Arch. Min. Sci., Vol. 52 (2007), No 3, p. 331–354

Arch. Min. Sci., Vol. 52 (2007), No 3, p. 331–354

Arch. Min. Sci., Vol. 52 (2007), No 3, p. 331–354
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JAN LIS*, PIOTR KIJEWSKI*
METHODOLOGY OF TESTING THE STRENGTH AND DEFORMATIONAL PROPERTIES OF ROCKS
FROM LGOM MINES UNDER TRIAXIAL STRESS CONDITIONS
METODYKA BADAŃ WŁASNOŚCI WYTRZYMAŁOŚCIOWYCH I ODKSZTAŁCENIOWYCH SKAŁ
Z KOPALŃ LGOM W TRÓJOSIOWYM STANIE NAPRĘŻEŃ
The paper presents the laboratory equipment used in triaxial tests carried out at the Rock Mechanics
Laboratory of the Mining Department, the KGHM CUPRUM Ltd Research & Development Centre. The
equipment for triaxial tests at confining pressures of up to 250 MPa which allows for the determination of
the strength and deformational properties of rocks is described. Using the results of the tests, the criteria
for rock failure under compression, shear, tension and bending were determined. An attempt to monitor
the propagation of cracks in rocks under hydrostatic pressure up to 60 MPa was also undertaken.
Keywords: deformational properties, failure criterion, rock strength, triaxial laboratory tests
W artykule przedstawiono wyposażenie pomiarowo-badawcze stosowane w trójosiowych badaniach
skał prowadzonych w Pracowni Mechaniki Skał, Laboratorium Górnictwa i Mechaniki Górotworu KGHM
CUPRUM. Badania te rozpoczęto w 1970 roku dla potrzeb budowy kopalń rud miedzi w nowo odkrytym
złożu w obszarze monokliny przedsudeckiej. Badania są wykonywane w pełnym profilu górniczym,
obejmującym skały złożowe i skały w najbliższym otoczeniu złoża (strop, spąg). Zakres prowadzonych
badań w warunkach trójosiowego stanu naprężeń obejmował:
• określenie wytrzymałościowych i odkształceniowych własności skał przy ściskaniu, ścinaniu,
rozciąganiu i zginaniu w warunkach wszechstronnego ciśnienia;
• wyznaczenie charakterystyk mechanicznych skał i określenie stałych sprężystości;
• określenie wytrzymałości skał w złożonym stanie naprężenia, w którym składowe tego stanu mają
różne wartości i znaki;
• opracowanie, na podstawie wyników badań trójosiowych, warunków kruchego zniszczenia skał;
• badania nad rozprzestrzenianiem się szczelin w materiale skalnym poddanym działaniu wszechstronnego ciśnienia.
*
KGHM CUPRUM LTD RESEARCH & DEVELOPMENT CENTRE, PL. JANA PAWŁA II, 50-136 WROCŁAW, POLAND
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Badania wytrzymałości i deformacji skał w warunkach trójosiowego stanu naprężenia realizowane
są według dwóch wariantów:
1. σ1 > σ2 = σ3 (wg tzw. schematu Kármána), tj. warunek, w którym na jednoosiowy stan naprężenia
nakłada się hydrostatyczny stan ciśnień.
2. σ1 ≠ σ2 ≠ σ3 – warunek złożonego stanu naprężeń, w którym składowe tego stanu mają różne
wartości i znaki.
Dla realizacji przyjętego programu skonstruowano stanowisko badawcze, z prototypową aparaturą
i urządzeniami, składające się z następujących elementów:
• układu hydraulicznego zasilającego komory ciśnieniowe;
• układu obciążającego z maszynami wytrzymałościowymi;
• aparatury z wyposażeniem pomiarowym i rejestrującym;
• zestawu komór badawczych (ciśnieniowych) o zróżnicowanych konstrukcjach stosownie do rodzaju
prowadzonych badań.
Badania skał w trójosiowym stanie naprężenia polegały na określeniu wytrzymałości badanej
próbki skalnej przy minimum trzech różnych poziomach ciśnień okólnych, zależnych od wytrzymałości
i zwięzłości skały.
Uwzględniając zmienne właściwości skał występujących w profilu górniczym złoża rud miedzi
LGOM badania prowadzone są przy następujących ciśnieniach okólnych:
• 5, 10, 15, i 10; 20; 30 MPa - dla skał słabych, o niskiej zwięzłości;
• 20, 40, 60 MPa - dla skał mocnych, zwięzłych i silnie zwięzłych.
Do badań próbek skalnych przy ściskaniu służą trzy typy komór ciśnieniowych:
1. komora typu M-23 własnej konstrukcji do ciśnień w zakresie 0÷60 MPa (rys. 3.1);
2. komora tłoczkowa typu M-25 własnej konstrukcji do ciśnień w zakresie 0÷120 MPa (rys. 3.2);
3. komora firmy Walter+Bai produkcji szwajcarskiej do ciśnień w zakresie 0÷250 MPa (rys. 3.3).
Wyniki badań wytrzymałościowych i odkształceniowych właściwości skał w trójosiowym stanie
naprężeń są wykorzystywane w obliczeniach inżynierskich, przy projektowaniu obiektów górniczych
oraz w modelowaniu procesów deformacji górotworu podczas prowadzenia eksploatacji złoża rud miedzi
w kopalniach LGOM.
Słowa kluczowe: badania trójosiowe, warunek kruchego zniszczenia, własności odkształceniowe,
wytrzymałość
1. Introduction
The intensification of rock-mass pressure symptoms in the vicinity of mine workings,
which are determined by the behavior of rock under complex stress, occurs as a result
of mining operations. In predicting the stresses and deformations induced in the rock
mass, the physical and mechanical properties of rocks under triaxial stress conditions,
particularly when the extraction is carried out at greater and greater depths play a crucial
role.
Mining the copper deposit in the Legnica-Głogów Copper District (LGOM), which
started in 1967 in the Lubin mine and later in the Polkowice mine, required the identification of the geomechanical properties of rocks occurring in these unfamiliar mine
strata. At first the tests determined the basic strength and deformational properties, but
as early as in 1970, in the Laboratory of Rock Mechanics in Copper Research and De-
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sign Office “Cuprum”, tests under complex stress condition were begun. The tests were
carried out for a full mining profile including the ore deposit and the surrounding rocks,
i.e. sandstones, carbonate rocks and sulphate rocks. The scope of tests under triaxial
stress conditions included:
• determination of the strength and deformational properties at compression, shear,
tension and bending under hydrostatic pressure conditions;
• determination of the mechanical characteristics of rocks and elasticity constants;
• determination of rock strength under complex stress conditions, when the components of the stress state are of different values and signs;
• elaboration of rock failure criteria based on the test results;
• investigation of the propagation of cracks in rock material under hydrostatic pressure conditions.
2. Methods and measures for testing rock samples
under triaxial stress conditions
Tests of strength and deformations under triaxial stress conditions were carried out
using two variants:
1. σ1 > σ2 = σ3 – axi-symmetric stress conditions, i.e. the conditions, in which hydrostatic pressure is superposed onto uniaxial stress (the co-called Kármán loading
scheme);
2. σ1 ≠ σ2 ≠ σ3 – asymmetric or general triaxial stress conditions, i.e. the conditions
where all three principal components of the stress tensor are different.
The programme of the experiments included tests of the impact of pressure on rock
strength and deformation under stress, shear, tension and bending conditions. To implement this programme, a test station with prototype testing apparatuses and devices was
designed and built. It consists of the following systems:
• hydraulic system feeding the pressure chambers;
• loading system (testing machines);
• instrumental system including measuring and recording equipment;
• system with testing pressure chambers having a structure adequate for the type of
tests (Fig. 1).
2 . 1 . Te s t s o f t h e s t r e n g t h a n d d e f o r m a t i o n a l p r o p e r t i e s o f r o c k s
under axi-symmetric compressive stress conditions
Tests of rock samples under triaxial compressive stress conditions consisted in determining the strength of the tested rock at a minimum of three different confining pressure
levels, depending on the strength and compactness of the rock. Taking into consideration
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the variable properties of rocks occurring in the mining profile of the copper deposit in
the LGOM area, the tests were carried out at the following confining pressures:
• 5, 10, 15, and 10, 20, 30 MPa – for weak rocks with low compactness;
• 20, 40, 60 MPa – for strong, compact and highly compact rocks.
According to the so-called Kármán scheme (σ1 > σ2 = σ3), the major principal stress σ1
in the sample is generated along its vertical axis, while minor principal stresses, induced
by confining pressure (where σ2 = σ3 = p), operate in radial directions (Fig. 1).
s1
s2 = s3 = p
s1
s2 = s3 = p
Fig. 1. Pattern of sample loading in the conventional triaxial compression test
(the so-called Kármán scheme)
Rys. 1. Schemat obciążania próbki w próbie na konwencjonalne trójosiowe ściskanie
(tzw. schemat Kármána)
A rock sample in an air-dry state is placed in a special rubber jacket to protect it against
oil penetration and is enclosed in a leak-proof pressure chamber (Fig. 2) where axially
symmetric stress conditions are induced in the sample. Confining pressure (p = σ2 = σ3)
is applied to the sample by hydraulic oil while the vertical stress (σ1) is generated mechanically by testing machine at a constant rate of 1.0 MPa/s.
The mechanical properties of the rocks tested under compressive conditions were
determined using solid cylindrical samples having a diameter of 44 mm and a slenderness
ratio of 1.0 (strength properties) and 2.0 (strength and deformational properties).
Three types of pressure chambers for triaxial compression tests on rock samples
were used:
1. M-23 chamber (designed by the authors) for 0÷60 MPa pressures (Fig. 3.1),
2. M-25 piston chamber (designed by the authors) for 0÷120 MPa pressures (Fig.
3.2),
3. Walter+Bai chamber (Swiss made) for 0÷250 MPa pressures (Fig. 3.3).
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piston
sample
cylinder
oil inlet
cover
sleeve
Fig. 2. Scheme of a pressure chamber for triaxial compression tests
Rys. 2. Schemat komory ciśnieniowej do badań trójosiowych przy ściskaniu
Fig. 3. High-pressure triaxial chambers for compression tests on rocks samples
Rys. 3. Wysokociśnieniowe komory trójosiowe do badań próbek skalnych na ściskanie
Compressive stresses produced in the sample under triaxial loading conditions are
as follows:
s1 =
P
> s 2 = s3 = p
Fo
(1)
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where:
σ1
P
Fo
σ2
σ3
p
—
—
—
—
—
—
vertical stress (major principal stress),
vertical load,
initial cross-sectional area of sample,
intermediate principal stress,
minor principal stress,
confining pressure.
Vertical stress σ1 reduced by hydrostatic pressure acting on the sample is called the
differential stress or deviatoric stress (σ1 – σ3), and the differential stress at failure is called
the differential compressive strength under triaxial stress conditions (σ1 – σ3)max:
(s1 - s 3 )max =
Pmax
-p
Fo
(2)
When represented by Mohr circles of principal stresses, the triaxial compression test
results allow the failure criterion for the rock tested in the form of a Mohr envelope of
the circles (Fig. 4) to be determined. In the simplest case, the Mohr envelope is a straight
line described by the Coulomb equation:
t = sn tan j + c
(3)
where:
τ
σn
φ
c
—
—
—
—
ultimate shear stress,
normal (compressive) stress,
internal friction angle,
cohesion.
The state of stress in the sample is also described by principal stresses: σz, σφ, σr acting
in the axial direction z, the perimeter (tangential) direction φ and the radial direction r,
respectively.
The plane perpendicular to the hydrostatic axis in the space of principal stresses is
called the octahedral plane (σz + σφ + σr = const). In this plane the normal stress (σoct)
operates:
soct =
sz + sr + sj
3
(4)
as well as the shear (tangential) stress τoct:
toct =
1
( s z - sr ) 2 + (s z - sj ) 2 + ( sj - sr ) 2
3
(5)
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Highly compact, fine-grained quartz sandstone
c = 40.0 MPa; j = 32.3°
Shear stress t, MPa
200
150
100
50
0
0
50
100
150
200
250
300
350
400
Normal stress sn , MPa
Fig. 4. Coulomb-Mohr rock failure criterion under triaxial stress conditions
Rys. 4. Kryterium zniszczenia skał wg Coulomba-Mohra w trójosiowym stanie naprężenia
Taking into consideration the condition where σφ = σr = p, the following is obtained:
soct =
2 p + sz
3
toct = 2 (soct - p)
(6)
(7)
An example of conventional triaxial test results for dolomitic rocks is presented in
Figure 5. This figure shows the dispersion of results and the average ultimate strength’s
dependence on confining pressure.
In order to determine the general criterion of rock failure, the compressive strength
chart in the octahedral plane (τoct – σoct) is plotted in Figure 6 for a selected package of
rock strata. This chart includes the trajectories of loading the sample until it fails. The
strength trajectory was approximated by a straight line using the least squares method.
The equation of this line is as follows:
toct = 0.7225 soct + 38, MPa
(8)
This equation is valid for normal stresses:
55 MPa £ soct £ 175 MPa
(9)
When analysing changes in the differential compressive strength of roof rocks, it was
confirmed that the rate of strength change decreases with a pressure increase within the
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420
400
380
360
340
Max. differential stress (s1 – s 3)max, MPa
320
300
280
260
240
220
200
180
160
140
120
Fig. 5. Ultimate strength of dolomites
under conventional triaxial compression conditions
100
80
60
0
10 20 30 40 50 60
Confining pressure p, MPa
Rys. 5. Wytrzymałość graniczna dolomitów w warunkach konwencjonalnego trójosiowego ściskania
range of applied pressures. Rock compressive strength at a pressure of p = 60 MPa is
three times higher compared to the uniaxial strength. Rocks, which under uniaxial stress
conditions have a high strength, also keep this feature under triaxial stress conditions.
Observations of a sample fracturing pattern during the tests showed that rocks under
uniaxial compression break into small fragments in a brittle manner with an audible
acoustic effect. This is because the elastic volumetric strain energy accumulated in the
sample during loading is released instantly, which is characteristic for rocks susceptible
to bursting. The moment of sample disintegration was difficult to capture as the rupture
occurred without any warning. In the case of homogeneous rocks with a low susceptibility to bursting, two chip-shaped cones were formed after failure.
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180
sz
160
sr = sj = p
Shear octahedral stress t oct, MPa
140
sz
120
p
p
p
100
p
80
60
40
20
0
20
40
60
80
100
120
140
160
180
Normal octahedral stress s oct, MPa
Fig. 6. Failure criterion of dolomites in the octahedral plane under axi-symmetric triaxial stress
conditions (σ1 > σ2 = σ3)
Rys. 6. Kryterium zniszczenia dolomitów w płaszczyźnie oktaedrycznej w osiowo-symetrycznym
trójosiowym stanie naprężenia (σ1 > σ2 = σ3)
Samples tested under triaxial stress conditions failed without audible acoustic effects.
The disintegration of samples after strength failure was not sudden but progressed slowly,
and the rupture of the sample was signalled by an increase of pressure in the hydraulic
system and a decrease in vertical load (a decrease in the load-bearing capacity of the
sample). The higher the confining pressure, the lower the sample disintegration rate
after the strength failure.
The result of each test of the deformational properties of rocks are the mechanical
characteristics of rocks in the form of stress-strain plots. These plots depict the mechanical behaviour of rocks under different confining pressures (Fig. 7).
The modulus of deformation does not change substantially with a change of confining
pressure. This is probably due to the diminishing influence of rock heterogeneity under
triaxial stress conditions. Thus, it can be assumed that the slope of linear sections of σz
– ε1 curves is independent of confining pressure; it is only the length of these sections
that increases with an increase of pressure.
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p=60
p=50
p=60 MPa
p=50
300
Axial stress s z , MPa
p=40
p=30
p=20
p=10
p=40
p=30
p=20
p=10
200
lateral strain
p=0
p=0
150
axial strain
100
50
-4
-2
0
4
2
6
8
Strain e, 10–3
Fig. 7. Deformational characteristics of dolomites under triaxial compressive stress conditions
Rys. 7. Charakterystyki odkształceniowe dolomitów w trójosiowym stanie naprężeń ściskających
2 . 2 . Te s t s o f t h e s t r e n g t h a n d d e f o r m a t i o n a l p r o p e r t i e s o f r o c k s
at tension under triaxial stress conditions
Tests of tensile strength were carried out using a pressure chamber of the M-17 type
(Fig. 8). The chamber allows for the conditions of pure tension as well as conditions of
compression and tension to be created. During such tests, deformation can be measured
by using electric resistance strain gauges.
Tension tests were performed on profiled (dogbone-shaped) samples with varying
diameter in the middle part (d = 24÷32 mm) and a constant D = 44 mm diameter in the
thick end parts. The samples were subjected to oil pressure in the M-17 chamber. Increasing hydrostatic pressure causes compression of a sample and at the same time tension
by a force equal to the product of pressure and the difference between area in the thick
part and in the middle part of sample. Combined lateral compression with axial tension
causes a fracture of the sample.
When the sample is extended by hydrostatic pressure (p), the axial stress is:
s z = p (1 -
D2
)
d2
(10)
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Fig. 8. M-17 triaxial chamber for tension tests of rocks
Rys. 8. Komora trójosiowa typu M-17 do badań skał na rozciąganie
On the octahedral plane, stresses are equal to:
– normal octahedral stress σoct
soct =
pæ
D2 æ
ç3 - 2 ç
3 çè
d çè
(11)
toct =
2 D2
p
3 d2
(12)
– shear octahedral stress τoct
The octahedral stresses were used when determining the failure criterion for the
rocks tested.
In order to determine the plastic strain at failure δz under tensile conditions, the method of calculating the sample necking at the moment of failure developed by Bridgman
was used. According to this method, the plastic strain at failure was calculated using
the formula:
d z = ln
Fo
F
(13)
where:
Fo — initial cross-sectional area of sample at the middle part,
F — cross-sectional area of the sample at the central middle part at the moment
of failure.
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e2
Axial stress s z , MPa
Figure 9 presents the deformation characteristics for a limestone sample subjected to
extension under hydrostatic pressure conditions. In addition, Figure 10 shows the plastic
strain of the extended rock samples at the moment of failure in the pressure chamber.
8
e1
6
4
2
-200
0
200
400
600
Strain e, 10–6
Fig. 9. Deformation characteristics of limestone at extension under hydrostatic pressure conditions
Rys. 9. Charakterystyka odkształceniowa wapienia przy rozciąganiu w warunkach
wszechstronnego ciśnienia
It was observed during the tests that rock samples undergo only slight deformation,
while the deformations decrease in the plane that is perpendicular to the direction of
tensile stresses.
The rupture of the sample is initiated at mineral inclusions and structural discontinuities. Necking of the sample at the moment of failure is δ = 0.02÷0.05%. Such
a small necking, as well as the orientation of the fracture surface evidences the brittle
character of rock failure. The rock sample breaks mainly as the result of chipping off,
thus in a different manner from a sample of a plastic material where failure occurs as
the result of sample necking or due to the formation of slip planes oriented obliquely
to the sample axis.
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800
dz = lnF0 /F, 10 –6
sz
600
sr = p
sr = p
400
sz
200
0
0
0.2
0.4
0.6
0.8
1.0
1.2
k = p/s z
Fig. 10. Plastic strain of limestones at the moment of failure under hydrostatic pressure conditions
Rys. 10. Graniczne odkształcenie plastyczne wapieni przy rozciąganiu w warunkach działania wszechstronnego ciśnienia
2 . 3 . Te s t s o f t h e s t r e n g t h a n d d e f o r m a t i o n a l p r o p e r t i e s o f r o c k s
at shearing under triaxial stress conditions
Testing the shear strength under triaxial stress conditions was performed using an
M-16 chamber equipped with a special shearing device (Fig. 11). The Jaeger method
that consists in forcing the slip of a rock disk along the cylindrical surface parallel to
the shearing force was employed.
Shear strength of rocks was determined using disk samples of 44 mm in diameter
and 10 mm in thickness. The magnitude of shearing stress τ at the moment of rupture
was calculated using the formula:
tz =
where:
Ps
p
F
d
h
—
—
—
—
—
Ps - pF
p dh
(14)
pressure of the punch on sample,
hydrostatic pressure,
cross-sectional area of the punch,
punch diameter,
sample thickness (height).
The carbonate rocks tested showed a high shear strength. Hydrostatic pressure enhanced this strength significantly. At a hydrostatic pressure of 60 MPa the shear strength
was about seven times higher compared to conditions where p = 0 (Fig. 12).
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Fig. 11. M-16 triaxial chamber with equipment for shear tests of rocks
Rys. 11. Komora trójosiowa M-16 z oprzyrządowaniem do badań skał na ścinanie
140
Shear strength t, MPa
120
100
80
60
40
20
Fig. 12. Shear strength of limestones under
hydrostatic pressure conditions
0
0
20
40
60
Hydrostatic pressure p, MPa
Rys. 12. Wytrzymałość wapieni na ścinanie
w warunkach wszechstronnego ciśnienia
345
180
160
140
p=60
p=50
p=40
120
Shear strength t, MPa
p=30
100
p=20
80
p=10
60
p
40
p
p
q
q=p+4P/p (D 2–d 2)
q
Fig. 13. Deformation characteristics of limestones subjected to shear under hydrostatic pressure conditions
20
p=0
0
0
0.4
0.8
1.2
1.6
2.0
Relative movement of sheared disc Dh/h, %
Rys. 13. Charakterystyki odkształceniowe
wapieni poddanych ścinaniu w warunkach
wszechstronnego ciśnienia
During the shear tests, measurements of displacement (∆h) of the sheared disk were
also done. The displacement was measured using an inductive sensor with accuracy
up to 10–6. The results were used to draw the τ – ∆h/h characteristics. The resulting
characteristics for carbonate rocks at different hydrostatic pressures are shown in
Figure 13. These are straight lines of a slope that increases with an increase in hydrostatic
pressure, meaning that the pressure inhibits the disk displacement. The value of the energy
before the final rupture of the sample as well as the friction forces on the fracture surface
can be determined based on the experimental characteristics τ – ∆h/h.
The results of shear tests show that rocks have a high shear resistance in the plane
perpendicular to bedding. At the same time, the shear strength is much greater than the
tensile strength.
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2 . 4 . Te s t s o f t h e s t r e n g t h a n d d e f o r m a t i o n a l p r o p e r t i e s o f r o c k s
at bending under triaxial stress conditions
In order to determine the properties of rocks at bending under triaxial stress conditions
a special pressure chamber to test up to 60 MPa (Fig. 14) was constructed. Bending
tests were carried out using beam-shaped samples with a rectangular cross-section of
the dimensions 20 mm × 30 mm. The length of beams was 195÷250 mm. Beams were
tested under bending conditions at hydrostatic pressures ranging from 0 to 60 MPa in
a series of tests.
Fig. 14. Triaxial chamber for testing rocks at bending under hydrostatic pressure
Rys. 14. Komora trójosiowa do badania skał przy zginaniu w warunkach wszechstronnego ciśnienia
During the tests, the measurements of beam deformation in the compressed and
extended zones using electric resistance strain gauges were taken. An example of the
deformation characteristics under bending conditions for a selected sample of anhydrite
is presented in Figure 15. The chart shows σg − ε curves for the compressed and extended
zones. They characterise the behaviour of rock during bending under given hydrostatic
pressure conditions.
When analysing the mechanical characteristics of rocks subjected to bending under
triaxial stress conditions, an increase in the strength and deformability of the tested
beams was observed (Fig. 16). In the extended zone the deformations of the beam were
greater than in the compressed zone.
At the initial stage of bending, the mechanical characteristics are the same for the
compressed and extended zones and have the form of a straight line. After exceeding
a certain value of the bending moment Mg = 0.6 Mg max, the characteristics become
concave downward and a more rapid increase in strain occurs in the extended zone.
The reason for this effect is presumably the growth of cracks in this zone. An important
conclusion derived from these tests is that the values of Young’s modulus are the same
for compression and tension in the direction perpendicular to bedding.
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ec
60
er
Bending stress s g, MPa
50
40
30
20
10
0
0
100
200
300
400
500
600
700
Deformation of beam in compressed e c and extended zone er, 10
–6
Fig. 15. Deformation characteristics at bending for a selected sample of anhydrite
Rys. 15. Charakterystyka odkształceniowa przy zginaniu dla wybranej próbki anhydrytu
22
20
Critical bending moment Mg, kNm
18
16
14
12
10
8
p
6
p
4
2
0
0
10
20
30
40
50
60
70
Hydrostatic pressure p, MPa
Fig. 16. Bending strength of anhydrite under hydrostatic pressure conditions
Rys. 16. Wytrzymałość anhydrytu na zginanie warunkach wszechstronnego ciśnienia
348
2 . 5 . Te s t s o f t h e s t r e n g t h a n d d e f o r m a t i o n a l p r o p e r t i e s o f r o c k s
u n d e r a s y m m e t r i c t r i a x i a l s t r e s s c o n d i t i o n s ( σ 1≠ σ 2≠ σ 3)
An M-23 Kármán pressure chamber (Fig. 17) was used to test rocks under asymmetric
triaxial stress conditions (σ1 ≠ σ2 ≠ σ3). This chamber was modified by introducing pressure
heads instead of loading pistons.
Fig. 17. Triaxial chamber with additional equipment for testing hollow cylindrical samples
Rys. 17. Komora trójosiowa z dodatkowym wyposażeniem do badań próbek cylindrycznych
Hollow cylindrical samples having an internal diameter of 21 mm, external diameter
of 44 mm and slenderness index of 2.0 were used in the tests. A special feature of these
samples is that vertical stress σz as well as the mean stress 1/3(σr + σφ + σz) and the sum
(σr + σφ) are the same for all internal points of the cross-section of the sample. Distribution
of stresses σr and σφ is, however, non-uniform in the radial direction. Due to this, the
concentration of stresses in the sample was eliminated.
The applied methodology of tests not only allowed the values of the principal stresses
to be changed but also their direction.
In traxial tests three schemes of sample loading were used:
a) first, where σz = 0, σr ≠ σφ. The test results were used for plotting the strength
characteristics of rocks under plane stress conditions for tension and compression.
b) second, where σz = const., σr ≠ σφ. The results obtained allowed the strength
characteristics of rocks on the octahedral plane (τoct, σoct) for average values of
the stress σoct up to a 150 MPa maximum to be plotted.
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c) third, where σr ≠ σφ ≠ σz. Mechanical parameters determined based on these
tests characterise fracturing of rocks at high values of stress σoct, reaching up to
120 MPa.
A complete strength characteristic of rocks under a triaxial stress state on the plane
(τoct, σoct) presented in Figure 18 was determined using extension test results (profiled
samples) and results of tests carried out using the hollow cylindrical samples. These
Shear octahedral stress t oct, MPa
120
100
80
60
hollow cylindrical samples
40
dogbone-shaped samples
20
0
0
20
40
60
80
100
120
Normal octahedral stress s oct, MPa
Fig. 18. Failure criterion for dolomites on the octahedral plane under asymmetric
triaxial stress conditions (σ1 ≠ σ2 ≠ σ3)
Rys. 18. Kryterium zniszczenia dolomitów w płaszczyźnie oktaedrycznej
w asymetrycznym trójosiowym stanie naprężenia (σ1 ≠ σ2 ≠ σ3)
characteristics have the form of two connected sections of intersecting straight lines
described by the following equations:
τoct = 1.1802 σoct + 3.7 for 1.7 MPa ≤ σoct ≤ 15 MPa
(15)
τoct = 0.7836 σoct + 10.3 for 15 MPa ≤ σoct ≤ 120 MPa
(16)
Results obtained on the hollow cylindrical samples showed a substantial difference
in strength in comparison with those obtained using solid cylindrical samples tested
according to the Kármán scheme.
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2 . 6 . Te s t s o f c r a c k p r o p a g a t i o n i n r o c k s u n d e r t r i a x i a l s t r e s s
conditions
The aim of those tests was to evaluate the rocks susceptibility to sudden and
uncontrolled rupture under triaxial stress conditions based on the measurement of
the critical opening of a crack (a notch) preset in beam-shaped samples subjected to
bending.
The crack opening was measured using a specially designed sensor with flat clip
springs and recorded automatically as a function of rock beam load at a constant
hydrostatic pressure. The obtained charts allowed a determination of whether a lowenergy, brittle failure under unconfined conditions may change into high-energy failure
under hydrostatic stress conditions.
Samples for these tests had the shape of a beam with a rectangular cross-section of
30 mm × 40 mm and a length of 220 mm. In the centre of the beam a chevron notch
2 mm wide with an internal angle of 30° was made. The depth of the notch was 10 mm
which was equal to 1/4 of the beam height.
Triaxial stress conditions in the beam were generated in a triaxial chamber (Fig. 19)
by using oil pressure ranging from 0 to 60 MPa. The bending load was applied to the
beam at two locations using the four-point loading method.
Using the existing measuring system with automatic recording of test results of the
bent beams, a series of plots was obtained showing the relationship between beam load P
and crack opening ∆f at different hydrostatic pressures. An example chart for a selected
sample is presented in Figure 20.
Fig. 19. Triaxial chamber for tests of crack propagation in beam-shaped samples of rock
Rys. 19. Komora trójosiowa do badań rozprzestrzeniania się szczelin w próbkach-beleczkach skalnych
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Bending load P, kN
6
4
p = 50 MPa
2
0
0
10
20
30
Crack opening Df, mm
Fig. 20. Example chart of the relationship between load and crack opening
in an anhydrite beam bent under hydrostatic pressure of p = 50 MPa
Rys. 20. Przykładowy wykres zależności pomiędzy obciążeniem a rozwarciem szczeliny
w zginanej belce z anhydrytu przy wszechstronnym ciśnieniu p = 50 MPa
Hydrostatic pressure causes an increase in critical stresses and in the size of the
crack opening (Fig. 21) exceeding which leads to an uncontrolled failure. High-energy
brittle failure under unconfined conditions changes into even more violent failure with
an increase of hydrostatic pressure.
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30
Critical crack opening Df, mm
25
20
15
10
p
p
5
p
sensor
0
0
10
20
30
40
50
60
Hydrostatic pressure p, MPa
Fig. 21. Critical opening of a crack in bent beams of anhydrite under hydrostatic pressure conditions
Rys. 21. Krytyczne rozwarcie szczeliny w zginanych belkach z anhydrytu w warunkach
wszechstronnego ciśnienia
3. Summary and final conclusions
The designed and constructed station for testing the mechanical properties of rocks
under triaxial stress conditions proved to be very useful for experiments carried out under
confining pressure of 0÷60 MPa. The tests allowed for the determination of the impact
of hydrostatic pressure on the strength and deformational properties of rocks from the
copper ore deposit (Zechstein carbonate rocks in particular) occurring in the immediate
and main roof of the orebody (Figs 22 and 23).
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Fig. 22. Types of samples used for tests under triaxial stress conditions
Rys. 22. Rodzaje próbek stosowanych w badaniach w trójosiowym stanie naprężenia
Fig. 23. Fractured samples after tests under triaxial stress conditions
Rys. 23. Zniszczone próbki po badaniach w trójosiowym stanie naprężenia
Fig. 23. Fractured samples after tests under triaxial stress conditions
Rys. 23. Zniszczone próbki po badaniach w trójosiowym stanie naprężenia
Based on triaxial test results, the following conclusions can be drawn:
1. The compressive strength of rocks at a pressure of 60 MPa is three times higher
in comparison with the uniaxial strength. Rocks, which under uniaxial stress conditions have high compressive strength, maintain this feature under triaxial stress
conditions. The deformation moduli do not change significantly with a change
of hydrostatic pressure. Critical deformations of the tested rocks are equal to
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0.15÷0.25% under uniaxial compression conditions and 0.6÷1.0 under compression
at a confining pressure of 60 MPa. It was also found that Poisson’s ratio does not
substantially change its values with a confining pressure within the range of up to
60 MPa.
2. Tests of the shear strength of rocks showed high strength in the direction perpendicular to bedding.
3. The so-called true fracture strain or plastic strain at failure of rocks under tensile
conditions is equal to 0.02÷0.5% on average. The direction of sample fracture that
is perpendicular to the tensile load direction and a small value of sample necking
at the moment of rupture is proof of brittle failure.
4. Test results of the strength and deformational properties of rocks under triaxial
stress conditions are widely applied in engineering calculations, in the design of
mine excavations and in numerical simulations of the mine strata deformation
processes during mining.
Received: 14 March 2007