In order to monitor elliptical fatigue crack growth in ferromagnetic steel using magnetic methods, analternative to the self magnetic flux leakage method must be derived as elliptical cracks can grow to significant sizes before they reach through the thickness of the plate material. An approach is sought by translating subtle changes in magnetisation back to the Villari effect, a phenomenon which depicts how applied stress induces changes in magnetisation in ferromagnetic objects. Since the magnitude of these changes in magnetisation is small, other nonlinear effects of similar order such as magnetic relaxation and hysteresis are identified, measured and quantified preliminarily.

The magnetic behaviour in this project is assumed to be quasi-static, and derivations of the expressions for the magnetic field around simple geometric shapes are provided in order to understand magnetic behaviour and verify the outcome of the numerical simulations. It is shown that the numerical simulations
produce identical magnetostatic induction fields as the analytically derived expressions when using a sufficiently refined mesh.

An attempt is made to measure long-term magnetic relaxation by subjecting a solid prolate spheroid to a continuous uniform background field for periods of an hour while trying to measure differences in the induction field at a fixed distance. Short-term relaxation, the time it takes for an object to reach a certain
magnetisation when the background field is abruptly changed, is also investigated. It is concluded that both effects could not be successfully measured using the current setup. In order to draw proper conclusions, further research into this topic should be conducted using more accurate equipment for extended periods of time.

Upon investigation it is discovered that it can not be assumed that the steel specimens exhibit a uniform permanent magnetisation. A self-developed method is introduced through which non-uniform magnetisation in three directions can be calculated by means of inversion using a set of magnetic induction field measurements in a plane below the specimen when the background field is zero. These measurements are translated to magnetisation using a set of higher order square Gaussian distribution functions that are spaced in a grid over the domain of the test specimen in order to vary the magnetisation locally.

Literature that shows comparable results regarding description of non-uniform permanent magnetisation using an array of induction field measurements has not been found. The concept of hysteresis is introduced and a method is presented through which the parameters of the Jiles-Atherton hysteresis model can be determined using parameter fitting in combination with a forward numerical model created in COMSOL. Closure of minor loops require modifications to the original JA equations which are implemented in the forward model. The numerical model is encapsulated within the Shuffled Leaping Frog parameter optimisation algorithm in order to compute the correct hysteresis
parameters. It is found that it is possible to successfully determine the parameters of multiple specimens using weak magnetic fields, and therefore minor loops, which is unparalleled in literature.

Eventually, the Villari effect is introduced and an attempt is made to measure and model the effect usingan extension of the Jiles-Atherton model proposed by Naus. Experiments have shown that using this methodology the magnetostriction parameters can be succesfully obtained. A recommendation is
provided into how these results can be implemented in crack-propagation models in future research.