Mixed-Mode Study of Rock Fracture Mechanics by using the Modified Arcan Specimen Test

Transcription

1 Vol:, No:5, 8 Mixed-Mode Stdy of Rock Fractre Mechanics by sing the Modified Arcan Specimen Test R. Hasanpor, and N. Chopani International Science Index, Geotechnical and Geological ngineering Vol:, No:5, 8 waset.org/pblication/6531 Abstract This paper stdies mixed-mode fractre mechanics in rock based on experimental and nmerical analyses. xperiments were performed on sharp-cracked specimens sing the modified Arcan specimen test loading device. The modified Arcan specimen test was, in association with a special loading device, an appropriate apparats for experimental mixed-mode fractre analysis. By varying the loading angle from to 9, pre mode-i, pre mode-ii and a wide range of mixed-mode data were obtained experimentally. Using the finite element reslts, correction factors applied to the rectanglar fractre specimen. By employing experimentally measred critical loads and the aid of the finite element method, mixed-mode fractre toghness for the limestone nder consideration determined. Keywords Rock Fractre Mechanics, Mixed-mode Loading, Finite lement Analysis, Arcan Test specimen. I. INTRODUCTION RACTUR mechanics can be applied to many F engineering fields inclding civil and mining engineering, where drilling, excavation, explosion and ctting of rocks are closely related to the strength, stability and fractre of rock materials and strctres. Obviosly the principles, methods and techniqes of fractre mechanics can play an important role in the analysis, design and constrction or prodction for many rock engineering projects [1-14]. In many sitations of rock engineering interest, complex stress states predominate in strctres and a potential defect may be shaped so that the failre propensity cannot be resolved by niaxial stress fields. Indced stress fields are mostly singlar and set the pper limit for the strength of strctres. Cases when the shear component, i.e. mode II, of the stresses dominates are particlarly difficlt to assess. In the theoretical analyses of mixed-mode fractre problem, one of the basic assmptions is that the stress field near the crack tip is solely determined by the linear elastic singlar soltion, i.e. linear elastic fractre mechanics (LFM) prevails [1, 15]. In this stdy, a modified version of the Arcan specimen was made for the mixed-mode fractre test of one of type rock specimens, which allows mode-i, mode-ii, and almost any combination of mode-i and mode-ii loading to be tested with R. Hasanpor, Postgradate Stdent of Rock Mechanics ngineering, is with Mining ngineering Department, Sahand University of Technology, Tabriz, Iran ( r_hasanpor@st.ac.ir). N. Chopani. Assistanct Professor, is with Mechanical ngineering Department, Sahand University of Technology, Tabriz, Iran ( Chopani@st.ac.ir). the same test specimen configration. Therefore, disadvantages presented in the previos mixed-mode toghness test methods can be avoided [15, 16]. This investigation seeks to extend nderstanding of the rock fractre behavior of a type of limestone nder mixed-mode loading conditions throgh nmerical and experimental methods. Using finite element reslts, correction factors were applied to the limestone specimen and a forth order polynomial fit was proposed to evalate the stress intensity factors of a modified version of the Arcan specimen with a crack sbjected to mixed-mode loading conditions. The main objective of this stdy was to determine the fractre toghness K IC and K IIC for the rock nder consideration for a wide range of mixed-mode loading conditions. Another goal was to stdy the relationship between the stress intensity factors and the fractre toghness. Based on those analyses, a mixed-mode fractre criterion for the limestone specimen has been determined. II. AN OVRVIW OF ROCK FRACTUR MCHANICS A. Backgrond Theory Linear elastic fractre mechanics (LFM) has been fond a sefl tool for investigation of cracks in rock materials. The prpose of fractre toghness testing is to determine the vale of the critical stress intensity factor, or plane strain fractre toghness K C. This material property is sed to characterize the resistance to fractre in the design of strctral members. ASTM standards 399 [17] and D 545 [18] give some gidance for plane strain mode-i fractre toghness K IC for metals and plastics. The stress intensity factor K C at the tip of the pre-crack in a compact tension specimen is given by: Pc π a Kc = f( a w) (1) wt where P c is the fractre load, a is crack length, w is the specimen width, t is the specimen thickness, and f ( awis ) a geometrical factor. Linear elastic fractre mechanics and plan strain conditions are the primary reqirements. The stress intensity factors ahead of the crack tip for a modified version of Arcan specimen were calclated by sing the following eqations [15, 16, 19-6]: P πa KIC f a w wt = c 1( ) () International Scholarly and Scientific Research & Innovation (5) scholar.waset.org/ /6531

2 Vol:, No:5, 8 Pc πa KIIC f a w wt = ( ) (3) In trn K IC and K IIC are obtained sing geometrical factors f1( aw ) and f( aw ) respectively, which are obtained throgh finite element analysis of Arcan test specimen. Also energy release rate for isotropic material with edge crack can be calclated from the following relationships: K G I K I =, II GII = (Plane stress) (4) (1 ν )K G I (1 ν )K I =, G II II = (Plane strain) (5) where is Yong s modls and ν is Poisson s ratio. the specimen, which were performed with ABAQUS nder a constant load of 1 N. The entire specimen was modeled sing eight node collapsed qadrilateral element and the mesh was refined arond crack tip, so that the smallest element size fond in the crack tip elements was approximately.5 mm. A linear elastic finite element analysis was performed nder a plain strain condition sing 1/r.5 stress field singlarity. To obtain a 1/r.5 singlarity term of the crack tip stress field, the elements arond the crack tip were focsed on the crack tip and the mid side nodes were moved to a qarter point of each element side. International Science Index, Geotechnical and Geological ngineering Vol:, No:5, 8 waset.org/pblication/6531 B. F Analysis of Mixed-mode Fractre The method sed to calclate the stress intensity factor was an interaction J-integral method performed in ABAQUS, and is reqired to separate the components of the stress intensity factors for a crack nder mixed-mode loading in conjnction of finite element analysis. The method is applicable to cracks in isotropic and anisotropic materials. Based on the definition of the J-integral, the interaction integrals J int can be expressed [7]: J int lim n. : ax I. = σ ε σ σax. Γ. qdγ Γ x ax x Where Γ is an arbitrary contor, q is a nit vector in the virtal crack extension direction, n is the otward normal to Γ, σ is the stress tensor and is the displacement vector, as shown in Fig. 1. The sbscript ax represents three axiliary pre mode-i, mode-ii, and mode-iii crack-tip fields for =I, II, III, respectively. The domain form of the interaction J- integral is: J int () sn. : A ax I. = λ σ ε σ σax.. qda x ax x Where λ(s) virtal crack advance and da is srface element. In the interaction J-integral method [7] the two-dimensional axiliary fields are introdced and sperposed on the actal fields. By jdicios choice of the axiliary fields, the interaction J-integral can be directly related to the stress intensity factors as: (6) (7) K= 4 π BJ. int (8) Where B is called the pre-logarithmic energy factor matrix, T Jint = J I, J II, J III int int int and T K = K, K, K I II III. In linear elastic fractre mechanics, the J-integral coincides with total energy release rate, J = GT = GI + GII + GIII, where G I, G II and G III are the energy release rates associated with the mode-i, mode- II and mode-iii stress intensity factors, respectively. Nmerical analyses were carried ot sing the interaction J- integral method. Fig. shows example of the mesh pattern of Fig. 1 Contor for evalation of the J-integral Fig. Finite element mesh pattern of the entire specimen and arond the crack-tip of limestone with crack length а= 3 mm III. ROCK MATRIAL The examined milky limestone is from a qarry near Khoramabad, Lorestan, Iran. It s related to Asmari formation where the majority of oil reservoirs located there. Also it s widely has been sed as bilding stone. The vales of nconfined Yong s modls,, and niaxial compressive strength, σ c, of the Khorramabad sandstone are =9±5GPa and σ c =15±14MPa. The average dry density is ρ=.3gr/cm 3. International Scholarly and Scientific Research & Innovation (5) scholar.waset.org/ /6531

3 Vol:, No:5, 8 International Science Index, Geotechnical and Geological ngineering Vol:, No:5, 8 waset.org/pblication/6531 IV. TST MTHOD AND STUP A. Specimen Geometry and Test Rig and Setp The test specimens were ct from blocks in both the longitdinal and transverse directions to the dimensions of mm 3. The dimensions of the modified Arcan test specimen sed are showed in Fig. 3. Three holes were drilled along the top and bottom edges of the specimen. Sharp crackshaped notches were made with a thin band saw of.8 1mm thickness with a/w=.45. For the testing of the rock specimens in pre mode I, pre mode II and mixed-mode loading conditions, then the crack-tip extended to a/w=.5. A simple and compact fractre mechanics specimen was sed for the determination of fractre toghness nder mode-i, mode-ii and mixed-mode loading conditions. The loading device is simply installed in the niversal testing machine and generates accrately repeatable mltiaxial loading conditions, Fig. 4. Fig. 3 Modified Arcan test specimen The tests were displacement-controlled; this was becase load-controlled testing was difficlt to achieve with the small loads that had to be applied on the specimen. All specimens were tested at the same displacement rate of.5mm/minte. Load and displacements were recorded with the bilt-in load cell dring the whole test. All tests were carried ot sing a kind of niversal tension testing machine (UTM). A total of 1 specimens were tested in this srvey. Tests were repeated 3 times for mode-i, mode-ii, 15, 3, 45, 6 and 75 loading angels. B. Test Method From the load-displacement crve, a fractre load P Q was defined according to the ASTM Standard 399 [17]. K Q was calclated from the following relationship; K Q vale compted from (9) is a valid K I or K II reslts only if all the validity reqirements were met [17]. K I and K II are obtained sing geometrical factors f (a/w), i.e. f I or f II respectively in (9). In trn f I or, f II are obtained from a finite element model of the modified Arcan test specimen. Fig. 4 Geometry of the loading and modified version of Arcan specimen V. NUMRICAL RSULTS A. Mixed-Mode Rock Fractre Specimen Calibration In order to assess geometrical factors or non-dimensional stress intensity factors fi ( aw) and fii ( aw) for Limestone, the a/w ratio was varied between.1 and.5 at.1 intervals and a third order polynomial was fitted throgh finite element analysis as (Fig. 5): 3 fi ( a w)( = ) = ( a w).87( a w) +.31( a w) f ( )( 9) ( ) ( ) II a w = = a w a w ( a w).167 Here a/w is the crack length ratio, where a is the crack length and w is the specimen length. The relationship between the non-dimensional stress intensity factor and the loading angle is shown in Fig. 6. It can be seen that for loading angles 6 o, the mode-i fractre is dominant and as the mode-ii loading contribtion increases, the mode-i stress intensity factor decreases and the mode-ii stress intensity factor increases. For 75 o mode-ii fractre becomes dominant. PQ π a K = f ( a/ w) (9) Q wt International Scholarly and Scientific Research & Innovation (5) scholar.waset.org/ /6531

4 Vol:, No:5, 8 International Science Index, Geotechnical and Geological ngineering Vol:, No:5, 8 waset.org/pblication/6531 Dimensionless stress intensity factor f(a/w) Fig. 5 Non-dimensional stress intensity factors verss crack length of limestone Non-dimensional stress intensity factors fi ( a w)( = ) a/w Loading angle [degree] fii ( a/ w)( = 9) Fig. 6 Non-dimensional stress intensity factors vs. loading angle of limestone for the crack length a= 3 mm B. ffect of Mixed-mode Conditions on Rock Fractre Characterizations The strain energy release rates were calclated sing qations 4. The relationship between the mixed-mode ratios of strain energy release rates and the loading angles is shown in Fig. 7. For loading angles close to pre mode-i loading, very high ratios of mode-i to mode-ii are dominant. The ratios of strain energy release rates close to pre mode-ii loading exhibit the opposite trend. As expected, it is confirmed that by varying the loading angle of the Arcan specimen, pre mode-i, pre mode-ii and a wide range of mixed-mode loading conditions can be created and tested. In Fig. 8, strain energy release rates G I and G II obtained by qations 4 and the total strain energy release rate obtained by G T = G I + G II are compared for a constant vale of the load. It is seen that for loading angles 6 o the mode-i strain energy release rate is maximm and as loading angle increases, G I decreases and G II increases. For 75 o mode-ii fractre becomes dominant. The total strain energy release rate nder mixed-mode loading condition decreases with the loading angle. Therefore, the increase of the mode-ii loading contribtion leads to a redction in the total strain energy release rate. ki kii Mixed mode ratio, log(g II/GI) Loading angle (degree) Fig. 7 The ratio of mode-ii to mode-i, G II /G I (in logarithmic scale), verss loading angle () for limestone with crack length a=3 mm nergy release rates (J/m ) Loading angle (degree) GI GII GT J INTGRAL Fig. 8 Strain energy release rates G I, G II, G T =G I +G II verss loading angle, for limestone with crack length a= 3 mm VI. XPRIMNTAL RSULTS A. Rock Fractre Mechanics Tests Fractre tests were carried ot sing a kind of niversal tension testing machine (UTM). All tests were condcted by controlling the constant displacement rate of.5mm/min in order to redce dynamical effects and the fractre loads and displacements were recorded. Tests were repeated 3 times for mode-i, mode-ii, 15, 3, 45, 6 and 75 loading angels. A total of 1 specimens were tested in this srvey. The load-displacement crves generated by the test machine were sed to determine maximm load and displacement. Fig. 9 shows the typical load-displacement crve of Limestone for pre mode-i. The average vales of critical fractre loads were sed to determine the critical mixed-mode stress intensity factors and strain energy release rates data. Also the fractre was fond to be completely brittle with the load-displacement crves. International Scholarly and Scientific Research & Innovation (5) scholar.waset.org/ /6531

5 Vol:, No:5, 8 TABL I AVRAG CRITICAL STRSS INTNSITY FACTORS (K) C [MPA.M 1/ ] FOR LIMSTON WITH CRACK LNGTH 3 MM loading angle (K I) C [Mpa.m 1/ ] (K II) C [Mpa.m 1/ ] TABL II AVRAG CRITICAL NRGY RLAS RAT (G) C [J/M ] FOR LIMSTON WITH CRACK LNGTH 3 MM loading angle (G I ) C [j/m ] (G II ) C [j/m ] G T [j/m ] International Science Index, Geotechnical and Geological ngineering Vol:, No:5, 8 waset.org/pblication/6531 Fig. 9 Typical Load-Displacement crve of Limestone B. Mixed-Mode Rock Fractre Toghness Rock fractre toghness was determined experimentally with the modified version of the Arcan specimen nder different mixed-mode loading conditions. The average vales of mixed-mode critical stress intensity factors for Limestone are smmarized in Table I. (K I ) C remains almost nchanged ntil =3 and then decreases and (K II ) C increases as the mode-ii loading contribtion, i.e. as increases from to 9. It is seen that for loading angles 6 the mode-i contribtion is greater than that of mode-ii and the openingmode fractre becomes dominant. For loading angles 75 there is an opposite trend and the shearing-mode fractre becomes dominant. From Table I, It can be seen that the shearing-mode (=9 ) fractre toghness is smaller than the opening-mode (= ) fractre toghness. This means that the cracked specimen is togher in tensile loading conditions and weaker in shear loading conditions. Also Fractre toghness measrements for the modified Arcan specimen nder pre mode-i loading show the average fractre toghness of K IC =(K I ) C =.94 [MPa.m 1/ ] for Limestone with crack length 3mm. For pre mode-ii loading sing modified Arcan specimen, the average fractre toghness for Limestone is fond K IIC =(K II ) C =.7 [MPa.m 1/ ]. The calclated critical strain energy release rate vales (G I ) C and (G II ) C sing experimental data nder varios loading conditions are smmarized in Table II. (G I ) C decreases while (G II ) C increases with an increase in mode-ii loading contribtion. The opening-mode and shearing-mode critical strain energy release rates were fond approximately 39.6 J/m and J/m, respectively. Table II also shows the total strain energy release rate, (G T ) C =(G I ) C +(G II ) C nder varios loading conditions, which decreases with the loading angle. Therefore, it is confirmed that the maximm fractre toghness occrs at mode-i loading condition. Also pre mode-i and pre mode-ii fractre toghness of Limestone specimens were approximately G IC =(G I ) C = 39.6 J/m and G IIC =(G II ) C = J/m, respectively. It can be seen that G IIC is smaller than G IC, indicating that the cracked specimen is togher in mode-i and weaker in mode-ii loading conditions. VII. CONCLUSION AND SUMMARY In this paper the mixed-mode fractre behavior of Limestone specimens was investigated based on experimental and nmerical analyses. A modified version of the Arcan specimen was employed to condct a mixed-mode test sing the special test loading device. The fll range of mixed-mode loading conditions inclding pre mode-i and pre mode-ii loading can be created and tested. It is a simple test procedre, clamping/nclamping the specimens is easy to achieve and only one type of specimen is reqired to generate all loading conditions. The finite element reslts indicate that for loading angles close to pre mode-ii loading, a high ratio of mode-ii to mode-i fractre is dominant and there is an opposite trend for loading angles close to pre mode-i loading. It confirms that by varying the loading angle of Arcan specimen pre mode-i, pre mode-ii and a wide range of mixed-mode loading conditions can be created and tested. Also, nmerical reslts confirm that the increase of the mode-ii loading contribtion leads to an increase of fractre resistance in the Limestone (i.e., a redction in the total strain energy release rate) and the increase of the crack length leads to a redction of fractre resistance in the Limestone (i.e., an increase in the total strain energy release rate). The fractre toghness was determined experimentally with the modified version of the Arcan specimen nder different mixed-mode loading conditions. The reslts indicated that the cracked specimen is weaker in shear loading conditions and togher in tensile loading condition. RFRNCS [1] L. Lim, I. W. Johnston, S. K. Choi, J. N. Boland, Fractre Testing of a Soft Rock with Semi-circlar Specimens Under Three-point Bending. International Scholarly and Scientific Research & Innovation (5) 8 7 scholar.waset.org/ /6531

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