Axial splitting of rocks under uniaxial compression
Ali Fakhimi n,Behzad Hemami
Department of Mineral Engineering,New Mexico Institute of Mining and Technology,Socorro,NM,USA
a r t i c l e i n f o
Article history:
Received26February2015
Received in revid form
15June2015
Accepted11August2015
Keywords:
Uniaxial compression test
Axial splitting
Rock–loading platen interaction
Discrete element
Rock fracture
a b s t r a c t
Although uniaxial compression testing of rock is considered a simple technique for evaluation of rock
strength,it induces a complicated fracture pattern in the specimen that is difficult to comprehend.In
particular,axial splitting is obrved in uniaxial compression tests.There have been some attempts to
interpret the failure pattern in compression testing of rock.The rock non-homogeneity and existence of
initial micro-cracks and pores have been considered the main caus for nucleation of axial cracks.Bad
on some physical uniaxial compression tests on Pennsylvania blue sandstone,we suggest another pos-
sible mechanism,which considers the rock–loading platen interaction for generation of axial cracks in
rock.In particular,this failure mechanism can occur in rock pillars which are loaded by the roof orfloor
that is made of a rock with different elastic properties.The non-uniform penetration of specimen in the
loading platens and the end friction produce shear strain that can be the cau of axial shear cracking at
the specimen ends.Dislocation along the axial shear cracks and the induced material dilation generate
radial stress which result in confinement of the core of the specimen and radial cracks in the vicinity of
the specimen end surfaces.As a conquence of axial and radial cracks,rock pieces in the form of
cantilever beams are generated around the specimen free surface.The rock pieces can break in tension
in mode I fracture due to the internal radial stress.Bad on the physical obrvations,a simple
mathematical model is propod to explain the fracture pattern obrved in the physical tests.Fur-
thermore,a bonded particle numerical model for rock,together with afinite element simulation of the
loading machine,is ud to show how the applied boundary conditions and loading machine stiffness
can affect the failure pattern.
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1.Introduction历史最长的朝代
Uniaxial compression is a popular test in evaluating the intact
rock strength.Even though the procedure needed for conducting
this test is straight forward,the state of stress within the specimen
and the failure pattern are much more complicated than what are
obrved in some other rock tests such as triaxial and Brazilian
tests.There have been some efforts in the literature to explain the
failure mechanism in uniaxial compression testing of quasi-brittle
materials such as concrete and rock.Wawersik and Fairhurst1
conducted some compressive tests on rock specimens using a stiff
loading machine and concluded that failure in uniaxial compres-
sion of rock can occur in two modes;local fracturing that is mostly
parallel to the specimen axis and both local and macroscopic shear
faulting.Horii and Nemat-Nasr2analyzed the phenomenon of
axial splitting in compression testing of rock.The rock was as-
sumed to include some pre-existing micro-cracks.Tensile cracks
were considered to nucleate at the tips of the pre-existing cracks
which can grow with increa in the applied compressive load.
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The induced wing cracks can eventually become parallel to the
direction of maximum farfield axial splitting can
occur.Holzhaun and Johnson3examined veral possibilities in
uniaxial compressive testing of rock which can result in induced
庭审笔录lateral tensile stress and axial splitting.The mechanisms in-
vestigated included buckling of vertical rock slices due to the
prence of internalflaws,induced axial cracks due to imperfec-
tion in the specimen geometry(such as testing of a barrel-like
shape specimen),sliding on an inclined internal crack which can
result in tensile cracking,and development of axial cracks due to
the prence of elliptical holes parallel to a free surface.Stress
inhomogeneity and cracks interactions have been described as the
cau for rock facture under compressive loading by Dey and
Wang4.Freddi and Royer-Carfagni5ud a fracture mechanics
model together with some heterogeneous inclusions within the
悲伤的歌曲一听就哭compressively loaded specimens to model the axial splitting.A
fracture mechanics bad model to study the compression failure
of quasi-brittle materials was propod by Bazant and Xiang6.The
crack was considered to be parallel or inclined with respect to the
specimen axis.The effect of specimen size on its strength was
investigated in their study as well.
Contents lists available at ScienceDirect
journal homepage:/locate/ijrmms
International Journal of
Rock Mechanics&Mining Sciences
/10.1016/j.ijrmms.2015.08.013
1365-1609/&2015Elvier Ltd.All rights
rerved.
n Corresponding author.
E-mail address:hamed@nmt.edu(A.Fakhimi).
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International Journal of Rock Mechanics&Mining Sciences79(2015)124–134
The mechanisms discusd above could explain the process of rock axial splitting in uniaxial compression testing,but they are not the only possible caus of axial splitting.In this paper,the interaction of the loading platens with the rock specimen is stu-died as another possible cau of axial splitting.It is known that the ratio of rock elastic modulus to the loading platen modulus can affect the measured uniaxial rock strength,7but as far as we know, no theoretical model has been propod to explain this interaction and the role that it has on the failure pattern in rock.Both physical and3D discrete element tests were conducted to investigate the rock–loading platen interaction and the developed cracks.We suggest that axial cylindrical shaped shear induced cracks in the rock followed by tensile cracking are another possible mechanism for axial splitting in uniaxial compressive testing of hard rock.It is shown that the axial cracks are created due to rock–loading platen interaction that caus a curvature in the loading platen at the contact surface with the specimen end.In addition to physical and numerical testing,an approximate theoretical model is pro-pod to explain the failure mechanism obrved in the actual tests.
2.Physical uniaxial compression tests
Uniaxial compression tests were conducted on Pennsylvania blue sandstone that has an elastic modulus of23–30GPa,a Pois-son's ratio of0.14–0.17,a uniaxial compressive strength of104–129MPa,a Brazilian tensile strength of8.8–10.6MPa,a friction angle of46.2°(within the confining pressure below5MPa)and a cohesion of24.1MPa.The reported uniaxial compressive strengths were obtained by testing cylindrical specimens4.1cm in diameter and8.2cm in height with no lubrication of the specimens ends. The compression tests of this study were performed on the sandstone specimens with different aspect ratios.Specimens with aspect ratios of0.25/1,0.5/1,1/1,2/1,4/1were prepared and tested using the MTS machine in the Mineral Engineering Department. All the specimens have a diameter of about 4.1cm,but their heights are variable.The uniaxial compressive strength of the rock was obtained by dividing the maximum applied load by the ori-ginal cross-ctional area of the specimen.
To study the role of specimen–loading platen interface friction (end-friction),tests were performed with end friction and with reduced end friction conditions.To minimize the interface friction, Stearic acid mixed with Valine was ud.8This lubricant can reduce the friction coefficient to0.02.For the situation with no specimen end lubrication,the friction coefficient between the polished rock surface a
nd the steel platen was measured in direct shear testing and was found to be0.18.9For all the tests conducted, the diameters of the loading platens were greater than that of the specimen.Fig1a and b shows the stress–strain experimental test results conducted with and without end lubrication,respectively. Note that while for specimens with low end friction(lubricated), the uniaxial strength values show no particular trend as the aspect ratio is changed,this is not the ca for specimens with frictional interfaces.The prence of high interface friction has caud the rock strength to increa for short specimens;the interface friction induces lateral confinement which results in greater rock strength. For longer specimens,the interface friction shows no impact on the rock strength as the middle part of the specimen receives no frictional confinement in this ca.It is interesting that for both specimens,with end friction and with reduced end friction,the post-peak rock behavior becomes more ductile as the aspect ratio is reduced.The possible reasons for greater rock ductility with reduction in specimen aspect ratio and the issue of size effect due to the existence of friction between the loading platens and spe-cimen ends have been discusd in Refs.7,9,10.
3.Numerical model
安全会议The bonded particle discrete element system was ud for numerical modeling of the rock.In the bonded particle model implemented in this paper,the rock is simulated using spherical particles that
can interact through normal and shear springs.The compliance of the springs provides the rock elastic behavior.To withstand the deviatoric stress,the particles are glued to each other at the contact points.The normal and shear bond strengths at the particles contact points are adjusted to calibrate the model for the actual rock strength.If the tensile or shear contact force exceeds the contact normal or shear bond strength,the contact los its cohesion.Following the cohesion loss,the contact point follows a Coulomb frictional behavior,if the contact normal force is compressive.In this latter situation,a micro-mechanical friction coefficient(m)is ud to put restriction on the maximum shear force that the damaged contact can carry.
The bonded particle model was calibrated according to the procedure propod by Fakhimi and Villegas11and Fakhimi12to simulate the mechanical behavior of Pennsylvania blue sand-stone.The micromechanical properties obtained by calibration are as follows:k n¼normal stiffness¼3.87Â107N/m,k s¼shear stiffness¼1.4Â107N/m,n b¼normal bond¼18.7N,s b¼shear bond¼63N,and m¼0.5.A genesis pressure of6.4GPa was ud for the sample preparation.The ball radii were assumed to have a uniform random distribution between0.8mm and1.2mm with an average radius of R¼1mm.After calibration,a numerical uniaxial compressive test was conducted on a cylindrical sample of4cm in diameter and8cm in height.The test resulted in an elastic mod-ulus of2
7GPa,a Poisson's ratio of0.15,and a UCS of118.5MPa which are in good agreement with tho of the physical specimen.
For numerical simulation of compression tests,two
loading
Fig.1.Physical stress–strain curves in uniaxial compression tests of Pennsylvania blue sandstone specimens with different aspect ratios,(a)with reduced end friction and (b)with end friction(no lubrication).
A.Fakhimi,
B.Hemami/International Journal of Rock Mechanics&Mining Sciences79(2015)124–134125
machines were modeled;the stiff and the soft machines.The soft machine includes a steel frame,while the stiff machine is only made of two identical small steel platens.The frame of the soft machine consists of a big upper platen (cross head),two small loading platens,and four bars or columns (Fig.2).The bottom ends of the columns are fixed in the x,y and z directions.The finite element method is ud for simulation of the loading machine as a linear elastic material.
The bonded particle specimen is accommodated between the two identical loading platens.The upper loading platen is glued to the machine cross head,while the lower one is assigned to move upward with a constant velocity of 9Â10À3m/s.Note that the stiff machine is made of only the two small loading platens shown in Fig.2b.To reduce the computational time,initially,a uniform traction was applied on the bottom surface of the lower platen.This was done in three or four steps;the applied traction was about 20%,40%,60%,and 75%of the peak strength of the speci-men.After reaching equilibrium in each loading step,the new traction was de fined,and the procedure was repeated.At the end of the last step (after equilibrium under a traction equal to about 75%of the unia
xial compressive strength of the specimen),the vertical velocity was applied and the rest of the test was accom-plished with this quasi-static boundary condition.With this loading machine,the simulated specimen is subjected to the uniaxial compressive load in a quasi-static condition,similar to what happens during a real physical test in the laboratory.Both the soft and stiff machines and the loading platens were assumed to be made of steel.
The results of the numerical tests together with the effect of machine stiffness on the complete stress –strain curve of the spe-cimens with different aspect ratios,and with and without rock –loading platen interface friction have been discusd in Ref.9.In this paper,only tho parts of our study that is related to the obrved failure mechanism are discusd.
4.Failure mechanism
Despite its apparent simplicity,inspection of rock failure pat-tern in a uniaxial compression test suggests a complicated me-chanism.Depending on the specimen aspect ratio,machine stiff-ness,loading platens elastic properties,rock brittleness,applied
force or velocity boundary condition,and the end platen friction coef ficient,the failure pattern can be changed.On the other hand,in some other rock tests such as the triaxial and Brazilian tests,the failur
e pattern is easier to understand and it is more predictable.According to the theory of linear elasticity,13when a cylindrical specimen is pushed into a loading platen,the induced contact stress and displacement may not be uniform at the interface between the platen and the specimen.14Consistent with the the-ory,if the z -direction (axial direction)displacement contours of the lower loading platen of the stiff machine (during testing of the specimen with the aspect ratio of 2:1)are plotted,it can be ob-rved that at the center of the platen,the axial displacement is smaller than that at the edge of the platen (Fig.3);the specimen shows bulging at its center line.The exaggerated deformed spe-cimen and the platens are shown in Fig.4a.In Fig.4b,the ex-aggerated shape of the top surface of the specimen is shown;the specimen is envisioned to be made of veral cylindrical thin shells (only two of them are shown in the figure)with distinct axial deformations.The vertical ction of the imaginary thin cy-linder #1is shown in Fig.4c assuming the specimen end frictional shear forces are capable to keep the material line ad approxi-mately parallel to the specimen axis.The higher local axial strain in the outer thin cylinder (#1in the figure)relative to
its
Fig.2.(a)The soft loading machine with the simulated rock specimen,which is shown in blue;(b)details of the two identical small loading platens in the soft machine.The upper loading platen is attached to
the machine cross head,and the lower one acts as a loading actuator.The stiff machine is made only of the two small loading platens with no frame.(For interpretation of the references to color in this figure legend,the reader is referred to the web version of this
article.)
Fig.3.z -Direction (axial)displacement (mm)contours of the lower platen of stiff machine at the applied average axial stress of 77%of peak stress.The perimeter of the specimen is shown with the black dashed line.
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B.Hemami /International Journal of Rock Mechanics &Mining Sciences 79(2015)124–134
126
surrounding material (cylinder #2in the figure)and the specimen end friction are the cau of induced vertical shear along the side ad of the shell.At higher axial stress,this shear stress can result in shear crack and conquently dilation at the interface between cylinders 1and 2.The interface surface ad is assumed to be ver-tical,even though due to the combined effect of shear and axial stress,it can be slightly tilted.As a conquence of dilation along the ad interface between the thin shells,normal stress along the ad line is generated which tends to push out the outer cylinder.Therefore,this thin shell will be subjected to internal radial ex-panding pressures (Fig.5).This
can cau generation of tensile radial cracks in the shell (parallel to the specimen axis and per-pendicular to the axial shear cracks)together with surface spalling (Fig.5a and b).The surface spalling can be viewed as a buckling phenomenon which is due to a combination of the radial and axial loadings.The radial loading provides the eccentricity that is nee-ded for buckling of the outer thin shell.Alternatively,the axial shear crack can extend as a tensile crack toward the specimen free surface,if the stress intensity factor at the crack tip becomes equal to the rock fracture toughness.This latter possible failure me-chanism is discusd in Section 5of the paper.
Inspection of the physical sandstone specimens after failure con firms that the above failure mechanism can occur in reality (Fig.6).Note that veral cylindrical thin shells are generated in the physical he failure mechanism described above is progressive toward the interior of the specimen,especially for specimens with lower aspect ratio.The cylindrical failure patterns similar to this study have been obrved in laboratory testing of hard rock,15and have been compared with spalling around un-derground openings.16With further loading and surface spalling,the ends of the specimen become short cylinders or cones with reduced radius that can punch through axially and create major axial cracks in the specimen.This latter situation can occur if the specimen is long enough such that its middle part is not affected by the end effects described above.For short specim
en (e.g.spe-cimen 0.5:1in Fig.6),the end effects can penetrate within the whole specimen height and therefore,more cylindrical shells are progressively created and detached which result in more ductile rock behavior.It is important to realize that the core of the spe-cimen at its ends suffers less damage due to the con fining effects provided by the thin shells (reaction of the radial forces in Fig.5)and the end platen frictional shear forces.
Progressive failure of thin cylindrical shells was obrved in the numerical simulations as well.Fig.7shows the developed micro-cracks for the numerical specimen with 1/1aspect ratio and end friction coef ficient of 0.18under different axial stress.The tensile and shear micro-cracks are shown in red and light blue,respec-tively.Note that similar to the physical tests,the damages evolve from the specimen free surface and gradually penetrate toward its central parts.For the ca of ideal situation with zero end friction,the crack pattern is not cylindrical in shape.The numerical results in this ca show that the cracks are developed initially clo to the specimen ends (Fig.8)and then they gradually appear in the middle part of the specimen;the abnce of shear frictional force not only leaves the specimen ends without con finement,it caus line ad in Fig.4c to rotate at point a to remain perpendicular to the line ab after deformation.As a conquence,the specimen top surface must expand laterally which results in radial tensile stress which provide a ground for induced tens
ile cracks.The development of specimen's end radial tensile stress is discusd in the next ctions of the paper.
The cylindrical failure pattern discusd above was obrved in all specimens,with different aspect ratios,without end lubrication and in some (low aspect ratio specimens)with end
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lubrication.
Fig.4.(a)The exaggerated deformed specimen and platens;(b)the deformed shape of specimen top surface;and (c)the vertical cross ction of the #1thin cylindrical shell showing the thickness of the thin cylinder and the internal stress after shear failure along the line
ad.
Fig.5.Generation of internal radial forces due to the non-uniform deformation of the specimen ends.Inner axial layers push outer cylindrical layers outward.(a)The axial and radial pressures;(b)the possible induced cracks and the spall region BCGF.The red arrow shows the rotation direction of the spall failure.(For inter-pretation of the references to color in this figure legend,the reader is referred to the web version of this article.)
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Note that higher end friction forces (for non-lubricated specimens)can partially negate the effect of the radial stress shown in Fig.5or at least postpone the generation of surface spalling.In addition,the generation and failure of cylindrical thin shells continue to-ward interior of the specimen with some regularity in the speci-mens tested with no end lubrication.On the other hand,in the specimens tested with reduced end friction,the induced damages are usually more erratic and show less symmetry (Fig.9).This reveals how the end friction can regularize the failure pattern.
5.Theoretical discussion
Inspection of the surface area of induced axial or slightly tilted
axial cracks in the vicinity of the loading platens shows that the cracks are indeed shear cracks as the cracks apertures are filled with lo rock grains and powder.Considering the contour values of axial deformation of the loading platen and the rock specimen (Fig.3)and the radial distance of the contour lines (about 1mm),the tangent of the angle θin Fig.4c (and Fig.10b),which is equal to γor shear strain,is about 0.002.Using Hooke's law,the induced shear stress is equal to
E E 21tan 211τγνθ
ν=
(+)=
制氧
(+)
()
Using the values E ¼27GPa,ν¼0.15,and γ¼0.002,the induced axial shear stress due to rock –platen interaction is about 23.5MPa (at the axial stress of about 90MPa in Fig.3)which is clo to the rock c
ohesion (24.1MPa).This simple calculation suggests that axial shear cracks parallel to the specimen free surface have po-tential to develop clo to the specimen ends.Since the specimen is cylindrical in shape,the axial cracks are cylindrical in shape dividing the specimen into a few thin cylindrical shells (Figs.4and 10).Following the development of axial shear cracks,radial stress are developed due to dilatation along the shear cracks.The radial stress push the thin cylinders outward;the reaction of the thin cylinders creates con fining pressure which together with the end friction help to create lateral stress for the central part of the specimen.Conquently,the central part of the specimen can carry higher axial load.Therefore,it is expected that the first thin cylindrical shell to develop clo to the specimen free surface where less lateral resistance and con finement exist.As the axial load increas,more cylindrical shells with smaller radii can develop.
The developed cylindrical shells can spall out only if radial cracks are created.For generation of radial cracks,the
innermost
Fig.6.Formation of damaged thin shells and axial cracks in the failed specimens in the physical testing.Failed specimens with aspect ratio of (a)0.5/1,(b)2/1and (c)4/1and tested without end lubrication are shown.The failure mechanism is progressive toward the interior of the specimen irrespective of the aspect ratio,even though it ems that it is discontinued prematurely with the development of major axial cracks for longer
specimens.
Fig.7.Progressive failure of the numerical specimen toward the specimen center.The specimen with end friction coef ficient of 0.18and aspect ratio of 1/1in the figure is subjected to axial stress of (a)74%,(b)90%,and (c)99%of peak stress.(For interpretation of the references to color in this figure,the reader is referred to the web version of this article.)
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128
