Anisotropy of magnetic susceptibility and plastic strain of rocks: A finsler geometrical approach

In the field research of plastically deformed rocks, empirical correlation between anisotropy of magnetic susceptibility (AMS) and plastic strain (permanent strain) has been reported, but there has been no general tensor equation for this empirical law. Recently, from the viewpoint of the physical interaction field, reversible thermodynamics and a flow law of plasticity, we clarified the general tensor equation for the empirical law. However, our previous theory does not have much physical basis yet. Because a magnetic substance (ferro-, para-, or diamagnetic one) in a rock has a spin magnetic moment on each material point, the physical meaning of our previous theory can be clarified by the concept of differential geometric theory of Finsler space which employs the structure of a "point equipped with its proper direction". Here, by this concept, we present a general tensor relationship between magnetic susceptibility and plastic strain of plastically deformed rocks. This tensor equation shows that the effects of plastic strain on AMS are a function of the secant modulus and the asymmetric tensor with irreversible property on the magnetostriction under plastic deformation. This result suggests that the effects (the symmetry of AMS) in macroscopic level may occasionally have the same or lower symmetry than the causes (the symmetry of plastic strain tensor) even in the isotropic rocks and is not consistent with Curie symmetry principles. A comparison of the existing experimental data suggests that our tensor equation is appropriate under the coaxial and homogeneous deformation history. Moreover, since the Finslerian concept of magnetic substances employs the concept of the continuum with the discontinuities, so called non-holonomic objects in differential geometry ("no-more continuum"), our tensor relationship is expected to be a powerful tool for the strain analysis not only of the coaxial and homogeneous naturally deformed rocks but also of the non-coaxial and inhomogeneous ones.