Lithospheric strengths derived from friction on pre-existing fractures and ductile flow laws show that the tensile strength of intact ice under applicable conditions is actually an order of magnitude stronger than widely assumed. It is demonstrated that this strength is everywhere greater than that required to initiate frictional sliding on pre-existing fractures and faults. Because the tensile strength of intact ice increases markedly with confining pressure, it actually exceeds the frictional strength at all depths. Thus, icy lithospheres will fail by frictional slip along pre-existing fractures at yeild stresses greater than previously assumed rather than opening tensile cracks in intact ice

Banerdt, W. B.

Golombek, M. P.

English

NASA Technical Reports Server

FAILURE STRENGTH OF ICY LITHOSPHERES 
M. P. Golombek and W. B. Banerdt (Jet Propulsion Laboratory, Caltech, 
Pasadena, CA 91109) 
It has been widely assumed in the literature that the stresses required 
for brittle failure of an icy lithosphere under tension are limited by the 
tensile strength of intact ice. This assumption has led to the widespread 
use of 2 MPa (20 bars) for the extensional strength of predominantly ice 
lithospheres (e.g. Ganymede's) , based on the unconfined tensile strength of 
ice near its melting temperature measured by Hawkes and Mellor (1). However, 
our understanding of the maximum stress levels in the earth's lithosphere 
are based on the frictional resistance to sliding on pre-existing fractures 
and not the strength of. intact rock in laboratory experiments, which is 
always greater. Similar friction relations for ice predict maximum stresses 
that are substantially greater than the apparent tensile strength of intact 
ice near its melting temperature. At first inspection, then, it appears that 
ice rich lithospheres would fail under tensional stress by tensile fracture 
of intact ice rather than sliding on pre-existing fractures. In this ab- 
stract we will introduce lithospheric strengths derived from friction on 
pre-existing fractures and ductile flow laws, derive these relations for icy 
lithospheres, show that the the tensile strength of intact ice under appli- 
cable conditions is actually an order of magnitude stronger than widely 
assumed, and demonstrate that this strength is everywhere greater than that 
required to initiate frictional sliding on pre-existing fractures and 
faults. 
The maximum stress levels found in the earth's crust are accurately 
predicted by Byerlee's law (2,3). This relation is based on laboratory 
measurements of the frictional resistance to sliding on pre-existing frac- 
tures, which occurs at stresses less than those required to break intact 
rock. Byerlee's law is of the form t=uSn+b, where t is the shear stress, Sn 
is the effective normal stress (normal stress minus pore pressure), u is the 
coefficient of friction, and b is a constant. In terms of S1 and S3, the 
maximum and minimum principal effective stresses (stress minus por 
2 5 pres- sure), Byerlee's law can be written S1=KS3+B, where K=[(u +I)'/~+u] , B is 
a function of b and u, and sliding is assumed to occur on the most advanta- 
geous slip plane. Laboratory friction measurements show that u and b are 
virtually independent of stress (except for a slight change for rocks at 135 
MPa), rock type, displacement, surface conditions, and temperature. For this 
application we will use the friction law for low stress determined by 
Byerlee (2) t=0.85Sn, which has been found to hold for a wide variety of 
geologic materials, because the appropriate low normal stress friction 
measurements on ice have not been made. At higher stress we use the friction 
data measured for ice, u=0.2 and b=10 MPa (4). The higher stress friction 
data for ice do constrain the applicable low stress friction law to be very 
close to that for rock, because the lowest normal stress measurements for 
ice are for Sn=17 MPa (5), which constrains the low stress friction law, 
that also must pass through the origin, to have a slope only marginally 
different (u>0.79) from that determined for rocks (u=0.85). In order for 
fluid pore pressure to decrease the friction on pre-existing faults and 
fractures, thereby changing the slope of the brittle yield stress versus 
depth curve, liquid water must fill the connected pore space. This is not 
likely to have occurred on the icy satellites given the extremely low 
surface temperatures (less than 1 0 0 ~ ~ )  and reasonable thermal gradients. 
Because the vertical stress is generally quite close to the lithostatic load 
(density times gravity times depth) in areas of low relief, this relation 
https://ntrs.nasa.gov/search.jsp?R=19870013924 2020-03-20T10:36:46+00:00Z
predicts a linear increase in yield stress with depth (Fig. 1). 
With increasing temperature, rock and ice deformation occurs by ductile 
flow. Flow laws for rocks, minerals, and ice have been experimentally deter- 
mined for stresses up to 1-2 GPa and strain rates down to 10-~/sec. These 
results can be extrapolated to geologic strain rates via creep equations 
which generally are of the form ~ ~ / ~ ~ = A ( S ~ - S ~ ) ~ ~ X ~ ( - Q / R T ) ,  where de/dt is 
the strain rate, R is the gas constant, T is absolute temperature, and A, Q 
(the activation energy), and n are experimentally determined constants. As a 
result the ductile strength is negligible at depths where T is high and 
increases exponentially with decreasing depth. We have used the experimen- 4 tally determined flow parameters (6,5) for pure ice Ih, ~=1.2x10-~~/sec-~a , 
4 Q-4.5~10 J/mole and n-4.0 extrapolated to geologic strain rates of 10-15/sec 
(about 3%/m.y.). A surface temperature of 1 0 0 ~ ~  and a thermal gradient of 
1.6O/km applicable to the early high temperatures likely in some of the 
larger icy satellites (e.g. Ganymede, 7) yields the strength envelope illu- 
strated in Fig. 1. It has been shown at high temperatures (i.e. polar 
conditions on the earth) that the inclusion of a small amount of silicates 
will greatly increase the creep strength of ice Ih (8). The amount of 
hardening from the inclusion of small amounts of silicates in the litho- 
spheres of the icy satellites is not known, but for lack of a better con- 
straint we will assume that the hardening resulting from the addition of 
less than a few percent of silicates in icy satellites lithospheres can be 
bracketed by an order of magnitude increase in creep strength (e.g. 9). 
The failure criterion for a given depth in the lithosphere is deter- 
mined by the weaker of the frictional or ductile strength at that depth. The 
yield stress increases with depth according to Byerlee's law until it ex- 
ceeds that calculated using the appropriate flow law, after which it de- 
creases exponentially with depth. The intersection of the brittle and duc- 
tile yield stress curves defines the brittle-ductile transition depth and 
also the peak stress needed to cause failure of the entire lithosphere. As a 
result the peak stress needed for lithospheric failure is dependent on the 
thermal gradient, which causes the ductile flow law to intersect the fric- 
tion curve at different depths (shallow for high thermal gradients, deep for 
low thermal gradients). These peak stresses are probably somewhat greater 
than those actually required to cause failure of the lithosphere because 
semibrittle and low temperature ductile processes tend to round off the 
intersection points between the brittle and ductile curves (10). 
As can be seen in Fig. 1, the peak stress needed to cause tensile 
failure of the lithosphere using the above parameters is on the order of 10 
MPa (100 bars), although the average stress in the lithosphere is only about 
half that value. This strength is applicable for large icy satellites (e.g. 
Ganymede). We have also determined strengths for the smaller icy satellites 
of Saturn using the appropriate gravities, densities, surface temperatures, 
and calculated thermal profiles (11) and found all strengths are substan- 
tially larger than this value. As a result this value is probably a minimum 
for the lithospheres of the icy satellites. 
Note that this peak stress (10 MPa) is significantly higher than the 
tensile strength of about 2 MPa used in virtually all previous studies, 
which is based on the the unconfined tensile fracture strength of ice near 
its melting temperature. Because failure should occur by the mechanism 
requiring the lowest stress, it would appear on first inspection that ten- 
sile fracture of intact ice is the relevant mode of fracture for icy satel- 
lites. However it can be shown using a Griffith failure criterion and taking 
the vertical lithostatic load into account that the stress difference for 
failure due to the tensile fracture of intact material at depth z is given 
by S1-S3=So+pgz for S1-S3<4So and by S1-S3=4[(Sopgz+So 2)1/2-~o] for 
S1-S3>4So,- where p is denslty, g is gravitational acceleration, and So is 
the unconfined tensile strength (e.g. 12). The first of these relations des- 
cribes the opening of tension cracks, which occurs when the confining pres- 
sure is relatively low. The second relation describes shear failure in 
tension (or compression), which is the mode of failure when the confining 
pressure is too great for open tension cracks to form. The depth of transi- 
tion between the two modes of tensile failure is given by z=3So/pg (about 5 
km for S0=2.5 MPa on a satellite with a gravitational acceleration similar 
to that of Ganymede). The curve for failure due to fracture of intact ice is 
plotted in Fig. 1, where it can be seen that frictional failure is preferred 
at all depths for the assumed material parameters of ice. This conclusion is 
borne out by the experimental results, in which ice samples failed due to 
frictional sliding along pre-existing saw cuts rather than initiating new 
fractures (4). Note also that if the tensile fracture strength of intact ice 
increases with decreasing temperature, as is the case for the compressional 
fracture strength (13), the curve for fracture of intact ice under tension 
will move to the left and frictional failure will be even more favorable. 
Similar results are also obtained for other choices of failure criteria. 
Thus the failure strength of an icy lithosphere is significantly greater 
than has been previously assumed. 
In conclusion, because the tensile strength of intact ice increases 
markedly with confining pressure, it actually exceeds the frictional 
strength at all depths. Thus, icy lithospheres will fail by frictional slip 
along pre-existing fractures at yield stresses greater than previously 
assumed rather than opening tensile cracks in intact ice. 
References (1) Hawkes & Mellor (1972) J Glaciol 11, 103. (2) Byerlee (1978) 
---- -
Pageoph 116, 615. (3) Brace & Kohlstedt (1980) JGR 85, 6248. (4) Beeman et 
al. (1984) EOS 65, 1077. (5) W. Durham, written com. (6) Durham et al. 
(1983) P14LPSC JGR 88, B377. (7) Golombek&Banerdt (1986) Icarus Nov. 86. 
(8) Baker & Gerberich (1979) J Glaciol 24, 179. (9) Friedson & Stevenson 
(1983) Icarus 56, 1. (10) Kirby (1980) JGR 85, 6353. (11) Ellsworth & 
Schubert (1983) Icarus 54, 490. (12) Jaeger & Cook (1976) Fund. Rock Mech. 
(13) Parameswaran & Jones (1975) J Glaciol 14, 305. 
Fig. 1. Brittle and duc- 
tile yield stress versus 
depth curve for compres - 
sion (to the right) and 
extens ion (to the left) , 
with a peak stress of 10 
MPa (see text for discus- 
sion). The dashed lines 
n 
E 
Y 
show the failure strength 
V of intact ice as a func- 
5 Q tion of depth assuming a 
23 Griffith failure criteri- 
on. Because stresses re- 
quired to break intact 
rock are greater than 
those required to initiate 
sliding on pre-existing 
-20 -10 0 10 20 fractures, lithospheric 
Stress Difference (MPa) failure occurs by fric- 
tional sliding on pre- 
existing fractures. 


Failure strength of icy lithospheres

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