ENGINEERING MECHANICS 2012 pp Svratka, Czech Republic, May 14 17, 2012 Paper #225

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1 . 18 m 212 th International Conferene ENGINEERING MECHANICS 212 pp Svratka, Czeh Republi, May 14 17, 212 Paper #225 EVALUATION OF WEDGE-SPLITTING TEST RESULTS FROM QUASI-BRITTLE PRISMATIC SPECIMENS USING THE DOUBLE-K FRACTURE MODEL I. Havlíková *, T. Pail **, V. Veel ***, D. Vo"ehovká ****, Z. Ker#ner ***** Abtrat:. The frature-mehanial parameter value of onrete, a quai-brittle ompoite material, are determined from reord of experiment on peimen with tre onentrator. One of the frature model appliable to onrete i the double-k model. Thi model ombine the onept of oheive fore ating on the effetive rak inrement with a riterion baed on the tre intenity fator. The output of the model are ritial rak tip opening diplaement and frature toughne value, inluding the initiation tre intenity fator value orreponding to the beginning of table rak propagation. In thi paper, a method of alulation by mean of the double-k frature model i verified uing publihed data and the reult of a pilot wedge-plitting tet performed by the author. Keyword: Double-K frature model, onrete, primati peimen, wedge-plitting tet. 1. Introdution Cement-baed ompoite are one of the mot widely ued building material. Conrete may be laified a a o-alled quai-brittle material. Studying the mehanial repone of peimen made of uh ompoite under tati and dynami/fatigue loading i ompliated due to their highly nonlinear nature. Numerial tool for modelling both elati (elati-plati) behaviour and alo the frature proe are ommonly ued to predit or ae the repone of truture fabriated from quai-brittle material. Suh tool often baed on the finite element method ("ervenka et al., 27) or phyial diretization of the ontinuum (Frantík, 27) are uually equipped by exploiting a type of nonlinear frature model imulating the oheive nature of raking of quai-brittle material (Ba#ant & Plana, 1998; Karihaloo, 1995; Shah et al., 1995). The parameter of thi frature model are determined from reord of frature tet; thi i arried out either uing evaluation method built on the priniple of the ued non-linear frature model, e.g. the work of frature method (RILEM, 1985) or the ize effet method (RILEM, 199), or uing invere analyi with the poible appliation of advaned identifiation method ($tafa & Frantík, 21). The utilization of exiting method for the evaluation of tet reord an reult in the obtaining of frature parameter value influened by both the ize and hape of the tet peimen and the tet geometry (the boundary ondition of the tet). Suh parameter annot be ued a relevant input to an analyi uing the above-mentioned numerial tool. A imilarly ditorted deription of the frature may be indiretly aued by utilization of the method for evaluation of the frature model parameter through the identifiation method ued if thi proedure i applied to the reult of only one type of tet and peimen ize/hape. The effet of the peimen' ize/geometry/free boundarie diretly affet the reorded load defletion or the load rak mouth opening diplaement diagram by mean of whih the invere analyi i arried out. Note that both group of method for determination of the parameter of quai-brittle frature model have been tudied by the author' team for everal year (ee e.g. Veel% et al., 27, 29, 21, and 211). * ** *** **** ***** Ing. Ivana Havlíková: Brno Univerity of Tehnology, Faulty of Civil Engineering, Intitute of Strutural Mehani, Veve&í 331/95; 62, Brno; CZ, havlikova.i@fe.vutbr.z Ing. Tomá' Pail: ditto, pail.t@fe.vutbr.z Ing. Válav Veel%, Ph.D.: ditto, veely.v1@fe.vutbr.z Ing. Dita Vo&ehovká, Ph.D.: ditto, vorehovka.d@fe.vutbr.z do. Ing. Zbyn(k Ker'ner, CS.: ditto, kerner.z@fe.vutbr.z

2 384 Engineering Mehani 212, #225 Reearh into the above-mentioned effet of the harateriti of the peimen or the tet ha been the ubjet of oniderable attention in reent deade. Variou method are ued for the determination of the harateriti of frature model for onrete tet geometrie on nothed peimen; the three-point bending of nothed beam or wedge-plitting of ompat nothed peimen (Brühwiler & Wittmann, 199; Karihaloo, 1995) are among the mot ommon. The model referred to a the "double K" (double-k or double-g ee Reinhardt & Xu, 1999, Xu & Reinhardt, 1999a, b, ; Xu et al., 23, Xu et al. 26, Zhao et al., 27, Xu & Zhang, 28; Kumar & Bara, 29; Zhang & Xu, 211) i imilar in thi repet. In priniple, thi model ombine the onept of oheive fore ating on the fae of the fititiou (effetive) rak inrement with a riterion baed on the tre intenity fator (SIF). Thi model an determine the ritial rak tip opening diplaement and the frature toughne and i apable of deribing different level of rak propagation: an initiation part, whih orrepond to the beginning of table rak growth (at the level of reahing the tre intenity fator, K I,ini ), and a part featuring untable rak propagation (after reahing the untable frature toughne, K I,un ). In thi paper, a eleted method of alulation exploiting the double-k frature model parameter i verified uing publihed data (Zhang & Xu, 211). Subequently, it i employed in proeing the reult of the author own pilot wedge-plitting tet performed on a primati onrete peimen. Note that the hape funtion of the wedge-plitting tet peimen ued in the evaluation proedure wa prepared from data publihed in Seitl et al. (211). 2. Evaluation of the wedge-plitting tet on onrete peimen The geometry of a primati-haped peimen for ue in wedge-plitting tet (WST) i hown in Fig 1, where D i peimen depth, 2H i peimen width, B i peimen thikne and a i the initial noth length. A keth of the loading fore deompoition i hown in Fig. 1 right and onit of applied vertial load P V, normal reation N, frition fore F f, horizontal fore applied on the peimen P H and wedge angle. The horizontal plitting fore P H an be alulated a: PV PH = (1) 2 tan where = 15 in our ae. The exeution of thi tet on a eleted onrete peimen i hown in Fig. 2. The input data are ummarized in Tab. 1 for two teted onrete peimen: i) data from the literature (Zhang & Xu, 211), denoted a Speimen 1, and ii) data from the author own afore-mentioned pilot wedgeplitting experiment (Speimen 2). Fig. 1: Speimen geometry and fore diagram.

3 Havlíková I., Pail T., Veelý V., Vořehovká D., Keršner Z. 385 Fig. 2: Wedge-plitting tet onfiguration and a detail of a raked onrete peimen (photo by Tá"a Holu#ová). Tab. 1: Input parameter. Parameter Symbol Unit Speimen 1 Speimen 2 Depth D mm 2 13 Width 2H mm 2 15 Thikne B mm Noth depth a mm 8 25 Thikne of holder H mm 2 5 Load from the linear part P i N Crak mouth opening diplaement for P i CMOD i mm Maximum load P max N Crak mouth opening diplaement for P max CMOD mm Calulation of double-k frature parameter The analytial method for WST wa developed (Xu & Reinhardt, 1999) a an alternative to the experimental approah (Xu & Reinhardt, 1999a) ued to ae double-k frature toughne parameter. Thi proedure i baed on a linear aymptoti uperpoition aumption (Xu & Reinhardt, 1999b) and require numerial invetigation of the oheive toughne K I,, whih repreent the growth of untable frature toughne K I,un above the level of the initiation frature toughne K I,ini due to oheive tre tranfer in the fititiou rak zone. Thi relation i given by the equation: K = K + K (2) un ini I I I and an be explained a an inreae in the rak reitane aued by aggregate interloking in the frature proe zone (FPZ) loated in front of the tre-free rak tip. One the oheive toughne K I, i olved analytially, the value of K I,ini an be evaluated by Eq. (2) and therefore it i only neeary to obtain a maller number of meaured parameter from a reorded P CMOD diagram in omparion with the experimental approah Determination of the untable frature toughne K I,un The untable frature toughne K I,un i defined a the ritial tre intenity fator reated by the maximum load P max at the effetive rak tip and an be expreed a the reitane to untable rak

4 386 Engineering Mehani 212, #225 propagation. To evaluate thi parameter one an ue the following linear elati frature mehani (LEFM) formula (Xu & Reinhardt, 1999a, b, ): K un I P " a max + H # = DF $ = % BD & D + H ' (3) F( ) = (2 + )( " " 5.6 ) 3/2 (1" ) (4) where the maximum load P max and the ritial effetive rak length a are the input parameter readily obtained from the meaured P CMOD diagram, H repreent the thikne of the lip extenometer holder and B, D are the peimen dimenion aording to Fig. 1. To evaluate Eq. (4) it i neeary to evaluate the ritial effetive rak length a at the untable loading ondition (P max ) by olving the following nonlinear equation propoed by Murakami (1987): CMOD P " a max + H # = V $ = % BE & D + H ' " 1+ # V ( ) = $ * & 2.65 & & % + ', 1& - where CMOD i the ritial rak mouth opening diplaement due to the maximum load P max and E i Young modulu, whih an be determined either through a ompreive ylinder tet or through the alulation method given in the RILEM reommendation (RILEM, 199). 2 (5) (6) Determination of oheive frature toughne K I, The oheive frature toughne an be deribed a the energy diipated by oheive fore ditributed along the ritial fititiou rak zone. To alulate the oheive frature toughne K I,, bilinear ditribution of the oheive tre i aumed. Generally, the oheive tre funtion $(w) expree the relation between the oheive tre ) and the fititiou rak opening diplaement w. Four parameter are neeary to uniquely define thi funtion, namely the tenile trength f t, the oheive tre $, the rak opening diplaement w at the lope diontinuity of the bilinear urve, and the ritial rak opening diplaement w at whih the oheive tre $ drop to zero. Thee four parameter an be determined uing expreion given e.g. in the CEB-FIP deign ode. The following parameter value of the bilinear oheive tre funtion were ued in thi paper: f t = MPa, $ =.599 MPa, w =.2363 mm and w = At the maximal load P max the rak beome untable and the orreponding opening at the tip of the tre-free rak (the origin of the fititiou rak) i termed the ritial rak tip opening diplaement, CTOD. Depending on the value of CTOD, two ae exit for the oheive tre ditribution along the fititiou rak length at the ritial tate (Zhang & Xu, 211). Cae I i uppoed to hold for CTOD < w, and the orreponding ditribution of oheive tre in the FPZ i linearly ditributed aording to the formula: x" a 1( x) = ( CTOD) + ft " ( CTOD) a " a where $(CTOD ) i the oheive tre at the tip of the initial rak with the length a at the ritial tate. Thi an be obtained uing Eq. (8) aording to the oheive tre funtion: w " CTOD ( CTOD) = ( w) + ( ft " ( w) ) (8) w Cae II i uppoed to hold when w < CTOD < w, and in thi ae, the orreponding oheive tre ditribution i bilinear and an be expreed for a x a a: (7)

5 Havlíková I., Pail T., Veelý V., Vořehovká D., Keršner Z. 387 or for a x a thi an be rewritten a: x" a 2( x) = ( CTOD) + ( w) " ( CTOD) a " a x a 3 x " = w + ft " w a " a (9) (1) where, in both equation, $ (w ) i the oheive tre orreponding to the rak opening diplaement w, f t i the tenile trength, a i the initial noth depth (initial rak length) and a i the effetive rak length orreponding to CTOD = w, whih an be obtained by olving the following nonlinear equation: w 1/2 2 2 a " a a a " "" # " # 1 $ # $ $ # # $ a & D' # a a $ $ = CMOD % + % % && ' & & ' '' (11) In Eq. (11) CMOD i the rak opening mouth diplaement at the ritial point, D i the peimen depth aording to Fig. 1 and a i the ritial effetive length mentioned above. One the ditribution of oheive tre in the FPZ i known, one an alulate the oheive frature toughne K I, by integrating the following expreion: K I 1 2 a = # ( U) F( U, V ) du (12) " a / a where 3/2 " 3.52(1 # U) 4.35 # 5.28U 1.3 #.3U FUV (, ) = # + $ +.83 # 1.76U% 1 #(1 # UV ) 3/2 1/2 2 1/2 (13) (1# V ) (1# V) $ ( 1# U ) % & ' [ ] and where the ubtitution U=x/a, V=a /D are ued: $(U) i the oheive tre aording to formula (7, 9, 1) and F(U,V) i the harateriti Green funtion. To evaluate Eq. (12) a peial numerial integration method i neeary to handle the ingularity at the integral boundary Reult of alulation Thi paper i primarily foued on the funtionality of the double-k frature model. The reult of the alulation of the value of eleted parameter of thi model an be found in Tab. 2. The value in parenthee for Speimen 1 are taken from Zhang & Xu (211). Tab. 2: The reulting value. Parameter Symbol Unit Speimen 1 Speimen 2 Critial effetive rak length a mm (119.84) SIF (untable frature) SIF (oheive toughne) SIF (initiation toughne) K I un K I K I ini MPa*m 1/ (1.557).918 MPa*m 1/2.868 (.885).141 MPa*m 1/2.689 (.672).777 Critial rak tip opening CTOD mm.3992 (.3938) Conluion The onept of the double-k frature model i not urrently ued in the Czeh Republi (exept for in a tudy reported in the paper Ker'ner & Mateová (21), whih foued on three-point bending of

6 388 Engineering Mehani 212, #225 nothed primati onrete peimen), but the worldwide ientifi and profeional publi interet in thi model ha reently been inreaing. A tehnial ommittee (TC) of the RILEM international fellowhip (International Union of Laboratorie and Expert in Contrution Material, Sytem and Struture), related to the double-k onept and headed by Prof. Shilang Xu and Dr. Shailendra Kumar, wa etablihed in Otober 211. The tak of the new tehnial ommittee i to examine the rak tability riteria related to the double-k frature model, whoe ue ha been hown to be independent of the ample ize, and on the bai of further extenive experimental and numerial verifiation, to attempt to prepare a RILEM doument whih will be uable for international tandardization ativitie in the field of onrete and onrete truture. Note that the double-k model riteria are ued to ae the afety of large onrete truture (dam) in Chinee national tandard No. DL/T , iued in 25. In thi paper, the utilization and reult of a method for alulation of the double-k frature model parameter were hown. The proedure wa programmed and verified uing publihed data and the reult of the author own pilot wedge-plitting tet. The appliability of the ued approah wa demontrated via the omparion of the evaluated reult of WST experiment on primati-haped onrete peimen with publihed data. Aknowledgement Thi outome ha been ahieved with the finanial upport of the Czeh Sientifi Foundation, projet No. P14/11/833, and of the Minitry of Eduation of the Czeh Republi, projet No. MSM Referene Ba#ant, Z. P. & Plana, J. (1998) Frature and Size Effet in Conrete and other Quaibrittle Material. CRC Pre, Boa Raton, Florida. "ervenka, V., Jendele, L. & "ervenka, J. (27) ATENA Program doumentation Part 1: theory. "ervenka Conulting, Prague. Brühwiler, E. & Wittmann, F. H. (199) The wedge plitting tet, a new method of performing table frature mehani tet. Engineering frature mehani, Vol. 35, Frantík, P. (27) FyDiK appliation, Karihaloo, B. L. (1995) Frature mehani of onrete. Longman Sientifi & Tehnial, New York. Ker'ner, Z. & Mateová, D. (21) Jak funguje model dvojí K u betonov%h vzork+. Sborník Problémy lomové mehaniky, Brno, 6 63, ISBN In Czeh. Kumar, S. & Barai, S. V. (29) Equivalene between tre intenity fator and energy approah baed frature parameter of onrete. Engineering Frature Mehani. 76: Murakami, Y. (1987) Stre Intenity Fator Handbook I, II, III. Pergamon Pre, Oxford. Reinhardt, H. W. & Xu, S. (1999) Crak extenion reitane baed on the oheive fore in onrete. Engineering Frature Mehani. 64: RILEM Committee FMT 5 (1985) Determination of the frature energy of mortar and onrete by mean of three-point bend tet on nothed beam. Mater. Strut., 18, RILEM Committee FMT 89 (199) Size effet method for determining frature energy and proe zone ize of onrete. Mater. Strut., 23, Seitl, S., Veel%, V. &,outil, L. (211) Two-parameter frature mehanial analyi of a near-rak-tip tre field in wedge plitting tet peimen. Computer and Struture, 89, Shah, S. P., Swartz, S.E. & Ouyang, Ch. (1995) Frature mehani of trutural onrete: appliation of frature mehani to onrete, rok, and other quai-brittle material. John Wiley & Son, In., New York. $tafa, M. & Frantík, P. (21) Model for high preiion approximation of load defletion diagram. In: Pro. of onferene Engineering Mehani 21, Svratka. ISBN Veel%, V., Frantík, P. & Ker'ner, Z. (29) Craked volume peified work of frature. In B.H.V. Topping, L.F. Cota Neve and R.C. Barro (ed), Pro. of 12th Int. Conf. on Civil, Strutural and Environmental Engineering Computing, Funhal, 29. Civil-Comp Pre. Veel%, V., Ker'ner, Z., N(me-ek, J., Frantík, P.,,outil, L. & Kuharzyková, B. (21) Etimation of frature proe zone extent in ementitiou ompoite. Chemiké Lity, vol. 14 (21), ISSN

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