Experimental and Numerical Study of the Split Tensile Test on a Silty Soil : Discrete Element Analysis

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1 Expermental and Numercal Study of the Splt Tenle Tet on a Slty Sol : Dcrete Element Analy Abdelkader Ammer, Mondher Nefar, Khaled Ibrahm, Mounr Bouada* Umm Al-Qura Unverty College of Engneerng at Al-Qunfudah Al-Qunfudah, Kngdom of Saud Araba * Unverté de Tun El Manar, École Natonale d Ingéneur de Tun, LR14ES0, Ingénere Géotechnque. BP 7 Le Belvédère 1002 Tun, Tuna. Abtract An analy of the plt tenle tet conducted on a lty ol. Th paper concerned wth the numercal and expermental analy for the bac aumpton: elatcty, brttlene and n-center crack ntaton. Numercal reult how that the Hertz tre feld vald untl falure, whch guarantee the elatcty and the brttlene. However the ncenter crack ntaton wa guaranteed only by a flattened dc. A vald loadng angle correpondng to the flat end wdth ha to be between 12 and 15. However, numercal reult how that the Hertz oluton no longer vald for the flattened dc. Therefore the calculated value of the tenle trength derved from the tet and baed on the Grffth falure crteron doe not correpond to the unaxal tenle trength. Keyword plt tenle tet; unaxal tenle trength; valdty; lty ol; dtnct element method I. INTRODUCTION The tenle trength of coheve ol, whch reman dffcult to obtan by drect tenle tet, depte attempt[1][2], the ubject of everal tude [][4][5]. Indrect tenle tet are the alternatve to the drect tenle tet and have been wdely ued [6][7][8]. Several ndrect tenle tet were ued. But the mot commonly ued the plt tenle tet alo known a the Brazlan tet thank to t mplcty of realzaton. Th tet already tandardzed for brttle materal a rock [9] and concrete [10], t ha been extended by everal author to coheve ol wth low platcty [7][12]. The valdty of the formula ued n the nterpretaton of the plt tenle tet baed on two aumpton, whch are charactertc of the materal : elatcty and brttle behavor. Note that the aumpton of elatcty requred to ue the tre feld oluton, developed by Hertz, whle the aumpton of brttlene allow extenon of the tre oluton to the pont of falure. Furthermore, the aumpton of the brttle behavor neceary for the materal to obey Grffth falure crteron. Hertz oluton and Grffth falure crteron are the ba of the theoretcal nterpretaton of the Brazlan tet a an ndrect tenle tet, leadng to the determnaton of the unaxal tenle trength. The Hertz oluton lead to the followng tre feld: σ x = 2F (R y) πh (x2 β4 + x2 (R+y) 1 β R ) σ y = 2F πh ((R y) β4 + (R+y) 1 β4 1 ) 2 2R σ xy = 2F πh (x(r y)2 β4 + x(r+y)2 1 β4 ) 2 where R y x β = R + y + x The Grffth falure crteron defned by: f 0 1 T f 0 1 T 1 1 where 1 the major prncpal tre (tenon), the mnor prncpal tre, and T the unaxal tenle trength of the materal. Thu n the center of the pecmen of the Brazlan tet (x = y = 0) we obtan F the tenle tre : x hr (8) (9) the compreon tre F y hr Aumng the Hertz oluton, the only pont where the hear tre zero, and 1 0 the center of the 929

pecmen. Therefore, the tate of tre at th pont excluvely the compreon and tenon.

The ncenter crack ntaton, attrbuted to tenle falure, repreent then a thrd bac condton to nterpret the Brazlan tet a an ndrect tenle tet.

2-Numercal verfcaton of Hertz oluton and nvetgaton of the falure mode durng the plt tenle tet. II. EXPERIMENTAL STUDY A.

2 pecmen. Therefore, the tate of tre at th pont excluvely the compreon and tenon. Thu, f the falure ntated at the center of the pecmen (frt crackng), 1 correpond to the value of the unaxal tenle trength. The ncenter crack ntaton, attrbuted to tenle falure, repreent then a thrd bac condton to nterpret the Brazlan tet a an ndrect tenle tet. The approach of the analy of th tet then baed on 1-Expermental tudy of the behavor of the lty ol n dfferent condton of the plt tenle tet. 2-Numercal verfcaton of Hertz oluton and nvetgaton of the falure mode durng the plt tenle tet. II. EXPERIMENTAL STUDY A. Sol charactertc[8] Tthe expermental work ha been conducted n collaboraton wth the cvl engneerng department of the Natonal Engneerng School of Tun. Expermental tet were carred out on a lty ol comng from a te that elected for a landfll project. The phycal and mechancal charactertc of the ol are gven n Table 1. Fg. 1. Loadng force veru plate dplacement durng the plt tenle tet The recorded behavor durng the tet eem brttle (Fgure 1). But the oberved falure doe not ft the aumed one (Fgure 2). Indeed we note an ntaton of crack n the vcnty of loadng platen. Thee crack propagate toward the center to produce a brttle falure. Phycal and Mechancal Charactertc % of ze gran < 2 m 5 % % of ze gran comprng between 2 m 67 % untl 80 m Specfc denty kn/m Vod rato e 0.47 Maxmum Dry denty (Proctor optmum) 17.9 kn/m dmax Optmum water content 14.5 % Platcty Index PI 1 % Undraned frcton angle u 17 Undraned coheon C u 68 kpa Table 1 Phycal and mechancal charactertc of the teted lty ol [8] B. expermental reult Sere of expermental qua-tatc tet were carred out on unaturated lty ol. Intal water content and dry denty were kept the ame durng the tet. In fact the ntal water content fxed to a tandard range from the optmum water content ( opt ) to opt +2% (the average of the water content 15.5 % 1 %, the degree of aturaton 95%). The dry denty almot 95% of the Proctor optmum value and the average of the dry denty kn/m. Fg. 2. Falure of the plt tenle cylnder A the frt crack are manly due to the concentraton of the hear tree n the vcnty of the loadng platen, we have created two dametrcally oppote flattened porton to facltate tenle falure (fgure ). The flattenng wa defned by the central angle decrbed n Fgure. 90

Loadng force (N) Fg.. Flattened cylnder of the plt tenle tet Fg. 6.

The curve of fgure 5 and 6 how the behavor durng the plttng tet at dfferent flattenng. A numercal nvetgaton wa conducted va a dcrete element model.

Numercal model [8] NUMERCIAL STUDY In order to nvetgate the crack ntaton and the falure mode, a dcrete element model (DEM) wa developed to mulate the plt tenle tet (fgure 7).

3 Loadng force (N) Fg.. Flattened cylnder of the plt tenle tet Fg. 6. Varaton of the falure loadng wth the flattenng angle For an angle 2 between 10 degree and 20 degree, t wa oberved that the falure ntated by a crack n the center (fgure 4). The curve of fgure 5 and 6 how the behavor durng the plttng tet at dfferent flattenng. A numercal nvetgaton wa conducted va a dcrete element model. After the calbraton of the model baed on the expermental reult, the Hertz oluton wa verfed, the falure mode and the n-center crack ntaton condton. III. A. Numercal model [8] NUMERCIAL STUDY In order to nvetgate the crack ntaton and the falure mode, a dcrete element model (DEM) wa developed to mulate the plt tenle tet (fgure 7). In fact, mulaton wa conducted wth Partcle Flow Code (PFC2D). Fg. 4. In-center crack ntaton n the flattened cylnder α=0 ; =00,00 w=14.47%,γgd=16.50 ;w=14,47;d=16,50kn/m kn/m w=14.6,gd=16.40 =00,00 ;w=14,6;d=16,40kn/m kn/m 2α =11,42 = 11,42 ; w=15,68%, ;w=15,68;d=17,4kn/m d= 1,74 2α =11,42 = 11,42; w=15,84%, ;w=15,84;d=17,82kn/m d= 1,782 2α =22,60 = 22,6 ; w=14,52%;d=1,87 ;w=14,52;d=18,70kn/m 2α =22,60 = 22,6 ; w=14,89%;d=1,87 ;w=14,89;d=18,60kn/m 2α =,40 =,4 ; w ;w=16,01;d=18,60kn/m = 16,01%; d =1,86 2α =,40 =,4 ; w ;w=15,7;d=18,90kn/m = 15,7%; d =1,89 2α =4,60 = 4,6 ; w ;w=15,99;d=18,90kn/m = 15,99%; d =1,89 2α =5,1 = 5,1 ; w ;w=16,4;d=17,00kn/m = 16,4%; d =1,7 2α =5,1 = 5,1 ; w ;w=16,28;d=17,00kn/m = 16,28%; d =1,7 2α =5,1 = 5,1 ; w ;w=16,02;d=18,80kn/m ; d =1, Vertcal dplacement (mm) Fg. 5. Loadng force veru vertcal dplacement at dfferent flattenng Fg. 7. Modelng of the Brazlan tet wth DEM In th tudy, the contact law are decrbed by fve rheologcal parameter: normal and tangental tffne (K n and K ), maxmum normal and hear trength (C n and C ) and frcton coeffcent (). It aumed that the parameter of two partcle n contact act n ere (Fgure 8). 91

4 F max max F n, C (19) Normal and hear trength and Frcton coeffcent Normal Stffne K n Tangental Stffne K B. Numercal mcro parameter [8] Quanttatve reult obtaned wth an ntal et of parameter how that the falure mechanm of the lty ol well reproduced by the numercal model (behavour law, crackng zone and falure mode). Fgure 9 and 10 how that the falure mechanm and the repone of the materal n expermental and numercal tet are dentcal. Fg. 8. The rheologcal model of the DEM The tffne model, n correlaton wth the elatc behavour materal, a functon of the normal and tangental tffne. The normal and tangental component of the contact force (F n and F ) are proportonal, repectvely, to the overlap between two dc n contact and to the tangental dplacement at contact. The correpondng bac equaton ued to defne the contact law between two partcle are: n n n F ( t ) K U ( t ) (10) F K U (11) F ( t ) F ( t t ) F (12) n Where F and F are repectvely the normal and tangental force at contact ( ) between two partcle. Alo, n U and U are repectvely the normal and relatve tangental dplacement at contact ( ). ( D t ) a tme tep durng the partcle movement. The tffne contact parameter are computed n PFC2D a: n nj n K K K (1) n n K K j K K K (14) j K K Where K n, K nj are the normal tffne modul and K, K j are the tangental tffne modul of the partcle and j. The falure behavor defned by a Coulomb lke lp model. The adheon at contact of two partcle defned by normal and tangental local adheon C n f and C f. The adheon erve to lmt the total normal and hear force that the contact can carry by enforcng adheontrength lmt. Hence, the tenle and hear trength, C n and C, of a contact between two partcle of dameter d and dj are computed n PFC2D a: n n C C mn( d, d ) (15) f j C C mn( d, d ) (16) f j The tangental component of the contact force lmted n magntude wth a Coulomb-lke lp model, wth frcton coeffcent μ. At each tep of computaton the relable contact are re-actualzed accordng to condton: F F (17) F max n n C (18) Fg. 9. Numercal and expermental falure Fg. 10. Numercal and expermental tet reult The fttng of the mcro parameter wa done n order to reproduce the whole expermental reult. Once the plt tenle tet well reproduced, the mcro-parameter retaned to ft the expermental reult are thoe gven n table 2. Mcro-parameter K K n n C f C f 14 MN/m 14 MN/m 150 kn/m 150 kn/m 0.4 Poroty 0.17 Denty of partcle 18 Partcle/cm² Table 2 Mcro-parameter ued n the numercal model C. Stree dtrbuton n the pecmen durng the plt tenle tet In the ample the tree are calculated n meaurement crcle unformly dtrbuted over the horzontal and vertcal axe a hown n Fgure 11. The average number of partcle 92

5 per meaurement crcle 80. Stree xx and yy were meaured durng the tet n the center of the crcle. Fgure 12, 1 and 14 how the evoluton, durng the tet, of the tenle and compreve tree along the horzontal and the vertcal axe. Fg. 11. Poton of the meaurement crcle n the numercal ample Fg. 14. Comparon between the copreve analytcal and numercal tree along the vertcal axe durng the tet The tre dagram preented n Fgure 12, 1 and 14 hghlght the elatc behavor and thu the utablty of Hertz tre oluton. Indeed, an error of le than % wa recorded between the value of the tran meaured n the ample and that calculated by the equaton of Hertz. Fgure 15 how the evoluton of compreve and tenle tree at the center of the pecmen durng the tet. Perfect overlap between the numercal value and the analytcal value of the Hertz oluton wa recorded untl falure when a dcrepancy occur. It, therefore, reult that the materal lnear elatc and obey the Hertz oluton untl ntaton of falure ( yy <1.%). Fg. 12. Comparon between the tenle analytcal and numercal tree along the horzontal axe durng the tet Fg. 1. Comparon between the compreve analytcal and numercal tree along the horzontal axe durng the tet Fg. 15. Analytcal and numercal tre - tran curve at the center of the pecmen D. Propagaton of crack and falure of the ample The locaton and the nature of the frt crack conttute the ba of the judgment crteron of the valdty of the Brazlan tet for determnng the unaxal tenle trength. Indeed, the falure mut ntate at the center of the ample and hall reult only from tenle load. Fgure 16 how the propagaton of mcrocrack n the ample durng the Brazlan tet. The mcrocrack are clafed nto two categore: normal crack due to tenle force and tangental crack due to hear force. 9

6 To examne the valdty of thee aumpton, we wrte the equalty between the two prevou expreon of the elatc tran energy, we get: E r D (22) 2 n f 2log( f ) log( / ) ( n )log( D) (2) Fg. 16. Evoluton of crack durng the plt tenle tet Fgure 16 how the appearance at an advanced tage of hear crack ( yy = 0.4%). Thee crack are at the upper and lower loadng plate (Fgure 9). Then after, tenle crack occur n the central area of the ample, but not necearly n the center. In order to tudy the mode of crack propagaton n the ample durng the Brazlan tet everal tet were conducted on ample of dfferent dameter. the maxmum tenle tre at the center of the ample wa then meaured for each dameter (Table ). In fact, the propagaton of crack n the ample can lead to, ether urface falure (Fgure 17) or volume falure (Fgure 18) or correpond to an ntermedate cae between the urface and the volume (eentally urface wth mcrocrack whch propagate n the volume urroundng the fracture urface). D (m) f(kpa) 0,09 17,05 0,12 15,6 0,15 1,2 0,18 11,7 0,24 10,9 0, 10,5 Table Maxmum tenle tre meaured at the center of the pecmen veru the dameter By adoptng the aumpton that for the qua - brttle materal, the pecfc elatc fracture energy proportonal to the quare of the plt tenle trength [1], we can wrte: E 2 f r (20) Carpnter and Ferro [14] aume that the pecfc elatc fracture energy proportonal to the pecmen dameter wth a certan power. Er n D (21) It noted that for n = 2 the lmtng cae obtaned whch correpond to the hypothe of falure of Grffth' theory. It aume that all of the elatc tran energy converted or tranformed nto energy of fracture urface (Fgure 18 ). For n =, one obtan the extreme cae whch aume that the elatc tran energy converted nto a contnuou chan of mcrocrack dtrbuted throughout the volume of the pecmen (Fgure 19). Fg. 17. Surface falure mode Fg. 18. Volume crack propagaton mode In other word f the expermental reult of Table 1 are reproduced n a logarthmc cale graph, a lnear curve fttng gve the followng equaton: log (f) log( f ) A q log( D) (24) The exponent n wrtten: n 2q 1, 1,2 1,1 1 0,9 0,8 y = -0,44x + 0,777 0,7-1,1-1 -0,9-0,8-0,7-0,6-0,5-0,4 log (D) Fg. 19. Splt tenle veru the dameter of the tet ample 94

7 Accordng to Fgure 19, the numercal reult of the plt tenle tet on the lty ol lead to the value n = 2,12. Thu wth n = 2.12, Numercal reult more n agreement wth Carpnter theory, whch aume that all the energy tored n elatc deformaton dtrbuted on an ntermedate dmenon between a urface and a volume. Th reult confrmed by the crack propagaton mode oberved n fgure 9. A the oberved ntal crack not located perfectly n the center of the pecmen, one can ay that, accordng to Grffth falure crteron, the maxmum tenle tre meaured at the center of the ample doe not correpond the unaxal tenle trength. Moreover, by analyzng the tre tate at the meaurement crcle at the vcnty of the load platen, one conclude that thee crack are produced excluvely by an exce of hear tre. Therefore, one could avod tre concentraton by expandng the loadng area. Whch to apply the load acro a wder urface (through two dametrcally oppoed flat and whoe effort evenly dtrbuted). The wdth of the flattenng defned by the angle at the center 2 (Fgure ). Thu, n order to defne the optmal flattenng whch lead to an ntal crack n the center, everal numercal tet wth dfferent flattenng were teted. The locaton of crack wa recorded. Fg. 20.c. Crack propagaton for 2 = 12 Fg. 20.d. Crack propagaton for 2 = 15 Fg. 20. a. Crack propagaton for 2 = 6 Fg. 20.e. Crack propagaton for 2 = 18 Fg. 20.b. Crack propagaton for 2 = 9 Fgure 20.a to 20.e how the propagaton of crack for flattenng angle between 6 and 18. For 2a le than 12 the frt crack are caued by hear at the vcnty of the loadng platen. Moreover, for 2a greater than 15 a hear band mode falure wa oberved. However, for 2a between 12 and 15 a frt tenle crackng wa oberved n the center of the pecmen, th fact ha been expermentally oberved (Fgure 6 ). Thu, a flattenng at both loadng generator wth a central angle between 12 and 15 can provde a lmt tenle tre of the materal. In contrat, the value of the tenle tre at the center not necearly the unaxal tenle trength. Indeed, Hertz oluton no longer vald for the new geometry of the flattened cylnder. Conequently, t proceeded wth a numercal check of the man condton:

8 Note that th condton eental to nterpret the maxmum value of the tenle tre derved from the plt tenle tet a the unaxal tenle trength. It hown n Fgure 21 the varaton of 1 at the center of a flat pecmen wth an angle of 12. A negatve value wa obtaned for the tran 1.9% yeld pont correpondng to the crack ntaton at the center of the pecmen. Therefore, the plt tenle trength doe not correpond to the unaxal tenle trength. However, t reman below the latter. Fg. 21. Varaton of 1 n the center of the pecmen durng the tet E. Concluon although the plt tenle the mot ued for determnng the unaxal tenle trength of rock and concrete, t hould be extended wth cauton to fne ol. Indeed t ha been hown n th work that aumpton of elatcty and brttlene are vald for the lty ol ubject of tudy. However the n-center crack ntaton wa guaranteed only by a lttle flattenng correpondng to a range between 12 and 15. For the flattened cylnder, t hown that the tre dtrbuton no longer correpond to that propoed by Hertz. Therefore, the ue of Grffth falure crteron baed on Hertz dtrbuton for th lty ol gve the plt tenle trength whch dfferent from the unaxal tenle trength. REFERENCES [1] Somak S., Ha-Rong L., Tong-Huan W., "Drect tenon tet and tenle tran capacty of concrete at early age". Cement and Concrete Reearch. Volume, Iue 12, December 200, pp [2] Mebah, A., Morel, J., Walker, P., and Ghavam, K. (2004). "Development of a Drect Tenle Tet for Compacted Earth Block Renforced wth Natural Fber." J. Mater. Cv. Eng., /(ASCE) (2004)16:1(95), [] Hebrock G., Zeh R. M., Wtt K. J., Tenle trength of compacted clay, Proc. of Int. Conf. From expermental evdence toward numercal modellng of unaturated ol, Germany September 200, Vol. 1, pp [4] Wang Q. Z., Ja X. M., Kou S. Q., Zhang Z. X., Lndqvt P., 2004, The flattened Brazlan dc pecmen ued for tetng elatc modulu, tenle trength and fracture toughne of brttle rock: analytcal and numercal reult, Internatonal Journal of Rock Mechanc and Mnng Scence, Vol. 41, N 2, pp [5] Km, T., Km, T., Kang, G., and Ge, L. (2012). "Factor Influencng Crack-Induced Tenle Strength of Compacted Sol." J. Mater. Cv. Eng., /(ASCE)MT , [6] Yong Y., Janmn Y., Zouwu Z., 2006, Shape effect n the Brazlan tenle trength tet and a D FEM correcton, Internatonal Journal of Rock Mechanc & Mnng Scence, Vol. 4, pp [7] H. Arlan, S. Sture, S. Batte, 2007, Expermental Smulaton of Tenle Behavor of Lunar Sol Smulant JSC-1, Materal Scence & Engneerng Vol. A, do: /j.mea [8] Ammer A., Jame M., Bouada M., Plé O., Vllard P., Gourc J.P, Numercal tudy of bendng tet on compacted clay by DEM: tenle trength determnaton. Internatonal Journal of Computer Applcaton n Technology, Vol. 4, N 1, 2009, pp [9] ASTM D967-08,Standard Tet Method for Splttng Tenle Strength of Intact Rock Core Specmen [10] ASTM C496 / C496M - 11, Standard Tet Method for Splttng Tenle Strength of Cylndrcal Concrete Specmen [11] Maher, M. and Ho, Y. (1994). "Mechancal Properte of Kaolnte/Fber Sol Compote." J. Geotech. Engrg., /(ASCE) (1994)120:8(181), [12] M.R. Moaddegh, M.A. Hajabba, H. Khadem, "Tenle trength of and, palygorkte and calcum carbonate mxture and nterpretaton wth the effectve tre theory", Geoderma Volume 14, Iue 1 2, September 2006, Page [1] Kakl K. N., Vardoulak I., 2004, An Expermental Invetgaton of the Sze effect n ndrect tenle tet on Donyo marble, 7 th Natonal Congre n Theoretcal and Appled Mechanc, Chana June. [14] Carpnter A., Ferro G., 1994, Sze effect on tenle fracture properte : a unfed explanaton baed on dorder and fractalty of concrete mcrotructure, Materal and Structure (RILEM), Vol. 27, pp

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