We examine the relaxed minimization problem for ferromagnetic bodies using the nonlocal exchange energy model. We show that the model possesses a wide range of phenomena including hysteresis, hysteresis subloops, Barkhausen effect, and demagnetization. The results are in three parts.

First, we examine analytically the problem of a unit sphere of ferromagnetic material. We show that when the exchange energy is zero we duplicate De Simone's model which has a wide range of measure-valued minimizers. As the exchange energy grows our model stabilizes at the saturated solutions of the Stoner-Wohlfarth model. Here, the measure-valued minimizers are eliminated.

Next, we examine numerically the problem of a body composed of several unit spheres of ferromagnetic material. We show that a constrained problem that focuses on the resultant field energy produces results similar to the unconstrained problem with considerable savings in time and resources.

Finally, we examine numerically the constrained problem on a moderately large body. It is shown that the constrained problem contains all the hysteresis phenomena mentioned above.