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1

Self-affinities of landforms and folds in the Northeast Honshu Arc, Japan

Kikuchi K., Abiko K., Nagahama H., Kitazato H., Muto J.

Acta Geophysica

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2013

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Vol. 61, no. 6

1642--1658

EN

A method to analyze self-affinities is introduced and applied to the large scale fold geometries of Quaternary and Tertiary sediments or geographical topographies in the inner belt of the Northeast Honshu Arc, Japan. Based on this analysis, their geometries are self-affine and can be differently scaled in different directions. We recognize a crossover from local to global altitude (vertical) variation of the geometries of folds and topographies. The characteristic length for the crossover of topographies (landforms) is about 25 km and is related to the half wavelength of the crustal buckling folds or possible maximum magnitude of inland earthquakes in the Northeast Honshu Arc. Moreover, self-affinity of the folds and topographies can be connected with the b-value in Gutenberg-Richter℉s law. We obtain two average Hurst exponents obtained from the self-affinities of folds in the Northeast Honshu Arc. This indicates that there are two possible seismic modes for the smaller and larger ranges in the focal regions in the Northeast Honshu Arc.

2

Geometry of stress function surfaces for an asymmetric continuum

Yajima T., Yamasaki K., Nagahama H.

Acta Geophysica

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2013

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Vol. 61, no. 6

1703--1721

EN

A two-dimensional stress field of dislocation or fault is geometrically studied for an asymmetric con tinuum. For geometric surfaces of the stress and couple-stress functions, the mean and Gaussian curvatures are derived. The mean curvature of couple-stress function surface is connected with the asymmetr ic of stress tensor. Moreover, the Gaussian curvature of stress function surface is characterized by bo th the stress and couple-stress. On the other hand, th e mean curvature of stress function surface is not affected by the asy mmetry of stress. Based on these geometric expressions, the Coulomb’s failure criterion and the friction coefficient are expressed by the curvatur es of couple-stress function surface. Moreover, geometric structures of st ress and couple stress function surfaces are shown for edge and wedge dislocations as faults. The curvatures of these surfaces show that the ef fect of couple-stress is constrained around the dislocations only.

3

Fractional-order derivative and time-dependent viscoelastic behaviour of rocks and minerals

Kawada Y., Yajima T., Nagahama H.

Acta Geophysica

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2013

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Vol. 61, no. 6

1690-1702

EN

A general constitutive equation for viscoelastic behaviour of rocks and minerals with fractional-order derivative is investigated. This constitutive law is derived based on differential geometry and thermodynamics of rheology, and the fractional order of derivative represents the degree of time delay. Analyzing some laboratory experimental data of high temperature deformation of rocks and minerals such as halite, marble and orthopyroxene, we propose how to determine the orders of fractional derivative for viscoelastic behaviours of rocks and minerals. The order is related to the exponents for the temporal scaling in the relaxation modulus and the stress power-law of strain rate, i.e., the non-Newtonian flow law, and considered as an indicator representing the macroscopic behaviour and microscopic dynamics of rocks.

4

Symmetropy of earthquake patterns: asymmetry and rotation in a disordered seismic source

Nanjo K., Nagahama H., Yodogawa E.

Acta Geophysica

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2006

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Vol. 54, no. 1

3-14

EN

We demonstrate that the idea of symmetropy can be used for quantification of earthquake patterns. The symmetropy can be considered as a measure of asymmetry. A pattern is richer in asymmetry when the symmetropy is smaller. The specific results of its applications are obtained as follows. In a discrete model of a seismic source with self-organized criticality, the spatial patterns of earthquakes during critical states and subcritical states are distinguished by the behaviour of the symmetropy: subcritical patterns show that the symmetropy is approximately a constant but this has various values during critical states. The critical patterns show asymmetric property without any asymmetric force from the outside and without asymmetric intracellular rule. We show that the emergence of asymmetric patterns is a generic feature of dynamic ruptures in our model. Such a generic asymmetry results from the model which is an inherently discrete system consisting of finite -sized cells. These cells may represent geometrical disordered fault zones. We further discuss rotational motions that generate seismic rotational waves. In micro-morphic continuum theory, such rotations are attributed to dynamic ruptures in dis-ordered systems. We note that the concept of disorder in this theory is expressed by a set of finite-sized microstructures and is consistent with the concept of disorder modelled in the present study. Thus, we suggest that the spatially asymmetric patterns of earthquakes might be related to the rotational motions, because both come from dynamic limit.

5

Seismic rotation waves: dislocations and disclinations in a micromorphic continuum

Nagahama H., Teisseyre R.

Acta Geophysica Polonica

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2001

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Vol. 49, nr 1

119-129

EN

Deformation of the micromorphic structure (continuum) induces the appearance of dislocations and disclinations. These structural defects are related to anholonomity caused by microdisplacement tensor and moment of microstrains. In this study, two cases of incompatibility sources are considered: microdisplacements and ressions moments of microdisplacements. For these cases the expressions for twist-bend tensor are derived.

6

Gauge theory of dislocational electromagnetic field in earthquake preparation zone

Nagahama H.

Acta Geophysica Polonica

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2001

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Vol. 49, nr 4

437-448

EN

From the view point of the continuum theory of defects, we consider how the generated electromagnetic field can be related to the deformation field due to dislocations and disclinations. Based on the theory of connections in the higher-order space (Kawaguchi space), the Finsler deformation theory of ferromagnetic substances is introduced and the relation between this Finsler theory and the gauge theory of dislocations and disclinations is pointed out clearly. Moreover, the various preferred directions as internal degrees of freedom of each geomaterial point (e.g., polarization, spin moment, directors) of crystal materials (geomaterials) are discussed and the "exciting" state of these various preferred directions is regarded as the electromagnetic field radiation from an earthquake preparation zone.

7

Micro-inertia continuum: rotations and semi-waves

Teisseyre R., Nagahama H.

Acta Geophysica Polonica

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1999

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Vol. 47, nr 3

259-272

EN

The rotation seismic waves become recently again a subject discussed from theoretical and observational points of view. Defects and internal structure contribute to processes of their generation and propagation. In theories of continua we can attribute generation of rotation waves due to defects of a disclination type or to micromorphic/micropolar structure of a source zone. However, these waves would become rapidly attenuated when propagating inside the continuum more close to that of ideal elasticity. Another possibility is provided by coupling between the seismic body waves and defects or micromorphic structures of medium just beneath and observation site; in this paper we discuss such a coupling in micro-inertia continua defined as the special cases of micromorphic/micropolar continuum.

8

Continuum theory of defects and gravity anomaly

Yamasaki K., Nagahama H.

Acta Geophysica Polonica

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1999

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Vol. 47, nr 3

239-257

EN

Based on the mathematical equivalence between the crack field and the continuous dislocation field, we briefly review continuum theory of defects from the view point of differential geometry. Then we derive a new differential geometric equation of static gravity change for anelastic effect due to the fault (dislocation) density. This equation shows that high gradient of dilatancy caused by the concentration of fault (dislocation) density accompanies high gradient of gravity change near the boundary between positive and negative gravity anomalies. This agrees with the characteristic distribution patterns; the distribution of short-wavelength gravity anomaly, active faults and shallow seismic activities overlap one another in the northeast Japan. Moreover, we discuss: (I) dynamic gravity anomaly related to earthquakes; (II) local gravity anomaly near the edges of an active fault; (III) differential geometric interpretation of gravity anomaly caused by the dislocation density; (IV) differential geometric relationship between gravity anomaly and magnetic anomaly (Poisson's relation).

9

Thermodynamics of line defects and transient electric current: electromagnetic field generation in earthquake preparation zone

Nagahama H., Teisseyre R.

Acta Geophysica Polonica

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1998

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Vol. 46, nr 1

35-54

EN

In thermal equilibrium the electrical neutrality of the charge dislocation is assured by a cloud of oppositely charged point defects around it. Here, from the view point of thermodynamics of line vacancies, we propose a possibility of constructing a model of transient electric current generation in an earthquake source zone which combines polarization processes and motion of charged dislocations under the influence of evolving field of stresses. This transient electric current differs essentially from the piezo-electric current in which the electric field is proportional to the stress itself. Then we review briefly the elements of stress evolution theory. Some examples of the stress evolution pattern with the electromagnetic field radiation are shown.

10

Microspheric continuum, rotational wave and fractal properties of earthquake and faults

Nagahama H.

Acta Geophysica Polonica

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1998

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Vol. 46, nr 3

278-294

EN

It seems that internal structures and discontinuities in the litosphere have an essential influence on the litocpheric derofmation like faulting or earthquakes. The micromorphic continuum providies a good framework to study the continnum with microstructure, such as earthquake structures. Here we briefly introduce micromorphic continuum to consider the earthquake structures (e.g., rotational wave). Then the equlibrium equation in terms of the displacements (the Navier equation) is derived from this micromorphic continuum. This equation is the generalization of Laplace equation in terms of displacements and can lead to some Laplace equation such as the local diffusion-like conservation equations. Moreover, these conservation fields bear the fractal properties of fracturing in the lithosphere in the form of facults or earthquakes. Finally some scalling laws of faults or eartquakes are discussed.

11

Dislocation field evolution and dislocation source/sink function

Teisseyre R., Nagahama H.

Acta Geophysica Polonica

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1998

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Vol. 46, nr 1

13-33

EN

The evolution of dislocation density field depends on the source/sink function; this function describes the nucleation of new dislocations and the coalescence processes (mutual annihilation of dislocations having the opposite signs which is equivalent to coalescence of two neighbouring dislocated elements or coalescence of two co-planar cracks). In this paper we discuss the choice of this function, we present the solutions for dislocation field and stresses and we discuss the obtained results, especially the short period perturbations (seismic events) related to the extrema and sign changes of the source/sink function. General discussion on the source/sink function for dislocation fields closes this paper.