Hollow cylinder tests for studying fracture around underground openings, RT Ewy, NGW Cook, LR Myer

Tags: stress path, internal pressure, Rotterdam, stress state, Sixth International Congress, hollow cylinders, external pressure, Lawrence Berkeley Laboratory, stress, California Mining and Mineral Resources Research Institute, The failure, axial stress, Earth Technology Corporation, free surface, Balkema Santarelli, Montreal, United States, UNDERGROUND OPENINGS, United States Government, University of California, University of California, Berkeley, Hollow cylinder, Lawrence Berkeley National Laboratory, U.S. Symposium on Rock Mechanics, laboratory test, failure mechanisms, Rock Mechanics
Content: Lawrence Berkeley National Laboratory Recent Work Title HOLLOW CYLINDER TESTS FOR STUDYING FRACTURE AROUND UNDERGROUND OPENINGS Permalink https://escholarship.org/uc/item/4xc9025s Authors Ewy, R.T. Cook, N.G.W. Myer, L.R. Publication Date 1988-05-01
Powered by the California Digital Library University of California
LBL-25308 (",q
Lawrence Berkeley Laboratory
Presented at the 29th Annual U.S. Symposium on Rock Mechanics, Minneapolis, MN, June 13-15, 1988, and to be published in the Proceedings
Hollow Cylinder Tests for Studying Fracture Around Underground Openings
R.T. Ewy, N.G.W. Cook, and L.R. Myer
May 1988
TWO-WEEK LOAN COPY ~--------~:c-lbis is a Librar;~-Cir;culating Copy __. which may be borrowed for two weeks.
Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098.
DISCLAIMER This document was prepared as an account of work sponsored by the United States government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of Califomia. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Govemment or any agency thereof or the Regents of the University of California.
To appear., 29th U.S. Symposium on Rock Mechanics, University of Minnesota, 1988. Hollow cylinder tests for studying fracture around underground openings Russell T. Ewy Department of materials science & Mineral Engineering, University of California, Berkeley, USA Neville G. W. Cook Department of Materials Science & Mineral Engineering and Earth Sciences Division, Lawrence Berkeley Laboratory, University of California, Berkeley, USA Larry R. Myer Earth Sciences Division, Lawrence Berkeley Laboratory, University of California, Berkeley, USA ABSTRACT: We are beginning a study of deformation and fracture around underground openings through experiments on thick-walled hollow cylinders of rock that incorporate several features: plane strain loading, the ability to impose different stress paths, and 'freezing' of the fracture geometry under load The_elastic moduli calculated from the - - - measured def<>rmatioiis are different than the moduli reported from uniaxial compression tests. ·The 'unconfined' strength of the rock surrounding the hole is two to three times typical uniaxial strengths, and a confining stress in the hole strengthens the rock. Although the load is axisymmetric, there is a preferred direction of failure. Failure is by britde spalling, resulting in triangular failed regions with pointed tips. The extent of failure is influenced by stress path and perhaps by strain rate. 1 INTRODUCTION For decades, laboratory tests on rocks have sought to achieve a homogeneous state of stress on a sample of rock, and to increase these stresses until the entire sample fails. The aim of this procedure is to define the strength of the rock as a function of the stress invariants and also to obtain a constitutive law describing the pre-failure and post-failure stress-strain behavior. The results of these measurements have often been applied to the design of underground·openings to determine, with varying degrees of success, the zones of rock subject to failure and the resulting defonnations. In many cases the predictions do · not match the observations, and the actual mechanism of failure is often different than expected. This is not surprising when one compares the geometry and loading conditions around an underground opening to those in a laboratory core test The rock next to an excavation is generally under polyaxial stress in a plane strain condition, but with only one surface that can dilate. It is subject to stress and strain gradients, and·is loaded by adjacent rock rather than a testing machine. The stress path imposed on the rock during excavation may be quite different than in a typical laboratory test. It is important to determine what effects these differences have on the strength, deformations, and failure mechanisms of the rock, and whether failure in this situation is controlled solely by the stress state. It is important also to determine whether stress-strain behavior and macroscopic failure mechanisms observed in core tests are inherent material properties or if they are influenced by features such as core geometry, loading geometry, and stress path. It is therefore instructive to perform laboratory experiments on model openings, such as hollow cylinders. Hollow cylinders have often been used to test the strength of rock under true polyaxial states of stress (Hoskins 1969). More recently, thick-walled hollow cylinders of rock subjected to axisymmetric loading on the external diameter, and sometimes axial loading as well, have been used to model boreholes or tunnels (e.g. Daemen & Fairhurst 1971, Santarelli & Brown 1987). Gay (1973) used hollow cylinders 1
with both circular and non-circular holes. The application of equal and unequal external
stresses to large rectangular blocks with pre-drilled circular holes has been investigated by
Mastin (1984), Haimson & Herrick (1985) and Kaiser et al (1985). Gay (1976) performed
similar tests on blocks with holes of large eccentricity. Recently, Bandis et al (1987) and
Kaiser & Maloney (1987) have drilled circular holes into externally stressed blocks of
artificial rock-like materials.
The aim of our current research is to carefully simulate and observe the deformation,
fracture, and failure around underground openings. We are accomplishing this through
experiments on thick-walled hollow cylinders of rock with the incorporation of several
important features: plane strain loading, the ability to impose different stress paths on the
rock using independent internal and external pressures, and 'freezing' of the fracture
geometry under load using a metal injection technique.
The rock samples are cored as coaxial hollow cylinders with an inner diameter of 1 inch, an
outer diameter of 3.5 inches, and a length of 5.75 to 6 inches. To provide a control on
-moisture content, the samples are dried in an oven at approximately 110 degrees C for 22
hours. After the sample is removed from the oven, both the surface of the hole and the
outer diameter are lined with rubber sleeves to seal the sample, and it is assembled into a
test cell (Figure 1). A vacuum is then applied to the sample to keep the moisture content
down.- The sample is constrained in the axial direction, as would commonly occur
underground. In this way the effects of the plane strain loading condition and of the
intermediate stress on the strength and failure process can be examined.
Uniform fluid pressure is applied to the outer diameter using a servo-controlled testing
machine, and a separate pressure is applied in the hole using a manually operated screw
pump capable of measuring precise volume changes. These two pressures are cycled
independently to determine the elastic deformation of the hole due to either changing
internal (in the hole) pressure or external pressure. These two independent pressures also
allow the imposition of various stress paths on the material adjacent to the central hole,
such as increasing tangential compression with constant (usually zero) radial pressure, or
increasing tangential compression with decreasing radial compression. ·In this way one can
explore the effect of stress path on the strength and failure process.
After room-temperature elastic testing is complete, the test cell is heated to 90 degrees C,
while continuing to vacuum the sample through the Woods metal port (Figure 1). Using a
system of valves, the evacuated pore space of the sample is then filled with Woods metal,
an alloy (Cerrosafe®) which at 90 degrees Cis liquid and resembles mercury. This liquid
metal pore fluid is kept at 5 MPa, which allows it to permeate openings down to the order
of 0.1 micron is size. This ensures that as the the failure loads are applied to the sample,
any fractures formed will be filled with Woods metal. After the flnalloads are reached the
sample is cooled while keeping the stresses constant, thus freezing the metal in the stress-
induced fractures in the state they exist under load. In this way one can carefully study the
processes of fracture formation and progression around the hole.
Observations that can be made from these tests include the elastic and inelastic
deformations of the opening, stress level at which yield and rupture occur, types of
fractures formed, extent of fractured zone, and stable shape achieved.
Preliminary tests have been performed on Berea sandstone, chosen because it is a welldocumented clastic rock, with no visually apparent anisotropy. Further tests will be performed on Indiana limestone and possibly other rock types. Aluminum specimens were used for calibration. Four samples have been tested using Woods metal porosimetry. The loading sequence after Woods metal injection for two of the samples was increasing external pressure with
Internal Pressure
Rubber (or Steel l Jacket Specimen Rubber Jacket "O"Ring Seals
-c. :E
...Ul Q)
-...iU c:: Q) 10 ..5
XBL 8712-5383
a~~~~~~~~~~~._~~ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume Change (%) XBL 8712-5384
Figure 1. Simplified scale drawing of
Figure 2. Elastic volume deformation of the
testing appar~tus;- ~- - - - · - ··- -- hole during cycling of iriternal pressure.
zero effective pressure in the hole (internal pressure minus Woods metal pressure). The loading sequence for the other two samples was decreasing effective internal pressure to zero with constant external pressure. This stress path is more similar to that occurring during excavation of an opening. In one case the internal pressure was decreased slowly and in the other it was decreased instantaneously. The final loads on all four samples were 75 MPa effective external pressure and zero effective internal pressure. After cooling, the samples were removed from the test cell and cut in half perpendicular to the axis. 3.1 Deformation and strength measurements Elastic volume deformation of the hole due to internal pressure was measured by cycling the internal pressure between zero and 35 MPa under a constant external pressure of 20 or 25 MPa. Typical results, after subtracting the deformation of the internal pressure system, are shown in Figure 2. Hysteresis is evident, with the sample behaving almost linearly with increasing internal pressure and less so with decreasing internal pressure. Although not evident in the figure, all the deformation is recovered at zero pressure. The elastic deformation of the hole due to external pressure was also measured. In all but one sample this deformation was greater than that due to internal pressure, in accord with the theory of elasticity. Preliminary measurements indicate that deformations from either internal or external pressure are less at high temperature than at room temperature. Hollow cylinders of Berea sandstone probably do not behave as homogeneous, isotropic, linear elastic materials, because the stresses are not homogeneous and there is evidence that the elastic moduli depend on the stress state. For example, Wilhelmi & Somerton (1967) found that the modulus of elasticity measured in triaxial compression tests increased with increasing values of constant confining pressure. If homogeneity and isotropy are assumed, however, the two elastic moduli can be calculated by combining the measured deformations with the equation for plane strain deformation of a finite hollow cylinder (Jaeger & Cook 1979). These calculations yield E = 16.5 to 19.1 GPa, and v = 0.35 to 0.47. Values from uniaxial compression tests reported in the literature are in the range E = 13.1 to 18.3 GPa, and v = 0.19 to 0.38. 3
80 ~ 30 ffi 20 0.0
~I \
i !
i i
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Volume Contraction (%) XBL 8712-5386
Figure 3. Volume deformation of the hole (contraction positive) during increase of external pressure, for the sample shown in Figure 6.
60 y I I
v /It
-1 .5
-1 .0
Volume Change (%)
XBL 8712-5385
Figure 4. Volume deformation of the hole (contraction negative) during.release of internal pressure, for the sample shown in Figures 2 and 5.
Two samples provided load vs. hole deformation measurements while loading to failure. Figure 3 shows the volume contraction of the hole in one of the samples as the external pressure was increased with zero effective internal pressure. The two deviations from linearity at pressures of _about 52 and 60 MPa may represent small failure events preceding large-scale rupture of the hole wall. In both cases the curve comes back to join a straight line passing through the lower part of the curve. Large-scale failure appears to occur at 64.5 MPa, which would give a tangential stress on the hole wall equal to 141 MPa. Figure 4 shows the volume change of the hole in another sample as the effective internal pressure was decreased to zero with constant external pressure equal to about 75 MPa (effective). The elastic part of the curve is non-linear, just as it is in the elastic deformation curves for decreasing internal pressure (Figure 2). Failure appears to occur when the internal pressure drops to 3.5 MPa. This load state would give a tangential stress on the hole wall equal to 159 MPa, and a radial stress equal to 3.5 MPa. When increasing the tangential stress with zero radial stress, the hole wall fails when its 'unconfined' strength is reached. A radial (confining) stress strengthens the rock. When releasing this radial stress the hole wall fails at the appropriate 'confined' strength. The failure stresses above yield a failure law <11 =141+ 5.1<13. In terms of a Coulomb failure law, the internal Angle of friction of the hole wall is 42 degrees, which is similar to that measured in triaxial core tests on many rock types. In both loading cases the volumetric strain associated with the onset of failure is 1.3% (Figures 3 and 4}, referenced to zero strain at zero internal and zero external load. This corresponds to a change in hole radius and a tangential strain at the hole wall of 0.65%. Uniaxial compressive strengths for Berea sandstone cores reported in the literature range from 46 to 74 MPa. The 'unconfined' strength of the hole wall (141 MPa) is thus 2 to 3 times as high. This high strength of the hole wall is in quantitative agreement with the experimental results of Mastin (1984), Haimson & Herrick (1985), Santarelli & Brown (1987) and many others as documented by Guenot (1987). · One possible explanation of this high strength is size effect; the vC'lume of overstressed rock next to a one-inch diameter hole is considerably less than that of a typical. uniaxial core specimen. A strength increase with decreasing uniaxial core ~ize has been found for many rock types (Hoek & Brown 1980), but the magnitude of this effect does not appear large enough to account for the full degree of strengthening observed in hollow cylinders. Another possible factor leading to the high strength is the presence of an intermediate (axial) principal stress. No axial deformation of the hollow cylinders is allowed, so an axial stress is developed of magnitude <1ax =v(Stress gradients could also result in a higher strength, but this actually may be the same mechanism as that leading to a size effect (Jaeger & Cook 1979). There could also be further geometric effects; there is only one surface allowing dilatational expansion, and this surface is curved. The influence of all the above factors on strength must be determined if these laboratory results are to be extrapolated to boreholes and large scale excavations.
3.2 Failure characteristics
Figures 5 and 6 illustrate the post-failure appearance of two samples subjected to different
loading paths. The sample shown in Figure 5 was failed by slowly decreasing the internal
pressure (hole deformation documented in Figure 4). The sample shown in Figure 6 was
failed by slowly increasing the external pressure (hole deformation documented in Figure
3). The large volume contraction upon failure recorded in Figure 3 is thought to not
include the flow of Woods metal into the hole, as seen in Figure 6, but this is not certain.
The failure zones in all samples consist of two diametrically opposite, symmetric regions,
consistently oriented along the length of the sample. In all four samples, the failure zones
are oriented in the same direction relative to the block from which the samples were cored.
Although the loading is axisymmetric, the failure is not, as has been observed by other
researchers using assumed isotropic materials (Daemen & Fairhurst 1971, Gay 1973,
Bandis et al. 1987).
The preferred failure direction is probably determined by slight anisotropy of the rock.
Obert (1946) found that the uniaxial compressive strength of samples of Berea cored
parallel to the bedding was 22% to 26% less than that of samples cored perpendicular to the
bedding. One could therefore expect that failure would first occur at locations where the
bedding is parallel to the hole wall. No obvious or consistent bedding has been found in
these samples, and oriented core tests have not yet been performed.
The failure process itself appears to be one of progressive spalling, in which intact slabs
of fairly uniform thickness are successively separated from the surrounding rock by
fractures oriented roughly parallel to the free surface (Figures 5 and 6). These fractures
probably start by an extensile splitting mechanism (e.g. Horii & Nemat-Nasser 1985) in the
highly stressed region near the hole wall. They could then propagate parallel to the
maximum (tangential) stress, fmally turning to meet the free surface, possibly by a
'pseudo-shear' mechanism (Maury 1987). This somewhat irregular intersection with the
hole wall is visible in Figure 5. The completion of a fracture creates a new free surface and
a new highly stressed region, allowing the process to repeat. Thus the final triangular
outline of a failed region might not represent two shear fractures, but might in fact be the
collection of the ends of several successively shorter splitting fractures.
This process of successive spalling due to extensile splitting has been simulated
numerically by Zheng & Cook (1985) and Ewy et al. (1987). They found that this process
will create a triangular failure region with a pointed tip and will then stabilize. These Berea
samples did stabilize, and they developed pointed tips in the failure regions. This feature
was distinctly evident for the samples failed by increasing external pressure, and was noted
upon microscopic observation of the sample failed by slowly decreasing internal pressure.
The microscopic observation of the samples is in the first stages, however, and the exact
process of fracture growth and coalescence is not yet clear.
One sample was failed by instantaneously decreasing the effective internal pressure to
zero. This sample did not form pointed tips in the failure regions, and these regions only
extended about half as deep as those for the sample unloaded slowly, with less fracturing
occurring in general. Although this sample was under failure loads for less time than the
one unloaded slowly, it is believed that sufficient time was allowed for failure to occur.
This sample may therefore indicate that the failure process is affected by strain rate,
independent of a direct time effect. The measured deformations of this sample in the elastic
range are different than those of the other samples, but they do not indicate that it should
have been any stronger than the others.
The extent of the failure zones in the circumferential direction appears to be affected by
the stress path imposed on the hole wall. The failure zones in the two samples that were
.- Fr.gure 5. Far·1 ure resu1tr·ng from decreasm· g m· ternaI pressure to zero with cXoBnBsta8n5t 5- 53 14 extern al pressure equal to 75 MPa. XBB 885 - 5313 Figure 6. Failure res ulting from increasing extern al press ure to 75 MPa wrth zero internal press ure . 6
failed by increasing the external pressure each encompass 70 to 75 degrees of the original hole wall, and extend to depths of a little over 1/4 inch (1/2 the original radius). In the sample failed by slowly decreasing the internal pressure, the depth of the failure zones is basically the same (a little under 1/4 inch), but they encompass 90 to 100 degrees of the circumference (Figure 5). The failure zones in the sample unloaded instantaneously on the inside show more variability, and encompass 75 to 100 degrees. Thus the stress path influences the extent of the hole wall subject to failure as well as the stress state at which failure begins (confmement vs. no confmement).
The rock surrounding an underground opening is much more similar in geometry and
loading conditions to a hollow cylinder than it is to a typical core sample. These
experiments, while just beginning, have shown that the deformations, strengths, and
failure processes observed in hollow cylinder tests are not necessarily the same as those
observed in core tests.
The elastic moduli calculated from the measured deformations are different than the
moduli reported from uniaxial compression tests, probably due to stress heterogeneity and
the dependence of the moduli on the stress state.
The 'unconfmed' strength of the rock surrounding the hole is 2 to 3 times that observed
in uniaxial compression tests. This strengthening is probably due to a combination of size
effect, intermediate principal stress, and additional geometric effects. A confining (radial)
stress strengthens the rQCk to much the same degree as confining pressure increases
strength in core tests.
· Although the load is axisymmetric, there is a preferred direction of failure, probably due
to slight strength anisotropy of the rock. Even within the failure zones the failure is not
uniform, but rather consists of intact pieces of rock separated by discrete fractures. Brittle
spalling, probably dominated by extensile splitting, appears to control the failure process,
even though the tests were performed on a porous, clastic rock. In fact, brittle fracture is
the most common mode of failure observed around underground openings subject to high
stress conditions (Hoek & Brown 1980, Maury 1987). These observations raise doubts
about the applicability of continuum theories describing the formation of a strain-softened
or plastic zone. Future tests on Indiana limestone may exhibit the same failure mechanism
as found in the Berea or may instead exhibit more true shearing behavior.
Increasing the external load on a hollow cylinder with constant internal pressure subjects
the hole wall to increasing mean stress and increasing stress difference, a stress path similar
to that used for most core tests. Decreasing the internal pressure with constant external
pressure subjects the hole wall to constant mean stress and increasing stress difference, and
this stress path is more similar to that occurring during the excavation of an underground
opening. It was found in these tests that the stress path can influence the stress state at
which failure begins and influences the size of the failure zone. Strain rate may also affect
the extent of failure, independent of the time-dependent effects which are generally
Further research along these lines should yield qualitative and quantitative information
that can be directly applied to the design of underground openings. Additional items of
importance include the effect of different axial stress magnitudes, the influence of pre-
existing discontinuities, and possible boundary effects in the laboratory experiments due to
the proximity of the outer diameter of the hollow cylinder. This latter issue will be
investigated shortly by using steel jackets of appropriate stiffness to ensure that the external
pressure acts effectively from infinity. The rock throughout the hollow cylinder will then
behave exactly like that surrounding a circular opening subject to uniform far-field stress.
ACKNOWLEDGEMENTS This work was partially supported by the Office of Geologic Repositories of the Office of Civilian waste management of the U.S. Departinent of Energy under Contract No. DEAC03-76SF00098. One of the authors (RTE) would also like to acknowledge suport from the U.S. Bureau of Mines for Grant No. G 1174106 to the California Mining and Mineral Resources Research Institute, and an award from the Earth Technology Corporation.
~ -·
Bandis, S.C., J. Lindman & N. Barton 1987. ·Three-dimensional stress state and
fracturing around cavities in overstressed weak rock. Proceedings Sixth International
Congress on Rock Mechanics, Montreal 2:769-776. Rotterdam: Balkerna.
Daemen, J.J.K. & C. Fairhurst 1971. Influence of failed rock properties on tunnel
stability. Proceedings 12th U.S. Symp. Rock Mech., p. 855-875. New York: AIME.
Ewy, R.T., J.M. Kemeny, Z. Zheng & N.G.W. Cook 1987. Generation and analysis of
stable excavation shapes under high rock stresses. Proceedings Sixth International
Congress on Rock Mechanics, Montreal2:875-881. Rotterdam: Balkema.
Gay, N.C. 1973. Fracture growth around openings in thick-walled cylinders of rock
subjected to hydrostatic compression. Int. J. Rock Mech. & Min. Sci. 10:209-233.
Gay, N.C. 1976. Fracture growth around openings in large blocks of rock subjected to
uniaxial and biaxial compression. Int. J. Rock Mech. & Min. Sci. 13:231-243.
Guenot, A. 1987. Stress and rupture conditions around oil wellbores (in French).
Proceedings Sixth International Congress on Rock Mechanics, Montreal!: 109-118.
Haimson, B.C. & C.G. Herrick 1985. In situ stress evaluation from borehole breakouts,
experimental studies. Proceedings 26th U.S. Symposium on Rock Mechanics, p. 1207-
1218. Rotterdam:Balkema.
Hoek, E. & E.T. Brown 1980. Underground excavations in rock. London: Institute of
Mining and Metallurgy.
Horii, H. & S. Nemat-Nasser 1985. Compression-induced microcrack growth in brittle
solids: axial splitting and shear failure. J. Geophys. Res. 90(B4):3105-3125.
Hoskins, E.R. 1969. The failure of thick-walled cylinders of isotropic rock. Int. J. Rock
Mech. & Min. Sci. 6:99-125.
Jaeger, J.C. & N.G.W. Cook 1979. Fundamentals of rock mechanics, 3rd Edition.
London: Chapman & Hall.
Kaiser, P.K., A. Guenot & N.R. Morgenstern 1985. Deformation of small tunnels- IV.
Behaviour during failure. Int. J. Rock Mech. & Min. Sci. 22:141-152.
Kaiser, P.K. & S. Maloney 1987. Factors influencing the stability of deep boreholes.
Proceedings Sixth International Congress on Rock Mechanics, Montreall:675-680.
Mastin, L. 1984. The development of borehole breakouts in sandstone. M.S. thesis,
Stanford University.
Maury, V. 1987. Observations, researches and recent results about failure mechanisms
around single galleries. Proceedings Sixth International Congress on Rock Mechanics,
Montreal 2: 1119-1128. Rotterdam: Balkema
Santarelli, F.J. & E.T. Brown 1987. Performance of deep wellbores in rock with a
confming pressure-dependent elastic modulus. Proceedings Sixth International
Congress on Rock Mechanics, Montreal 2:1217-1222. Rotterdam: Balkerna. Wilhelmi, B. & W.H. Somerton 1967. Simultaneous measurement of pore and elastic
properties of rocks under triaxial stress conditions. SPE Journal, p. 283-294.
Zheng, Z. & N.G.W. Cook 1985. Generation and analysis of stable wellbore cross-
sections (abstract). EOS Transactions, AGU 66:1056.
!'~ ~ "- .., 2-0t.~.. ...~~ LAWRENCE BERKELEY LABORATORY OFFICE FOR PLANNING AND DEVELOPMENT University of California Berkeley, CALIFORNIA 94720
~ ~,.. ::;

RT Ewy, NGW Cook, LR Myer

File: hollow-cylinder-tests-for-studying-fracture-around-underground.pdf
Author: RT Ewy, NGW Cook, LR Myer
Author: Kodak Capture Software
Published: Fri Feb 15 18:36:42 2013
Pages: 12
File size: 2.09 Mb

Financial risk management, 4 pages, 0.04 Mb

111 islands, 11 pages, 1.33 Mb
Copyright © 2018 doc.uments.com