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

Yamasaki K., Yajima T.

Acta Geophysica

2012

Vol. 60, no. 1

4-23

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.

Remarks on Plane, Steady, Creeping Flows Generated by Singularities

Klonowska-Prosnak M.E., Cześnik P.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2002

Vol. 50, Nr 1

45--69

Two properties of plane, steady, creeping flow generated in the infinite domain around a given contour by a single singularity are investigated in the paper. The first property concerns velocity field. It is shown, that stream uniform at infinity must occur automatically in this case, and formulae for velocity of the stream are derived. The second property concerns similarity of streamline pattern of such flow, and the potential one, generated around the same contour by the same singularity. Investigation of the both properties concerns - in particular: a disc, an ellipse and the airfoil sections NACA 0012, RAE 2822.