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1

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.

2

Differential geometric approach to the stress aspect of a fault: Airy stress function surface and curvatures

Yamasaki K., Yajima T.

Acta Geophysica

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2012

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Vol. 60, no. 1

4-23

EN

We considered the two-dimensional stress aspect of a fault from the viewpoint of differential geometry. For this analysis, we concentrated on the curvatures of the Airy stress function surface. We found the following: (i) Because the principal stresses are the principal curvatures of the stress function surface, the first and the second invariant quantities in the elasticity correspond to invariant quantities in differential geometry; specifically, the mean and Gaussian curvatures, respectively; (ii) Coulomb's failure criterion shows that the coefficient of friction is the physical expression of the geometric energy of the stress function surface; (iii) The differential geometric expression of the Goursat formula shows that the fault (dislocation) type (strike-slip or dip-slip) corresponds to the stress function surface type (elliptic or hyperbolic). Finally, we discuss the need to use non-biharmonic stress tensor theory to describe the stress aspect of multi-faults or an earthquake source zone.

3

A quantum particle motion and thermodynamics in faults-defects field: Path integral formulation based on extended deformation gradient tensor

Yamasaki K.

Acta Geophysica

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2009

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Vol. 57, no. 3

567-582

EN

We consider the effect of the faults-defects (FD) field on the following quantum phenomena: (i) the motion of a particle expressed by the Green function; (ii) thermodynamic phenomena expressed by the partition function. We use the path integral formulation based on the extended deformation gradient (EDG) tensor. This formulation connects the Green function of (i) with the partition function of (ii) to describe the thermodynamics in terms of a quantum particle motion. We obtain the following results: (a) The Lagrangian in the Green function includes the new potential consisting of stress functions that shift the path of the free particle from the shortest distance; (b) The solution of the partition function in one-dimensional space makes it possible to deduce the thermodynamic relations in the FD field. Such results could not be obtained by taking the traditional mechanical and quantum approaches, so the path integral formulation based on the EDG tensor is a useful tool.

4

Tensor analysis of dislocation-stress relationship based on the extended deformation gradient

Yamasaki K.

Acta Geophysica Polonica

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2005

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Vol. 53, nr 1

1-12

EN

The dislocation density used to estimate the magnitude of paleostress in rocks has been expressed in terms of a scalar quantity. Dislocations are classified into two types: edge dislocations and screw dislocations. However, the scalar expression of dislocations does not contain information on the type of dislocations. Therefore, we cannot see the effect of stress on the type of dislocations. In other words, we can extract the information related to the magnitude but not the orientation from previous dislocation-stress relationship. Then, we attempted to derive the tensor equation for dislocation-stress field. For this analysis, we introduced the extended deformation gradient tensor, that is, a differential geometrical expression of the ordinary deformation gradient tensor. We assumed that: (1) the higher order terms and spatial derivatives of dislocation density can be ignored; (2) the material is isotropic. We found that our tensor equation for dislocation-stress field is the square root expression of the equation derived from the experimental data of aluminum under static tension. Moreover, we found that the type of dislocation affects the stress field through the difference in the value of coefficients of the dislocation-stress relation-ship.

5

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).