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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

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.

3

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.