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The stress fields surrounding cave in karst terranes exert important controls on passage shape that are ubiquitous and sometimes dominant. Compressional stresses may enhance dissolution of lateral cave walls below the water table, causing the characteristic oval sections of phreatic tubes. Tensional stresses probably cause the ceiling cracks that lie along cave passages, the arcuate fractures of breakdown rooms, the high vaulted shapes of old, inactive caves, and breakdown effects near entrances. Breakdown is common near cave entrances, particularly in temperate zones, where freeze-thaw magnifies transient stresses due to temperature variations. Finally, arched shaped cross sections, especially when coupled with breakdown, illustrate the structural response to ovoid cavities to rectify their dissolutionally efficient, yet structurally poor design by forming a vaulted shape.
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Rock Mech. Rock Engng. (2006)
DOI 10.1007/s00603-006-0106-7
Technical Note
Effects of Stress on Cave Passage Shape in Karst Terranes
By
E. M. Criss, R. E. Criss, and G. R. Osburn
Washington University, St. Louis, MO, U.S.A.
Received January 18, 2006; accepted May 18, 2006
Published online August 17, 2006 # Springer-Verlag 2006
Keywords: Rock stress, tensional failure, speleology, caves, pressure solution.
1. Introduction
The shapes and orientations of caves in karst terranes are long-standi ng enigmas.
Palmer (1991) recognizes four basic passage shapes-branchwork, network, ramif orm
and anastomosing, that he argues are due to different water chemistries, bedrock and
caprock characteristics, and hydraulic controls including inundation histories. Jennings
(1985) attributes the shape and orientation of caves to preexisting rock structure and
the type of water flow inside the cavern. Veni (2005) argues that local lithologic, struc-
tural and hydrologic factors govern passage patterns, such as bedding, joints, and dip
and strike. Tall caves have been attributed to formation along joints, whereas low,
wide caves are commonly thought to result from preferential dissolution of particular
strata. Although such explanations are correct or at least plausible in many instances,
disagreement exists regarding many mechanisms. For example, joint orientations are
clearly important in network caves, yet in many areas little correlation is found
between the orientations of joints and cave passages (Orndorff et al., 2001). Thus,
although special controls are clearly important in given instances, their significance is
not universal.
We suggest that the stress fields surrounding caves exert important controls on pas-
sage shape that are ubiquitous and sometimes dominant. Compressional stresses may
enhance dissolution of later al cave walls below the water table, causing the charac-
teristic oval sections of phreatic tubes. Tensional stresses probably cause the ceiling
cracks that lie along cave passages, the arcuate fractures of breakdown rooms, the high
vaulted shapes of old, inactive caves, and breakdown effects near entrances.
2. Stress Concentrations around Caves
The stress concentrations around a circular opening can be approximated with the
biaxial equations (e.g., Obert and Duvall, 1967):
2
r
¼ð
x
þ
z
Þ
1
a
2
r
2
þð
x
z
Þ
1
4a
2
r
2
þ
3a
4
r
4
cosð2Þð1aÞ
2
¼ð
x
þ
z
Þ
1 þ
a
2
r
2
ð
x
z
Þ
1 þ
3a
4
r
4
cosð2Þð1bÞ
2
r
¼ð
z
x
Þ
1 þ
2a
2
r
2
3a
4
r
4
sinð2Þð1cÞ
These equations give the radial, tangential, and shear stresses (
r
,
,
r
), acting on a
stress element as a function of its distance from the cavity center (r), its angle from
horizontal (), the cavity radius (a) and the nominal horizontal and vertical stresses
(
x
,
z
), as illustrated in Fig. 1a, b. The horizontal and vertical stresses are inter-
related but their ratio depends on local conditions. Our calculations assume negligible
Fig. 1. a Orientation and direction of stresses for a stress element in a cylindrical coordinate system, after
Obert and Duvall (1967). b Field relationship of stress element to a cylindrical cavity of radius ‘a’ and
depth H
E. M. Criss et al.
nominal horizontal stresses, and a vertical stress related linearly to depth (e.g. Obert and
Duvall, 1967):
z
¼ H; ð2Þ
where
z
is the nominal vertical stress, is the rock’s unit weight, and H is the depth
(Fig. 1b). This equation when combined with Eq. (1a) indicates that the highest com-
pressional stress occurs on the side walls of caves, while the largest tensional stresses
are in the ceiling and floor (Fig. 2).
This useful, closed form solution of the stress field around a circular opening is
symmetrical in both the horizontal and vertical directions, as it neglects asymmetries
due to gravity. Certain inaccuracies arise in detail. The actual gravitational gradient
increases tension in the ceiling but decreases it in the floor, so in real caves the roof
and floor behave differently. These tensional stresses are also larger than would be ex-
pected under lithostatic o r hydrostatic conditions where significant horizontal pressure
exists. Additionally, Eqs. (1a), (1b), (1c) neglect certain stresses that can be significant
Fig. 2. Vertical section showing contours of the tangential stress concentration factor (
) near a cylindrical
cavity, defined as the ratio between tangential stress (
), calculated in Eq. (1b), over the nominal vertical
stress (
z
). Positive values indicate compressive stresses. The nominal horizontal stress is assumed to be 0.
Note the high compressive stress concentrations along the side walls of the cavity, with
approaching 3,
and the large tensional stress near the ceiling and floor where
approaches 1
Effects of Stress on Cave Passage Shape in Karst Terranes
in caves. Pore pressures have a significant effect on the stress field. Sudden draining of
cave passages can cause collapse, due to transient effects and loss of support.
Temperature variations, particularly freeze thaw, are similarly not addressed in the
above equations, but will also promote tensional failure. Temperature variations affect
various sections of the cave differently, particularly near entrances where breakdown
is accordingly common. In contrast, cave interiors have uniform temperatures and lack
such boundary effects.
3. Discussion
3.1 Compressional Stresses and their Effects
Compressional stresses can influence cave shape during dissolution due to their thermo-
dynamic effect on carbonate solubility. Both dissolution and precipitation of calcium
carbonate occur in karst regions, as a sensitive response to environmental conditions
that affect the dominant equilibrium relationship between calcite, dissolved carbon
dioxide, and calcium and bica rbonate ions:
CaCO
3
þ H
2
O þ CO
2
¼ Ca
þþ
þ 2 HCO
3
: ð3Þ
A well-known thermodynamic equation (e.g., Nordstrom and Munoz, 1986) relates the
pressure derivative of the equilibrium constant (K) of a reaction to the net volume
change (V
r
), the gas constant R and temperature T.
@ðln KÞ
@P
T
¼
V
r
RT
: ð4Þ
Fig. 3. Cross sectional sketches (ae) and longitudinal section (f) of cave passage shapes illustrating
different controls. a Oval shape of phreatic cave tube with elongated horizontal axis due to pressure
dissolution. b Tall cave passage formed along a preexisting vertical joint. c Oval cave passage with
subsequent tensional ceiling crack. d Tall, vault-shaped passage formed by upward stoping and collapse
of ceiling. e Breakout dome. f Longitudinal section of cave entrance on a hillside, showing large pile of
breakdown so that entry is near cave ceiling and far above bedrock floor
E. M. Criss et al.
ThecommongeologicoccurrenceofpressuresolutionprovesthatV
r
for calcite
dissolution is negative. That increased pressure drives calcite into solution is empiri-
cally shown through dissolution on grain to grain contacts (e.g., de Boer, 1977) and
can be observed in limestone at depths as shallow as 2030 m (Railsback, 1993).
Fig. 4. Photos of cave passage shapes (cf. Figs. 3a– f). a Oval phreatic tube, Pohl Avenue, Mammoth Cave
KY. b Structurally-controlled cave formed along preexisting vertical joint, Catacomb Cave, MO; the cave
continues for 100 m, mostly as a 1 m-wide, 510 m-tall slot that trends due eastwest. c Intersection of
1 m-high, 1.5 m-wide oval tube with main cave passage, showing jagged tensional ceiling crack formed
subsequent to passage, Rankin Cave, MO. d Enlarged, vault-shaped entrance of Milburn Cave, MO; en-
trance breakdown has been efficiently removed by outflowing cave stream. e Breakout dome where stoping
along arcuate ceiling fractures has modified the oval tube of Pohl Avenue (cf. Fig. 4a). f Entrance of Mertz
Cave, MO, showing large, sloping breakdown pile; the cave stream is inflowing so breakdown is not
efficiently removed, in contrast to Fig. 4d
Effects of Stress on Cave Passage Shape in Karst Terranes
Stylolites provide another example of pressure dissolution and have been described as
a type of failure, an ‘anticrack’ in the rock (Fletcher and Pollard, 1981). In areas of
high compressive stress (Fig. 2), the additional pressure will enhance already occur-
ring dissolution. Although this effect may be small, it will be significant near equili-
brium. Were the ceiling of a phreatic cave at equilibrium with the ambient ground
water, the side walls could continue to dissolve, as they are under higher pressure.
Furthermore, as the cave cross section becomes elongated horizontally, the stress on
the lateral walls will increase even more, accentuating the effects of pressure dissolu-
tion. The oval shape of phreatic tubes (Figs. 3a, 4a) could be due to pressure effects,
rather than to the influence of bedding planes, the preferential dissolution of particular
rock strata, or partial lling of cave passages with water.
3.2 Tensional Stresses and their Effects
Tensional stresses have great relevance to cave behavior. Even if hosted by rock with
great compressional strength, such as limestone, caves and tunnels can collapse due to
tension. Ceiling cracks provide an obvious example; common ly, one dominant crack
will run for great distances down the very center of a passage ceiling. Such cracks are
usually considered to be joints, and sometimes even referred to as the ‘lifeline of a
cave’’, the assumption being that the cave formed along this preexisting ‘master’
joint (Fig. 3b). Whereas such examples exist (Fig. 4b), caves more commonly do not
follow regional joint sets, yet ceiling cracks are almost universal, forcing the conclu-
sion that the so called ‘lifelines’ of caves are evidence of their tensional failure rather
than a cause of their formation (Figs. 3c, 4c). This explanation better explains the
discontinuous, jagged, irregular, and even arcuate character of most ceiling cracks and
their contrast with joints which are very straight. Such cracks generally occur in the
very center of the ceiling where tensile stresses are greatest.
Generally crack formation is but an initial step in cave collapse; caves tend to fail
incrementally through progressive crack formation, culminating in ceiling failure due to
the unsupported weight of roof rock. Presence of bedding planes will often result in a
characteristic ‘stair step’ (Figs. 3e, 4e) as the roof incrementally collapses. Engineers
have long observed that the collapse of unsupported materials will eventually form a
vault (Lauchli, 1915). This shape can be observed, in different stages of completion, in
various caves (Figs. 3d, 4d). This shape is also attained in three dimensions, through the
production of ‘beehive’ domes, as observed in the collapse of large rooms (Figs. 3e, 4e).
Breakdown is common near cave entrances, particularly in temperate zones where
freeze-thaw magnifies transient stresses due to temperature variations. In many caves
large passages are almost completely choked where they intersect Earth’s surface, and
entry is achieved by crawling beneath the ceiling (Figs. 3f, 4f). However, if breakdown
removal is efficient, such processes can greatly enlarge the cave entrance relative to
the size of the interior passage (Fig. 4d).
4. Conclusion
Stress phenomena inside caves are universal, but their importance is variable. Certain
cave shapes clearly reflect influencing or dominant stresses. The oval sections of
E. M. Criss et al.
phreatic tubes may indicate preferential dissolution of the lateral walls where com-
pressive stress is maximized. Ceiling cracks attest to the destructive power of roof
tensional stresses. Both large cave entrances and entrances choked with breakdown
indicate the influence of transient stresses due to temperature variations. Finally, arch-
shaped cross sections, especially when coupled with breakdown, illustrate the struc-
tural resp onse of ovoid cavities to rectify their dissolutionally efficient, yet structurally
poor design by forming a vaulted shape.
Acknowledgement
We thank Professors R. L. Mullen and S. A. Hauck for reviewing a preliminary version of the
paper, and Professors J. I. D. Alexander and S. Sridharan for valuable discussion.
References
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Fletcher, R., Pollard, D. (1981): Anticrack model for pressure solution surfaces. Geology 9,
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Jennings, J. N. (1985): Karst geomorphology. Blackwell, New York.
Lauchli, E. (1915): Tunneling. McGraw-Hill, New York.
Nordstrom, D. K., Munoz, J. L. (1986): Geochemical thermodynamics. Blackwell Scientific, Palo
Alto, CA, 477 pp.
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Orndorff, R. C., Weary, D. J., Sebela, S. (2001): Geologic framework of the Ozarks of south-
central Missouri Contributions to a conceptual model of karst. U.S. Geological Survey,
Water-Resources Investigations Report 01-4011, 1824.
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Veni, G. (2005): Passages. In: Culver, D. C., White, W. B. (eds.), Encyclopedia of caves, Elsevier,
New York, 436440.
Authors’ address: Dr. Robert E. Criss, Washington University, 1 Brookings Drive, Campus
Box 1169, St. Louis, MO 63130-4862, U.S.A.; e-mail: criss@wustl.edu
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Parts of a solid under stress show a higher solubility than stress-free parts. In porous, granular formations this effect causes dissolution of grain-to-grain contacts because stresses are concentrated here. This pressure-solution reaction leads to closer packing of the grains, resulting in porosity decrease. The dissolved material will reprecipitate in the remaining pores, thus further reducing porosity. The pressure-solution process comprises three steps: dissolution of the grain at stressed spots, diffusion to stress-free parts, and precipitation.In this article four different mechanisms are compared and evaluated: 1.1. dissolution caused by compressive stresses inside grain-to-grain boundaries, followed by diffusion through an adsorbed water layer to free pores (Weyl, 1959).2.2. Dissolution due to shear stresses at or just outside the rims of grain-to-grain contacts (Bathurst, 1958, 1975).3.3. Dissolution of tiny particles abraded from the original grains.4.4. Dissolution of plastically deformed parts.A general equation for the effect of anisotropic stress on the solubility of a solid is derived and some sources that led to deviations in earlier derivations are spotted. This equation indicates that high supersaturations may occur inside grain-to-grain contacts. Supersaturations due to stresses at or just outside the rims of grain-to-grain contacts, as well as those of abraded particles, are too low to cause effective pressure solution. This result strongly supports the mechanism proposed by Weyl (1959). It is in disagreement with Bathurst's proposal (1958). The mechanism mentioned in (1) is also supported by geological arguments, in contrast to the hypothesis of plastic deformation causing pressure solution.