Tectonomagnetic Modeling on the Bases of the Linear Piezomagnetic Effect
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Modeling of tectonomagnetic changes is formulated in a unified way. First the linear relationship between magnetization and stress is obtained for a general 3-dimensional stress state. It is deduced by applying the principle of superposition to experimental results of uniaxial compression tests. The isotropic piezomagnetic law thus obtained has two independent parameters, which is equivalent to the law proposed by ZLOTNICKI et al. (1981). The ordinary piezomagnetic law (STACEY, 1964; NAGATA, 1970a) with a single parameter, i.e. the stress sensitivity, is a particular case of this extended formula. Because of its simplicity and validity as an average for aggregates of various rocks, the single parameter formula is applied in the following calculations. The basic equation is derived by connecting the Gaussian law for the magnetic field and the Cauchy-Navier equation for static elastic equilibrium through constitutive relationships, i.e. the piezomagnetic law and the Hooke law for isotropic elasticity. It is a Poisson's equation with a source term expressed in terms of the displacement. The representation theorem is obtained for the solution : the tectonomagnetic field is given by surface integrals of the displacement and its normal derivatives over the strained body. Applying the theorem to a medium including a dislocation surface within it, we find that the dislocation surface behaves as a magnetic sheet. For Volterra dislocations, the magnetic sheet becomes simply a double layer, of which the moment is given by the inner product of the displacement discontinuity and the magnetization vector. The seismomagnetic moment thus defined is useful to intuitively realize coseismic magnetic changes. In the following calculations, the model earth considered is the simplest one: a homogeneous and isotropic elastic half-space having a uniformly magnetized top layer with a constant stress sensitivity. The piezomagnetic field associated with the Mogi model is investigated in detail. Two mathematical techniques are introduced, which are frequently used throughout this study: i.e. the double Fourier (or Hankel) transforms and the Lipschitz-Hankel type integrals. The piezomagnetic field associated with the inflation of a finite spherical pressure source is solved with the aid of these two methods. The point source solution is also obtained, and subsequently used as a Green's function for the multiple Mogi model in Chapter 5. The way we obtained the point source solution becomes the prototype of constructing Green's functions in the following chapters. In the case of integrals containing a singular point of the stress field, we must take a limit in a special way: i.e. to enclose the singular point with a closed surface which satisfies the boundary conditions and then to shrink the surface to that point. A variety of tectonic models can be formed by superposing the displacement field solutions of single forces acting at points in an elastic half-space. Piezomagnetic changes associated with the same models are given as well by the linear combination of fundamental piezomagnetic potentials, which arise from stress-induced magnetization produced by unit single forces acting at points. The method is adaptable to surface load and volume source problems. As an application example, piezomagnetic change is calculated for a uniform circular load. Comparing the calculations with some observations of the dammagnetic effect, we suggest that the in situ value of the stress sensitivity of the upper crust is an order of magnitude greater than that of stiff rocks which are usually tested in rock-magnetic experiments. Finally the dislocation problems are considered. The same integral representation as the Volterra formula for the elastic field is derived for the piezomagnetic field. The elementary piezomagnetic potentials are defined as the potentials produced by a point dislocation. The effect of divergent stresses around a point dislocation is evaluated as follows: we enclose the point dislocation with a small thin disk parallel to the infinitesimal dislocation surface, diminish the thickness of the disk and then its radius. Elementary potentials consits of dipoles and multipoles at the position of point dislocation and their mirror images with respect to the Curie depth. However, some types of strain nuclei lack magnetic source equivalents at the dislocation position. Hence the seismomagnetic effect accompanying some kinds of fault motion becomes much weaker than that anticipated from the seismomagnetic moment. An important application of the theory is the multiple tension-crack model, which is a versatile model for crustal dilatancy or crustal deformation of volcanic origin. Another application is the piezomagnetic change associated with faulting. Formulas for a vertical rectangular fault with shearing as well as tensile fault motion are presented.地震や火山活動,あるいはゆるやかな地殻変動による地殻応力に伴って,地磁気が変化する.これは岩石が応力によって磁化を変えるピエゾ磁気効果を,その原因としている.本論文では様々な地殻活動に伴うピエゾ磁気変化を統一的に扱う方法を定式化する.この理論は基本的には, STACEY (1964)が｢地震地磁気効果｣を見積る計算で行ったやり方を一般化したものである.
- 1992-03-25
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