Stress-deformed state of cylindrical specimens during indirect tensile strength testing

Levan Japaridze

G.Tsulukidze Mining Institute,E.Mindeli Street,Tbilisi,0186,Georgia

Full length article

Stress-deformed state of cylindrical specimens during indirect tensile strength testing

Levan Japaridze*

G.Tsulukidze Mining Institute,E.Mindeli Street,Tbilisi,0186,Georgia

A R T I C L E I N F O

Article history:

Received 17 December 2014

Received in revised form

21 June 2015

Accepted 24 June 2015

Available online 12 August 2015

Brazilian test method Analytical solution Elliptical contact stresses Curved bearing plates Tensile strength

In this study,the interaction between cylindricalspecimen made ofhomogeneous,isotropic,and linearly elastic material and loading jaws of any curvature is considered in the Brazilian test.It is assumed that the specimen is diametrically compressed by elliptic normal contact stresses.The frictional contact stresses between the specimen and platens are neglected.The analyticalsolution starts from the contact problem of the loading jaws of any curvature and cylindrical specimen.The contact width,corresponding loading angle(2θ0),and ellipticalstresses obtained through solution of the contact problems are used as boundary conditions for a cylindrical specimen.The problem of the theory of elasticity for a cylinder is solved using Muskhelishvili’s method.In this method,the displacements and stresses are represented in terms of two analytical functions of a complex variable.In the main approaches,the nonlinear interaction between the loading bearing blocks and the specimen as well as the curvature of their surfaces and the elastic parameters of their materials are taken into account.Numericalexamples are solved using MATLAB to demonstrate the in fl uence of deformability,curvature of the specimen and platens on the distribution of the normal contact stresses as well as on the tensile and compressive stresses acting across the loaded diameter.Derived equations also allow calculating the modulus of elasticity,total deformation modulus and creep parameters of the specimen materialbased on the experimentaldata of radial contraction of the specimen.

?2015 Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.Production and hosting by Elsevier B.V.All rights reserved.

1.Introduction

The indirect tensile testing method,known as the Brazilian test, provides an alternative to direct tensile testing of cylindrical rock specimens and other brittle materials clamped between two loading fl ats or arcs.The International Society for Rock Mechanics (ISRM)(1978)of fi cially suggested the Brazilian test as a method for determining the tensile strength of rock materials.The standard test method can be followed according to ASTM D3967-08(2008). The European standard for testing the tensile strength of concrete specimens was approved by European Committee for Standardization(CEN)on 18 February 2000(EN 12390-6,2000).The history of development and applications of the Brazilian test in rock mechanics has been reviewed and investigated most recently by numerous scientists,e.g.Daemen et al.(2006),Ye et al.(2009),and Li and Wong(2013).

Carneiro and Barcellos(1953),the pioneers of the Brazilian test method,used Hertz(1895)analysis of the action of concentrated compressive forces P in diametrically opposite points of a cylindrical disk with radius R1and length L(Fig.1a)as the theoretical basis of the method.According to this analytical model,the principal tensile stressesσxalong the vertical diameter are uniformly distributed and given by

The normal principal compressive stressσyis given by

where r is the distance from the origin,and-R1＜r＜R1.In the center(r=0),σy=-3P/(πR1L),and it is in fi nite on the periphery points(r=±R1)of the disk.

The uniform distribution of tensile stresses(Eq.(1))as well as the preparation of cylindrical specimens from core material with ease stimulated the wide application of the Brazilian test method as standard for brittle materials.Such an approach can have an adequate accuracy and the error is smallcompared to direct tensile testing,when the experimental installation provides the action ofalmost concentrated(abstract)forcesP(Fig.1a).In practice,this model could be approximated to reality in case by using sharp wedges or small-diameter steelrods(Fig.1b)between the loading plates and the specimen.But in such a case the line load creates extremely high contact stresses,causing different modes of failure if destruction of the specimen starts at the contact points due to highly intensive shear stresses,rather than in the center because of tension.Therefore,it is assumed that a wider contact strip can reduce these problems signi fi cantly and that an arc of contact smaller than 15°causes no more than 2%error in the principal tensile stress while reducing greatly the incidence of premature cracking(ASTM D3967-08,2008).

Fig.1.Typicalschemes of the Brazilian test:(a),(b)diametricalcompression over symmetrical points,(c)test with thin(e.g.wood,cardboard)bearing pads or strips,(d)test with fl attened specimen,and(e)curved jaws.

When the specimen is placed directly between fl at platens,or between indenters with nonzero radius of curvature of contacting surfaces,the application of Eq.(1)can cause a certain error,the magnitude of which should be estimated.

Anumber ofscholars have paid attention to this problemshortly after the popularization of the Brazilian test method,e.g.Hondros (1959),Jaeger and Cook(1964),Wijk(1978),Amadei(1983), Amadei et al.(1983),Chen et al.(1998),Lavrov and Vervoort (2002),Ye et al.(2009).Marion and Johnstone(1977),Procopio et al.(2003),Markides et al.(2010,2011),Markides and Kourkoulis(2012),and Li and Wong(2013)suggested different analytical and numerical solutions and improved schemes,generalized for different kinds of anisotropy and homogeneity of testing rocks,concretes,glass,and many other brittle and not quite brittle materials(e.g.nuclear wastes(ASTM C1144-89,1989),asphalt concrete).

Jaeger and Cook(1964)and Jaeger et al.(2007)have given an analytical approach for estimating the stress-strain state of the cylindrical specimen of elastic,isotropic rock.The distribution of tractions(Fig.2)on the specimen surfaces corresponding to some central loading angle(2θ0)is uniform(Fig.1b).The radial,tangential normal and shear stress components can be expressed respectively in the polar coordinates as

Along the x axis,whereθ=0,τrθ(θ=0)=0,-a≤r≤a,andρ=r/ a,the series in Eqs.(3)and(4)can be summed in the closed form (Hondros,1959)to give

Amadei(1983)and Chen et al.(1998)developed an analytical approach and computer program to determine the stress at any arbitrary point in the disc made of a transversely isotropic medium under diametrical loading uniformly distributed over a strip of the same contact angle(2θ0=15°).It should be noted that such uniform distribution of contact stresses may be most characteristic of pretreated fl attened cylindricalspecimens(Fig.1d)(e.g.Wang et al., 2004;Dave et al.,2011).

At present,most researchers usually assume a uniform distribution of contact stresses when using Brazilian tests.The uniform distribution of contact stresses seems highly unlikely.But many authors,e.g.Fairhurst(1964),Colback(1967),Vardar and Finnie (1975),Dan et al.(2013),have adopted this assumption.They argued that the details ofthe distribution ofcontact stresses should not be particularly relevant,given that the prime interest focuses on the characteristics of the rock failure at the center ofthe disk,i.e. far away from the applied(boundary)stresses.

As is noticed by Daemen et al.(2006),the distribution ofcontact stresses between the specimen and loading platens in a Brazilian test is a typical contact problem,which remains dif fi cult to be modeled numerically(e.g.Hills et al.,1993).Since it is impossible for the real test conditions to meet all assumptions made for the theoretical development,they tried to determine the distribution of contact stresses experimentally using the pressure fi lm of SPI Corporation.

Fig.2.(a)Diametrical compression over two symmetrical arcs of a cylinder with radius a.(b)Stresses along the loaded diameter,for the case of 2θ0=15o,normalized against P (Jaeger et al.,2007).

Similar opinions are stated by Ye et al.(2009),who remarked that there is no way to know exactly the speci fi c variable in the processing of loading of the contact angle 2θ0of the stress distribution.The value of 2θ0varies despite using the same concave loading plates and under the same loading,due to the type of rock. These factors make it very dif fi cult to calculate the contact angle. Accordingly,it is also very dif fi cult to calculate exactly the stress fi eld in the disc subjected to distributed loads over an arc when the disc is in the elastic stage.

There is recent work by Kourkoulis et al.(2012,2013),Markides and Kourkoulis(2012)and others.They conducted intensive research on the problems of stress-deformed state of cylindrical specimens under parabolic contact loading employing the complex potentials method(Muskhelishvili,1963).The author also paid attention to this problem,and some problems of stress-deformed state of cylindrical specimens under elliptic(parabolic)contact loading were investigated using the complex potentials method (Japaridze,1971,1972).

Thus,the problem of the shape of the distribution of normaland tangential contact tractions for the standardized indirect tensile splitting(“Brazilian”)test,and its in fl uence on the stress concentration,tensile strength and other mechanical parameters of the rocks and other hard materials still remain topical.

The goal of this paper is to present some new insights into the Brazilian test.It provides a solution of a nonlinear interaction between the fl at and curved loading platens and cylindrical specimens in the Brazilian test.The contact width,corresponding contact angle and elliptical contact stresses obtained from the solution of the contact problems are used as boundary conditions for the cylindricalspecimen.The problem ofthe theory of elasticity for a cylindrical specimen is solved using the complex potentials method developed by Muskhelishvili(1963).

Numerical examples are shown to determine the in fl uence of deformability and curvatures of specimen and bearing platens on the normal contact stresses,tensile and compressive stress concentration factors across the loaded diameter.

Suggested equations also allow calculating the modulus of elasticity,the totaldeformation modulus and the creep parameters of the specimen materialon the basis ofexperimentaldata of radial contraction of the specimen.

2.Theoretical background

A cylindrical specimen with radius R1and length L,and the loading jaws with radius of the contact faces R2compressed(Fig.3) by forces P,touch each other on surfaces with width 2a.

According to the developments of the contact problem of the theory of elasticity(Hertz,1895;Shtaerman,1949;Muskhelishvili, 1963;Timoshenko and Goodier,1970),the half width of the contact surface,a,is given by

where v1,v2and E1,E2are the Poisson’s ratios and moduli of elasticity of specimen and loading jaws,respectively;are the second derivatives of the functions describing the surfaces of a compressible bodies at the point of initialcontact.For the case of cylindrical surfaces,this equation takes the following form:

Consequently,the half loading angle can be written as

Fig.3.Compression of a cylindrical specimen made of homogeneous,isotropic and elastic material between two elastic jaws.pmaxis the maximum contact stress,z1and z1are the boundary points of the contact surface.

Eq.(9)relates the case of plane strain that would occur if L＞R1. When L＜R1,the thin cylindrical specimen will be closer to the“plane stress state”and Eq.(9)will be more correct if v1=v2=0 is used.

When considering the joint deformation of the contacting bodies,the components of displacements of their surface points may be fully identical(“no slip”)and partially identical(“full slip”) in relation to limiting frictionaland/or cohesion contact stresses.In the fi rst case,there will exist normal and tangential contact stresses,which may be interesting as an example in the rocksupport interaction problems.In the second case,when friction and cohesive forces could not provide fully identical joint deformation, tangential contact stresses may be ignored.

Regarding the Brazilian test,the in fl uence ofdry friction stresses on disc-jaw contact surfaces is doubtful,because ofseveralreasons, such as smallcontact width and possibility of the relaxation of the surface friction stresses.At last,the use of various types of soft loading pads(ASTM C1144-89,1989),bearing strips(ASTM C496/ C496M-11,2004)and others,practically exclude the appearance of signi fi cant tangential surface stresses.Thus the consideration of tangentialcontact stresses at the disc-jaw interaction for the more exact boundary conditions will cause an unnecessary complication of the analytical study of the Brazilian test.

If the tangential stresses are neglected(Shtaerman,1949),the elliptic function ofthe normalcontactstresses in polar coordinates is

Eq.(11)does represent an ellipse with smallsemi-axis as a halfwidth of contact surface(Eq.(9))and large semi-axis 2P/(πaL) presenting the maximum contact stress pmaxat the pointsθ=π/2 andθ=3π/2,respectively.

Using series expansion,Eq.(11)with satisfactory accuracy for technical purposes can be expressed in the complex formulation as follows:

The following notations will be used:

The points z and z1on the planeζ=z/r correspond to the points σ=e iθandσ1=ei（π/2-θ1）.Consequently,the boundary conditions can be written as

Corresponding analytical functions of a complex variable,Φ(ζ) andΨ(ζ),which are holomorphic inside the contourγof the circle for pointsζ=rσ,are given by

Returning to the previous variable z=ζR1from these equations yields:

The combination of stress components at any point of the disk can be found by substituting these functions into the well-known equations of Muskhelishvili(1963):

The equations of the internal stresses at points on the y axis can be written in the analog form as Eqs.(1)and(2):

where K and C are the normalized concentration factors for tensile and compressive stresses,respectively(Chen et al.,1998),and can be expressed as

In the particular theoretical case of absolutely rigid platens and disk(E1=E2→+∞),or sharp indenters(R2=0),according to Eqs. (3)-(5),a→0 and Eqs.(24)and(25)converge to the Hertz equations,i.e.Eqs.(1)and(2).

The components of displacements at any point of the cylindrical specimen may be determined by inserting the integrals of analytical functions(Eqs.(20)and(21))into Eq.(24)and separating the real and imaginary parts of the resulting expression.

It is also interesting to determine an explicit measure ofthe rock stiffness,e.g.in terms of its Young’s modulus,on the basis of the simplest possible indirect tensile testing(e.g.Chen et al.,1998;Ye et al.,2009).If such a determination could be made reliably and consistently,it would provide an idealtoolto investigate the spatial variability of the rock stiffness,because it requires far less time for sampling,specimen preparation,testing,and data analysis than compressive testing(Daemen et al.,2006).

The radial and tangential displacement components u and v at any point of the disk can be found using the analytic complex variable functions,Φ(z)andΨ(z)by the well-known equations of Muskhelishvili(1963):

After simpli fi cations of the bulky functions obtained by the use of boundary condition(Eq.(31)),we can derive an equation for determination of the relative contraction of a specimen radius:

Consequently,after substitution of the half width of contact surface(Eq.(9))and the experimentally de fi ned value of contraction ofa specimen radius,the modulus ofelasticity of the specimen material can be obtained from Eq.(33):

Using these equations,it is possible to determine the modulus of elasticity of the material if the half width of the contact surface,a, and the radial contraction of a specimen,ΔR1,are experimentally measured up to 50%-80%of ultimate magnitude of the applied force,P,so as to decrease the possibility of the development of plastic deformations in the specimen.Hence,if the contraction of the specimen radius develops with time,t at constant loading,it is possible to determineΔR1(t)using Eq.(35)and coef fi cientsγandδ, which are the parameters of creep(Rabotnov,1966):

3.Application of proposed functions to practical problems

In this section,an analysis of equations derived above will be conducted in order to show how the practically existing elastic and geometrical parameters of the specimen and curved loading platens in fl uence the width of the contact surface,the maximum of the elliptical contact stresses,the contraction of radius and fi nally, the normalized stress function K,which could be used as the correcting factor for tensile strength of materials calculated from the Hertz equations used now for the Brazilian test method.

Concrete practicalexamples with differentinitialdata willbe used for analysis ofthe proposed functions ofstress factors.For a start,this study considers the case of compression of a cylindrical specimen with the radius R1=2.5 cm andthe thickness L=2.5 cm,between fl at (R2→+∞)and curved(R2=8 cm)steelplatens.The Young’s modulus and Poisson’s ratio ofthe specimen are 20 GPa and 0.25,respectively, and the load P is 50 kN.We will calculate the normalized Hertz stresses using Eqs.(26)and(27)ofthe principalcompression(K)and tension(C)functions along the compressed diametrical line in parallelusing Hondros’solutions(Eqs.(6)and(7))for 2θ=15°.

The analysis of these functions can be easily made using the computer program MATLAB.Results of calculations are given graphically in Fig.4.

Fig.4 shows that the magnitudes of tensile and compressive stresses in the internal part of the diameter of high-modulus specimens and fl at loading platens are close to each other, whereas in the external areas the difference between the magnitudes obtained from Hondros’equations and the proposed analytical solution increases signi fi cantly.This difference is much larger in the external areas as well as in the internal part and even in the center ofthe disk,especially when the curved loading platens are used.This con fi rms the comment in the ISRM suggested methods(Brown,1981)that a criticaldimension of the apparatus is the radius of curvature of the jaws.

To determine the in fl uence of elastic and geometric parameters on the loading angle and the stress concentration factors K and C acting across the loaded diameter,let cylindrical specimens with radius R1=5 cm,length L=2.5 cm,Young’s modulus E1=500 MPa, 1000 MPa,2500 MPa,5000 MPa,15,000 MPa,and Poisson’s ratio v=0.3 be compressed diametrically by force P=40 kN.

The specimen is compressed between planar(R2→+∞)steel (E2=200 GPa)platens without pads(Fig.1a).Appropriate values of the half width of contact surface,a,loading angle,2θ0,maximum contact stress,pmax,contraction of radius along the y axis,ΔR1are respectively calculated by Eqs.(9)-(11)and(33),and the stress concentration factors(Eqs.(26)and(27))are given in Table 1.

Fig.4.Comparison of stress concentration factors K and C along the compressed diameter(-R1≤r≤R)from the developed analytical modeling results and Hondros’solutions for(a)fl at(R2→+∞,E1=20 GPa,2θ=15°)and(b)curved(R2=8 cm, E1=20 GPa)bearing platens.

Ifthe specimen is compressed between curved(R2=8 cm)steel (E2=200 GPa)platens(Fig.1e),appropriate values of half width of contact surface,a,contact angle,2θ0,maximum ofellipticalcontact stresses,pmax,contraction of radius,ΔR1,calculated by Eqs.(9)-(11) and(33),and the stress concentration factors(Eqs.(26)and(27)) are given in Table 2.

From Table 2,one can see that in the center of the specimen (r=0)there always acts the tensile normalstress.But the deviation

Table 1 Numericalresults ofa halfwidth ofcontact surface,a,loading angle,2θ0,maximum contact stresses,pmax,contraction of radius along the y axis,ΔR1,and stress concentration factors K and C in the center(r=0)and on the contour(r=R)ofspecimen for the given values of applied load,P,geometric and elastic parameters of a specimen and planar loading platens(Fig.1a)in function of Young’s modulus E1of a specimen(P=40 kN,R1=5 cm,R2→+∞,v=0.3). ofthe stress concentration function K approaching-1 for the given initial conditions and for the low-modulus specimen reaches 21%. However,this deviation decreases according to the increment of its rigidity and for the specimen of R1=5 cm,L=2.5 cm,P=40 kNand E1=15,000 MPa,the deviation equals only 1%.

?

Table 2 Numericalresults ofa halfwidth ofcontactsurface,a,contact angle,2θ0,maximum contact stresses,pmax,contraction of radius,ΔR1,along the y axis,and stress concentration factors K and C in the center(r=0)and on the contour(r=R)ofspecimen for given values of applied load,P,geometric and elastic parameters ofa specimen and curved loading platens(Fig.1e)in function of Young’s modulus E1ofa specimen (P=40 kN,R1=5 cm,R2=8 cm,L=2.5 cm,v=0.3).

The in fl uences of variation up to the theoretical limits of Poisson’s ratio of the specimen on the loading angle and stress concentration factors are shown in Table 3.

The behaviors of the stress concentration factors K and C calculated using Eqs.(26)and(27)in function of distance from the center,r,and Young’s modulus,E1,are shown in Figs.5 and 6.

Fig.5 illustrates that using Eq.(1)is more reasonable for the rigid materialofa specimen at points close to the center,where the stress concentration factor K approaches-1.The factor K deviates from-1 depending on the distance fromthe center,changes sign at some distance r1,and increases rapidly near the boundary.This value of R1-r1is smaller when the modulus of elasticity of the specimen and the curvature of the bearing platens are larger.Both factors have less in fl uence on the characteristics and magnitudes of the compressive stress and stress concentration factor C,as shown in Fig.6.

The relative distance r1/R1for the initial data is given in Table 4. In the internal diametrical part of the disk(-r1＜r＜r1),negative stress,i.e.tensile normal stress,acts,and in the external parts r1＜r＜R1,positive stress,i.e.compressive stress,acts.

The normalized vector sum F of the stresses acting on all three parts(Fig.7)of a disk diameter(-R1＜r＜R1)was obtained from integration by parts of Eq.(24): where Ftensand Fcompare the normalized vector sums of the tensile and compressive stresses,respectively.

The magnitudes of the normalized vector sum F dependent on Young’s modulus and type of loading platen calculated using MATLAB are also listed in Table 4.The results of speci fi c numericalexamples show that the vector sum ofthe normalstresses acting on the whole disk diameter at the elastic stage are positive,i.e. compressive.This is because,on the two relatively small external parts(r1≤r≤R1)of the diameter,the magnitude of the compressive forces 2Fcompis greater than the sum of the tensile stress Ftenson the internal(-r1＜r＜r1)diametrical part.

Table 3 In fl uence of Poisson’s ratio of the specimen on the loading angle and the stress concentration factors(E1=1 GPa,P=40 kN,R1=5 cm,R2=8 cm).

Fig.5.Dependence of the stress concentration factor K on Young’s modulus,E1,and distance from the center,r,ofa specimen on the y axis for(a)planar(R2→+∞)and(b) curved(R2=-8 cm)platens.

The action of the high intensity compressive stresses in the externalareas of the diameter may somewhat impede the spreading of the cracks from the centralpart ofa disk due to tensile stresses.At the same time,the developmentofthe oncoming cracks due to shear stresses in the peripheral parts can complete the splitting of disk. These cases have been observed in experiments and are described in numerous publications,more recently by Ye et al.(2009),Li and Wong(2013)and others.Such“tensile-shear”kind of the failure mode willcause a decrease of the indirect tensile test accuracy.

When the fracture deviates from the center too much due to shear stress effects,for example,in most of the more than a hundred experiments,Daemen et al.(2006)considered other modes of failure as invalid for determination of tensile strength.This should be considered to improve the modes of failure of specimens in the Brazilian test.

Fig.6.Dependence of the stress concentration factor C on Young’s modulus,E1,and distance from the center,r,of a specimen on the y axis for(a)planar(R2→+∞)and(b) curved(R2=8 cm)platens.

4.Conclusions

The analyticalsolutions of the plane stress-strain state problem are developed for the elastic,homogeneous,isotropic cylindrical specimen compressed between jaws of any curvature.The contact width,corresponding contact angle,2θ0,elliptical contact stressesobtained from the solution of contact problems from the theory of elasticity are used as boundary conditions for cylindrical specimens.The problem is solved using the method developed by Muskhelishvili(1963)according to which the stresses and displacements are represented in terms of analytic functions of a complex variable.

Table 4 The relative distance r1/R1from the center to the zero tensile stress point along the loaded diameter and normalized vector sum F in function of Young’s modulus E1and curvature radius of the loading platens.

Fig.7.Disc with normalized vector sums Ftensand Fcompof tensile and compressive stresses on the external(1)and internal(2)diametricalparts.

The resulting relations allow estimating the stress state in the specimens and the tensile strength of the cylindrical specimens of rock,concrete and other brittle materials taking into account the shape and rigidity of the specimen,and stamps and indenters of different curvatures in the Brazilian test method.The derived equations can be used also for determination ofthe modulus ofelasticity or total deformation modulus and the creep parameters of specimens based on the experimentaldata ofcontraction ofspecimen diameter.

The main solutions takes into account the nonlinear interaction between the loading bearing platens and specimen,curvatures of the jaw surfaces,and elastic parameters of the materials.

Numerical examples solved using computer program MATLAB are presented(a)to determine the in fl uences of Young’s modulus, Poisson’s ratio and radii of curvatures of specimen and jaws on the loading angle 2θ0of the nonuniform normal contact stresses,and stress concentration factors K and C acting across the loaded diameter,and(b)to illustrate the possibilities of using the obtained equations to compare with the existing analog formulae derived on the basis of the concentrated or uniformly distributed normal stresses on the contact surface of given width or loading angle 2θ0.

The application of models of concentrated loads and corresponding equation of tensile stresses always gives a greater or smaller overestimation ofthe maximum tensile stress in the center of disk where the initial crack is generated.Consequently,this causes the overestimation of the tensile strength of a specimen in the Brazilian test.Greater overestimation occurs for low-modulus material especially at the curved platens,and smaller overestimation occurs for a rigid,brittle specimen atthe sharp indenters.

The models with more realistically distributed elliptical contact stresses always give somewhat less tensile stress in the center of the disc,and therefore,the overestimation of tensile strength of a specimen is less than that obtained from the models with a concentrated load.But this difference depends on the rigidity and strength of the specimen material.It is greater for soft materials with low modulus and high strength,and smaller for rigid and brittle specimens with low strength.

These relationships are demonstrated above in tabular and graphical representations and given in the Appendix.Also the ready-made M-fi le for MATLAB is included,with which one can easily calculate the values of stress concentration factors,the loading angle,etc.for real laboratory test and compare with other predictions of the tensile strength.

Con fl ict of interest

The author con fi rms that there are no known con fl icts of interest associated with this publication and there has been no signi fi cant fi nancial support for this work that could have in fl uenced its outcome.

Acknowledgments

The author thanks Professor Jaak Daemen for help in the acquaintance with experimental and normative materials,and for review of this paper and valuable comments.Also,the author expresses his gratitude to anonymous reviewers for excellent and valuable comments and suggestions that have helped materially in improving this manuscript.

Appendix

This appendix includes calculation examples of tensile strength of the specimens by indirect method.Fig.A1 shows the graphical representations of the stress concentration factorKfor Tests 1 and 2.

Test 1.The specimen of high rigidity in the fl at loading platens. Parameters of Test 1

·Radius of cylindrical specimen,R1=3 cm

·Radius of loading platens,R2→+∞

·Thickness of the specimen,L=1.5 cm

·Peak load,P=20 kN

·Young’s modulus of the specimen,E1=20 GPa

·Young’s modulus of the platens,E2=200 GPa

·Poisson’s ratio of the specimen,v1=0.16

·Poisson’s ratio of the platens,v2=0.3

Test 2.The specimen of low rigidity in the curved loading platens. Parameters of Test 2

·Radius of cylindrical specimen,R1=3 cm

·Radius of curved loading platens,R2=4.5 cm

·Thickness of the specimen,L=1.5 cm

·Peak load,P=20 kN

·Young’s modulus of the specimen,E1=2 GPa

·Young’s modulus of the platens,E2=200 GPa

·Poisson’s ratio of the specimen,v1=0.16

·Poisson’s ratio of the platens,v2=0.3

MATLAB code for Test 2

The code is the same as that for Test 1,with the replacement of the value of E1=2*10^4 and R2=4.5.

Results

When r=0.01,0.02,0.03,…,2.98,2.99,3 cm,we have:

k=-0.9164,-0.9163,-0.9163,…,6.2465,6.3843,6.5239; a=0.867 cm;2θ=33.5°;σx(r=0)=KP/(πR1L)=-0.916× 141.47=-129.6 kg/cm2.

Fig.A1.Graphicalrepresentations ofthe stress concentration factor K for(a)Test1 and (b)Test 2.

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Levan Japaridzereceived his Master of Design and Construction of Underground Structures at St.Petersburg Institute of Mining Geomechanics and Surveying(VNIMI) in 1968 and PhD at Moscow Mining Institute in 1978.He worked as Professor of Mining Engineering atthe Georgian Technical University since 1972 in Rock Mechanics,Theory and Design of Tunnels and Underground Constructions. Prof.Levan Japaridze is now a member of the Georgian National Academy of Sciences,the Georgian Engineering Academy,and the Russian Academy of Mining Sciences.

*Corresponding author.Tel.:+995 32 2325831.

E-mail address:levanjaparidze@yahoo.com.

Peer review under responsibility of Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.

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http://dx.doi.org/10.1016/j.jrmge.2015.06.006