initially put forward by [PERSON] in 1997. This technology is not limited to the previously mentioned uses, and also allows the evaluation of deficiencies in ferromagnetic structures, including the detection of the position of the defect, microcracking, welded joints at the production stage, plastic deformation, etc. [22; 23]. Moreover, the MMM method offers distinctive features; among which, the simplicity of the operating principle, the low cost of the measuring equipment, the need for no external sources of the magnetic field, and no treatment of the surface of the structures nor the premagnetization or demagnetization of the target specimens [5; 24; 25; 26]. Over the past twenty-five years, the MMM and its application to the investigation of defects in ferromagnetic structures is among the nondestructive magnetic testing methods that have attracted considerable interest from many researchers around the world [4; 27; 28; 29; 30; 31; 32]. [PERSON] et al. [33] and [PERSON] et al. [30] performed fatigue bending experiments under various stress levels to study the magnetomechanical effect. The results reveal that the metal magnetic memory signal can be utilized to assess the damage degree of materials in the elastic and plastic deformation phases, and the degree of stress concentration. [PERSON] et al. [34] conducted a tension-tension fatigue test of ferromagnetic steel with different stress concentration factors, and they concluded that the stress concentration factor has a significant impact on the variation of magnetic signals in a dynamic stress fatigue test. [PERSON] et al. [35] presented a method for the evaluation of residual stress in ferromagnetic steels based on residual magnetic field (RMF) measurements. The findings show that the values and the distributions of the gradients of the RMF components correlate well with the values and distributions of the residual stress. [PERSON] et al. [36] measured the variation of the surface magnetic field intensity during rotary bending fatigue experiments; of which, the outcomes indicated that the magnetization under tensile stress was different from that under compressive stress.