# Optically Pumped Magnetometer Measuring Fatigue-Induced Damage in Steel
## 3 Description of the Experimental Observables  
The magnetic field \(\overrightarrow{B}(t)\) at the magnetometers' location is a sum of several contributions:  
\[\overrightarrow{B}(t)=\mu_{0}\left(\overrightarrow{H}(t)+\overset{\rightarrow}{ M}(t)\right)\,, \tag{1}\]  
where \(\overset{\rightarrow}{M}(t)\) is the macroscopic magnetization of the specimen and represents the observable of interest. The field \(\overrightarrow{H}(t)\) is composed of the Earths' magnetic field, magnetic fields generated by the coils of the setup, and other environmental sources.  
In ferromagnetic materials without an external load, the magnetization \(\overset{\rightarrow}{M}(t)\) exhibits a hysteresis due to irreversible pinning between domain walls, which are driven by the field \(\overrightarrow{H}(t)\). Inhomogeneities in the crystal, such as dislocations, regions of inhomogeneous strain, and any precipitates or nonmagnetic inclusions within grains, contribute to the hysteresis behavior as well [8]. In piezomagnetic testing, the magnetization is changed by the additional application of a mechanical stress \(\sigma\) via magnetoelastic coupling. This is the so-called Villari effect. The applied stress acts as a strong field \(H\), shifting domain walls and causing similar pinning effects with dislocations [27]. Thus, the magnetization of a ferromagnetic specimen depends on the evolution of the mechanical parameters applied to the specimen as [28]:  
\[\overset{\rightarrow}{M}(t)=\int_{V}\overset{\rightarrow}{m}\left( \overrightarrow{x},\,\sigma\left(\overrightarrow{x},\,t\right),t\right)d \overrightarrow{x}, \tag{2}\]  
where \(\sigma\) is the mechanical stress applied to the specimen and \(\overset{\rightarrow}{m}\) the magnetic moment at location \(\overrightarrow{x}\). If the applied field \(\overset{\rightarrow}{H}(t)\) is constant, as can be achieved in this piezomagnetic testing setup, changes in \(\overset{\rightarrow}{B}(t)\) are proportional to the magnetization \(\overset{\rightarrow}{M}(t)\).  
The present experimental setup aims to find early indicators of fatigue damage in materials. The setup produces both mechanical (\(F-\varepsilon\)) hysteresis curves and piezomagnetic (\(B-\varepsilon\)) hysteresis curves. An important aspect of such hysteresis curves is the area delimited by the curve [5]. By measuring the area delimited by these curves, we expect to be sensitive to the energy involved in the deformation process of the specimen. For a non-oriented ferromagnetic material with homogeneous magnetostriction, the magnetoelastic energy, \(E_{\sigma}^{\varepsilon}\), has the following proportionality:  
\[E_{\sigma}^{\varepsilon}\propto-\frac{2}{3}\lambda\sigma, \tag{3}\]  
where \(\lambda\) is the effective magnetostriction constant of the material and \(\sigma=F/A\) is the uniaxial mechanical stress applied by a force \(F\) to a specimen with cross-section \(A\)[27]. Both the outer magnetic field \(H\) and mechanical stress \(\sigma\) cause domain wall motion in ferromagnetic materials. Specific points of interest on the piezomagnetic hysteresis are the so-called \"Villari reversals\" near the transitions from tensile to compressive stress. For uniaxial tensile stress, magnetic domains with moments parallel to the stress grow at the cost of those with perpendicular directions, whereas compressive stress increases domains with moments perpendicular to the stress. Therefore, significant amounts of domain wall motion and defect interaction can be expected at these points.