JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. E7, PAGES 14,691-14,702, JULY 20, 1994

Observations, models, and mechanisms of failure of surface rocks surrounding planetary surface loads

R. A. Schultz

Geomechanics-Rock Fracture Group Department of Geological Sciences, Mackay School of Mines, University of Nevada, Reno, Nevada

M. T. Zuber

Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Maryland Geodynamics Branch, NASA Goddard Space Flight Center, Greenbelt, Maryland

Abstract. Geophysical models of flexural stresses in an elastic lithosphere due to an axisymmetric surface load typically predict a transition with increased distance from the center of the load of radial thrust faults to strike-slip faults to concentric normal faults. These model predictions are in conflict with the absence of annular zones of strike-slip faults around prominent loads such as lunar maria, Martian volcanoes, and the Martian Tharsis rise. We suggest that this paradox arises from difficulties in relating failure criteria for brittle rocks to the stress models. Indications that model stresses are inappropriate for use in fault- type prediction include (1) tensile principal stresses larger than realistic values of rock tensile strength, and/or (2) stress differences significantly larger than those allowed by rock-strength criteria. Predictions of surface faulting that are consistent with observations can be obtained instead by using tensile and shear failure criteria, along with calculated stress differences and trajectories, with model stress states not greatly in excess of the maximum allowed by rock fracture criteria.

Introduction and Summary

Normal faults and reverse (thrust) faults are commonly observed modes of fracture on solid planets and satellites in the solar system. Both sets of structures, but particularly normal faults or grabens, are commonly found in association with surface loads such as flooded impact basins and volcanoes. Although strike-slip faults are abundant on the Earth, they are presently observed only in limited occurrences on Mars [Forsythe and Zimbelman, 1988; Schultz, 1989], various icy satellites such as Europa [Schenk and McKinnon, 1989] and perhaps Venus [Solomon et al., 1991, 1992]. Comparatively few strike-slip faults are known to occur in association with surface loads.

Faulting of planetary lithospheres in response to surface or subsurface loads has commonly been studied by comparing predictions of theoretical models to observed structures. For example, concentric arrangements of normal faults associated with impact basins on the Moon, Mars, and Venus have been used to evaluate the predictions of flexural loading models [e.g., Melosh, 1978; Solomon and Head, 1979, 1980, 1990; Wichman and Schultz, 1989, 1993]. Similarly, patterns of Martian normal faults and wrinkle ridges, the latter of which are thought to be associated with thrust faulting and folding [e.g., Lucchitta, 1976], have been related to loading by volcanoes [Comer et al., 1985], to caldera dynamics [Zuber and Mouginis-Mark, 1992],

Copyright 1994 by the American Geophysical Union.

Paper number 94JE01140. 0148-0227/94/94JE-01140505.00

and to models for the formation of the Tharsis rise [e.g., Banerdt et al., 1982, 1992; Janes and Melosh, 1990]. Surface-breaking structures have also been used in conjunction with geophysical models to constrain the regional tectonics of Venus [e.g., Williams and Gaddis, 1991; Grimm and Phillips, 1992; Bindschadler et al., 1992; Janes et al., 1992; Smrekar and

Solomon, 1992].

Many models of axisymmetric surface loads have been interpreted to predict a concentric region of strike-slip faulting about the load [e.g., Melosh, 1978; Pullan and Lambeck, 1981; Willeman and Turcotte, 1982; Comer et al., 1985; Golombek,

1985; Janes and Melosh, 1990]. This prediction is at variance with the apparent absence of strike-slip faults in the corresponding locations about real planetary loads.

In the same vein, Tanaka et al. [1991] have pointed out a related inconsistency between observations and predictions of faulting in association with the Tharsis rise on Mars. They note that radial graben sets apparently were formed during one episode of faulting over their entire strike length, in apparent conflict with a two-stage sequence required by isostatic and flexural loading models that also predict strike-slip faults where normal faults are observed [Banerdt et al., 1982]. A similar age relationship was found for Valles Marineris by Schultz [1991], and certain stress models predict strike-slip faults there, particularly in the eastern and western parts of the trough system [e.g., Banerdt et al., 1992], instead of normal faults. The contradiction between observed faults and model predictions has motivated a search for alternative models of Tharsis evolution

that are consistent with the tectonic record.

14,691

14,692 SCHULTZ AND ZUBER: FAILURE OF SURFACE ROCKS

Figure 1. Concentric grabens associated with Mare Humorum on the Moon. Basin diameter --- 600 km; basin centered near 61 øN, 84øE. Lunar Orbiter IV photograph 143M.

Discrepancies between observations and model predictions of faulting surrounding planetary surface loads have origins in one or more of three sets of components. These are (1) consideration of stress trajectories (map view orientations) only; (2) neglect of stress magnitudes; and (3) identification of initial failure mechanism. Although many previous workers may at times have included some of these, it is shown below that

explicit consideration of all three of these components is critical to fault-type predictions on any planet or satellite.

To illustrate our rationale, we calculate elastic stresses due to

an axisymmetric Gaussian surface load to approximate loading of the lunar lithosphere by a mare basin (Figure 1). We emphasize the importance of several caveats in the interpretation of these elastic stress states. First, near-surface bending stresses far exceed rock strength, indicating that they are inappropriate for predicting the onset and type of faulting. Second, stress geometries commonly taken to indicate strike-slip faulting are nonunique and are consistent instead with either strike-slip faults or joints (tensile fracture), depending on the stress difference and rock strength. Third, consideration of stress state at the onset of failure indicates that concentric jointing and normal faulting, not strike-slip faulting, are predicted in the locations beyond the load where concentric grabens are observed. The use of stress orientations or trajectories only, with or without stress differences, can lead to incorrect fault-type predictions such as strike-slip faulting if final, "post-failure" elastic stresses are used.

Our results demonstrate that concentric normal faults and

joints, no.t strike-slip faults, are predicted to occur at or near the surface about lunar mare basin loads. These results are also

directly applicable to impact basin loads on Mars and other planetary bodies, as well as to volcano, coronae, and other surface and subsurface loads. Further, our results are not

restricted to axisymmetric surface loads, but apply to the general problem of fault-type prediction due to a load on silicate or icy lithospheres. Future models of regional and global tectonics should take rock strength, stress difference, and load evolution into account explicitly in order to more accurately predict styles of faulting.

Fracturing about a Lunar Mare Basin

Stresses due to an Axisymmetric Surface Load

We use the formulation given by Melosh [1978] to calculate the normal and shear stresses in an elastic lithosphere subjected to an axisymmetric Gaussian surface load (Figure 2) which represents a lunar mare basin. The lithosphere is treated as a thick elastic plate that overlies a strengthless fluid of equal density. Confining pressure-n:pcgz, where tc is the ratio of horizontal to vertical overburden (pretectonic) stresses, is included in the calculations so that total stress, not just deviatoric stress, is used in the prediction of fault types. The value of tc for hydrostatic or isotropic conditions (tc = 1), which we adopt in this paper, is considered to be an appropriate reference case by McGarr [1988] and Banerdt [1993] and will decrease for nonzero pore-fluid pressure and certain other factors.

The stresses in the plate are given by

t•zz = - Pc gZ

+ Jo(kr) [(A + Cz) cosh kz + (B + Dz) sinh kz]

O'rr = -- If pc gz

•:rz = - J•(kr) [(A + Cz + •) sinh kz + (B + Dz + •) cosh kz] •rO = •zO = 0

where Ji(kr) are Bessel functions, r is radius from load center, z

.... iiii•i?`..!i.:i•iiii!i!iiiiii•iiiiiiii:...:.:.i!i•.::.:.iiiii!iiiiii..`..iiii!•ii•iii•!•i!!!:.:.i.::.!i.:i!iiiii. ' • .... i½!!!i!•!!i½!!!!!i!!!!!i!i!ill!!ii!i•!..".-!i!i!!!i•!!!i•i!!!i!!!i•i•i!!!!! "•

ß .':!'•:: ....... .:• .... ...., ..... =============================================================== ii?•::•i•iiiiiiiiiii?:•i::ii::•::•ii•?:?:i::i::?:?::•?::•ii::?:•i•iii•iii::::::::i::•::•:•:•:• .........

Figure 2. Geometry of the surface load problem. An axisymmetric Gaussian mass of half-width a, density Pm, and central thickness T loads an elastic lithosphere with density Pc, thickness H, Young's modulus E, and Poisson's ratio v that floats on fluid asthenosphere having equal density. Radial stress O'rr, circumferential (hoop) stress o'00, and vertical stress O'zz are calculated in cylindrical (r, O, z) coordinates.

SCHULTZ AND ZUBER: FAILURE OF SURFACE ROCKS 14,693

is depth in the plate, and k is wavenumber. Tensile stress is positive in these calculations. Using the boundary conditions

Crzz (z = O,r) = b Jo (kr) (2)

rrrz (z = O,r) = O'rz (Z -- H,r) = rrzz (z = H,r) = 0

the constants in equations (1) are

A=b

B= - b [ sinh kH cøsh kH + kH 1 sinh2(kH}-(kH} 2 C = -kB

ksinh2{_ kH} ] D: - b sinh2(kH) - (kH} 2]

(3)

where H is plate thickness and

_a • k• ] b(k} = q a2 k + F(k) l --•- 1

F(k) = •a [cøsh kH sinh kH + kH [' •-n•k-• • •-l• l

E Pcga

(4)

(5)

(6)

Tension

80-

40-

I

Compression

...;# i•:::;" HB

O'mi n 40 O'max O'

Figure 3. Mohr diagram showing fracture criteria for near- surface rock. MG, Modified Griffith; assumes tensile strength of -14 MPa and friction coefficient of 0.85. HB, Hoek-Brown criterion for basaltic rock mass showing uncertainty in properties; parameters given in text. BL, Byerlee's law; assumes friction coefficient of 0.85 and no cohesion. Maximum (most compressive) and minimum principal stresses denoted O'max and rrmin, respectively.

in which b is a dimensionless buoyancy/stiffness constant that characterizes the size of the load, E is Young's modulus of the plate, and Pc is crustal density. Load parameters contained in coefficient b are q = pmgT, where Pm is load density, g is gravity (1.61 m s -2 for Moon), and Tis the load thickness at r= 0; and a, the load half-width. The stresses are obtained by numerically integrating the expressions in (1) from k = 0 to k = oo using a Romberg routine [Press et al., 1986]. The three principal stresses rrl, rr2, and rr3 are calculated from •zz, O'rr, 0'00, and •'rz following Timoshenko and Goodier [1970, pp. 223-224] in order to interpret potential brittle deformation using the tensile and shear fracture criterion discussed below.

In this example, loading of the lunar lithosphere by mare basalts is simulated by a load having a maximum thickness T = 2 km, radius a = 300 km, and load density Pm= 3000 kg m -3 on an H = 50-km-thick lithosphere with crustal density Pc = 2800 kg m -3, so that H/a = 0.17 and load-size parameter b = 0.025. For the elastic properties of the lithospheric plate we choose Young's modulus E = 100 GPa [e.g., Kulhawy, 1975] and Poisson's ratio v = 0.3. Stresses are calculated as a function of

normalized radial position with respect to the load, r/a, and depth, z.

Rock Fracture Criteria

The criterion adopted in this paper to relate stresses to rock strength is a nonlinear rock mass envelope (Figure 3). Developed by Hoek and Brown [1980], this envelope combines a Griffith-type criterion for tensile normal stresses with a more linear curve for compressive normal stresses (similar to the Coulomb frictional slip criterion) into a single equation. The Hoek-Brown criterion is analogous to a Modified Griffith criterion [e.g., Secor, 1965; Scholz, 1990, pp. 15-16] but

provides a more compact description of rock strength in both tension and compression. Thus the Hoek-Brown criterion can be used to predict tensile failure and frictional faulting for a wide range of stress states. In addition, this criterion incorporates explicit corrections for the reduction of rock strength due to scale and fracturing as is commonly experienced in large (>l-m) samples [e.g., Hoek and Brown, 1980]. It should be noted that jointed rock masses are considerably weaker in tension than the -10 to -50 MPa range of values quoted in the literature and used in failure criteria for intact rock (e.g., MG in Figure 3), so Griffith, Modified Griffith, or Coulomb criteria may not represent adequate brittle strength criteria for near-surface rocks. The versatility of the Hoek-Brown criterion makes it useful for predicting various types of brittle fracture of shallow planetary lithosphere due to a surface load.

The Coulomb criterion has been used in previous studies of stresses associated with axisymmetric surface loads [Melosh, 1976, 1978; Hall et al., 1986; McGovern and Solomon, 1993], trench flexure [Goetze and Evans, 1979], and flexure associated with normal faulting [Buck, 1988; Melosh and Williams, 1989]. McGovern and Solomon [1993] used a linear Coulomb criterion having values of cohesion and friction (slope) comparable to the Modified Griffith criterion shown in Figure 3. The choice of a Coulomb, Griffith, or Modified Griffith criterion, or bilinear

friction law for compressive normal stress [Brace and Kohlstedt, 1980], however, along with appropriate values for the strength parameters, would not lead to significantly different results than those we present in this paper. For convenience, compressive stress is taken to be positive when using this criterion.

The Hoek-Brown criterion is given by

O'ma x = O'mi n + 4mO'cO'min + SlYc 2 (7)

14,694 SCHULTZ AND ZUBER: FAILURE OF SURFACE ROCKS

in which O'max and O'min are the greatest and least compressive principal stresses at the initiation of fracturing, Crc is the unconfined compressive strength of the intact rock, and m and s are empirical parameters that reflect the degree of block interlocking and fracturing of the rock mass. For intact rock, s = 1, and for pervasively fractured rock, s = 0. The parameter m decreases with the degree of fracturing or blockiness of the rock mass. We choose values of O'c = 262 MPa, m = 6.3, and s =

0.021 appropriate to a typical jointed rock such as basalt [Schultz, 1993] and in accord with current thinking on rock mass strength [e.g., Brady and Brown, 1992]. Associated values of bulk tensile strength and cohesion are-0.85 MPa and 2.5 MPa, respectively [Schultz, 1993]; values of friction coefficient obtained from this criterion are comparable to values cited in the literature for basalt (e.g., 0.85) (Figure 3). Although these values may differ for various rock types, depths, and/or timescales of loading, the elastic stresses associated with the lithospheric loads may exceed the rock (or rock mass) strength by such a large degree that differences in the strength parameters in (7), or choice of fracture criterion, do not significantly alter the results presented in this paper.

Jointing or faulting in the model is predicted when the calculated stresses, expressed as a Mohr circle in Figure 3, approximately equal those required for fracture, expressed as the Hoek-Brown envelope (equation (7)). Comparison of stress state and rock strength is facilitated by using the concept of "proximity to failure" illustrated by Melosh and Williams [ 1989] and $egall and Pollard [1980] in their studies of faulting. At any given point in the model, calculated values of cr• and cr3 (corresponding to-O'min and-O'max, respectively, in the Hoek- Brown criterion) are converted to stress differences (AO'model '- O'max-O'min). For each value of O'max the limiting value of O'min is

also calculated from equation (7); the associated difference between these stresses is AO'Hoek-Brown. Model stress differences are then compared to the largest stress difference allowed from rock strength (AO'Hoek-Brown) using equation (8). The critical stress ratio is given by

½O')model O'critical = {AO')Hoek-Brow n (8)

Stresses that equal or exceed those given by the Hoek-Brown envelope (Crcritical > 1) are associated with brittle deformation (jointing or faulting), whereas stresses smaller than this limit (O'critical < 1) remain elastic and no fracturing is expected. This approach to predicting brittle fracture is identical to that employed in strength envelopes for the brittle regime [e.g., Goetze and Evans, 1979; Brace and Kohlstedt, 1980]. "Fracture" is used here instead of "failure" because complete continuity of the lithosphere is not necessarily lost, as implied in failure, when a fracture or slip surface nucleates.

Fracture at some point in the lithosphere is taken to occur when stresses at that point first exceed the normalized fracture criterion (8). Once fracturing has occurred, the stress state in the vicinity of the fracture (or fractured region) is no longer homogeneous [e.g., Pollard and Segall, 1987], or even elastic for extensively fractured rock [Melosh, 1977], so the assumption of an isotropic, homogeneous, linearly elastic plate on which the calculations are based is no longer valid for that fractured region. If the rock mass strength is exceeded (O'critical >- 1), more sophisticated methods such as discontinuum theory or plasticity must be used to represent the inelastic stress state in that region. We also recognize that the processes of fault nucleation and slip, jointing, and brittle strength envelopes based on these processes, may not be adequately characterized simply by the Hoek-Brown criterion defined above [e.g., Rice, 1983; Segall, 1991], but we

80

60

-40 0

RT SS : CN

1 2 3 4

Figure 4a. Elastic stresses (compression positive) at the surface due to the final load showing apparent prediction of strike-slip faults based only on stress geometry. Asterisk indicates position of maximum bending stress and approximate location of observed concentric grabens. Bar shows width of final load. RT, radial thrust faults inferred; SS, strike-slip faults; CN, concentric normal faults.

Fr• r/a LOAD 1 , 2 3 4

• 10 --

20 I I I

0

0.2 •

0.4

Figure 4b. Contour map of normalized brittle fracture criterion (critical stress ratio) for lunar lithosphere subjected to a surface load. Shaded area indicates potentially failed lithosphere. Asterisk indicates position of maximum bending stress. Plot assumes instantaneous emplacement of final load and no inelastic deformation in potentially failed zone. Contours for elastic region may also change for incremental emplacement of load.

SCHULTZ AND ZUBER: FAILURE OF SURFACE ROCKS 14,695

retain the simpler formulation as an aid to comparing our results to those of previous workers.

Results

Stresses calculated from the mare basin loading case for the lunar surface (z = 0 km) are comparable to analogous values of the normal stresses obtained in previous studies of lunar basin loading [e.g., Solomon and Head, 1979; Pullan and Lainbeck, 1981 ], as illustrated in Figure 4a. The maximum bending stress Crrr occurs in our model at a normalized radius r/a -- 1.7-1.8. The prediction of strike-slip faults from these stresses, as shown in the figure for this location, is based on stress geometry or trajectories alone and does not consider rock strength in tension or compression.

Application of rock-strength criteria to the stresses calculated for the final load (Figure 4a) shows regions of jointing and/or faulting due to the mare basin load that might be inferred from these stresses (Figure 4b). Contoured values of the critical stress ratio O'critica• in the figure divide the lithosphere into two main regions: one stable and elastic (O'critical < 1), and the other potentially fractured and inelastic (O'critical > 1). This representation of yielding in a flexed lithosphere is similar to that used by Chappie and Forsyth [1979] for oceanic lithosphere and illustrates the region in which elastic stresses are invalid (shaded in Figure 4b) [Turcotte and Schubert, 1982, pp. 341- 344; Forsyth, 1980; Price and Audley-Charles, 1983; McNutt, 1984; Sandwell and Schubert, 1992; McGovern and Solomon,

1993]. Radial thrust faults are predicted to occur under the load only

for a thin lunar lithosphere (H = 25 km). Stresses beneath the load for thicker lithosphere, such as 50 km used above, are slightly smaller than that required by the rock strength criterion. We infer that radial thrust faults may nucleate in the lithosphere beneath mare basalt loads under certain conditions.

Failure of Near-Surface Rock

In apparent agreement with previous studies, the orientations of principal stresses at the planetary surface due to the entire, final load appear to predict strike-slip faulting interior to the position of maximum bending stress (r/a < 1.8). However, these stresses substantially exceed the rock fracture criterion and predict a fault type (strike-slip) that is not observed. Instead, smaller values of the load were investigated in order to identify stress states comparable to the fracture criterion (equation 8) so that a more realistic prediction of fault types could be made. We evaluated a succession of loads, having constant aspect ratios (appropriate also to volcano growth), of 1/10-, 1/4-, and 1/2- size in addition to the full, final load. Stresses from the 1/10 load case were too small to be included in the following figures and were not associated with significant rock failure. Successive growth of loads having constant radii equal to the final load radius was also investigated. On the other hand, the elastic modulus used in the calculations, 100 GPa, may be too large to represent the modulus of near-surface rocks [Golombek, 1985]. A modulus value of 30 GPa, appropriate to an average jointed basalt [Schultz, 1993], was also used to investigate the modulus effect by suitably reducing the magnitudes of stress calculated for the surface.

The stress state due to an axisymmetric surface load on the Moon is illustrated in Figure 5. The stress difference calculated

from model stresões, shown in Figure 5a, is not equivalent to the

• 60

40

• 20

• 60

40

• 20

30

• 20

10

_FINAL__• LOAD

Model

a

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

inal

i

Required

I

b

Figure 5. Stresses required for failure prediction at the surface (z = 0) versus normalized radial position r/a. Decreasing load sizes indicated by final, 1/2, and 1/4 in all panels; curves for 1/2 and 1/4 loads plotted as function of final normalized radius r/a. (a) Stress difference calculated from elastic model. (b) Minimum stress difference required for tensile or shear failure, given by Hoek-Brown criterion, equation (7). (c) Critical stress ratio for fracture, equation (8). Dotted line indicates O'critical - 1; greater values imply fracturing.

0 1 2 3

14,696 SCHULTZ AND ZUBER: FAILURE OF SURFACE ROCKS

minimum level of stress difference required for rock fracture, as shown in Figure 5b. Neither of these quantities is sufficient to predict faulting although both are necessary, as indicated by the values of critical stress ratio plotted in Figure 5c. As the load grows in thickness and radius, the local position of maximum bending stress (peaks in Figure 5a) moves outward. Figure 5c clearly shows for the assumed example that loads larger than about 1/4 of the final load should cause significant fracturing of surface rocks, and that the greatest intensity of fracturing (peaks in Figure 5c) sweeps outward as the load grows. Comparable results are found for the other load geometry although surface fracturing may not migrate outward as in the previous case.

Mohr diagrams of the stresses for the lunar surface near the position of maximum bending stress are shown in Figure 6. At the local, transient position of maximum bending stress due to the smaller loads (see Figure 5a), 1/4 to 1/2 load is sufficient to produce rock fracture at the surface (Figures 6a, 6c). However, the predicted fracture type will be a joint (tensile crack) because the Mohr circles for stress become tangent to the strength criterion at or near the tensile strength intercept (arrow, Figure 6a) on the • axis [e.g., Suppe, 1985, p. 154].

Map-Scale Structures and Initial Mixed-Mode Cracking

An interesting case arises for stress states that intersect the Mohr envelope with a tensile normal stress and nonzero shear stress. Examples of this phenomenon are variously termed "transitional tensile joints" [Suppe, 1985, p. 154], "hybrid extension/shear fractures" [Price and Cosgrove, 1990, p. 44; Hancock, 1985; Engelder, 1987], "echelon breakdown joints" [Pollard et al., 1982], or "mixed-mode cracks" [e.g., lngraffea, 1987]. Such fractures (mixed-mode I and II) typically propagate out-of-plane, producing wing cracks [e.g., lngraffea, 1987; Scholz, 1990, p. 26].

The map pattern of structures predicted to occur under tensile normal stress and nonzero shear stress can be evaluated

by examining the stability of fractures subjected to this stress state. Such fractures may be preexisting or newly formed. The most favorable cracks to propagate into map-scale structures will be those whose orientations are given by the intersection of the appropriate Mohr circle and the Hoek-Brown envelope (Figures 3, 6). The initial normal and shear stresses acting on these cracks are given by the intersection point on the envelope. If the Mohr circle intersects the envelope at the c• axis, no shear stress is resolved on the most favorable fracture and a Mode I crack

will grow perpendicular to the trajectory of the (tensile) least compressive remote principal stress. In all other cases, nonzero shear stresses on the crack, in addition to tensile normal stress, will produce oblique dilation and out-of-plane growth.

Mixed mode (Modes I and II) fractures characteristically propagate at angles to their original crack planes, as shown in experimental and theoretical studies [e.g., Brace and Bornbolakis, 1963]. The angle of initial crack extension 0under mixed-mode stresses is given by [lngraffea, 1987, pp. 90-93]

0 ---- -2tan-l[•g•(-2Ki+24Ki2+8Ki•/] (9) in which the stress intensity factors are Ki=•,r'• and K n = ,,r-• where a is crack half-length. Cracks will propagate at this angle if the total stress intensity (Mode I plus Mode II) equals or exceeds the fracture toughness of the rock, Kic. The mixed-mode stress intensity gi,ii is given by

Ki, n = cos (•)[KlCOS 2(•)__• Knsi n 0] (10)

so the condition for crack propagation is given by gI,II _> gIc. Predicted angles of initial crack extension due to the normal

and shear stresses associated with the Hoek-Brown envelopes are illustrated in Figure 7. Cracks can extend at high angles (0 > 75 ø to the crack plane) for most nonzero values of shear stress if Kic is also exceeded. Given plausible values for fracture toughness of basaltic rock of 1 < Kic < 3 MPa m 1/2 [e.g., Schultz, 1993], preexisting or newly formed cracks would propagate at the angles given by (9).

However, growth of an array of mixed-mode cracks can either produce a fault zone oriented obliquely to the remote stress directions or an array of cracks normal to the remote tensile stress trajectory. Horii and Nemat-Nasser [1985] demonstrated that the sign of the least principal remote stress strongly influences the subsequent evolution of the mixed-mode crack array. They showed that a tensile principal stress promoted continued crack growth in such a way that the wing cracks (Figure 7) curved and propagated normal to this stress trajectory. The resulting crack array would be a set of generally parallel tensile cracks oriented normal to the tensile remote stress. On the other hand, a compressive least principal stress can be associated with linkage of mixed-mode cracks into a fault

1/4 I I I -/':;:;:1 I I i I

-5 0 5 •Y

Tension

20-

Final ':':':'

I : • I :i::/] I -20 -10 0 IY

b

Figure 6. Comparison between calculated elastic stresses (Mohr circles) at the lunar surface and brittle strength of jointed rock. Shaded regions, upper limit to strength of basaltic rock mass from Schultz [1993]. (a, c) Stress states for transient positions of local maximum bending stress. Arrow shows intersection between Mohr circle and rock strength envelope, indicating point of initial brittle failure. (b, d) Stress states for final position of maximum bending stress, r/a -- 1.8. Upper diagrams, E = 100 GPa; lower, 30 GPa. Stresses are given in MPa; small unlabled circles represent 1/4 loads.

Final

•r -5 0 •0 -5 0 O' c d

SCHULTZ AND ZUBER: FAILURE OF SURFACE ROCKS 14,697

ß

o

• 75

I .

•0 25

,

ø "1" .... /' ' 'i' ' .... / / • /10m / 1 • sF / • : / q

i / • b • • / -

/ • 4k /lm

Crack Growth ................................................................................ Kic

0 ß .... i .... i .... I , , , -2 -1.5 -1 -0.5 0

Least compressive stress, MPa

zone [e.g., Mastin and Pollard, 1988; Olson and Pollard, 1991] oriented obliquely (e.g., 30 ø) to the greatest compressive stress. The latter case would predict faults rather than joints in map view. We note that circumferential crack arrays, not strike-slip faults, are indicated for the situations considered in this paper because 0.rr is tensile in the region of maximum bending stress (Figure 8).

The transition from cracking to faulting occurs when the normal stress resolved on favorably oriented fractures changes from tensile to compressive. On the Mohr diagram (Figure 3), this point corresponds to intersection of Mohr circle and failure envelope on the ß axis. For a given value of maximum compressive principal stress (compression positive) o'1, the transition can occur for values of least principal stress 0'3trans of

¬ 2) (11) 0'3 trans = (m0'c - •m20'c 2 + 160' 1

Jointing can be predicted for model 0'3 at failure less than 0'3trans; faulting can be predicted if 0'3 exceeds 0'3trans.

Fracturing in Other Locations

Tensile cracking is also inferred as an initial fracture mechanism for the position of maximum bending stress defined by the final load (Figures 6b, 6d). As noted above, stress states that significantly exceed the strength criterion, such as the 1/2 load in Figures 6a and 6b, and the final load in Figures 6b, 6c,

Figure 7. Stability of cracks under resolved tensile and shear stresses. Curves in upper panel calculated by using ordered pairs (0'n, •') for failure obtained from Hoek-Brown envelopes in tensile regime. Ordered pairs represent loads on cracks (inset) having most critical orientations. Predicted angle of initial crack extension 0 (upper panel) oblique to crack for I• > 0; cracks > 1 m long will propagate (lower panel), producing wing cracks for I'rl > o.

I j

a I b

(Yrr

I

c

Figure 8. Progressive map pattern of joints resulting from resolved tension and shear. (a) Initial crack network. (b) Small extensional strain, showing both in-plane and out-of-plane (wing crack) growth. (c) Large extensional strain showing development of concentric joint sets about the load.

14,698 SCHULTZ AND ZUBER: FAILURE OF SURFACE ROCKS

and 6d, do not provide reliable indications of stresses during the onset of fracturing. In no case is strike-slip faulting predicted to occur as the initial failure mode in association with the smaller

loads. This prediction of radial extensional deformation (concentric jointing and normal faulting), which follows directly from the elastic model and Figure 6, agrees with the photogeologic evidence for such deformation about surface loads throughout the solar system.

Stress states for positions closer to the respective loads also predict tensile fracture, whereas those for the surface at greater distances (beyond the outer rise) predict concentric normal faulting, based on Mohr diagrams similar to those shown in Figure 6. These results suggest that a planetary surface can become progressively fractured by arrays of concentric joints and normal faults as a load grows in size. It is likely that bending strains in the near-surface region induced by lithospheric flexure about the load could also impose normal fault displacements onto previously formed joints, as shown experimentally by Mastin and Pollard [1988] for graben formation above dikes, so that normal faults could nucleate at

the surface within the jointed region. Mechanical interaction between normal faults, in the manner described by Melosh and Williams [1989], could produce shallow grabens without necessarily invoking the additional influence of subsurface layering as suggested by Golombek [1985], although layering, if it exists, could play some role in near-surface deformation.

The stress state at 1-km depth was also investigated for the same radial positions. At both local and final positions of maximum bending stress, stress states due to final loads are potentially fractured ("postfailure"), whereas those due to 1/2 and 1/4 loads are elastic. Stresses at depth below r/a = 1.8 unambiguously predict radial extension (concentric normal faulting, although O'critical < 1). The "postfailure" stress state for the final load is particularly interesting, however. At this location, the vertical and tangential (circumferential) stresses are equal and compressive, and the radial stress is tensile. This stress state is consistent with either strike-slip or normal faulting although neither is unambiguously predicted.

Elastic stresses calculated for the example above assume a ratio of horizontal to vertical pretectonic stress of •c = 1.0. However, this value of •c represents a worst case scenario because the associated stress state is most conducive to strike-

slip faulting. If a smaller value were used, such as 0.4 (fixed radial displacement) or 0.2 (geometric constraint: see Banerdt [1993]; assumes Poisson's ratio of 0.3), then the circumferential stress would be reduced and normal faulting, instead of strike- slip faulting, would be predicted unambiguously for all load sizes at the stated depth.

It is likely that concentric normal faults could nucleate either at the surface (on cracks) or at depth in the region of maximum bending stress. An extensive zone of jointing and/or normal faulting about the final load is, in fact, required by the levels of stress obtained in the elastic models. The depth of faulting beneath the observed concentric grabens could reach 5 km on the Moon (Figure 4). Surface fracturing could extend approximately from the edge of the load (r/a = 1) outward to beyond three load radii. The progressive development of surface extensional strain with increasing load size implied in these results may explain the range of radial positions for concentric grabens about Martian volcano loads noted for example by Comer et al. [1985].

In a parallel effort, McGovern and Solomon [1993] investigated stresses in the Martian lithosphere due to an axisymmetric volcano load which was emplaced incrementally. They used a finite element grid having a 5-km mesh, a linear Coulomb criterion, and strength properties appropriate for intact basalt. McGovei'n and Solomon found that concentric normal faults should form about the load first, as stresses attained levels

appropriate to rock strength; further increases in stress due to the larger load altered the stress magnitudes in such a way as to predict strike-slip faults in the same location. They suggested that strike-slip faults either should not form, due to the presence of previously formed normal faults, or might occur and become buried.

Our analytical model is generally consistent with the numerical results of McGovern and Solomon [1993] but we focus instead on failure mechanisms of shallow near-surface

rocks. In particular, we include a provision for tensile failure in a more complete rock strength criterion and find that tensile cracking might be important in the near-surface region. Prediction of strike-slip faults in the same region by McGovern and Solomon ignores the small tensile strength of near-surface rock masses. We also investigate different values of Young's modulus for surface rocks as well as the effect of stress

differences on fault-type prediction. Our detailed examination of the stress state from the surface to 1-km depth provides us with additional insight into the mechanisms of shallow brittle deformation.

Previous studies have estimated the effective elastic

thickness of lithosphere in the vicinity of major axisymmetric surface loads such as volcanoes and filled basins using the radial distance of concentric grabens and masses of the loads as observational constraints [Comer et al., 1979, 1985; Hall et al., 1986]. Some of these studies assumed instantaneous imposition of the final load on the surface and did not include an explicit criterion for shear failure. Our analysis demonstrates that consideration of failure criteria, stress magnitudes and differences, stress ' •:'• onen ..... n, and the importance of evolution of the stress state during loading, profoundly influence the interpretation of model stress states in the context of the development of surface faults and fractures. Future estimates of effective elastic thickness based on spatial distributions of tectonic structures should take these factors into account. It

should also be noted that in the case of volcanoes and basins on

Venus and Mars, the effective elastic thickness can be

independently calculated using flexural deflections of topography and power spectral relationships between gravity and topography. Improved data for both of these planets are currently available or may be from the Magellan and Mars Global Surveyor missions [Saunders et al., 1992; Ford and Pettengill, 1992; Tyler et al., 1992; Zuber et al., 1992].

Discussion

Stress Trajectories

Results of stress models of planetary surface loads are frequently displayed as trajectories of the principal horizontal deviatoric stresses. Only two of the three principal stresses need be evaluated for quasi two-dimensional theories of faulting such as that of Anderson [1951]. Previous interpretations of the "strike-slip stress state" (tensile and compressive horizontal

SCHULTZ AND ZUBER: FAILURE OF SURFACE ROCKS 14,699

Table 1. An Algorithm for Fault Prediction From Stresses

Step Procedure

Calculate normal and shear stresses

Calculate principal stresses, stress trajectories Choose rock strength criterion Compare calculated stress state to rock strength (note sign conventions for each) Calculate critical stress ratio 0.critical: (1) If stresses greatly exceed strength

(0.critical >> l), fracturing likely but characteristics uncertain

(2) If 0.critical < l, then no fracturing predicted

(3) If 0.critical • l, then evaluate jointing or faulting (1) If 0.3 < 0.3trans: jointing//0.1 predicted (2) If 0'3 > 0'3trans: faulting predicted.

Use principal stress orientations to define style.

stresses) have motivated a variety of explanations for the apparent lack of strike-slip faults including stress perturbation by layering or overburden [Golombek, 1985], magma transport [Sleep and Phillips, 1985], preexisting structure or depth [Williams and Gaddis, 1991], and unyielding lithosphere [Tanaka et al., 1991 ].

However, the stress state sometimes used to predict strike- slip faulting (Figure 4a) does not require this style of deformation. The same pattern of stresses is consistent with either strike-slip faulting or vertical jointing (tensile failure), depending in part on the magnitude of confining pressure. Numerous studies have documented jointing as a process associated with the so-called "strike-slip stress state" including Secor [1965], Horii and Nemat-Nasser [1985], Dyer [1988], Hancock and Engelder [1989], Olson and Pollard [1988], Pollard et al. [1990], and Cruikshank et al. [1991 ]. Because pore-fluid pressure is not required for jointing to occur in association with this stress state, these findings are also relevant to interpretation of stress states on anhydrous bodies such as the Moon or Venus. Thus the stress trajectories commonly taken as indicative of strike-slip faulting are not unique and should not be used in isolation to predict this type of deformation.

Stress Magnitudes

Stress trajectories have been combined with stress difference in order to predict fault orientations and locations. While the level of deviatoric stress (stress difference) is necessary for the prediction of faulting, it is not by itself sufficient. Models of plate flexure that assume the properties of elastic or viscous materials can result in values of stresses that are unphysically large. This is because ideal elastic and viscous materials contain no provision for failure, so in principle they have infinite strength [e.g., Lawn and Wilshaw, 1975, p. 73].

Explicit consideration of rock strength is necessary to predict faulting. For example, large values of bending stress have been obtained for the surface in elastic models of flexure near

Figure 9. Idealized comparison of model stresses and fault-type prediction. Contours show values of critical stress ratio. Dark shading, stresses exceed normalized failure criterion; applicability to fracture prediction questionable. Light shading, failure criterion met; prediction of brittle fracture valid, stress trajectories useful in predicting fracture orientation. No shading, elastic strain only; fault type prediction and stress trajectories irrelevant to fracturing. Note that accurate prediction contingent on deformation of "failed" region (dark shading) for elastic models.

terrestrial trenches and seamounts (e.g., >100 MPa tension at the surface), volcano loads [McGovern and Solomon, 1993] and coronae on Venus (-1.5 GPa tension for Artemis [Sandwell and Schubert, 1992]).

Given values of tensile strength for fractured rock masses such as basalt noted above, however, bending stresses in ideal elastic plates are often unrealistically large. This level of stress implies that the real lithosphere would have fractured well before the stresses built up to their final (elastic) levels. If stresses in the potentially fractured region were limited by using a rock strength criterion or by assuming an elastic-plastic plate rheology in the initial calculations [e.g., Chapple and Forsyth, 1979; Goetze and Evans, 1979; McNutt, 1984], the state of stress in this region would have been "frozen" at or near values of stress when the lithosphere first fractured and yielded due to the load-induced flexure [Melosh, 1977; Chapple and Forsyth, 1979; Beer and Johnston, 1981, pp. 182-183; Kanninen and Popelar, 1985, pp. 122-124]. Further, nonlinear evolution of stress states in a fractured medium [e.g., Chinnery, 1963; Lachenbruch, 1961; Jaeger and Cook, 1979; Pollard and Segall, 1987] implies that final elastic model stresses in excess of rock strength cannot be extrapolated linearly to a prefailure state in order to predict the type of initial failure. This limitation was recognized early by Melosh [1977], who emphasized that stresses which exceeded a failure criterion were taken to indicate

that the faulted lithosphere should be treated as a plastic material, because the elastic equations would no longer apply to fractured lithosphere.

14,700 SCHULTZ AND ZUBER: FAILURE OF SURFACE ROCKS

Initial Failure and Final Stresses

Many previous analyses have implicitly assumed that the stresses calculated from an elastic model and using the entire load are the appropriate ones to use to predict initial failure [e.g., Banerdt et al., 1982, 1992; Golombek, 1985; Janes and Melosh,

1990; Williams and Gaddis, 1991], although several workers have noted problems inherent in this approach [e.g., Melosh, 1977, 1978; Comer et al., 1985; McGovern and Solomon, 1993].

With the exception of a recent study of load growth associated with volcanic construction [McGovern and Solomon, 1993],

fault-type predictions assumed that fracture of the lithosphere to produce the observed structures did not occur until sometime after the entire, final load had been emplaced. In this approach, the final stresses calculated in the models are those which are

compared to failure criteria in order to predict faulting or to invert for lithospheric thickness. This assumption must be met for approaches such as those used by Golombek [1985] and Sleep and Phillips [1985] among others to be valid. However, because final elastic stresses typically exceed the rock strength envelope by a considerable amount, this assumption should be called into question.

Various workers have noted that prediction of faulting styles must be made when elastic stresses just equal the Coulomb (or other appropriate criterion for) rock strength [e.g., Sibson, 1974; Jaeger and Cook, 1979, pp. 425-429; Rice, 1983; Nur et al., 1986; Angelier, 1989; Scholz, 1990, p. 14]. Elastic stresses which greatly exceed the limits imposed by rock strength criteria implicitly indicate that the rock should have been allowed to fail. In a model that explicitly admits brittle fracture, the stress state obtained in the corresponding elastic model probably would not be achieved. This suggests why the prediction of strike-slip faults about axisymmetric surface loads is associated with "postfailure" elastic stresses. Instead of using this stress state to predict fault type, we suggest that stresses which are at or near failure should be used.

Synthesis

The results discussed in this paper are appropriate to a variety of axisymmetric surface loads such as mare basins, volcanoes, and coronae. Loads and associated structures on

silicate or icy lithospheres can also be investigated using this approach. An algorithm that documents important steps in the prediction of fault types from calculated stresses is given in Table 1. Application of this procedure to fault prediction is shown in Figure 9. The most reliable predictions can be made when calculated stress magnitudes are comparable to the rock mass or large-scale strength of near-surface rock.

Acknowledgments. Discussions with W.B. Banerdt, M.L. Blanpied, M.P. Golombek, S.J. Martel, P.J. McGovern, R.J. Phillips, and especially H.J. Melosh are much appreciated. Reviews by W.B. McKinnon and S.C. Solomon improved the presentation. Work contained in this paper was begun while R.A.S. was a postdoctoral fellow at NASA Goddard Space Flight Center. The study was supported in part by the Mars Geologic Mapping Program, NASA grant NAGW-2347 to R.A.S. and by a NASA Planetary Geology and Geophysics Program grant to M.T.Z. Lunar Orbiter image was provided by the National Space Science Data Center.

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R.A. Schultz, Geomechanics-Rock Fracture Group, Department of Geological Sciences, Mackay School of Mines/172, University of Nevada, Reno, NV 89557-0138. (Internet: schultz@mines.unr.edu)

M.T. Zuber, Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, MD 21218, and Geodynamics Branch, Code 921, NASA Goddard Space Flight Center, Greenbelt, MD 20771.

(Received August 9, 1993; revised March 30, 1994; accepted April 29, 1994.)