Environnement - The Accommodation of Large Strains in the …king/Geoffrey_King/Publications_files/1983... · 2014-05-06 · The mechanics of the zones between the plates are less

PAGEOPH, Vol. 121, 5/6 (1983) 0033-4553/83/050761-5551.50+0.20/0 © 1983 Birkh/iuser Verlag, Basel

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

G E O F F R E Y K I N G 1

A b s t r a c t - 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 f u n c t i o n 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

' Builard Laboratories, Department of Earth Sciences, Cambridge University, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, England.

762 Geoffrey King

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.

Key words: Earthquake; Faulting; Prediction; Fractal.

1. Introduction

The purpose of this paper is to discuss the mechanics of the Earth's lithosphere and the geometry of faulting. The ideas presented originate in a number of recent studies of earthquakes; but the paper addresses some broader questions, in particular, the geometrical and mechanical relations between large and smaU faults and the relation between continuous and discontinuous deformation of rocks of the lithosphere. To do so we shall consider scales that range from that of continents to that of microfracture in rocks and seek to show that discovering rules that relate deformation at one scale to that at another is crucial to understanding how the lithosphere behaves mechanically.

The discussion applies the ideas of fractals developed by MANDELBROT (1977) who suggests that geometrical self-similarity is a widespread feature of the morphology of landscapes. He supports this view with some synthetic landscapes generated using simple geometrical rules. Two processes are obvious candidates for creating such fractal geometry. One is the action of faulting, and the second is that of river erosion.

While in general it is not possible to separate these two effects, under some circumstances a separation is straightforward. We start this paper with three examples of fault-determined geometry. They each indicate that fault or fault-controlled geometries can be much less random than the unfiltered Brownian form that Mandelbrot assumes and are chosen to exclude the possibility that fluvial processes are associated with their creation. This provides a justification for what may initially appear to be an oversimplification in the theoretical faulting structures described here. The simplicity of the faulting structures proposed is also why the discussion differs from other considerations of fractal behaviour `of faulting (e.g., ANDREWS, 1980a,b; KAGAN, 1982). The simple geometries that are discussed can readily be made more complex by adding

Accommodation of Large Strains in the Upper Lithosphere of the Earth 763

randomness, but it does not appear that nature demands such complexity.

Figure 1 shows a rock exposure. Although a range of rock face sizes is visible, there are only a small number of orientations. Many exposures of hard "rock that have not been greatly affected by weathering exhibit this behaviour. Figure 2a is a lineament map of the surface of the planet Mercury, and Figures 2b and c are directional analyses of the observed features using Fourier transform methods (THOMAS and MASSON, 1983). The observed features presumably arise from faulting in the brittle crust of the planet and any influence of rivers on Mercury can be discounted. In this case three orientations are present. An examination of lineament and joint orientation studies on Earth shows similar results (e.g., TCHALENKO, 1970; HODGSON, 1961 ; OVERBEV and ROUGH, 1968; BABCOCK, 1973). Patterns which show three orientations similar to Figures 2b and c are most common. Figure 3 shows fault plane solutions for the aftershocks of an earthquake in the Pyreenees (GAGNEPAIN- BEVr~XX et al., 1982). The solutions are well controlled by a dense array of stations and again the events fall into three groups. There is partial but not complete spatial separatioh in the types of events so they cannot be regarded as being on three discrete faults but must represent intermixed faulting in different orientations, as the authors suggest. The faulting mechanism we will discuss suggests that for two-dimensional (biaxial) deformation faults should occur in sets of three, and for three- dimensional (triaxial) deformation in sets of five. As we shall see later, a horizontal section is likely to show three fault-strike directions; the condition commonly observed.

Since earthquakes are a major manifestation of fault motion, certain features of earthquakes underpin the following discussion. The most striking is the self-similar behaviour of faults (TCHALENKO, 1970; KING, 1978) particularly as manifested in the earthquake frequency-magnitude relation. A search for a geometrical means of understanding this relationship has been a major impetus to this work.

In a discussion of Structural Geology, DE SXTTER (1956) stated that there is no real difference between faulting and folding. To some extent it is possible to show how this arises. Ductile behaviour, or what might be described as pseudoductile behaviour, is not necessarily due to microscopic crystalline mechanisms. (Here deformation due to the latter is referred to as plasticity.) What appears to be ductile at one scale can be entirely brittle when examined more closely. Thus whether a process is described as brittle or ductile may depend on the scale at which it is observed. Scale is very important. A concentration on a limited scale range leads inevitably to confusion, which is a primary reason why a marriage between laboratory studies of rock samples and the observed field behaviour of rock has not been achieved.

764 Geoffrey King

Figure l(a). Shear planes in Jurassic recta volcanic rocks beside Route 132 near CoulterviUe, California. The figure shows a general view of a roadside outcrop with planes of various sizes visible. The outcrop has five directions of plane. These are identified in figure 1 (b).

Under conditions where fractal fault behaviour operates, neither a brittle nor a ductile description is satisfactory although either can be applied with limited success. This explains the alternation o f the geological theorists between block models (e.g., MCKENZIE, 1972; HILL, 1982) and continuum models (e.g., TAPPONmR and MOLNAR, 1976; JACKSON and MCKENZIE, 1983) of continental deformation. In theory, fractal systems are self similar at all scales, but in practice fractal behaviour only operates over a range of scales between upper and lower fractal limits. Defining what these limits are, and why they occur is important to appreciating the physical significance of a natural fraetal.

Following a brief discussion of b-value, we consider large scale deformation at Plate Boundaries. This scale provides the upper scale limit of fractai fault behaviour. The behaviour of triple junctions is then discussed, and we point out that it is the geometrical requirements of deformation at triple junctions that requires that fractal fault behaviour occur. A fractal geometry for faults at such a junction is proposed, and it is noted that more faults can exist at such junctions than are required to provide any specified amount of deformation. We then consider various rules that determine why some possible faults are exploited rather than


Figure l(b). A detail of the outcrop in figure l(a) with the planes identified. Note the slicken- siding visible on some planes.

others. These occupancy rules become complex when rupture propagation effects are considered, but a combination of an appropriate three-dimensional fractal fault behaviour with a very simple rule demanding continuous deformation using the minimum of fault area predicts that the b-value of fault movement events is unity: the globally observed average.

We then outline the application of these ideas to some model fault systems and consider the inner limit of fractal fault behaviour. It is shown that the inner fractal limit demands that solids subject to shear deformation must expand and that this ultimately is the reason for frictional laws and dilatancy prior to failure. Finally the problems of initial fault formation are considered, although it is explained that the

766 Geoffrey King

a)

o=

~n o ,

+ 2 5 ° / . , a t .

4.~<>-. -'.. . . . . t.' ( Y~ - z'" | ' ' > " ~ ' < ~ . " ' . < ' k ' ~ . . . . . '( , , ,. \ . / l i , % I ~q\, +. ',',

' ~ tt ~ ! " " , ' ' / '''''>' ' " \" " " " " ; ' " ' ~ " ' '~" ( " 3 " ~ i ¢ " / " ' + " ' " ; , " .-;,"-.',,"../,~¢,7" ,,k,!;',L~-., ",. ' /

,.~ ~l ,.; , I , ' .";.4, I q 1 ; + 5 < 1 • i I

M-~;. .7 . . , ; ~ " t % ~ ' , ; ~ , ''~" .~I ' ; :" z . ,~ / ~ / , , ;;, h "~

/..13j ~,,,~51 ,),1~3~\ ,.¢ ~, ,£ ' r< . , I ' \ t . ~ " ~ i , , ~ - ,'1 "~ v,~-,,).l,,~ , D';i~'[, ,~2 13'.~4.~,'U'. ~%..'" . % , ' j -

, .7 "k "~ ~ , ~ 5 ; , ,~ ,~ ' ' " ~ ~x I J , / ~ > . ; ,~ " , / , ~i

V ' ,.."r ,.i.~ "t ~-'.~,~.7; , ' z ' ~ l

"" , ; i ~ % ' ~, ' " " • " ~ , , , ; ' ~ ..; ' , . , , , " , ~ ' I t / %,,r';w , , ' ~ . . , ' , , \ 7 ,~ --~

I " il ', - '. ¢,/#11 / M# ~ I ~'-li#.4 ") ii 7/~ ~ '1 4 I

- 2 5 " I .~ l t .

b) N c)

¢

sun light

I

Figure 2. Alinement map of the surface of the planet Mercury taken from THOMAS and MASSON (1983) is shown in (a). Diagrams of the orientation of linements are shown in (b) and (c). The diagrams in (b) are for the four quadrants of the map, and (c) is for the map taken as a whole.

discussions in the paper describe the relationship between complex fault systems once they exist but do not apparently explain how they initiate. This is analogous to plate tectonics describing the large-scale motion of the Earth's lithosphere without describing the driving mechanism.


i : l i i i r I i

20 12 ! 3Q ~ 27

~ V

2 4 4 T

3 $

o .,,. 0 " 2 5 ' ' ° ' = " ~ " "' " = 0 ' ~ 2 0 L L L . L 1

Figure 3. Aftershocks of the February 29, 1980 Pyrenean earthquake. The events range in depth from 2 to 7 km. The upper hemisphere fault plane solutions shown were constructed using upward going rays. The events fall into groups according to mechanism, but the groups are not spatially separated, indicating that different fault mechanisms are intermixed,

The figures presented in the paper show how processes at different scales relate. Drafting and printing problems, however, limit the range of detail that we can present here. The reader is urged to imagine the effect of including in some of the figures much greater detail than that shown. It is only when this is done that the apparent oversimplifications will be seen to disappear.

2. The magnitude-frequency relation o f earthquakes

The magnitude-frequency behaviour of earthquakes satisfies the empirical relation

logN r = a - bm (1)

where N r is the total number of shocks of magnitude m or greater (IsIJIMOTO and IOA, 1939; GUTENBERG and RICHTER, 1949). This holds

768 Geoffrey King

not only for earthquakes but also for microfracture in rock sample studies (MoGI, 1962; SCHOLZ, 1968). It does not hold for volcanic earthquakes (e.g., OKADA et al., 1981; TAKEO, 1983), a feature o f great interest that will be discussed later.

The constant a is of little importance since it is determined only by the number of events in a sample. The value of b or "b-value" is of greater importance. It is widely observed to be close to 1 and does not usually fall below 0.7 or rise above 1.3. Various explanations of b-value have been offered in terms of inhomogeneous stress distributions. These demonstrate that the magnitude-frequency relation can be attributed to suitable stress distributions, as indeed they must be, but they do not explain why such distributions exist. Such discussions transform the question of "why do earthquakes obey the Gutenberg-Richter relation?" to one of "Why do suitable stress distributions exist to cause earthquakes to obey the Gutenberg-Richter relation?" We shall find here that b-value is a direct consequence of long-term uniformity of finite deformation under certain geometric constraints and is a necessary consequence of the fractal nature of faulting. A consideration of stresses is unnecessary and, in practice, is very confusing.

For the purposes of the later discussion, the magnitude-frequency relation can be recast in more useful forms using the concept of seismic moment (M0). This latter is the product of fault area (A), mean fault displacement (d) and shear modulus (#) (M 0 =/~Ad). Since both fault area and fault displacement are vectors, seismic moment is in fact a t ensor (Mij). (For a recent description see MOLNAR, 1983). All o f the discussion here concerns only the geometric attributes of the seismic moment so we will use a quantity. ~0 = A d , which is again a tensor ( "~u) and we will refer to this as the moment.

The scalar moment (.//0) is empirically related to magnitude (m) by

log ~0 = cm + constants (2)

and thus

b logN r = - - - log. ~0 + constants (3)

¢

The empirically determined c varies over the magnitude range and according to the definition of rn (magnitude) employed but may be taken to be 1.5 (AI,:I, 1981; KANAMORI and ANDERSON, 1975).

The quantities A and d, which comprise the moment, are also related empirically. The relation is normally noted by the statement that earthquakes have relatively constant stress drops (e.g., SCHOLZ, 1982). This is attributable geometrically to a constant ratio of fault displacement to fault width (L- the shortest dimension of the fault plane). Thus if


we restate constant stress drop as constant strain drop (s), we find that

d = sL. (4)

If we further assume that all the faults in a system have the same aspect ratio, then

J¢" = sGL 3 (5)

where G is a geometric constant close to unity for faults with low aspects ratios.

The result that..~" is proportional to L 3 is widely quoted and implies three-dimensional self similarity (AKI, 1981). It is, however, worth noting that the aspect ratio of faulting could vary systematically with moment over a range of scales. We shall see that three-dimensional self similarity implies triaxial deformation. Biaxial deformation occurs when one fault dimension remains fixed for a wide range of fault areas. Self similarity only describes a two-dimensional cross-section and moment is proportional to L z under these circumstances.

By combining the foregoing results, we find for three-dimensional self similarity that

3b log N(L) = - - - - log L + constants (6)

c

and for two-dimensional self similarity

2b log N(L) = - - - logL + constants (7)

c

In each case N(L) corresponds to the number of events with fault length greater than L and is equivalent to Nr in the tables.

3. Deformation at a continentat plate boundary

In contrast to typical oceanic plate boundaries, the deformation of continental boundaries occurs over a broad zone. This can readily be seen by comparing the seismicity of the western United States or the Alpine-Himalayan belt with the seismicity of ocean basins. The reason for such broad zones of deformation may be understood using the basic geometrical ideas of plate tectonics.

In the oceanic environment three types of boundary are permissible: transform faults, ridges, and trenches. The second two types do not conserve area and generate and destroy plates. They exist because oceanic lithosphere can be created from or returned to the mantle reservoir. The narrow belts of seismicity associated with these features

770 Geoffrey King

result from the faulting necessary to change horizontal plate motion into the vertical motion that carries material either into or out of the mantle. This can be accomplished in a width not much greater than the plate thickness. The greater apparent width of subduction zone seismicity is due either to deep earthquakes in the subducting slabs or to deformation of continental material associated with the subducting system.

The situation in a continental environment is very different. Continental lithosphere cannot be derived from or returned to the mantle, and thus the simple plate tectonic mechanisms for the creation and destruction of plate area cannot operate. A normal fault can accommodate some extension and a reverse fault some shortening, but these features are limited to a total throw comparable to the crustal thickness. In convergent or extensional regions greater motion can be accommodated by the development of a series of parallel or subparallel features that distribute crustal thinning or thickening over a wide area. Other mechanisms such as overthrusting of one continental unit over another or motion within the crust along detachment surfaces also apparently occur to produce crustal thickening (e.g., WERNICKE and BURCHFIELD, 1982; BURCHFIELD, 1983; FROST and MARTIN, 1982) and major strike-slip faults form to transport material laterally to avoid overthickening of the crust in particular places (TAPPONIER and MOLNAR, 1976).

Whatever the mechanisms involved, it is clear that in the absence of the mantle as a source or sink of material, deformation must spread over a wide region. The ultimate cause of the difference between this behaviour and the behaviour at ridges and trenches is the low density of continental crust and not its rheological properties.

Some aspects of the nature of a continental plate boundary may be understood by considering the motion in a zone between plates as if it were purely ductile. Figure 4 shows the form of flow lines associated

const.

V

Figure 4. The compression of a hypothetical ductile crust over a ductile mantle. The flow lines to produce the observed deformation must curve and vary in spacing. A system of faults in a brittle medium required to accommodate the same overall deformation must be complex. The largest faults can only approximate the deformation, and many smaller faults at different orientations are needed to approach the continuous ductile deformation field.


with an idealized form of crustal thickening. Whatever the rheological properties of the crust may be, the flow lines must bend. If we now consider that the finite deformation is accommodated by faulting, then the fault system must evolve with time and incorporate many faults of different orientations. This is true whether the faults are straight or curved. Only straight faults or faults which are arcs of circles can accommodate motion without the creation of new structures, and the type of deformation that can be accommodated by a small number of such simple structures is very limited. Hence, in general, whatever system of faults is actually created, that system must both envolve with time and include faults in various directions.

A similar conclusion can be found by considering horizontal flow in a hypothetical ductile region between two plates. This is shown in Figure 5. (Ductile models of this sort are discussed by MCKENZm, 1979; and

a} (V2)o t . (Vt)o

1 / I / / I ~ i l l / i l l

/ / / / / / / / / / / / /

fixed

b) > c) ,,,,-77T,,,,

_ / - (

/IIIIII[1" IIIIII/II

/ / * r l / / / l l / l / 1 1 1 1 1 1 1 [

Figure 5. Ductile deformation in a zone between rigid blocks modelling a continental plate boundary. One block is taken to be fixed (lower) and the other can move as shown in (a). In (b), the upper block displaces parallel to the block boundaries to produce a simple shear flow. This can, in principle, be relieved by a simple shear fault (c). In practice it is not, for reasons discussed later in the text. A component of motion perpendicular to the block boundaries results in a much more complex flow. Under these boundary conditions a brittle material can only respond with complex faulting.

772 Geoffrey King

MCKENZIE and JACKSON, 1983.) One plate is taken to be fixed, and the second moves in the plane of the paper. The velocity in the ductile band is given by:

vl = allxl + al2x2 (8)

V 2 = a 2 1 x I + a 2 2 x 2 (9)

In general (v:) 0 :~ 0 and (vl) 0 4= 0 and under these circumstances the av are nonzero and are functions of position. A simple case occurs when (v2) 0 = 0 (Figure 5b). The coefficient a12 is a constant and the other a o. are zero. The deformation is simple shear as shown in Figure 5b. This deformation can be accommodated, in principle, by a single simple strike-slip fault (Figure 5c). In general the flow system is not simple. Figure 5d shows one possible flow pattern. Figure 5e shows a possible arrangement of faults accommodating it. It again follows that a system of faults is needed to accommodate such a flow whether the faults are straight or curved, and that they must evolve with time and include active features in different directions.

The foregoing discussion has two purposes. One is to show that multidirection faulting must occur under the conditions normally found at continental plate boundaries; it is a consequence of finite motion and the geometrical constraints placed on the system. Further- more, the faulting will include more than one fault direction whether viewed in plan or in section. Second, we will find that the fault systems, required to accommodate finite deformation, form fractai sets, and that the geometrical systems that we have described provide the upper fractal limit to these sets and hence determine the upper bound on fractal behaviour.

Before discussing fractal fault behaviour we note that, in general, finite motion on two faults that meet at an angle requires the creation of a third fault. This can be seen by referring to Figure 6. The third fault can be in one of two positions, but the orientation and slip rate is specified by the orientation and slip rate of the other two planes. The junction between the three faults is a triple junction similar to those discussed in connection with the evolution of oceanic plates and shall be discussed in that context in the next section. It should be noted at this point and in relation to Figure 6 that only motion in the plane of the paper is important. This assumption is implicit in some later figures but wiU not be restated.

4. Con t inen ta l triple j u n c t i o n s

MCKENZtE and MORGAN (1969) discuss the stability of triple junctions between plate boundaries that mix one or more of ridges,

Accommodation of Large Strains in the Upper Lithosphere of the Earth

(a )

- - c

A ~ A \

773

(c) A t3 (d) \" "1 , / \

\

C

E l

\ \ \

A r B' Figure 6. Two faults meeting must produce faulting in a third direction. Two possibilities are shown (a) and (b), with their corresponding vector triangles (c) and (d). The faulting near the junction of the faults (shaded) must be complex.

trenches and transform faults. They show that, provided certain geometric rules are obeyed (demanded by the need for ridges to spread at right angles and for trenches to consume on one side only), then all junctions that include one trench or one ridge can occur without the feature changing form. The inclusion of one or more structures, out of three, that create or destroy material permits stable behaviour.

The only triple junction that is unconditionally unstable is that formed by the junction of three strike-slip faults. We argued, however, in the last section that such junctions are the only features that can occur on a large scale in continental deformation; therefore how they behave is critical.

First we note that the relative displacement on the three faults must drop to zero at the actual point of the junction. This is shown schematically in Figure 7. The faults have a slip function that reaches a maximum at their centers and drops to zero at their ends. The faults are consequently effective at producing deformation near to their centers, but ineffective at their ends. The amount of deformation that must be accommodated by other structures is indicated.

To understand how this takes place it is convenient to think in terms

774 Geoffrey King

a)

B / I

i , i i ¢ ¢ r ¢

. 1 1 - f ~

\ \ C

b) DISPLACEMENT ON FAULT A

c)

J DISPLACEMENT

DISPLACEMENT ON FAULT C

+ ~ EQUIVALENT

• • | DISPLACEMENT EQUIVALENT T ~ OFF FAULT B DISPLACEMENT OFF FAULT A

EQUIVALENT I DISPLACEMENT

OFF FAULT C

~ b TOTAL / DISPLACEMENT

~, • ~ AND EQUIVALENT I ,~ DISPLACEMENT

~ TOTAL DISPLACEMENT ~qJlll AND EQUIVALENT

"%,1 DISPLACEMENT FOR FAULT

o

d) a" TOTAL DISPLACEMENT AND EQUIVALENT DISPLACEMENT FOR FAULT A

Figure 7. Processes where three faults meet. In (a) at the junction where the faults meet their displacement is zero. The displacement on the faults increases away from the junction but will come to zero again at the next junctions (not shown). The displacement as a function of position along the faults is shown in (b). The area under the fault displacement curve is proportional to the moment released by the fault (the figure is for a two-dimensional representation so that moment is proportional to the product of fault length and displacement). The equivalent displacement that must be accommodated by off-fault structures to permit main fault displacement to come to zero at the barrier is shown in (c). This sums with the fault displacement to give the equivalent total displacements shown in (d). Total displacements must sum to zero in a vector fashion as shown.

of moment. To illustrate this the area that is hatched in Figure 7b is the moment that can be accommodated by the main faults, and the hatching in Figure 7c indicates the moment to be accommodated elsewhere such that the total effective deformation is the same as if the faults actually met with constant displacement (Figure 7d). At this point we shall consider faults of infinite extent perpendicular to the direction of slip and


therefore regard the moment as proportional to the fault length squared (biaxial deformation). The point of intersection of the faults we shall term a "barrier" as defined by KING and YIELDING (1983) following work of AKI (1979). We shall find later that the junction only creates the conditions that produce a barrier and is not itself a barrier.

In order to proceed further we must make assumptions about the slip function (i.e., the distribution of displacement on the fault plane). For ease of visualization we take it to be triangular for the time being. Under these circumstances 50 percent of the deformation (or moment release) takes place on the main faults and 50 percent must be taken up elsewhere. The choice of a triangular function and the 50:50 ratio is simply for convenience of discussion in the absence of any better alternative distribution. It might be argued that the slip distribution for a crack in an infinite medium (similar to that shown in Figure 7b) is more appropriate. However, later discussion will show that elastic approxi- mations are not necessarily correct because the medium cannot be regarded as a continuum at any scale. Thus the lack of clarity introduced by considering a nontriangular slip function is not justified.

Before looking at an intersecting junction such as that shown in Figure 7a we can visualize some of the features to be discussed later by considering a single fault with a triangular slip function coming to zero at two barriers (Figure 8). This slip function is shown by the shaded area in Figure 8b. The rest of the deformation is taken up by an infinite number of smaller faults. These are shown in Figure 8a but their position is unconstrained. The smaller faults fall into a series of orders. Each order has faults of half the length of the next highest order, and each order is composed of twice the number of faults of the next lower order. Order 1 is taken to be the main fault. These features are summarized in Table 1. The moment of faults of each order is shown on the assumption that the faults are constant in length in the third dimension (into the paper) and hence moment is proportional to the square of fault length (the fault of order 1 is assigned a length of 1).

Examining the right-hand column shows that the sum of the moment of all the faults of order less than 1 equals the moment of the main fault (2 = I + ½ + ¼ + ~ + ~6 + ..., etc.). Using Table 1, a fault length frequency relation can be plotted (Figure 9) and the slope compared with relation 7. A b-value of c/2 is found for small events. It is important to appreciate that this b-value does not define the relative amount of deformation on and off the main fault. This can be seen if we consider a system of faults with the behaviour shown in Table 2, which also gives a b-value of c/2 for small events (log L < 0), but in this case the proportion of deformation accommodated off the main fault is half the previous case (1.5 = 1 + ¼ + ~6 + ~2 + .... etc.). (The slip function is no longer triangular.)

776 Geoffrey King

a) 2

f . . e , , . . , . . .o ~ ~ A A

b) Stip

Figure 8. Brittle deformation off the main fault plane. In (a) a single fault is shown with two barriers at which displacement comes to zero. Schematically a set of smaller faults is indicated, each ending in barriers. These faults accommodate deformation that the main fault cannot accommodate. The way in which this takes place is shown in (b). The slip on the main fault is indicated by the shaded triangle. The slip on the smaller faults is indicated by the smaller triangles. The slip behaviour of all faults is required to be self similar (i.e., the slip functions are similar triangles). The sum of the slip on the main fault and an infinite series of smaller faults is required to give constant overall slip along the fault zone. This figure illustrates the principles to be used to ensure self similarity and determine the relative proportion of off-fault and on-fault deformation in later, more complex models. Notice that the triangular slip function requires that 50 percent of the deformation occur on the main fault and 50 percent off the main fault. This ratio is chosen for much of the subsequent discussion because it makes it easy to visualize how moment is distributed between different scales of faulting when the function is triangular. However, the basic arguments do not depend on the slip function being triangular.

Table 1

Cumulative number Total moment

including all Moment of of all faults Order Length Number lower orders each fault of given order

(n) (L,,) (N.) (Nr) (.4".) ( N . . / / . )

1 1 1 1 1 1 2 1/2 2 3 1/4 1/2 3 1/4 4 7 1/16 1/4 4 1/8 8 15 1/64 t /8 5 1/16 16 31 etc. 1/256 1/16

The origin of the difference can be seen by examining Figure 9 and arises because true self-similar behaviour requires that fault size relations be described by geometric series and not by continuous functions. Before it is possible to determine how deformation is partitioned between larger


tog N

0 -3

ble 1

-2 -1 o tog L

Figure 9. The logarithm of the total number of faults with length greater than L plotted against the logarithm of fault length L. The upper line is for Table 1, which corresponds to faulting with 50 percent of the deformation occurring on the fault and 50 percent off the fault (defined in Section 9 as ~# = 1). The lower line is for Table 2 where the amount of off-fault deformation is halved ( # = 0.5). Both lines have similar slopes and hence the b-value of both systems is the same except near to log L = 0. Thus b-value alone does not define the ratio of on-fault to off-fault deformation. Other geometrical information is needed.

Table 2

Cumulative number Total moment

including all Moment of of all faults Order Length Number lower orders each fault of given order

(n) (L) (N,,) (Nr) (. #',,) (Am, ~, ,)

1 1 1 1 1 1 2 1/4 4 5 1/16 1/4 3 1/16 16 21 1/256 1/16 4 1/64 64 85 1/4098 1/64

etc.

and smaller earthquakes, it is necessary to have information about where secondary faults are located and what their dimensions are. It is for this reason that the result based on b-value derived by WYss (1973) suggesting that half the deformation in a region occurs on the largest faults is not necessarily correct.

When a linear fault with barriers is considered, there are no constraints placed on the locations of minor faults. However at a triple junction the possible positions of secondary faulting become more tightly constrained by the requirement that faults do not cross (i.e., where two faults meet, one must have zero displacement) and the requirement that the entire system must be self similar. Before this is discussed it is necessary to consider fractal geometries.

778 Geoffrey King

5. Self-similar geometry and fractals In some recent books and articles Mandelbrot has drawn attention to

the wide distribution in nature of geometrical patterns that look the same at whatever scale they are examined (i.e., they are self similar). These self-similar figures he terms fractals and discusses their properties. Natural fractals include trees, rivers, coastlines and the shapes of mountains.

A basic property of fractals is illustrated by posing the apparently innocuous question, "How long is the coast of Britain?" The question has no simple answer, since the length increases indefinitely as the interval at which it is measured decreases. It is, in fact, unreasonable to pose the question unless a scale is specified at which further detail may be ignored. It is possible, however, to ask how much the length increases as the level of scrutiny increases. A plot of the logarithm of the length of a coastline against the logarithms of the length of the yardstick used to measure the coast yields a straight line. The slope of this line is 1 - D where D is the fractal dimension of the coastline. The west coast of Britain has a fractal dimension of 1.25 (i.e. length ~ (yardstick)-l/4).

The significance of fractal dimension that relates behaviour at one scale to behaviour at another is most readily understood by considering features that are less random than coastlines. Figure 10 shows an example of the development of a fractal shape and the way in which it is constructed. The shape is based on a triangle, which is termed the initiator. This is then operated on at each successive scale by the generator indicated at the bottom of the figure. The form of the generator determines the fractal dimension D of the final figure

log N D = ~ or Nr ° = 1 (10)

log 1/r

where N is the number of equal segments in the generator and r is their length as a fraction of the direct end-to-end length of the generator. The example has N = 4 and r = ]. D is therefore log4/log 3 = 1.26.

The fractal dimension relates in an interesting way to Euclidean dimension. A generator with a dimension of 1 that connects its two end points will produce a straight line out of a straight line, and an initiator operated on by such a generator will therefore remain unchanged. A generator with a dimension of 2 will completely fill an area if no overlaps occur, and a generator with a dimension of 3 will fill space. A generator with a fractional dimension between 1 and 2 will partially fill an area, and a generator with a dimension between 2 and 3 will partially fill a volume. Not all generators are continuously connected and hence do not produce figures that are continuously connected. Thus a generator with D = 1 can turn a straight line into a dust. Higher dimension generators

Accommodat ion of Large Strains in the Upper Lithosphere of the Earth 779

Figure 10. An example of a simple fractal figure. The initiator is shown at the top of the page and the generator is at the bottom. The intermediate figures are formed by successive operations of the generator on the initiator. The generator has N -- 4, r = 3, D = log4/log3 ~_ 1.26 Note that because the elements of the generator are non-parallel the fractal dimension of this figure will change if it is distorted. This is not true for the faulting figures (11, 13).

can exhibit the same property including the generator that we adopt to describe fault behaviour. The fractal dimension of figures produced by such generators cannot be determined empirically in the same manner used to determine the lengths of coastlines. Other methods must be adopted, since a set of faults forms a disconnected rather than a connected system.

It is important to appreciate that, while it is simple to produce a generator of any chosen dimension, it is difficult to find generators that

780 Geoffrey King

by not creating overlaps produce "real" figures. Once a generator is found, the construction of a figure is very simple and results from repeated application of a simple rule. Computer algorithms to d raw such figures are therefore very compact. Mandelbrot points out that if understanding a phenomenon is no more than having a simple w a y to describe that phenomenon, then fractals permit a large range of phenomena to be understood. It is well worthwhile pondering the significance of this statement.

Although a perfect fractal must exist at all scales, in practice there are inner and outer limits at which fractal behaviour ceases (the inner and outer fractal limits). It is clear that the fractal behaviour of the British coast will cease at the atomic level if not at the crystalline level, and at the upper scale limit, all coastal scales are limited by the circumference of the Earth if not by smaller-scale tectonic processes.

Appreciating the nature of the fractal limits of a fractal process can be the key to understanding why the system exists. This can be illustrated by considering a tree. The upper fractal limit of a tree is the scale of the trunk, which connects the tree to the root system below the ground. The lower fractal limit is the scale at which twigs connect to the leaves. One purpose of the fractal system is to carry fluid, which is best transported in large conduits (the trunk and larger branches), and spread it over a surface (the leaves). The fractal geometry of the branches connects from the one-dimensional trunk to the two dimensions of the leaf canopy in an approximate way. The need to do this arises because photosynthesis operates most effectively in approximately two-dimensional structures, whereas fluid transmission is most readily accomplished in one- dimensional structures. The root system operates in a slightly different way, since fluid in the soil is collected from a volume and not from a surface.

A fractal remains a fractal when it is incomplete. Trees grow by adding new twigs at regular intervals. Most of these are lost as the tree grows. Thus an algorithm to produce a tree must include rules to both create and delete features. The fractal shape that we will use to describe faulting includes more elements than are needed, and some of the simplicity of the complete figure is not apparent because only a part of it ever comes into existence. Natural fractals in general may be expected to behave in this way. Thus there is an important distinction between the fractal that describes all the possible faults that could accommodate a certain type of deformation and the actual fractal fault system that occupies part of the theoretical figure. The two need not even have the same fractal dimension.

A fundamental property of fractal shapes is that they cannot be differentiated. The generator form appears when a small element is examined, and a tangent is never the result of close inspection. Calculus


can be used at the inner fractal limit if Euclidean behaviour is restored, or at any scale at which the features may be blurred such that a fractal description is unimportant.

At any scale at which a fractal description is needed, and this applies to some of the most important aspects of fault behaviour, calculus cannot be employed. The earlier discussion concerning Figure 8 illustrates the type of problem to be encountered. Thus calculus-based concepts such as stress and strain must be abandoned.

6. A fractal forrn for two-dimensional faulting

Provided that only a single-fault orientation is considered, all manner of generators may be constructed to produce secondary faulting to permit deformation around barriers. Where three faults intersect and no faults are permitted to cross or duplicate, however, the fractal forms are severely constrained. It is not clear to the author just how constrained. One satisfactory structure is discussed but others may exist.

All the figures here are drawn for faults that meet at a 120 ° or 60 ° angle. This is not a limitation since any system that operates for these angles will operate for any other angles. This may be appreciated by imagining all the forthcoming figures to be drawn on highly deformable rubber membranes.

Figure 11 shows part of a system of faults based on a three-fault initiator system (which can extend to a hexagonal grip). The generator produces four subfaults for each main fault, N = 4, r = 0.5. The fractal dimension of this system is therefore 2. The fault system covers area without any overlapping or fault crossings. Note that the figure as presented is only part of the ideal figure for two reasons. First, some of the faults that should result from the first application of the generator are not shown. Second, only five applications of the generator are shown.

The deformation that such a system can accommodate is indicated in Table 3. It is apparent from the table that, for this geometry, if all the possible faults exist, then equal amounts of total moment are contributed by all scales of faulting, and that the deformation provided by all the smaller faults is greater than that provided by the largest fault. However, the fault length that must be created to produce the same deformation doubles with each increase in order. Thus, if the energy required to deform the body depends only on the total length of faulting that must move, then lower-order faulting will be produced in preference to higher-order faults. This is the generally observed behaviour of faults.

If we adopt the triangular slip function used earlier, we can now discuss how the possible fault positions can be partially occupied in a self-similar fashion. The triangular slip function requires that all faulting

Geoffrey King

J£ J

782

0.5

05

05 ~ J

t-0 ~ /

0.5

z j z b J

Figure 11. A fractal geometry for two-dimensional faulting corresponding to Figure 6a. The initiator consists of the faults a, b, and c, and the generator is shown at the bottom of the figure. Five orders of faulting are shown corresponding to four applications of the generator. The figure has the property that no fault intersects any other fault. The generator has N = 4, r = ½, D = 2. This remains true if the initiator is uniformly distorted, provided that the generator is subject to the same deformation. The same is not true, in general, for fractal figures (e.g. figure 11).

at higher orders than a specified order accommodate the same deformation as that order. Such a system is described by Table 4.

At each successive order both the proportion of subfaults utilized and the moment relieved is halved. The total fault length at each scale is ~he same, but the effectiveness of faulting to produce deformation reduces as

Accommodation of Large S t r a i n s in the U p p e r L i t h o s p h e r e of the Earth

Table 3

783

Total fault Moment Cumulative Proportion length at of each Total

Order Length Number number of positions given order fault moment (n) (L) (n) (N r) used (N,L,) (,¢',) (N, .¢',)

1 1 I 1 1 1 1 1 2 1/2 4 5 1 (all) 2 1/4 1 3 1/4 16 21 I (all) 4 1/16 1 4 1/8 64 85 I (all) 8 1/64 1

etc.

Table 4

Total fault Moment Cumulative Proportion length at of each Total

Order Length Number number of positions given order fault moment (n) (L) (n) (N r) used (N~L,) ('¢'n) (N~ If,)

1 1 1 1 I 1 1 1 2 1/2 2 3 1/2 1 1/4 1/2 3 1/4 4 7 1/4 1 1/16 1/4 4 1/8 8 15 1/8 1 1/64 1/8

etc.

the order decreases. This is a direct consequence of moment being proportional to the square of fault length.

The arrangement utilizes only a small proportion of the fractal pattern shown in Figure 11, and with N = 2 and r = ½ the fractal dimension becomes equal to 1.0. This arrangement can be considered to be the fault system that will be produced in a completely rigid, brittle body in which no deformation can be taken up in any other way. Any displacement on the first-order fault must be accompanied instantly by motion at all scales. If continuum deformation is permitted, then some fault motion can occur without requiring further faulting at smaller scales. We shall consider how this can operate in the next section.

7. Simple geometrical implications for fault behaviour

To allow for continuum deformation we shall examine the consequences of another rule. We shall permit any fault to move one displacement unit (inversely proportional to fault length) without motion on any other faults. For the same fault to move a second time, however, either two associated faults of the next highest order must move, or the

784 Geoffrey King

equivalent deformation must occur on even higher-order faults. This can be thought of in the following way. The deformation associated with one unit of fault motion can be accommodated elastically in the surrounding media, but this elastic strain must be relaxed by secondary faulting before the fault can move again. The fractal system is only required to extend down to a scale at which elasticity can accommodate the remaining deformation.

The simplest sequence that obeys the above rule is shown in Figure 12. We move the main fault (order 1) three times, the first time without the creation of secondary faulting (Figure 12a). Two second-order faults then appear to allow it to move a second time (Figure 12b). Two more second-order faults permit it to move a third time (Figure 12c). The sequence is shown schematically in Figure 12d. It is helpful to appreciate that each of the second two increments of faulting has a fractal dimension of 1 (N = 2, r = ½), but the resulting faulting has a fractal dimension of 2 (N = 4, r = !) and illustrates in a primitive way how the 2 fractal dimension of the process that can create a figure may differ from that of the final figure.

An alternative arrangement to allow the main fault to move three times is shown in Figure 13. The first two stages (Figures 13a and b) are the same as before, but in the third stage, third-order faulting appears, and the second-order faults move for a second time. In this case both the process and the figure have fractal dimensions of 1.

Figures 12 and 13 are both so simple that they may appear unrealistic. It is, however, straightforward in principle to construct fault sequences in which the simplicity although present is much less obvious. This is illustrated in Figure 14. All second-order faulting is absent, and the compensating deformation at higher orders of faulting is distributed randomly. In this example, the main fault has still moved only three times. If we now imagine the effects of the other two main faults and of more increments of movement, a very elaborate pattern of faulting can result.

It should be pointed out that failure to fully exploit the lowest orders of faulting requires that a greater total length of fault move or be created than the minimum required to produce the necessary deformation. Presumably, in terms of the energy of fault surface creation alone, irregular systems are less efficient. In other words the creation of the largest faults of the fractal pattern first (i.e. the scheme of Figure 12) is optimal. However, other criteria also determine the overall energy involved in faulting. We shall return to this question in a qualitative sense when discussing examples of observed fault behaviour, but one geometric figure deserves examination here.

This is shown in Figure 15. The figure is similar to Figure i 1 except that fault c has been shifted to lie between a and b, and only faults that


b) 2 2

lr2

c)

2 2

1r2f3

3 3

d) Order

Figure 12. Sequences of movement on a fault and subfaults. The faults should be regarded as one orientation set from a system such as that shown in Figure 11. In (a) the main fault moves in the first increment of deformation. This is indicated by the 1 marked beside it. The rest of the deformation is accommodated elastically. In a second increment of deformation, (b), two second-order faults move and the main fault moves again. This is shown by marking the second-order faults and the main fault with the number 2. The final increment of motion is shown in (c) and two more second-order faults (marked 3) move. The main fault, which has moved each time, is marked 1, 2 and 3. The process is summarized in (d) which indicates a "family tree" of fault orders and indicates which have moved and in which increment of deformation.

deform the region of the bend between faults a and b are included. We can construct tables for the secondary faulting of this system noting that a and b behave in the same way (Table 5) but differently from c (Table 6).

786

a)

Geoffrey King

1

2 2

b) 1,2

c)

2,3

3 3

lf2,3

2,3

3

d) Order 1

2

3

1,2,3

33 33

Figure 13. A figure similar to Figure 12. The same deformation occurs for the first two increments shown in (a) and (b). However, in (c) a third increment of motion occurs on third-order faults and possible second-order faulting is not exploited. The "family tree" for the sequence is shown in (d). Only faults marked with deformation increment numbers have moved. The other lines correspond to possible but unexploited fault locations. The total fault length created for three increments of motion on the main fault is the same as for Figure 12, but the second-order faults must move twice.

With a system of the sort shown in Figure 15 it is apparent that the system cannot repeat indefinitely (on a hexagonal grid) in the same way as Figure i 1. The junction region can therefore be regarded as falling under the influence of only half of each fault. Considered in this way, the figures in brackets are appropriate for fault numbers and total moment since the other numbers imply an unrealistic infinite repetition. However, since the ratios remain the same whichever set is chosen, this does not affect the following discussion. This sort of problem inevitably results at


a)

2 2 2

2 lj2 Z 2 - 2- ,, b)

2 , 2 2 2_..

2 -

I 3 r /

1,3 z~ z3 --2 _ L _ L 2 2 3

If2~3 2 3 2 3 2 3 3

3 .2. 2--

z 3 3

Order

1

2

3

i d) 1,2,3

2 33 23 32 23 3 2 32 3 3 2 3 3 2 3

Figure 14. A more complex fault evolution than those shown in Figures 12 and 13. No second-order faults move to permit a second increment of main fault motion. Instead, six third-order and eight fourth-order faults move. The third increment of motion is accompanied by motion on two further third-order faults and further motion on the six that moved in the second increment of deformation. To permit this, motion also occurs on 12 fourth-order faults. The "family tree" is shown in (d). Many possible fault positions are unoccupied, including all possible second-order faults. The total length of faulting created is greater than for Figures 12 or 13.

7 8 8 Geoffrey King

a

C

b Figure 15. A fractal form for faulting corresponding to Figure 6b. With deformation restricted in the way shown, some faulting orders are only partially available or are absent.

the upper limit of fractal behaviour. The outer fractal limit boundary conditions require that some parts of an idealized fractal figure cannot exist.

It can be seen from Table 5 that a deformation system can proceed that includes all orders of faulting for faults a and b, but for c, the second-order faults generated by c are absent. There are also insufficient third-order faults to permit repeated motion on the main fault. For repeated motion, all faults to the fifth order must move for every movement of the main fault, and only at higher orders can partial motion occur. Thus the system requires more fault length to be created than the system in Figure 11 for a given amount of total deformation, but the deformation is more localized. Geometries of this type occur in the upper


Table 5

Moment Order Length Number each fault Tota moment

(n) (L~) (N,) (~',) (N~ ,r,)

1 1 1 (1 /2 ) 1 1 (1 /2 ) 2 1/2 2 (1) 1/4 1/2 (1/4) 3 1/4 6 (3) 1/16 3/8 (3/16) 4 1/8 32 (16) 1/64 3/8 (3/16) 5 1/16 96 (48) 1/256 3/8 (3/16)

etc.

Table 6

Moment Order Length Number each fault Total moment

(n) (Ln) (N~) (d"~) (N n ~n)

1 1 1 (1 /2 ) I i ( 1 /2 ) 2 I/2 0 (0) 1/4 0 (0) 3 1/4 4 (2) 1/16 1/4(1/8) 4 1/8 16 (4) 1/64 1/4 (1/8) 5 1/16 64 (32) 1/256 1/4 (I/8)

etc.

block of low angle dip-slip faults, and pairs of similar structures occur at offsets in strike-slip faults.

8. Fraetal faulting in three dimensions

The discussion so far has concerned two-dimensional faulting. These are systems in which all faults have one common dimension along which no deformation occurs. The equivalent continuum behaviour is described as plane strain. Such systems give a b-value of 0.5 c. Taking c to be 1.5, this produces a b-value of 0.75, which is lower than that normally observed. A more realistic b-value is obtained with three dimensional deformation.

Drawing three-dimensional figures is a difficult, time-consuming process. The nature of triaxial fractal deformation, however, can be understood as a generalization of the previous discussion. We note first of all that while three sets of planes are needed to permit biaxial deformation, five are required to permit triaxial deformation. This commonly quoted result is a consequence of von Mises' criterion (YON MISES, 1928). It is the ability of metal crystals to permit dislocation

790 Geoffrey King

motion on five or more planes that allows metals to behave in a ductile fashion. The rock deformation that we are discussing is, in effect, an enforced ductile behaviour. Many rock faces show a rhomboidal structure cut by two other planes. An example was shown in figure 1.

To discuss the geometry, we shall consider plane orientations that have the angular relations of a square-based pyramid (Figure 16a) and a generator of the form shown in Figure 16b. The figures produced by the foregoing combination are very difficult to visualize. The generator must be applied to the boundaries of the initiator planes in such a way that a series of self-similar planes with the same angular relations is created. As an aid to visualizing the three-dimensional geometry, the author constructed a large number of pyramids similar to that shown in Figure 16a with sizes varying by factors of two.

It is immediately clear that however the five planes are chosen, they cannot all be the same shape. Some can be rectangular and some triangular. They cannot be either all triangular or all rectangular. The implications of this are not considered.

For the new initiator we can construct a table (Table 7) that indicates the deformation available at different scales in the same way that we have done for two-dimensional systems. The moment is now proportional to the cube of the length of a fault.

Again we find that more small faults exist than are required, if 50 percent of the deformation is to be accommodated by higher-order

Figure 16. ½, D = 3.

a) b)

I

The initiator (a), and generator (b), for three-dimensional fractal faulting. N = 8, r =

Accommodation of Large Strains in the Upper Lithosphere of the Earth

Table 7

791

Cumulative Moment of Fault Total Order Length Number number each fault area moment

(n) (L,,) (N.) (Nr) ( / / . ) (a.) (N. ~'.)

1 1 1 1 1 1 1 2 1/2 8 9 I/8 2 1 3 1/4 64 73 1/64 4 1 4 1/8 512 585 1/512 8 1 5 1/t6 4096 4681 1/4096 16 1 6 1/32 32768 37449 1/32768 32 1

etc.

Table 8

Cumulative Total Moment Total Order Length Number number fault area each fault moment

(n) (L,,) (N,,) (Nr) (a,,) ( / / . ) (N. . / / . )

1 1 I I I 1 1 2 1/2 4 5 1 1/8 I/2 3 1/4 16 21 1 1/64 1/4 4 1/8 64 85 1 1/512 1/8 5 1/16 256 341 1 1/4096 1/16 6 1/32 1024 1365 1 1/32769 1/32

etc,

faulting. Again, the fault area required to produce the same deformation using higher fault orders is greater and increases rapidly as fault dimensions are reduced. The three-dimensional system can therefore behave in a similar way to the two-dimensional systems described earlier. If we apply the 50 percent rule (or any other constant ratio rule) and only partially occupy possible higher-order fault sites as shown in Table 8, we find an interesting result. A plot of log N r against.~", has a slope of 2/3. Taking c to have a value of 1.5, the resulting b-value is 1.0. This is the value observed most generally for earthquakes. The square-based pyramid of Figure 16a is deliberately placed on its side. This is the orientation that conforms to the ANDERSON (1951) fault criterion. The square vertical plane corresponds to strike-slip faulting releasing simple shear. The other two vertical planes release pure shear as conjugate planes, and the conjugate planes at an angle to the vertical are normal or reverse faulting. Insofar as the Anderson criterion is obeyed and the generator pyramid is in the orientation shown, horizontal sections through any fault system will exhibit three fault-strike orientations.

792 Geoffrey King

9. Fractal dimension of active faults and the origin of b-value

Using the foregoing discussions as a basis, we can now general ize the relation o f fractal dimension (D), b-value, and the p ropor t ion of deformat ion taken up on the main fault and higher orders o f faulting. First we note the following relations.

1 N = - - (11)

where N is the number of subsegments associated with a change o f scale r, and D is the fractal dimension. The expression is a res ta tement of equat ion (10).

At any specified order n the length of each fault L,, is

L, = r ~ (12)

and the corresponding moment

/ / . eL (L. ) d (13)

where d relates fault dimension to moment. For a two-dimensional fault d = 2, and for three-dimensional faulting d = 3.

The total number of faults up to a given order n is given by

N r = l + N I + N 2 + . . . N n (14)

= 1 + N~ + (NO 2 + ... (N~)" since N = N~

N ~ - I N r = (15)

N - - 1

Am ~ _ - - for N ~ > I (16)

N - - I

F rom 11, 12 and 13 A m =. / / -~n/a so that the total number of faults with moment greater than or equal t o . / / , is

N T ~- (17)

D logN r = - - - - log. ~" + constants (18)

d

Thus by compar ison with equation (3) we find

D b - - = ( 1 9 ) d c


This result agrees with that of AKI (I 981) provided d = 3. This fractal dimension is however that which is associated with faults active at any given time and must not be confused with the fractal dimension of the fault system produced over a period of time, which has a higher dimension (see Section 7).

Equation (19) suggests that the b-value is determined by the ratio of fractal dimension to the parameter d which, in turn, is determined by whether the deformation is biaxial or triaxial. This result must be treated with caution since the empirical constant c could be similarly related.

The total moment ( # r ) released by the fault system can also be calculated. L e t / / r = 1 + . d where the main fault has a moment of 1 and the ratio of off-main fault to on-main fault deformation is .~, then:

/ / r = 1 + , ~ ( 2 0 )

= 1 + (r a-°) + (r~-°) 2 + ... + (r a-D) (21)

1 - - - ( 2 2 )

1 - r a - °

i • ~ = - - ( 2 3 )

r ° - a - 1 1

1

Nr a

We find that . d is a function of (l/r), N and D. The values that these may take are constrained by the geometry of fractal faulting, and the remainder of this section is concerned with exploring what these constraints may be.

For the fault model we have outlined it appears that ( l / r ) must equal 2, and for triaxial deformation d = 3. We assumed that N = 4 but any integral value between 1 and 7 is apparently possible. Table 9 shows .~, D and b-value for varying values of N.

Although choosing values for N not equal to 4 is possible, alternatives are less satisfactory in certain respects. Any odd value of N gives more

Table 9

N .,¢ D b

1 ~ o o

2 } 1 0.5 3 ] 1.58 0.79 4 1 2 1.0 5 1} 2.32 1.16 6 3 2.58 1.29 7 7 2.8 1.4

794 Geoffrey King

secondary faults at one end of a main fault than at the other. Thus strict self similarity is not followed. From the possible even values, N = 2 gives a very low b-value, and N = 6 a high b-value. Thus unless fractional values of (I /r) are possible, the geometry with N = 4, ( l / r ) = 2, D = 2 and b = 1 is the most plausible. The slip function has the property tha t as much deformation occurs off the main fault as on the main fault ( ~ ' = 1). In the two-dimensional examples the slip function was triangular and obviously unrealistic. For three dimensions the form of the slip function can be visualised by referring to Figure 17. The figure caption describes possible analytic functions to approximate the slip and notes tha t one with a sine squared form gives ,~ = 1. This is a physically plausible slip function.

Despite the pleasing simplicity of the N = 4, (I / r) = 2 system we should note that b-value is an average of many events and under such circumstances it is possible to relax the requirement that ( l / r ) and N be integers. The same effects as giving them fractional values can be created by missing out a certain proportion of faults at all scale dimensions.

This can be illustrated by considering a two-dimensional system with .;~ = 1 that has r = ½ (N = 2) and gives

. z / r = I + ½ + ¼ + - ~ + ~ + .... (25)

We now miss out alternate orders

/ ¢ ' r = l + 0 + ¼ + 0 + ~ 6 + ... . (26)

a) stip / m'y

b sip f Y

. . . ~ ×

Figure 17. The approx ima te form of the slip function with , ~ ~ 1 for a square faul t p l ane : (a)

A c i rcular or elliptical form that does not extend to the corners of the p lane gives a~ ~ 0.9. (b) A

function ex tending to the corners can be approximated as u = sin x sin y, which gives , ~ ~ 1,5 or u = sin 2 x sin 2 y, which gives . ~ = 1.0.


and find .9~-- 0.5 and an effective r = ¼ ( N = 4). Repeated faulting, however, can utilize the intermediate scales. The rule that was proposed earlier that any fault can only move a second time if two faults move at the next order (i.e., N = 2 always), or the equivalent at higher orders, is relaxed. The new condition is that N and hence r can take any value to give the correct slip distribution. Fractional values of N are to be interpreted as the result of orders being randomly or systematically missed. The long-term average gives the appropriate N and r. A corollary is that the slip function on the fault will not be identical for each event.

It is interesting to note that in general most fractional values of N will result in apparently irregular temporal behaviour at high but not low orders of faulting. Regularity depends on N being an integer. This has implications for earthquake repeat time. At low fault orders (i.e., near to the upper fractat limit), earthquakes will recur at regular intervals if the plate boundary motion is constant. At higher orders the behaviour will apparently be irregular because regularity appears only at periods that are long, compared to the repeat time of the main events.

10. Some other considerations in the distribution of secondary faulting: The relation to fracture mechanics

In the earlier discussion it was noted that not all of the possible fault positions need to be occupied to accommodate the deformation that must occur as secondary faulting. The discussion then proceeded as if the only consideration were to minimize total fault area or to satisfy a simple geometric rule such as confining deformation to one sector of a fault junction.

Another consideration is likely to be important, and a discussion of it indicates how the geometrical description presented so far can be related to ideas of engineering fracture mechanics (e.g., KNOTT, 1973; BROECK, 1974; LAWN and WmSHAW, 1975). We proceed from the suggestion that a fault can move once without the creation of secondary faulting. Once an appropriate amount of secondary faulting has occurred, the fault can move again. For this to occur the secondary faulting must, in some approximate sense, relieve the strain produced by the first fault motion. An alternative view is to say that the secondary faulting must accommodate the off-fault moment produced by the first fault increment. Previously we implied that this could be accomplished by any suitably-sized fault or sequence of faults in any position in the fractal distribution. Now we suggest that there is a spatial constraint on where these can be.

Near a fault tip the elastic strains have amplitudes of the form:

K~ e = (27)

X/'2 zra

796 Geoffrey King

where K E is a strain equivalent of the stress intensity factor and a is the distance from the fault tip. All the strain components obey this expression but with different azimuthal distributions. For our purposes we will simply note that strain is inversely proportional to the square root of distance from the fault end.

The strain can be thought of as moment per unit area in two dimensions (moment per unit volume in three dimensions). This is the moment that must be released by secondary faulting. The moment ~¢', to be released in any specified radius a0 is therefore

Ot 0

• ~ a f e d A a a3o/2. (26) 0

From Figure 18 we can calculate the moment that may be released by the largest faults within any given a 0. This is

/ / I ct ao 2. (27)

The two functions are plotted in Figure 19. If we equate a 0 with the zone within which secondary faulting occurs, then we see that as the zone expands the available fault moment to relieve the applied stress increases rapidly. An equilibrium between the stress intensity factor and the area of faulting can clearly be achieved with faulting preferably occurring in the region where the main fault repeatedly produces most stress. The balance, however, will be more subtle than Figure 19 suggests because, for any given radius, several scales of subfaulting will operate and not simply the largest. Furthermore, the distribution will be modified by the azimuthal distribution of strain. We should also note that after many fault movements the distribution of strain (moment) to be relieved may no longer be similar to that for a single fault in an elastic medium. Nonetheless, the description provides a way in which to understand some further aspects of fault behaviour. Some aftershock distributions do show maxima in regions where the main fault modelled as a simple crack predicts further deformation (STEm and LISOWSKI, 1983). It should be noted that faulting can only match the applied strain within a 0 (critical) if each fault has a larger than usual number of subsidiary faults (i.e., D > 2). There are simply not enough positions for big faults near the crack tip to accommodate all the deformation by low-order faulting. Outside the region of a 0 (critical) the reverse is true, and fewer than the usual number of large faults are needed (i.e., D < 2). Thus the b-value associated with relieving crack-tip strains that are approximately elastic in form will have a b-value greater than 1 near the end of the crack and less than 1 well away from it. In the next section we suggest that the temporal change of b-value associated with the earthquake cycle can be associated with spatial changes in earthquake distribution in relation to the main faults.


f ~

/ / \ . a/\

b,

cl / < . /

Figure 18. The faulting within a distance a 0 of a main fault. In (a) four third-order faults are included; in (b) four fourth-order faults, and in (c) four fifth-order faults. The deformation that can be accommodated by the fractal set is proportional to a~.

11. Rupture termination and initiation: The geometrical nature of barriers and asperities

The relations described in the last section enable us now to discuss the way in which a fault triple junction evolves and to discuss how it can act both as a barrier to terminate rupture and also create the asperities that rupture to initiate the next earthquake (see AKI, 1983). We can at the same time relate the fractal description of faulting to the engineering concepts of fracture toughness, critical stress intensity factor and Griffith's energy release rate.

Figure 20a shows a sequence of earthquake events on a set of faults. One event is shown to trigger the next event in a sequence that

798 Geoffrey King

ao 2

8J a 0 criticat

Figure t9. The deformation that may be accommodated by fractal faulting (~ ao z) and required by crack tip deformation (~ a03/2). A radius a 0 (critical) always exists at which more large faults exist than are required to relieve the crack tip strains. This radius will scale with the length of the main fault.

propagates from left to right. The junction that creates the barrier for the first rupture is also associated with creating the asperity or asperities associated with initiating the second rupture, and so forth. We assume for the discussion that the events are separated by a time interval sufficient for us to talk about aftershocks, periods of quiescence and foreshocks. The events however, could proceed rapidly one after another to form a multiple event.

Figure 20a indicates the first rupture propagating away from Asperity 1 towards the triple junction associated with Barrier 1. When it reaches the triple junction, rupture commences on one or both of the other two fault branches (shown) and on their subsidiaries (not shown). The result is simultaneous motion of sets of interlocking faults with a wide range of scale sizes. The motion distributes deformation over a zone and locks the large faults. The dimensions of the zone spread to the point where dynamic rupture terminates.

The multiple-scale faulting behaviour is analogous to ductile behaviour at a crack tip in an engineering material. The plastic zone inhibits the extension of the crack and contributes to the fracture toughness of the material. In a general way the zone over which subsidiary faulting spreads can be equated with a plastic zone size and the relation can be used to discuss the fracture toughness of a triple junction.

KING and YIELDING (1983) discussed this concept in relation to the E1 Asnam, Algeria earthquake and noted that for an ideal elastic-plastic


material and a mode III crack, the crack tip plastic zone radius is given by:

a = - ( 2 8 ) 8

where K is the stress intensity factor and ay is the yield stress at the edge of the plastic zone (e.g., RuI~NICm, 1980). If K is the critical stress intensity factor, Kcrit, then acrit is the critical plastic zone (sometimes called process zone) size that inhibits rupture.

In Figure 19b as rupture starts to extend from the first rupture plane to other planes the effective plastic zone size grows until rupture is suppressed. The process is independent of scale since a will grow to provide an acrit that prevents further rupture, and this will scale with the length of the first fault. Thus the square of the critical stress intensity factor of a fault is proportional to its length. Noting that the Griffith's energy release rate is proportional to the square of the critical stress intensity factor, we find that the Griffith's energy is proportional to fault length. This is a corollary of the often-quoted statement that earthquake faulting is associated with constant stress drop (for example, Scr~oLz, 1982) and can now be seen to have a simple geometrical reason.

The final stages of an earthquake are associated with simultaneous motion on a set of different planes, and it is therefore to be expected that the last part of the energy release will be different from earlier parts. If the deformation is triaxial then it will not have a double couple character, and this can explain why the long period behaviour of some earthquakes has a clear nondouble couple component (DzmwoNsKI and WOODHOUSZ, 1983). When the main rupture ceases, the "plastic zone" size is gradually extended by aftershocks that represent continued motion on many different planes, but in a more gradual way.

In Section 10 it was noted that near to the triple junction the deformation will be associated with a high b-value and that this decreases with distance from the junction. With time and tectonic loading, the deformation will spread and increase a o so that late-stage aftershocks may be expected to have a lower b-value than earlier ones.

The process associated with the termination of rupture involves interlocking faulting, and the process disrupts the continuity of the larger fault planes. For rupture to continue in the second direction, asperities in these planes formed by the termination process of the previous event must be broken. Before the tectonic loading has reached a sufficient level to break asperities, it is reasonable to expect a period of quiet, particularly near the future epicenter. The number of asperities is greatest nearest to the barrier of the last event.

When faults start to move again, those with fewest asperities, which

8 0 0 Geoffrey King

a)

..~ ,,.. Asperity 1

~ , Rupture 3 .,~

A j#Barrier 2 ~ Barrier 3 % . ~ ~ Asperify3

I Barrier 1

b)

Rupturel

/

v - - - °d°'rup ure "-k

\ \ /

Figure 20. Event sequence on a fault system with fractal subfaults. The overall sequence is shown in (a). Rupture initiates at asperity 1 and is stopped by barrier 1 associated with the first fault junction. This same junction creates the asperities to be broken before the second event that then propagates to the barrier associated with the next fault junction. The third event initiates from this junction. The stopping of the first rupture is shown in (b). Motion on the first fault causes motion on one or both the faults beyond the barrier. This, in turn, causes fractal faulting in the zone around the junction. For clarity only, one of the orientations of the fractal faulting, that parallel to the first main fault, is shown. The faulting produces a "plastic" zone sufficiently large to stop further rupture. The process of forming the fractal plastic zone creates asperities in the plane where the second rupture will occur. Attention is drawn to Figure 2 in AKI (1983). There will be more asperities near to the barrier than far from it. Thus, as the tectonic load increases (c), the first faulting to move will be faults parallel to, but at a distance from the nucleation site of the second rupture. When the second rupture initiates (d), it may be expected to start close to, but not at the fault junction that stopped the first event.

are those furthest from the triple junction, will move first, giving rise to a Mogi doughnut (e.g., MOGI, 1981). Because of their spatial position these events may be expected to have a low b-value.

The second main rupture occurs when one of these events lies close


c) /

/

/ J

/ , / /

d) H ~J ~ initiation of _.~/ rupture

/' ~" \Small /ptastic

\

\ \ \

enough to the second fault plane to trigger motion on it. The reason for low b-value for foreshocks can also be seen from a probabilistic argument as well as a spatial one. Each of the new events may trigger a larger or a much larger event, and this is the opposite behaviour to the termination process where each event will trigger smaller events. A situation in which an infinitesimal event triggers a finite event has a b-value of zero. If the triggering is likely and not certain and the events are both of finite size, the b-value will be low but not zero. (See also KING and YIELDING, 1983). The final process of rupture initiation is shown in Figure 20d with the main rupture being initiated a small distance from the junction and rupturing back towards it as well as towards the next barrier.

Figure 20 is two dimensional, and the processes in real earthquakes

802 Geoffrey King

are apparently three dimensional. Thus it should be appreciated that the systems shown in the figures can be complicated by normal or reverse faulting. In other words, they can include components of motion perpendicular to the plane of the paper. It is also emphasized again that for clarity only a small number of the total number of faults taking part in each stage of the process is shown in Figure 19. The processes are all self similar and involve motion on fault systems with three orientations (five in three dimensions). When one direction predominates, it is a partial rather than complete predominance and only temporary.

Figure 20a also indicates that all the faults have equal lengths. Thus a unilateral propagation from left to right is plausible. However, this will not be the general situation. Figure 21 shows a system with faults of different lengths. Beside each fault a moment (.A'0) is shown indicating the deformation that it can accommodate in one increment of faulting. Thus it may be seen that the segments b and c must move twice for each single motion on a and e, and segment d must move four times. Short segments whose lengths are comparable to the plastic deformation zone size of an adjacent segment will not move independently of the main segment but will move with the larger segment in a multiple event.

In fact, the main segments may more usefully be considered to be composed of systems of long and short segments that have this property of moving collectively. The frictional behaviour is then related to the angular distributions and lengths of these segments. The behaviour of straight faults or, more correctly, nearly straight faults is considered in the next section.

12. Nearly straight faults with barriers and asperities

In the previous sections we have described geometries with large angles between the fault systems. It was pointed out, however, that these

a, c. \ ,v

5zR= 1/l, \

Figure 2 I. The length of fault segments need not be equal. Their repeat time (if they are two-dimensional) may be expected to be inversely proportional to their length. Thus the single propagation process indicated by Figure 19 will not occur. Small fault segments may be constrained to move when a larger adjacent segment moves. If the plastic zone for one segment is comparable in size to a shorter fault segment, then the short segment will never move independently. A large fault system can be regarded as a set of small fault systems that have this property of moving collectively.


systems were topologically identical to any other system, and the reader is again invited to imagine distorting the preceding figures as if they were drawn on rubber sheets. When figures are distorted in this way, fault angles can be made small and circular ductile zones become elliptical. The behaviour, however, remains the same. A bend in a fault causes fractal behaviour whether the bend is large or small.

Strike-slip fault systems such as the San Andreas can be described by distorted versions of the preceding figures. Thus the commonly observed en echelon offsets are describable by one or more pairs of triple junctions (such as those in Figures 22a and b) interacting in various ways (Figures 22c and d).

Two implications of this view are noteworthy. The first is that the fault-plane solutions from events on all the faults will be similar and thus the existence of different planes will not be very obvious from the data. The second is that since the secondary faulting results from the combined effect of a series of subparallel lozenge-shaped zones, it will lie close to the mean direction of the fault zone. The narrowness of the earthquake belts in California is presumably the origin of the belief that aftershocks lie on a single fault plane. The data, however, are not inconsistent with deformation over a zone of definite width and with slip vectors that are not parallel to the fault strike. The observed zone of shattering associated with faults such as the San Andreas is to be expected from the mechanical processes caused by triple junctions associated with fault off-sets. The offsets in a strike-slip fault system can operate to initiate and terminate rupture in the same way as the single junctions described earlier.

Figure 23a shows the form that a system may take at any particular time. Bear in mind that each junction of three faults is the center of a fracture system of the types shown in Figure 11 or 15 extending to the smallest scale sizes. For comparison, Figure 23b shows the assumed orientation of fault planes for aftershocks of the Coyote Lake earthquake of 1979 (REASENBERG and ELLSWORTH, 1982). Note that all the events are shown to have planes oriented in the a- and b-directions (of Figure 24a), but some could quite plausibly be in the c-direction.

13. Dilatancy and the inner fractal limit

In the discussion of fault development in Section 7 it was suggested that a fault of any given order could move once without the need for higher order faults to move; the deformation associated with one increment of displacement could be accommodated elastically. Under these circumstances, the inner fractal limit is elastic behaviour and the scale at which it occurs is determined by the total amount of

804 Geoffrey King

B D

A F E

,t - - i " - \ C ~ ) ~ c F

"~,

Ductile zone

b

Combined ductile zone

Figure 22. An offset in a strike-slip fault may be regarded as being composed of two or more triple junctions, each with its own "ductile zone" of fractal faulting. These zones can be elongate, and systems of paired junctions can create long zones that behave in an apparently ductile fashion to control fault rupture. If the angle between the faults marked (a) and (b) is small, then all of the faults will give similar focal mechanisms. Strike-slip faults produce elongated ductile zones and all the fault plane solutions are similar. For this reason the geometric features of the barriers on strike-slip faults are less readily visible than regions where volume deformation is more obvious.

deformation. The more deformed a region becomes, the more smaller- scale faults are created. At a small enough scale the region remains unfractured.

It was pointed out in Section 4 that triple junctions where three faults meet are unconditionally unstable in completely rigid material, because none of the three elements permits a volume Change. For limited motion, elasticity can perform the role of stabilising fault junctions by allowing volume change. At low confining pressure, however, or in the presence of fluid under pressure fault surfaces can separate over small areas. Thus at

Accommodation of Large Strains in the Upper Lithosphere of the Earth 8 0 5

a)

b)

Figure 23. An idealized fractal fault offset geometry, (a), to compare with the aftershock locations of REASEr~BERG and ELLSWOR~ (1982), (b). The planes are assumed to lie in the b- and a-directions of (a), but some could lie in the c-direction. Some planes suggest dip-slip motion, which is required if the system is to have a component of triaxial deformation.

this small scale, if one of the fault directions permits opening, stable triple junctions can form and repeated motion can occur without fracturing more material. This is shown in Figure 24 where the b

806 Geoffrey King

direction is permitted to accommodate some opening. The figure m u s t be understood in a schematic way. In Figure 24a and b, the two possible positions of large-scale shear faulting are shown, and Figure 24c and d indicate that at some scale b-direction fissures can open. Figures 24e and f emphasize that this occurs in a volume. The b-direction fissures can have both conservative and nonconservative motion simultaneously. The situation may reasonably be expected to be more complex than that shown with opening occurring either simultaneously along differently oriented planes, or more probably in some sequence controlled by the evolution of the large-scale deformation process.

When fault surfaces may separate, motion can occur without the creation of more fault surface, but the deformation must be accompanied by a net expansion. There is consequently a trade-off between the energy of creating more fault surface and the elastic energy associated with opening fissures. The latter energy can clearly be modified dramatically by the presence of fluids under pressure, and, as we shall see, fluids can modify the scale of the lower fractal limit.

In circumstances in which a volume increase must occur to permit deformation, even simple shear boundary conditions will cause complex behaviour. Thus the simple situation depicted in Figure 5b will never occur and all boundaries will behave to some extent like more complex boundaries. The complex behaviour of strike-slip faults may be attributed to this effect. Thus biaxial deformation will never occur in practice. All deformation will have a triaxial component, and all systems will have long-term b-values close to unity. In a more general sense, friction itself can be thought of as owing its origin to dilatant behaviour at the lower fractal limit. It has been noted by a number of authors that at the ductile-brittle transition, an absence of dilatancy and an equality of frictional strength and fracture strength occur at the same confining pressure (e.g., BRACE, 1977). The foregoing discussion provides a means of visualizing why this should occur. When dilatant fissuring is suppressed, an applied simple shear can be relieved by simple shear structures alone. Thus the multiple direction deformation that creates triple junction barriers and asperities is removed, and friction and seismicity consequently disappear.

Looking again at Figure 24, it might seem that of the two alternative processes for accommodating motion associated with c, neither is more favourable. The system in Figure 24c however allows a greater volume to be dilatant than that in Figure 24d. There is some evidence that on strike-slip systems, off-fault deformation favours the outside rather than the inside of fault bends (Figure 25). The opposite is evidently true however for listric faulting. The hanging-wall block is invariably more distorted than the footwall block (e.g., WERNICKE and BURCHFIELD, 1982; FROST and MARTIN, 1982). In this case, the presence of the

Accommodat ion of Large Strains in the Upper Lithosphere of the Earth 807

stress-free surface can explain the greater susceptibility of the upper block to dilatant, and hence shear, deformation.

a

a) b) /

/ ¢/

d) c) c

c.,~/

f)

Figure 24. Dilatant structures at the lower fractal limit of faulting. Diagrams (a) and (b) show the two basic generator systems. At the lower fractal limit dilatant structures exploit the b-direction. This is shown in a schematic way in (c) and (d) where it is also indicated that the region behaves as if the boundary conditions were changed to include some dilation. Figures (e) and (f) indicate that the dilatant fault structures can exploit the outside or inside of the fault bend.

808 Geoffrey King

37"30' , ,

C S C ~ AJST

AJSS • JAL

JEC~ 37"OO' AJRG JTGA AJE

&JPL ~ H HPF

CZ HCl &HCO HA2 H0L • MONTEREY I

KILOMETERS

BPC

36"30 ' I I 122"00'

JOB&

JRR

A.GW

CAO&

\ L ~. HSP

HJG,A HSF

HFP A

es~ A

20 J

OF HP /~! A"Fe

HKR HLT A

k ~sA

,HOR

BVL

&HSL

.8EH m 8 E M A

] 1 I 121"30' 121"00'

Figure 25. Figures suggesting that deformation occurs on the outside-of bends in strike-slip fault systems (look along the faults to see the bends). The upper figure (a), is the aftershock zone for the Coyote Lake, California, Earthquake of August 6, 1979 (REASENBERG and ELLSWORTH, 1982), and the figure (b), on the opposite page is the aftershock zone for the Bear Valley, California, Earthquake sequence of February and March 1972 (ELeswoRxn, 1972).

14. The inner fractal limit and b-value

At the inner fractal limit the b-value must change. In tectonic environments, this has not been observed. Nor has it been observed in laboratory experiments (MOGI, 1962; SCHOLZ, 1968). This is presumably because in both cases the scale of faulting at the fractal limit is too small to be detected. Under volcanic conditions, this is not the case. Volcanic


• ",,, . • . \ -. . .-" ",,, ,, . - : . L . . : .... "--~, - .

"' , , i :':,

• ' ~ , ' ~ "" '~,~-L" ".: " ~ . - \ \ . . : . . ~%~...-.~-~.~..~...... ~.~ -

9 1 1 1 1 9 7 0

j 6 ° 20'

Figure 25(b).

36 ° 50' _

t o 0 5 : 2 7 2 1 2 2 1 1 9 " 7 2 )~. . : . " "

\ 0 5 K M E 5 5 ~ E E ~ E 3

"\ \ ,

earthquakes behave in a completely different fashion and exhibit great regularity of behaviour and characteristic scale lengths. In the volcanic environment, large dilatant fissures in the form of magma or gas-filled dikes are possible, and regular nonfractal geometries are possible.

The behaviour is discussed in papers by OKADA et al. (1981) and TAKEO (1983), who report observations of two types of earthquake families associated with the Usu volcano. One family has repeating events of identical size and waveform, and the other has varying size and identical waveform. The way in which events of this kind can come

8 1 0 Geoffrey King

about is shown in Figure 26. Shear faults on which the earthquakes occur end at open fissures and not at fault junctions where fractal smashing would occur to create smaller events. A system that produces events of changing size is shown in Figure 26a and one that produces events of the same size in Figure 26b. Many different geometries having similar seismic properties to those illustrated are possible. A system similar to Figure 24a is described from field observations by SEGALL and POLLARD (1983) in an exhumed granite terrain in the Sierra Nevada.

a)

b)

,,g\~.x.~.

Figure 26. Repeating events on faults that terminate on fluid filled fissures. The upper diagram (a) and (b), is for a fault ending on two short fissures. The fissures grow significantly with each fault displacement resulting in a change in behaviour. The fault length remains unchanged; only the amount of displacement varies with each event causing events of identical waveform but different amplitude. The events are not self similar because the displacement to length ratio of faulting changes with each event. The lower diagram (b), is for longer fissures. The change of fissure length is unimportant and each event can be identical. Note that the foregoing are only examples. The fluid fissures allow the faults to terminate without the fractal subfaulting that produces self-similar behaviour. Many geometries can be stabilized to produce nonfractal behaviour if dilatant structures are allowed. Similar behaviour may be expected from dip-slip faults that cut through the brittle crust from the stress free ductile zone to the Earth's stress free surface.


The features represented as dykes in Figure 25 could also be provided by the Earth's stress free surface (e.g., TAKEO, 1983) and the base of the brittle zone. Thus some variations in the fractal behaviour of earthquakes and faulting may be expected at such scale dimensions. This may explain the observations of AKI (1983) that earthquake spectra and seismic scattering appear to be associated with structures having scales of the order of the thickness of the brittle crust.

15. Problems with the formation of fractal fault systems

Throughout our discussion it has been assumed that fault surfaces will automatically form in the appropriate directions to accommodate the applied finite deformation. Field observation does not contradict this assumption. Examination of road cuts such as that shown in Figuire 1 commonly show simple planar surfaces except where human activity to create the road cut has produced curved surfaces.

Our understanding of fracture mechanics, however, suggests that only purely tensile fractures form in a planar fashion, and all other fractures will curve (KNOTT, 1973; BROEK, 1974; LAWN and WmSHAW, 1975). Man-made features in road cuts generally show the predicted curved geometry. To explain planar features, SE~ALL and POLLARD (1983) argue convincingly that the early stages in the development of simple fault systems in the Sierra Nevada rely on the formation of new structures initially in tension, with the structures only later being exploited as shear planes. Some of their more complex zones, however, appear to have geometries similar to those shown here (such as Figure 23), but they are unable to demonstrate how the more complex systems formed. The discussion in this paper describes how deformation can proceed but does not explain how features come into existence. Although it seems likely that the faults in Segall and Pollard's simple systems existed by chance formation in an earlier stress system, it is difficult to avoid the conclusion that, to explain the more complex zones, some process exists to form planar faults in an environment that is subject to shear on a large scale.

16. Conclusions

The ideas presented here are, in effect, an extension of plate tectonics to the general problem of the finite deformation of brittle elastic solids. Finite motion is taken up on sets of planes with differing orientations. Because these planes cannot intersect and have nonzero displacements, intersections are associated with multiple faulting. This is described in terms of a fractal geometry that allows faulting at one scale to be related

812 Geoffrey King

to faulting at the next smaller scale. Two fundamental constraints are placed on the fault systems. One is purely the geometric constraint that faults cannot cross, and the second-is the constraint that sufficient faults must exist to accommodate the applied deformation. The latter constraint is more severe than the former. Fewer faults are required at any phase of deformation than those that exist in a theoretical pattern of nonintersecting faults, or than those that can come into existence over a period of time. A system of active faults therefore can have a lower fractal dimension than the system of fossil faults that a period of activity will leave.

On the basis of some simple assumptions the fractal dimension of an active fault system can be directly related to earthquake b-value. For a triaxial system with a fractal dimension of 2 the b-value is 1.0. For biaxial deformation with a fractal dimension of 1, the b-value is less than this. Variations in b-value can therefore be attributed to variations in deformation type, but it is argued that because of dilatancy at the lower fractal limit all systems will behave in a triaxial fashion. Fault systems evolve with time, and it is possible to show that this evolution will result in b-value variations within a long-term average of unity. Within the same context, other features of the observed seismic and geometric behaviour of fault systems can be explained and their relation to fracture mechanical concepts indicated. The stopping processes involve motion on planes with different orientations that spread deformation over a volume. If this deformation is triaxial it will contribute a nondouble couple component to the earthquake mechanism. The stopping processes create the asperities that must break before the next main event initiates. An important consequence of the behaviour described here is that the fracture toughness of rocks subject to finite deformation is not a material constant or even a quasi constant. At the most basic level this is because for fracture toughness to be a natural constant requires that the material involved have a built-in length scale. The self-similar behaviour of earthquakes shows that the earth does not have this property. This result will actually apply to the deformation of all solids, and similar conclusions emerge from rock testing (WONG, 1982). The behaviour, however, is more evident in the deformation of the Earth's crust where the boundary conditions demand finite strain. In engineering situations "large strains" result in "catastrophy", and instability conditions associated with stress boundary conditions are more important.

The use of a Fractal description denies the use of calculus and hence calculus-based concepts such as stress and strain. The significance of this cannot be overemphasized. Calculus can be employed at the upper and lower fractal limit but the "patching" together of fractal and continuous deformation descriptions has to be an approximate process, because at some scale level the assumptions of discontinuous and


continuous behaviour are contradictory. At the scale at which this problem exists no alternative seems to be available except to examine faults as they actually appear in the field and to explain their behaviour on the basis of their observed geometry. The fractal description does offer some support to the system of analysing Neotectonic deformation adopted by some workers (e.g., ARMUOU et aL, 1982). The system uses a large number of measurements of fault slickenside directions to calculate the causal stress orientation. The self-similar behaviour of faulting described here suggests that random sampling of faults over a wide range of scales can be an acceptable process. The deformation behaviour we describe, however, suggests that the quantities employed in Neotectonic discussions should be redefined as strains rather than stresses. Also the problems associated with approximating a fractal system by a continuum demand that the procedure contain inherent errors.

Throughout the paper we have not specifically considered rotations. These will, in general, have amplitudes similar to the effective strains that the mukiple fault systems relieve.

Acknowledgments

I would like to thank Keiiti Aki for inspiration and for supporting my visit to M.I.T. I would also like to thank Bob Wallace who assisted me in a visit to the U.S. Geological Survey where much of this paper was written. John Beavan, Keiiti Aki and Robert MacDowell read the manuscript, and I greatly appreciate their advice and comments. I particularly appreciate careful reviews that have been helpful and stimulating by Joe Andrews, Mike Ashby and Charlie Sammis. I have enjoyed discussions with Earl Williams, and I thank Ann Rayner for typing the manuscript.

The work has been supported by the Royal Society, the Natural Environment Research Council (GR3/3904), the National Science Foundation (CEE 8206456), and the United States Geological Survey. This is Cambridge Earth Sciences Contribution number 446.

This paper is dedicated to the memory of Raffy Freund, a geologist of great vision.

REFERENCES

AKI, K. (1979), Characterization of barriers on an earthquake fault, J. Geophys. Res. 84, B 11, 6140-6147.

AKI, K., A probabilistic synthesis of precursory phenomena, In Earthquake Prediction--An International Review (Maurice Ewing Series 4, American Geophysical Union,"Washington, D.C. 1981).

814 Geoffrey King

AKI, K. (1983), The use of a physical model of fault mechanics for earthquake prediction (Preprint). Lecture delivered at U.S.G.S., Menlo Park, California.

ANDERSON, E. M., The Dynamics of Faulting (Oliver and Boyd, Ltd., London 1951). ARMIaOU, R., CAaEY, E., and CISTERNAS, A. (1982), The inverse problem in microtectonics and

the separation of tectonic phases, Tectonophysics 82, 145-160. ANDREWS, D. J. (1980a), A stochastic fault model 1. static case, J. Geophys. Res. 85, B7,

3867-3877. ANDREWS, D. J. (1980b), A stochastic fault model 2. time-dependent case, J. Geophys. Res. 86,

Bll , 10821-10834. BABCOCK, E. A. (1973), Regionaljointing in southern Alberta, Canada, J. Earth Sci. 10, 1769-

1781. BRACE, W. F. (1977), Volume changes during fracture and friction: a review, Proceedings of

Conference II. Experimental Studies of Rock Friction. United States Geological Survey, Office of Earthquake Studies, Menlo Pk., California 94025.

BROEK, D., Elementary Engineering Fracture Mechanics Martinus Nijhoff 1982, Hague. BUaCHFIELD, B. C. (1983), The continental crust, Scientific American 249, 3, 130-142. DE SITtEr, L. U., Structural Geology (McGraw-Hill, 1956). DZIEWONSKI, A. M., and WOODHOUSE, J., (1983), An experiment in systematic study of global

seismicity: centroid-moment tensor solutions for 201 moderate and large earthquakes of 1981, J. Geophys. Res. 88, B4, 3247-3271.

ELLSWORTH, W. (1975), Bear Valley, California, earthquake sequence of February-March 1972, Bull. Seismol. Soc. Am. 65, 483-506.

FROST, G. F., and MARTIN, "D. L., Anderson-Hamilton Volume: Mesozoic-Cenozioc tectonic evolution of the Colorado River region, California, Arizona, and Nevada (Cordilleran Publishers. 6203 Lake Alturas Ave., San Diego, California 92119, 1982).

GAGNEPAIN-BEYNEIX, J., HAESLER, H., and MODIANO, T., (1982), The Pyrenean earthquake of February 29, 1980: an example of complex faulting, Tectonophysics 85,273-290.

GUTENBURG, B., and RICHTER, C. F., Seismicity of the Earth and Associated Phenomena (Princeton University Press, 1949).

HILL, D. P., (1982), Contemporary Block Tectonics: California and Nevada, J. Geophys. Res. 87, BT, 5433-5450.

HODGSON, R. A. (1961), Regional study of jointing in Comb Ridge-Navaja Mountain area, Arizona and Utah, Bull. American Assoc. Petrol. Geologist 45. 1, 1-38.

ISHIMOTO, M., and IDA, K. (1939), Observations sur les s~isms enregistr& par le microseismograph construit derni~rement (1), Bull. Earthq. Res. Inst. 17, 443-478.

JACKSON, J., and MCKENZIE, D. (1983), Active tectonics of the Alpine-Himalayan Belt between Western Turkey and Pakistan, Geophys. J. R. Astra Soc. Lond. (submitted).

KANAMORI, H., and ANDERSON, D. (1975), Theoretical basis for some empirical relations in seis,nology, Bull. Seism. Soc. Am. 65, 5, 1073-1095.

KAGAN, Y. Y. (1982), Stochastic model of earthquake fault geometry, Geophys. J.R. Astr. Soc. Lond. 71,659-691.

KING, G. C. P., (I 978), Geological faults:fracture, creep and strabT, Phil. Trans. R. Soc. Lond. A, 197-212.

KING, G. C. P., and YIELDING, G. (1983), The evolution of a thrust fault system: processes of rupture initiation, propagation and termination in the 1980 El Asnam (Algeria) earthquake, Geophys. J.R. Astr. Soc. 77, 915-933.

KNOTT, J. F.. Fundamentals of Fracture Mechanics (Wiley, New York. 1973). LAWN, B. R-, and WlLSHAW, T. R., Fracture of Brittle Solids, (Cambridge University Press,

1975). MANDELBROT, B. B., The Fractal Geometry of Nature (W. H. Freeman and Company, San

Francisco. 1977). MCKENZIE. D, P., and MORGAN, W. J. (1969), Evolution of triple junctions, Nature 224, 5215,

125-133.


MCKENZlE, D., (1979), Finite deformation during fluid flow, Geophys, J.R. Astr. Soc., 58. 689-715.

MCKENZIE, D., and JACKSON, J. (1983), The relationship between strain rates, crustal thickening, paleomagnetism, finite strain and fault movements within a deforming zone, Earth Plant. Sci. Lett. 65, 182-202.

MCKENzm, D. (1972), Active tectonics of the Mediterranean region, Geophys. J.R. Astr. Soc. 30, 109-185.

MOGI, K., (1962), Study of the elastic shocks caused by the fracture of heterogeneous materials and its relation to the earthquake phenomena, Bull. Earthq. Res. Inst. 40, 125-173.

MOGI, K., Seismicity in western Japan and long-term earthquake forecasting, In Earthquake Prediction, Maurice Ewing Series 4. (ed. Simpson, D. W., and Richards, P. G.) (A.G.U., Washington, D.C., 1981)pp. 635-666.

MOLNAR, P. (1983), Average regional strain due to slip on numerous faults of different orientations, .t. Geophys. Res. 88, B8, 6373-6394.

OKADA, H., WATANABE, H., YAMASHITA, H., and YOKOYAMA, I. (1981), Seismological significance of the 1977-1978 eruptions and the magma intrusion process of Usu volcano, Hokkaido, Volcanol. Geotherm. Res. 9, 311-334.

OVERBEY, W. K., and ROUGH, R. L. (1968), Surface studies predict orientation of induced formation fractures, Producers Monthly 32, 16-19.

REASENBERG, P., and ELLSWORTH, W. L. (1982), Aftershocks of the Coyote Lake, California, earthquake of August 6, 1979: a detailed stud),, J. Geophys. Res. 87, B 13 10,637-10,655.

RUDNICKI, J. W. (1980), Fracture mechanics applied to the Earth's crust, Ann. Rev. Earth Plant. Sci. 8,489-525.

SCHOLZ, C. H. (1968), The frequency magnitude of microfracturing in rock and its relation to earthquakes, Bull. Seism. Am. 58, 399-416.

SCHOLZ, C. H. (1982), Sealing laws for large earthquakes: consequences for physical models, Bull, Seism. Soc. Am.-72, 1, 1-14.

SEGALL, P., and POLLARD, D. (1983), Nucleation and growth of strike-slip faults in granite, J. Geophys. Res. 88,555-568.

STEIN, R., and LxSOWSKI, M. (1983), The 1979 Homestead Valley Earthquake Sequence, California: Control of Aftershocks and Postseismic Deformation, J. Geophys. Res. 88, B8, 6477-6490.

TAKEO, M. (1983), Source mechanisms of Usu volcano, Japan, earthquakes and their tectonic implications, Phys. Earth and Planet Int. 32, 241-264.

TAPPONNIER, P., and MOLNAR, P. (1976), Slip-line field theory and large-scale continental tectonics, Nature 264, 319-324.

TCHALENKO, J. S. (1970), Similarities between shear zones of different magn#udes, Bull. Geol. Soc. Am. 81, 1625-1640.

THOMAS, P. G., and MASSON, P. (1983), Tectonic evolution of Mercury; comparison with the moon, Annales Geophysicae 1, 53-58.

VON MISES, L. Z. (1928), Mechanik der plastischen Formh'nderung yon Kristallen, Zeitschrift ffir Angewandte Mathematik und Mechanik 8, 161-185.

WERNICKE, B., and BURCHFIELD, B. C. (1982), Modes of extensional tectonics, J. Struct. Geol. 4, 2, 105-115.

WONG, T. (1982), Shear fracture energy of Westerly Granite from post-failure behaviour, J. Geophys. Res. 87, B2, 990-1000.

WYSs, M. (1973), Towards a physical understanding of the earthquake frequency distribution, Geophys. J.R. Astr. Soc. 31,341-359.

(Received 15th November 1983, revised 1 lth April 1984, accepted 14th April 1984)

Documents

Environnement - The Accommodation of Large Strains in the …king/Geoffrey_King/Publications_files/1983... · 2014-05-06 · The mechanics of the zones between the plates are less