Magnetic susceptibility and plastic strain of rocks by the differential geometric theory of the physical interaction field

From the differential geometric theory of the physical interaction field between the deformational field and the magnetic field and thermodynamics principles, we can derive a new non-linear equation on the piezomagnetic effects of plastically deformed rocks without using special knowledge of material properties. Moreover, from von Mises' yield condition (plastic potential), Onsager's theorem (non-linear phenomenological equation) and a new flow rule of the plasticity theory generalized by the theory of the physical interaction field, we lead to a new theoretical relationship between the magnetic susceptibility tensor χ_mn^Pl on the plastic deformation and the plastic strain tensor ε_ij^Pl of plastically deformed rocks given by χ_mn^Pl = 2/3E_Sω̃_mn^ijε_ij ^Pl where ω̃_mn^ij is the fourth-rank asymmetric tensor with a non-linear property on the physical interaction coefficient and E_S is the secant modulus referred to plastic strain. Let χ̌_mn be an initial magnetic susceptibility tensor, then the second-rank asymmetric tensor (3/2E_S)χ̌_mnω̃_ij ^mn is equivalent to Borradaile-Alford's empirical matrix M_ij relating strain to susceptibility change. We are developing this relation to infer the strain of plastically deformed rocks from magnetic susceptibility changes.