Relationship between displacements on a free surface and the stress on a fault

A simple relationship exists between the change in the displacement at a free surface of a solid in the vicinity of infinitely long faults or cracks and the stress acting across the plane of the crack or fault. The plane of the crack or fault is taken to be perpendicular to the free surface. Let Δτ( x ) represent the change, at a distance x from the free surface, in the shear stress minus the applied stress. If Δ w ( y ) is the change in the displacement at the free surface, measured parallel to the surface, y is the distance on the surface from the crack or fault plane, and Δ w ′( y ) = d Δ w ′( y )/ dy , the relationship is: Δτ( x ) = Real part of [-μΔ w ′( ix )], where μ is the shear modulus. This expression has application to the study of the flow stress in the material ahead of the tip of a freely slipping crack or notch cut into the surface of a test sample. It also has application to the study of the frictional stress on earthquake faults in the earth9s crust.