The accommodation of large strains in the upper lithosphere of the earth and other solids by self-similar fault systems: the geometrical origin of b-Value

Abstract

Plate tectonics has provided a method of visualizing the geometry of the deformation of the Earth’s lithosphere on a large scale. The description is so concise that for many purposes it provides an explanation of geological processes that overshadows the need to understand the driving processes. The mechanics of the zones between the plates are less well understood, particularly in continental regions where large areas are subject to deformation. Both continuous and discontinuous models have been tried but both have obvious drawbacks.

In this paper concepts of geometrical self-similarity are adapted to provide a description of the multiscale faulting that must occur in such environments. The fractal geometry of Mandelbrot is applied to the problem of continental triple junctions and it is shown that certain arrays of faults can “stabilize” a junction where three faults meet. The conditions required to do this indicate that earthquakes of different sizes must occur in certain proportions. For simple assumptions and conditions of triaxial deformation the proportion is that which is observed globally for earthquakes. Thus, the b-value of unity found empirically by Gutenberg and Richter and others can be regarded as a consequence of three-dimensional self-similar fault geometry.

The geometric description can be used to understand the way in which fault systems evolve. Earthquakes initiate and terminate in regions where fault systems bend, because the bends become zones subject to multiscale faulting. Movement on many faults in these regions distributes the stress concentration of a propagating rupture front and terminates motion. The multiple faults create offsets in the next fault to move. These offsets are the asperities that must break before a new earthquake occurs.

The self-similar fault geometry requires that a substantial proportion of the deformation in a fault system occur on minor faults and not on the main faults. The proportion of the deformation taken up off the main fault depends on the form of the slip function on the main fault.

The off-fault deformation produces aftershock sequences and forms background seismicity and foreshocks. The geometric relation of aftershocks and foreshocks to the main faults suggests that the former will tend to have b-values greater than unity and the latter b-values less than unity.

The geometric description can be compared to the ideas of fracture mechanics, and it is shown that for earthquake faulting and brittle deformation of the lithosphere in general, fracture toughness, critical stress intensity factor and the Griffith Fracture Energy are not material properties but properties of the geometry of fault systems.

Examined in terms of self-similar behaviour, concepts such as ductility can become ill defined. What may be treated as ductile behaviour viewed at a large enough scale is seen to be brittle when examined more closely. Clear-cut boundaries between the two phenomena do not necessarily exist.

At very large scales and very small scales self-similar behaviour breaks down. Intermediate scales occur but are not discussed at length. The upper scale limit is the Plate Tectonic scale, which provides a very wide range of deformational boundary conditions, and the lower limit is the scale of opening fissures. The latter process can be microscopic or macroscopic. At whatever scale it occurs it is responsible for the confining pressure dependence of shear failure that is responsible for frictional or dilatant behaviour of rocks. Although we do not discuss the relations between fault motion and rotation, the large strains that can be accommodated by the mechanisms we describe will, in general, be accompanied by rotations of similar magnitude.

This paper is concerned with the deformation of the brittle upper part of the Earth. Many aspects of the descriptions, however, are also appropriate to the brittle deformation of any solid.