Stress and temperature in the bending lithosphere as ...

Geophys. J . R . ash. SOC. (1979) 59,463-478

Stress and temperature in the bending lithosphere as constrained by experimental rock mechanics

Christopher Goetze* 'and Brian E v a n S D e p a r t m e n t of Earth and Planetary Sciences, ~ Massachusetts Institute of Technology, Cambridge, Massachusetts 021 39, USA

Received 1979 April 23 ; in original form 1978 November 13

Summary. Previous attempts to deduce the stress distribution in the bending lithosphere near a consuming plate margin have relied on the observed bathymetry and an assumed constitutive relation for lithospheric behaviour, e g . perfectly elastic, viscous/perfectly plastic, or elastic perfectly plastic. From the point of view of rock mechanics, each of these approximations fails to describe one or more of several basic phenomena, including brittle failure of rock, temperature dependence of elasticity, and temperature and/or strain rate dependence of ductile behaviour. In order to formulate a more realistic constitutive relation, a limiting yield strength curve, which is primarily a function of temperature, is constructed from data from brittle failure and ductile flow experiments. The moments which can be supported by plates with this constitutive behaviour are compared to the moments calculated from bathymetric profiles. The comparison indicates that moments required by the bathymetric data are consistent with moments supported by plates with experimentally determined constitutive laws as extrapolated to geo- logically reasonable temperatures and strain rates. The stresses developed in such models are required to reach values greater than 100 MPat in the depth range 25-45 km. Geotherms necessary for strength curves consistent with moments calculated from the bathymetric data match those derived from heat flow data for the Aleutian, Bonin, Mariana and Tonga trenches. Of the trenches studied, only the geotherm inferred from the Kuril trench data is significantly different, perhaps implying that the Kuril plate is weaker than the others. The strength curves show that as a first approximation it is better to assume that bending moment is independent of curvature of the plate than to assume that bending moment and curvature are linearly related.

1 Forward

Most of this paper had been written when C. Goetze died suddenly on 1977 November 21. Many aspects of his principal work are represented in this study: the careful measurement *Deceased. + 1 MPa = 10 bars.

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of creep properties of olivine, the fitting of data to appropriate theoretical flow laws, their extrapolation to geologic strain rates, and finally the application to a specific geologic problem. B. Evans completed the manuscript. It is hoped that the publication of this study will stimulate increased communication between workers in rock mechanics and tectonic modelling, in addition to providing an estimate of the stress profile in the earth.

C. Goetze and B. Evans

2 Introduction

Outer topographic rises on the seaward sides of consuming plate margins are well established features at most island arcs (eg. Hanks 1971; Watts & Talwani 1974). Gravity data show that the outer rises are isostatically uncompensated, as they would be if due to the bending of a plate (Watts & Talwani 1974). The existence and simple quantitative aspects of outer rises have been investigated assuming a perfectly elastic plate (Walcott 1970; Hanks 1971 ; le Pichon, Francheteau & Bonnin 1973; Caldwell et al. 1976; Parsons & Molnar 1976), a viscuous plate (de Bremaecker 1977), and an elastic-perfectly plastic plate (Turcotte, McAdoo & Caldwell 1978; Forsythe & Chapple 1978). Examples of the fits of a bent elastic plate (from Caldwell et al. 1976) and of a viscous plate (from de Bremaecker 1977) to a topographic profile of an outer rise are shown in Fig. 1. The deflection of the plate as a function of position relative to the geoid w(x) is directly observable from bathymetry data.

If the topography is due to bending, using the quantity w(x), as determined from bathymetry, we can easily compute the moment about the trench axis (see Fig. 1 and Caldwell et al. 1976 for definitions and details of the geometry) which is necessary to cause the outer rise. Under the assumption that the moments caused by forces at the trailing edge of the sub- ducted plate have little effect on the outer rise system, then the moment distribution can be

c

wb n

(b)

Figurel. (a) Bathymetric profile of the Mariana trench with the elastic curve (dashed) of Caldwell et al. (1976) and the curve (solid) for a viscous rheology (de Bremaecker 1977) shown for comparison. (b) Definition of the parameters wb, xb, w(x) of trench geometry following the terminology of Caldwell et a / . (1976).

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Stress and temperature in the bending lithosphere 465 computed directly from w(x) from statics alone, without any assumptions about the rheology of the material, viz.,

rn(xo) =J’ w(x) Apg (x-xo) dx

where Ap is the difference in the densities of the overlying and underlying fluids (water and aesthenosphere) and g = 9.8 m s - ~ , and xo is the location of the trench axis. We call this quantity the actual or observed moment. Since the integral is very sensitive to irregulari- ties at large distances from the trench axis, one must smooth the trench profile. Any curve which fits the profile could be used, but in this paper we use the elastic profile to represent the smoothed curve. In so doing we do not intend to imply that the plate behaves as a perfectly elastic body and, in fact, we intend to show later that it does not. However, use of the elastic solution allows one to compute the moment analytically. Furthermore, this moment differs insignificantly from that computed using the viscous (de Brernaepker 1977) or elastic perfectly plastic rheologies (Turcotte et al. 1978). The shape of the bent plate, and therefore of the outer rise, can be characterized by two independent parameters x b and wb as shown in Fig. l(b) (Caldwell e t al. 1976). x b is related to the wavenumber, a, by x b = n / k , while the maximum moment, which occurs at x = - x b , is given by

ca

XO

M = maximum moment 1

Another useful parameter is

K = maximum curvature 1 6wb/xg.

The maximum moments for a number of trench profiles are given in Table 1. The differential stress distribution which supports such a moment remains completely

unknown. Several alternative stress distributions are shown in Fig. 2(a)-(c). The elastic case, 2(a), requires the largest stresses (- 0.9-1.1 GPa, Caldwell et al. 1976) to be present at the top and bottom of the plate. Fig. 2(b) shows how this profile would be modified if the material had a plastic yield strength, up, which for simplicity is assumed to be the same in tension as compression. Notice that when the plate is perfectly plastic, the moment nd longer increases without limit as in the elastic model but reaches a peak value of %up h2 for large K (h is the thickness of the plate). The yield stress up is reached at the point of inter- section of the yield strength envelope (which is, in this case, simply a straight line) with the

Table 1 . Maximum bending moment computed from curves matched to bathymetric data (actual moment).

Trench name Wb (km) Max. moment ( loz2 dynes cm/cm of trench)

Mariana 55 1.4 Bonin 53 1 .o Aleutian 50 0.8 Kuril 42 0.4 Tonga 6 0 0.7

Data from Caldwell ei al. (1 976) and Turcotte et al. (1978). Maximum moment was computed using the elastic formula given in the text.

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466 C. Goetze and B. Evans

0 0;- 9;

K

-q '0,

=--f I 1 I (c>

K

1 K

Figure 2. Stress versus depth and moment developed as a function of curvature for the perfectly elastic (a), elastic perfectly-plastic (b), and brittle, elastic and ductile (c) rheologies. The solid lines interior to the bounding strength curves in (b) and (c) are stress distributions for different curvature.

elastic curve. Curvatures for which the differential elastic stress does not reach up remain completely elastic (Fig. 2(a)). Notice that a region of yield in relative tension overlies a region of yield in relative compression.

Previous authors have used perfectly elastic, perfectly plastic, and elastic perfectly plastic models to describe the deflection of the bending plate at any given distance along the trench profile Although these models allow one to formulate and solve the differential equations appropriate to a given rheology, the stress distributions consistent with the models seem to be unrealistic when viewed in the light of experimental evidence from rock mechanics.

Fbr instance, the large differential stresses obtained for the elastic case in the upper portion of the plate are unrealistic in view of the abundant experimental evidence of the weakness of earth materials in relative tension (see Barton 1976 for a review). Likewise the perfectly plastic and elastic perfectly plastic rheologies do not account for the dramatic variation of the ductile yield stress with temperature and hence depth in the earth. A more realistic estimate of the strength of the lithosphere, based on experimental rock mechanics and discussed in some detail below, is shown in Fig. 2(c). The vertical stress is assumed to be the lithostatic pressure (pgz). In the normal faulting case when the horizontal stress, UH, is less than the vertical stress, uv, the strength is lower than for the case when UH> uv (compressive stiengths > 0). In both cases the strength increases with depth and reaches a peak in the middle of the lithosphere, before decreasing as temperatures increase. At first the corresponding moment increases rapidly with increasing curvature, but later the moment approaches an asymptotic limit.

Thus we do not attempt to completely describe the plate profile as a function of distance from the axis, but use the results of experimental rock mechanics to formulate a more realistic yield curve. We then compute the resulting moment, called the derived moment, as a function of maximum curvature, and compare the derived moment to the actual moments of

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Stress and temperature in the bending lithosphere 467

Table 1. Since the yield curve is a function of temperature the derived moment wiU also be a function of temperature. Therefore, comparison of derived and actual moments can be used to constrain the geotherms in different regions.

3 Formulating a yield envelope

Rock mechanics experiments indicate that three separate regimes of deformation are involved in the definition of the strength envelope: cataclastic, where the stress rises rapidly with pressure; elastic, for the narow band around the neutral plane of the lithosphere; and ductile, for the region of descending strength near the bottom of the lithosphere (see Paterson 1978 and Goetze 1978 for reviews). In each of these regimes, large extrapolations from laboratory conditions to geologic conditions are required, and each regime is also a subject of continuing research. Consequently some aspects of the envelope are the subject of controversy and discussion. In this paper, only the basic aspects of each regime wih be taken into consideration. Notwithstanding the uncertainties in the treatment of the detformation mechanisms, a number of important conclusions may be reached and the stress distributions derived from the envelope can be expected to approximate those actually present in the lithosphere. Below we discuss each regime and define the strength envelope.

3.1 C A T A C L A S T I C F A I L U R E A N D J O I N T S L I D I N G

At low temperatures and pressure, fracture strengths, as determined in the laboratory, can exceed the stress needed for slip on a pre-existing fault by several kilobars (see Fig. 3). How- ever, the latter are believed to be the relevant strengths for the lithospheric problem for several reasons:

(a) Direct and indirect observations indicate that jointing is widespread at shallow depths (Birch 1964; Brace 1972). (b) Direct observations of in situ stresses to 4 km depth shown that in the areas sampled, stresses, and therefore presumably strengths, approach zero f 5-10 MPa at the Earth’s surface (for a review see McGarr & Gay 1978). (c) Increasing temperature causes the fracture strength of a large number of common rock types to decrease to the point where the strength is controlled by sliding on the fracture that formed (Stesky et al. 1974; Tullis & Yund 1977). (d) The fracture strength is known to be very dependent on the quality of the specimens, in particular, the length of the cracks initially present (Barton 1976). A reduction in fracture strength is commonly observed when testing larger specimens (Brace 1974; Pratt et al. 1972; Bieniawski & Van Heerden 1975), simply because the maximum crack length cannot exceed the specimen width. On the other hand, the sliding characteristics are the same for granite and sand and do not depend on sample volume, and therefore sliding experiments can be extrapolated to large rock masses with more confidence than fracture strength. (e) Finally, faults clearly exist in the oceanic lithosphere in trenches (Ludwig et al. 1966) and earthquakes are common in these regions (Stauder f 968). Although exceptions exist, the radiation from these events does not imply stress drops much larger than those of earthquakes occurring on pre-existing faults (Molnar & Wyss 1972; Wyss 1970). Therefore, although some of these events seem to rupture deep into the lithosphere (Abe 1972; Kanamori 1971), the brittle strength of the oceanic lithosphere is probably that for faulted, not unfaulted material.

One of the notable advances in rock mechanics in recent years has been the observation that joint sliding strengths for a wide variety of rock materials fall within a remarkably

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10

c, (IO'PA)

5

0

'< ,>' W G 5 W

, ,,y W G 6 0 C

'>/ WG 900

(IO'PA)

Figure 3. Shear stress (uT = u, -u3,,, u l , uz, u3 are the principal stresses) as a function of confining pressure for Westerly granite (WG, Tullis & Yund 1977) and San Marcos gabbro (SMG, Stesky 1975). The granite experiments were done at several different temperatures which are indicated. The strength shows a large pressure dependence in the initial portion of the curve when the deformation processes are primarily brittle. The strength becomes relatively pressure-insensitive in the second portion of the curve when the deformation processes are primarily ductile. The shaded band represents a strength envelope for the brittle portion of the curves, the slope of which is 30".

narrow range shown by the dashed band in Fig. 3 (Stesky et al. 1974; Byerlee 1975; Byerlee & Summers 1976; Zoback & Byerlee 1976; Barton 1976; Summers & Byerlee 1977). A wide range of plastic strengths and fracture toughnesses are represented by the materials whose sliding characteristics lie within the dashed band. This suggests that sliding strength is controlled by some fundamental frictional property of rubble rather than any aspect of the mineralogy; the sliding strength, in situ, is therefore anticipated to be very insensitive to strain rate or rock type, and to temperature until the plastic strength is reached and the failure envelope levels off.

3.2 B R ITT L E - D u C T I L E T R A N s IT 10 N

Comparatively few measurements have been carried out in the temperature/pressure regime of@ the transition from brittle to ductile behaviour. Until recently only calcite-rich rocks had been investigated in the region of transition. Fig. 3 shows the results of experiments done by Tullis & Yund (1977) on this transition in Westerly granite for a range of temperatures. Notice also the high temperature sliding experiments of Stesky et al. (1974) on San Marcos gabbro. The single slope that best matches these data appears to be 30". Higher values are typical of low normal stress and lower values occur near the transition to a plastic regime.

3.3 D U C T I L E R E G I M E

Since we attempt to describe the bending of an oceanic lithosphere and upper mantle, the ductile strength of interest here is assumed to be that of olivine. In contrast to the problem of creep in the mantle, the yield strength of the oceanic lithosphere will be much greater due to the diminished temperatures. Numerous authors have commented on the fact that a power law does not fit the data well in the range > 2 kb (among them Stocker & Ashby 1973, and Post 1977). Since the stresses in the lithosphere may exceed this value, it is important not to assume a power law rheology over the whole range of stress.

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Stress and temperature in the bending Zithosphere 469

Goetze (1978) has summarized the mechanisms of flow in olivine, including hardness data from Evans & Goetze (1979). The flow law can be expressed analytically as

9.3 x 109pa > u > 2 x 108Pa

tol = 5.7 x 10” up = 8.5 x 109Pa el= 128 kcal/mole

i = Pozu’’ exp-Q2/RT (Climb cross slip and glide region (lb))

2 x 108Pa > u > 1O6Pa

Pm = 7.0 x n = 3 Q, = 125 kcal/mole.

For comparison with other data, Fig. 4 shows the experimental data for dry olivine obtained within the range 10-4/s < P < 10-6/s; the data were corrected to a common P = lO-’/s (see Evans & Goetze 1979, and Goetze 1978 for details). Also shown are curves

of equations using the values given. The strain rate appropriate for the bending lithosphere was estimated to be - lO-”/s by

assuming that strains of a few per cent develop over 10 million years. Changes in strain rate are later found to have a very modest effect on the computed moments.

3.4 ELASTIC R E G I M E

Near the neutral plane of the bending lithosphere, the stresses will be supported by the elastic stiffness of the plate, E(l - v 2 ) , where E is Young’s modulus and v is Poisson’s ratio.

O L I V I N E

O G - B rn C + A O K I R I R + K . B O P A E+G

‘ \ I

I , I 1 I I

500 moo 1500

T (DC)

0

Figure 4. A comparison of experimental data on ‘dry’ polycrystalline olivine aggregates in the ductile regime. Data are from Carter & Ave’Lallemant (C+A) 1970; Raleigh & Kirby (R+K) 1970; Black (B) 1972; Kirby & Raleigh (K +R) 1973; Post (P) 1977; Evans & Goetze (E +G) 1979. For comparison, flow laws for the high and low stress regions (equation la,b) are plotted and labelled Peierls and climb respectively. The solid curve is intended to represent the trend of the data. For a discussion see Goetze (1978).

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The values of E and v may be obtained directly from seismology for the period range 1 -1 00 s. The seismologically determined parameters are

p = 0.74 x 10" MPa K = 1.4 x 10"MPa v = 0.27 (unitless) E = 1.9 x 10"MPa and

E - = 2.0 x 10"MPa 1 - 2

for depths in the range 20-40km. However, one might well question whether parameters determined from stress waves of period lO+*s are valid for the description of changes in stress over periods of 10"s.

The initial dispersion within the seismic band is less than - 0.1 per centldecade of frequency; extrapolated over 13 decades this implies about a 1 per cent drop in modulus. But is this a valid extrapolation? The dispersive correction to the moduli will be greatest when the elastically supported stresses are of the same order as the yield stress (i.e., near point B in Fig. 2(c)) where the transition is made to steady state creep. The only two studies of microplasticity in rocks (Goetze 1971 ; Goetze & Brace 1972) indicate that as steady state creep is established, transient anelastic strains have not exceeded the instantaneous elastic strains. This indicates that for the rocks studied (granite, diabase and peridotite) the effective modulus is still greater than half of the high frequency value obtained from seismic data.

We provide a 'best guess' based on the limited experimental evidence as a dashed line near point A in Fig. 7 , but wish to emphasize that this estimate may be quite inaccurate. However, because of the strong thermal activation of the transient creep effects (Goetze 1971 ; Goetze & Brace 1972) the effective modulus is expected to rapidly return to near its seismic value with diminishing temperature. Uncertainties in the stress distribution so near the neutral plane are comparatively unimportant to the determination of the total moment, as we show in the next section. In fact even a four-fold reduction in modulus has almost negligible effect on the computed moment.


3.5 T H E S T R E N G T H E N V E L O P E

We may now proceed to use the various constitutive laws for the elastic, ductile and brittle regimes to define a strength curve for the lithosphere. The failure criterion is conveniently portrayed using a Mohr's diagram (Fig. S), where the vertical stress (by presumption a principal stress) has been set equal to the lithostatic pressure, pgz. In the uppermost part of the plate, which is in relative tension, the least principal stress at failure is horizontal and given by OH TENSION. In the lower part of the plate, the stress state is in relative compression and the applicable failure stress is uH COMPRESSION. The cataclastic Coulomb law (Fig. 3) is sensitive to increasing pressure but is not temperature or strain rate sensitive. The ductile portion of the envelope (Fig. 4) is relatively pressure-insensitive (Stocker & Ashby 1973; Goetze & Brace 1972) but is sensitive to strain rate and temperature. Strength decreases quite rapidly with depth (see equation la , b), reaching values whch generate negligible moment at 50 km. At shallow depths, the maximum stress achievable is limited by the Coulomb law. At a depth in the plate set by the flow laws, the strain rate and the geotherm, the ductile yield envelope intersects the Coulomb envelope and the maximum stresses obtainable are limited by the plastic yield envelope. It is important to notice that once the

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Stress and temperature in the bending lithosphere 47 1

Figure 5. Mohrb diagram showing the transition from brittle to ductile regimes. The sloped portion of the envelope is the cataclastic Coulomb failure criterion (see Fig. 3 ) while the level portion of the envelope is the ductile failure criterion which is pressure-insensitive. uT necessary for ductile flow depends, of course, on temperature and strain rate (see Fig. 4).

geotherm and the ductile and brittle constitutive behaviour are specified, the transition from brittle to ductile is completely determined for a given strain rate. The actual stresses developed in the plate as a result of the bending at the trench increase in magnitude on either side of the neutral plane, according to the elastic constitutive law, until the elastic stresses approach the appropriate failure envelope, i.e. brittle on the extensional side of the neutral plane and ductile on the compressional side (see Figs 2(c) and 7). The moment is then computed from the stress profile

M ( K ) = (OH - a") (z - z,) dz lo- where t, is the depth to the neutral plane. The brittle and elastic portion of the strength criterion are relatively independent of temperature. However, the plastic portion strongly depends on temperature, and therefore the computed moment depends on the portion of the geotherm above 650°C (approximately the point of transition between elastic and ductile behaviour) as well as the curvature of the plate. In the elastic analogy, this amounts to saying that the maximum moment is dependent on the deflection of the plate and the flexural stiffness of the plate.

For the purpose of computing moments from the strength curve, we use geotherms of the family of curves,

T(K) = 350 t OLZ.

Each individual member of this set of linear approximations to the geotherm could simply be specified by its slope. Instead for reasons of computational convenience, we arbitrarily choose to describe each other particular geotherm by the depth at which the temperature is 750°C. For instance, for geotherm A, do = 40 km (Fig. 6). Thus, as stated above,

where do determines the strength envelope, and K determines the slope of the elastic portion of the stress-depth curve. Fig. 7 shows the computed stress distribution along geotherm A (do = 40 km) of Fig. 6. The assumption has been made that no external force is applied to the plate, and this results in slight shifts of the neutral plane because of non-linearities in the plastic strength envelope. This assumption is discussed further in the next section.

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1000

v

F 500

0 50 km 100 DQpth

Figure 6. A comparison of previously published geotherms (C+R, Clark & Ringwood 1964; M+B, MacCregor & Basu 1974; M+C, Mercier & Carter 1975; S + F , Sclater & Francheteau 1970) with the linear approximation (A) used to compute the strength envelope in Fig. 7. For this approximation the depth to 750°C (do) = 40 km.

o,-gM (IO'PA)

-15 -10 -5 0 5 10

c ::

Figure 7. The strength envelope, uv -q+ versus depth, for a linear approximation to the geotherm A (shown in Fig. 6). The dot-dash lines show the maximum stresses, both in relative tension uv - UH < 0 and in relative compression, uv -OH > 0, which may be supported. Any larger stress imposed would be relieved by brittle or ductile flow. The solid lines interior to the strength envelope are the stress distributions for various curvatures of bending (K is given in cm-'). The moment associated with each curve is shown at the end of each stress distribution line where the transition from the elastic regime to the brittle regime occurs. The very light dotted lines show the effect of uncertainty of Q on the ductile flow law. The interior dotted line corresponds to Q = 100 kcal/mole and the outer line is for Q = 130 kcal/mole. The dashed line, as discussed in Section 3.4, represents our guess at the decrease in the elastic modulus at the elastic-ductile transition.

4 Results and discussion

Fig. 8 shows the moment as a function of the depth to 750°C for several curvatures

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1c

2 10

2 10

/ 0 10 2 0 3 0 40 50 60 70

Depth to 750°C

Figure 8. The moment developed in the bending plate is used to predict the depth to 750°C for the five plates considered here (see text). The solid lines correspond to different curvatures which are indicated by the number next to each curve (each number has been normalized by multiplying by l@, i.e. 1 indicates a curvature of 1 X 10-8/cm). Shown for comparison are fhe depths at which the temperature is 750°C as predicted by previously published geotherms.(see Fig. 6 for sources).

(normalized according to Caldwell et al. 1976) as well as the values of do for several geotherms in the literature. The observed trench profiles have been parameterized by values of M and K and, by assuming that the stress distribution of Fig. 7 is applicable, the depth to 750" (do) may be predicted (Fig. 8).

The material where the temperature is below 800°C is likely to be part of the lithosphere where heat is transferred only by conduction. Therefore, the temperature as projected from surface heat flux should be comparatively well constrained. Nevertheless, the range of uncertainty in temperature deduced from the moments is probably larger than that suggested by the spread of curves in Fig. 6 and probably remains the largest uncertainty in comparing observed and computed moments. However, it is comforting that the predicted depths to 750°C for the Aleutian, Bonin, Mariana and Tonga trenches (Fig. 8) agree with the depth predicted by the geotherms derived from the heat flow data (Fig. 6). The Kuril trench data, however, predict a weaker lithosphere and, presumably, therefore a steeper geotherm than the other trenches. In order to test the significance of this prediction, we now examine the sensitivity of the moment curves to uncertainties in the various input parameters.

4.1 U N C E R T A I N T I E S IN ELASTIC M O D U L U S

If the elastic portion of the stress profile, K(E/l -v*), were diminished by a factor of 4 (at K = 1 x 10-8/cm), the computed moment would be lowered by only 15 per cent. In the terminology used earlier one can interpret this lowering of the effective modulus, E/1 - vz, at constant K as illustrating the insensitivity of the computed moments to variations in the assumed moduli. For curvatures near lod8 (cm-') the portion of the plate that remains elastic is simply too thin and too close to the neutral plane to develop an appreciable fraction of the total moment.

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4.2 U N C E R T A I N T I E S IN P E A K S T R E S S D E V E L O P E D

Likewise, because of the proximity to the neutral plane, changes in magnitude of the peak stresses at A and B (Figs 2(c) and 7) do not exert a strong influence on the computed moments. Assuming a perfectly elastic plate, Caldwell et al. (1976) compute a peak stress of 970 MPa for the Mariana trench profile (K = 1 x 10d/cm,M = 1.4 x lo'' dynes). The failure criterion used in generating Fig. 7 predicts a peak stress at A of - 860MPa. But the dashed line representing our best guess at the elastic-ductile transition is lower than this number by - 200-300MPa. With minor adjustments in do the strength profile probably need not exceed 200-300MPa in order to match the known moment and curvature. Although peak stresses are evidently not well determined by this method, the moments observed cannot be developed with reasonable geotherms if the peak stress is lowered by an order of magnitude (up = 10 MPa).


4.3 F A I L U R E CRITERION

The effect of modifying the slope of the Mohr's diagram from 30 to 20" results in a moment reduction of about 15 per cent or an increase of do of about 2 km. A reduction of 10" slope would result in a 9km shift in the depth to 750°C. Even at these lower failure angles, stresses of several kilobars are required to achieve the observed moments. The view that differential stresses cannot exceed a few hundred bars, as is still held by many seismologists, requires the computed moments to be more than an order of magnitude too low. The same may be said of using flow laws for hydrolytically weakened olivine (Post 1977). In our view, these two possibilities have been excluded for these regions of plate flexure by the observational data in hand.

4.4 V A R I A T I O N S I N ASSUMED S T R A I N R A T E

The rate sensitivity of the moments of the bending lithosphere is small, at least within the range of strain rate derived from plate velocities. Because the strength above one kilobar becomes progressively less dependent on temperature (more stress activated and less tbermally activated) (for a review see Goetze 1978), the strain rate dependence of the yield stress is decreased. Within the range of variables considered here, an order of magnitude increase in strain rate corresponds to a 1-2 km increase in do.

4.5 U N C E R T A I N T I E S IN T H E DUCTILE FLOW L A W

Goetze (1978) has discussed the uncertainties involved in the determination of the flow law of olivine and in this paper we use the flow law given by him to describe the deformation of dry olivine by dislocation mechanisms. Investigation of the Coble creep or superplastic regime and the effect of water on dislocation motion are ongoing projects in several rock mechanics laboratories. Pending the complete resolution of these equations, we regard the use of equation (la,b) as the best approximation for the following reasons:

(a) The experimental uncertainty of the' Coble creep/superplastic flow law is quite large (Twiss 1976; Schwenn & Goetze 1978). (b) Such comparisons of Coble creep/superplastic flow laws and dislocation flow laws as are possible indicate that dislocation flow will be dominant in the upper mantle (Goetze 1978).

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(c) Preliminary calculations of the stress distribution similar to Fig. 7 but using the hydrolytically weakened flow law given by Post (1977) require unacceptabw large stresses C> 1000 MPa) to generate moments which are required from the bathymetry.

The effect of experimental uncertainties in the parameters of equation (la, b) can be assessed by examining the effect of the scatter in the experimental data at P = lO-'/s and the effect of uncertainty in the determination of the activation energy (Q in equation lb). The former may be judged from Fig. 4 to be k 50°C for a given differential stress in the range of ~ 1 4 3 = 100-500MPa. Along geotherm A, for instance, this corresponds to approximately f 3 km in do. A range of activation energies that includes all results on dry olivine is

100 kcal/mole < Q < 130 kcal/mole

Fig. 7 shows the effect on the strength envelope of varying the activation energy. Increasing the activation energy increases the extrapolated strength for a given strain rate and temperature. Thus to maintain the original strength of each depth that will result in a given moment and curvature, an increase in activation energy requires an increase in temperature at each depth, namely an increase in geotherm steepness and therefore a decrease in the depth to 750". In this case, recomputing self-consistent flow laws for 100 and 130kcal/mole gives an increase of d o of 7 km and a decrease of do of 2 km respectively.

4.6 T H E E F F E C T O N I N T R A P L A T E STRESS

McAdoo, Caldwell & Turcotte (1978) have suggested that the bathymetry of the trench profile requires a 2-3 kb compressive stress in the plane of the flat-lying portion of the plate. The effect of adding a compressive stress to the lithosphere may be investigated by computing moments for stress distributions having an imbalance in the areas above and below the neutral plane. An imbalance corresponding to a net compressive stress of 20 MPa over the approximately 50 km lithosphere results in only a 3-7 per cent increase in the computed moment, or an increase of 1-2 km in the depth to 750OC. Compressive stresses in the kilobar range would alter Fig. 7 substantially, although, in principle, an appropriate strength curve could be computed using the experimental data, provided that the magnitude of the compressive stress were known. There is, of course, no need to appeal to such stresses in order to support the observed moments.

4.7 CONCLUDING REMARKS

A number of other points may be made about these computed moments. Firstly, for the range of curvatures typically encountered, K = 0.3-1 .O x 10-8/cm, the moment has nearly reached its peak value (Figs 2(c) and 9). For an elastic plate, moment (M) and curvature ( K ) are related linearly

M = D K

where D is the flexural stiffness of the plate. The trench profiles examined here are very far from this condition. A more reasonable first approximation would seem to be that moment is independent of K . Fig. 9(b) illustrates the variation of stiffness with curvature along the geotherm A. Even at low curvature and hence low stresses, the stiffness is still a non-linear function of curvature. This introduces complications into the interpretation of the data gathered by Walcott (1970) on flexural stiffnesses. The enormous difference in stress level

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32 r

0 5 10

Figure 94a) The moment developed as a function of curvature for the elastic theory (broken line) and the elastic-brittle-ductile strength envelope (Fig. 7). The moment has been normalized by the moment developed by a curvature of lO-'/cm. (b) A plot of the stiffness (0) of the plate shows that D is a non- linear function of curvature (K) even for very low curvature.

and curvature between the profile of Lake Agassiz and the depression surrounding Hawaii suggests that flexural stiffness for these two regions should differ by over an order of magnitude on the basis of non-linearity alone, a fact not taken into account by Walcott since he used an elastic approximation. The rate dependence which Walcott's stiffnesses suggest is also vastly in excess of the rate dependence suggested above.

Secondly, the good overlap between the predicted geotherms and the geotherms based on heat flux seems to indicate that the method outlined here can provide a measure of the geotherm which is accurate to perhaps +20 per cent. The fact that the stress profiles were computed using experimentally obtained material constants and the added constraints of agreement with an observed moment and an independently predicted geotherm encourages us to believe that the stresses have been estimated to & S O per cent and perhaps better, although that is, of course, supposition.

Thirdly, the abnormally low moment displayed by the Kuril profile could be due to an abnormally warm geotherm, an abnormally weak failure criterion, or net tensile stresses in the lithosphere in excess of 1 kb. At present, the first reason is considered to be the most plausible, but the other reasons are not ruled out by any current observational evidence. , Finally, Kanamori (1971) has suggested that, in the case of at least one large earthquake,

fracturing may have taken place across the entire thickness of the lithosphere. Such a situation is inconsistent with bending regardless of the assumed rheology unless super- imposed stresses substantially lower the neutral axis of the bent plate. Thus, if large-scale deep fracturing is proven to exist, one may be able to place many useful constraints on the stress state in the plate by using methods similar to the ones outlined above.

5 Conclusions

(1) The experimental results of rock mechanics and previously published geotherms can be used to formulate curves of strength versus depth which in turn generate moments that agree well with those observed from bathymetry. (2) The computed moment depends most strongly on the details of the geotherm; hence, the moment-curvature relation of observed trenches may give information on the geotherm under the trench profile. ( 3 ) As a first approximation it is probably better to assume that moment is independent of curvature than to assume that they are linearly related.

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Acknowledgments

Peter Molnar and William F. Brace read the manuscript at several different stages and contributed many useful comments which improved the paper. The editor, J. W. Dewey, and the reviewers of the paper are thanked for their critical appraisals.

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