Numerical simulation of shear fracture evolution in laboratory-scale samples
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- Melissa Day
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1 Sy Numl mul f h fu vlu lby-l ml by D.. Rb* h vg md mul h xm fmd by N Gy (Gy, N.C Fu gwh ud g lg blk f k ubjd uxl d bxl m. Il Jul f Rk Mh d Mg S d Gmhl Ab, vl ) fu gwh ud g blk f k ubjd uxl d bxl m, ug h Elf d lm d. h ul f h hyl xm dd umb f d h w ufully ld h uml mul, ludg uhg hd f h f, h fm f ub-vl l fu -, h fm f gu f fu gg fm h uhd f z, h fm f lg -- fu whh d ml flu, d lkg f l d ml fu ldg ml flu. I dd, h fu w h w d by Gy. I blvd h h k h fm f jug h fu ( d by Gy vu ) d h fu h my hv b ld hbd by h f -xg uu h hyl ml. I w fud h h ml, hugh mlly md u f h m k y, mu hv hd gfly dffg. h f xg uu my hv bud h dy. S mum fm h hyl xm w bly wll mhd, x wh h gug ly dly h h f dvg fu. Kywd h fu, mdllg, Mh-Culmb. Idu N Gy dud f k blk ubjd uxl d bxl ldg (Gy, 1976). Eh blk d l, whh m w xdd whl ud ld mul mg. h ulg fu ud h l w xmd dl d d w uggd f h flu f, ml y, d -xg l f wk. h m f h x l h ul bd by Gy ug uml mdllg. Gy d h f h fu h lgu h h u h dg f d-lvl gld mg xv. h fm f h fu f d wh dmgg my (Lhd d Hg, 199) Ulmly, w wh d h u d vy f uh v ug -l uml mdllg. h wk m dm h uh mdllg bl f lg h fug bd h lld g vdd by Gy xm. Rul md m f h fu d d bvd by Gy. S gug ld ud h l l vd d f m. Gy xm Exml u A f w dud m f d d quz. h m w mhd dm f 1 mm 1 mm 75 mm. Eh d l mhd ll h wy hugh h ml (lll h h x). h l w 2 5 mm wd d f vu lgh, wh d vg lgh f 25 mm. I m f h ml, dgd mg, h z f h l w xdd by flg wy h f mlly. I h, h l g w h 5 mm dm hl 5 mm 1 mm l. h ml w d ud ld. h w fmd ml wh l f xmly h m z. h ml w ldd l fm. A l fm w uld by ghg f w g l l dj h ml. h full ll ld w ld by uzg fl-jk * AglGld Ah, Suh Af. h Suh Af Iu f Mg d Mllugy, 212. ISSN h w f d h, Suh Hmh Il Rk Mh Symum (SHIRMS) 212, My 212, Su Cy, Suh Af. h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 112 AUGUS
2 Numl mul f h fu vlu lby-l ml ld bw h fm d h l. Vl ld w ld by h f mv g mh. M f h w fmd d ml. h ml f quz w d dff fg : C13 Ufd ( ) C17 33 M fm ( ) C18 27 M fm (mg ). Gy m h f bddg l d y bd wh h quz ml, bu h db hm ly m f h flu ul. Exml ul A yl - hugh -d ufd quz ml (C13) d Fgu 1. Cuhg bvd dj bh f f h l. A ughly vl fu xd dwwd fm h f h l. Fml f fu xd fm h l ubvlly v h ld, m f whh uv bk v h l. h bddg l bv h l lly ff h fu, vdg hl f h fm f ub-hzl fu, d g h k whh uv v h l, bly mg h bddg l bv. Sml fu w bvd h ul fm ml C17 (Fgu 2). I h h bddg l hd v m fud flu h fug. A mj fu xdg fm h gh v h l u dj bddg l, ug d lg h fu fmly gg h lf. F uhg d h l l fu bvd. Cuvg f h fu v h l vd, bu my l b d wh udfd bddg l h bm f h ml. Fu m f h bxl mg (C18) d Fgu 3. Gy h h f fug h quz ml lld mly by h bddg l. Cm f h ul w hf dfful. h d ml w gly ffd by -xg fu d uld hf b d m dly. Fgu 4 hw h flu f hzl h fu f vu mg mdl, uvg v h l hgh fm () d, lw fm, fmg fu whh ulmly xd ll h wy h mdl (). h fu vlu f d ml md u fm l dll hl hw Fgu 5. h ml w lghly mll h md bv (75 mm 75 mm 75 mm) d w ldd uxl m. h vl ld w 4 kn (71.1 M). h mj fu bvd w llg f h f d ub-vl fug, whh d v h l f, d h v h ld muh lg. Cmm fu f h fu Fgu 6 hw h l d f h fu xd h uml ul. h fu b ummzd fllw: Cl ub-vl k h fu l fu whh d u h bv blw h l. I h mg mdl, h k bv d blw h l. Fgu 2 hgh d d - hugh ml C17. G fug h lf d d wh h fl ll f h ml d hf ludd h () (b) Fgu 3 Fu m f h f () d bk (b) d f ml C18 (bxl mg ) () (b) () Fgu 1 hgh d d - hugh ml C13. Fu, uh z, d bddg l (dhd l) dd Fgu 4 Fu d f dff fg : σ v /σ h f () 1.95, (b) 2.5, d () AUGUS 212 VOLUME 112 h Jul f h Suh Af Iu f Mg d Mllugy
3 Numl mul f h fu vlu lby-l ml m Fgu 5 Fu vlu ud md-u l d ud uxl m. d Cl l fu m Fmly f fu vu Fu uvg v h l hgh fm Cuhd z f Fu xdg h ml lw fm d lg vu mul f Gy xm d ml mdllg Sll (1994) d Sll d Shl (1996) muld h f l d h dmg z Gy xm ug dmg mdl wh h ABAQUS f lm d. Mdlld l w l wll-mhd wh h mud h xm. Cmu lm h m d h ly f, ml mdl by dy dd, d w bl mdl h fg d wh m-fug. Nvhl w dmd h h m fu f Gy xm uld b ld ug uml mdllg hqu. N d Hldyd (1992) dud ml x ug h budy lm d DIGS, wh dlm du (DD) dud mul fu. Ck gwh (x f h DD) xud bd h mgud d f h mxmum h hd f h k. h l l fu w llwd dvl lly, ju wh h f f f h h fu gwg fm h xv. h m (l h f h mdl) fu w md gw f, fllwd by h mddl fu, h by h um fu. A h g, fu gwh gd ul vu w fd, ul hy w ully hbd by hg h fld. h ul hwd h gv dvlm f h fu fm h xv. Fu w d bvd Gy xm: v h l hgh fm, dg wd vl h fm w dud, vully dg ly v h ld. h ff f bddg l w l udd, d w hw h h bd ld h d x f h dvlg fu. Fgu 6 Cmm fu f h xml ul Numl mdllg F uhg Sllg u h f, ulg z f uhd ml h g. h ub-vl fmly Gu f fu d ub-lll h mj ld ud h l uhd f z. h k d uv v h l hgh fm, d m vl d v h ld lw fm. I h mg mdl hy dd gly lw d h fml lg. Dml d ub-hzl fu A hgh fm, fu z fm h l l, fmg ub-hzl dml fu z v ud h l. C-- h fu A lw fm d lg, h fu xd fm h l h f h ml, ffvly flg h ml d dg h. Ah h Elf f/d lm d w ud mul h. Elf u xl g hm ml h mlyd by FLAC d UDEC. Elf hly dym, d qu h dmg ud b qu lu. Dmg mdlld ug mbd Mh- Culmb/g k -fg. h ml yld h, dg h Mh- Culmb. Wh uff l h umuld xhu h fd fu gy, d fu gd wh h mh. h f f h fu h v ly-bd lw (Klk, 2). Cuhd ml ffvly bm mblg f d bd g ully, hugh h fuh dfm ffd by h bd dmd by h h d f h fg uv. h dud fu k y, bu f h mdl d lm bud ly. h du u m d mv h bly f h ly, bu f h fu fm h mh. I N d Hldyd (1992), h uh h h h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 112 AUGUS
4 Numl mul f h fu vlu lby-l ml f h gwh m mlx ub-uu f lkd fu fm h z f f hk d d h hy umd h h m-z b mdlld dlm duy u h mdum. I h h h z d by f bly d -lm fu, hugh h fu u vy dff l h l ubuu f lkd fu. h qu h, h, whh h h fug dquly muld gv h fg f dvdul fu d h ulg l dy. m f h -fg mdl w bd by mulg lby fmd v ml. h d mg xm w h muld dl d h uml ul md wh h xml. Ml lb I m wh ug -fg ml mdl u h h lm z ly ld, f z h fg g, d m ll mdl (Vm d d B, 1984; d Cudll, 1989). h llg dm h mdl h f lgh (2 mm). vu x wh Elf h hw h 5 1 lm lg h llg dm uff u flu. h mh dg lgh w h.2 mm, whh gv 1 lm lg h lgh f h f. Mdl d ud hlf mll lm d u m w f h d f 2 35 hu f h lg mdl. A ml v f yl quz w d bd f mv d Bzl (Ch, 28). Mbug quz w h v f quz hwd h m yl d bhvu. h vg l l d dy : Yug mdulu: 7 G :.23 Dy: 27 kg/m3 h ul fm h UCS d xl fmd h ml d bl I. h UCS dmd fm b f Mh-Culmb xm 224 M (vg: M, dd dv: M) d h l f h σ 1 -σ 3 lh (β ) 6.5. h vlu fm h b f d mxmum h f 45.5 M d f gl f UCS, xld Bzl w muld umlly. h ml m w f-ud ul gd mh w bd bw uml d lby ul. h -fg ml uv w gd ug h gud f Ryd d Jg (22). S- uv f h uxl d mul md Fgu 7. h σ 1 -σ 3 lh, bd fm xl d uml mul, hw Fgu 8. h fg ml mdl ud b h ul hw Fgu 9. Fm h Bzl ul, h l gh w dmd 14 M, wh fu gy (G f ) f 5 J/m2. Lmd fm w uld h d ud Gy xm. A ulv d ml w gd wh bd fm Klk l. (24). S (M) Ll Axl S (m) 3B Ll 3C Axl 3B Axl 3C Ll 3A Axl Mdlld xl 3A Ll Mdlld ll Fgu 7 Cm f mdlld uxl - d h lby ul Axl y = 5.892x y = 8E-5x Cfg k Axl flu g xl flu L (Axl flu) g k Ll flu g ll flu L (k ) Fgu 8 Cm bw h d d h mdl ul S bl I k d fm UCS d xl Mbug quz Cfg [M] k gh [M] Axl flu Ll flu Equvl ll flu f l AUGUS 212 VOLUME 112 h Jul f h Suh Af Iu f Mg d Mllugy
5 Numl mul f h fu vlu lby-l ml Ch (M) l Ch F gl Dl gl Fgu 9 S-fg m f h Mbug quz mdl h d mdlld h hd ml h qud by Gy (E = 35 G, υ =.19 d UCS 1 M). Klk gv lghly lw UCS, h h w d dgly. Mdlld d vdd bl II. S-fg uv fu f l d Fgu 1. Mdl d h mdl gmy, ldg, d budy d f h llud Fgu 11. Ll ld w ld u ld ff l lg h d f h mdl (wh w). Vl w ld h l h f h mdl (blk w) wh h d d h vl d (l). C vl w ld h l bud. Aly w u dg ldg ul ul ( m f dlm,, d gy h; d m f h ulg fu d muld mum) w bd. Sgd ly (lyg ld, kg h ld, d llwg h mdl hv dy ) w l u d w fud h h ul w vy ml h mdl lyd wh vly. Wh mg w mdlld, blk w mvd fm h mdl v gl m d h mdl h u dy (dmd by mxmum mbl vlu f vly d gy h), mm. Gy d quz ml uxlly (C13) d wh ll f 33 M (C17). b vg h ff f fm, mdl wh ll ld f 1 M d 1 M w l lyd. Gy dud mg xm quz m (C18), d umb f d mdl. I dd C18, f h b-dumd d ul l muld h wk (C9 - Fgu 5). Exm C18 w fmd quz ml wh -u l f 25 mm 1 mm. h l w xdd mululy bh f, lll lg x. h ll (lll h d f mg) w 27 M d h xl w 5 m g (135 M). h f mg md by Gy w vbl, d lf-mg w d. Sml C9 w lghly mll h h h (75 mm 75 mm). h gl xv w dll-hl f 5 mm F d dl gl ( ) dm. h wdh f h l w mud fm Gy Fgu 7, d md 2 mm. Mg w muld 1 mm m. h ml w d uxlly wh md xl ld f 4 kn (71.1 M). Ch (M) l Ch F gl Dl gl Fgu 1 S-fg m f h d mdl bl II Sd dg Gy, Klk l., d h vlu ud h mul y Gy Klk l. Mdlld Yug mdulu 35 G 28 G 35 M Dy 27 kg/m3 28 kg/m3 27 kg/m3 UCS 1 M 87 M 135 M Mxmum h - 2 M 27.6 M Mxmum f gl Mxmum dl gl l gh - 4 M 4 M Fu gy (G f ) - 1 J/m2 1 J/m2 2 mm 1 mm 25 mm 1 mm 2 mm 2 mm Fgu 11 Mdl gmy,, d ldg f mul f h, ludg l ll bud F d dl gl ( ) h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 112 AUGUS
6 Numl mul f h fu vlu lby-l ml Smul ul Cfm Sml l mdl w lyd wh vyg lvl f ll fm. h xl ld w lly d h. Mdl w l lyd wh ll fm d wh ll dlm lg h bud (fd). h ulg fu hw Fgu 12. h u w k dff g f ldg. h ul hw d wd d umb f d fu hgh fm, d dy f h k uv v h l. I h fd h v ld h fm f ub-hzl (ublll h l) fu. h lgh f h l l fu dud wh d fm, x f h fd wh h l fu b d wh h ub-hzl fu. C13 Sl quz ubjd g xl ld C13 w muld uxlly wh l dbd bv. Rul f h ly wh l d Fgu 13. C fu f h lby ud h ul: A f l l fu F uhg Fu vu h l Lg h fu xdg fm h ml hgh ld. Oh bl fu: Sub-hzl fu xdg fm h l l k Fm f jug h wh h ml h fu Ex d bfu f h l- d x fu. h ub-hzl fu w bvd h lby ul ly wh bddg l w. I kw whh h fu wuld f h bddg l w b, hugh h mdl ul ly ugg h flu wuld u flly lg y hzlly d fu d by h l fu. h f h fug h d f h ly hw fu h w bvd C13. Hwv, h b f -- h fu Fgu 1 ugg h h ml w ldd h m lvl h Elf mdl. h fl fu ugly ml h ul f ml C17 (Fgu 2), wh h fug h lf f h l uld fl flu f h ml. h lby ul f C17 d h mdlld C13 ul md Fgu 14. h 95.5 M 19.3 M M M M M Fgu 13 Fu f mul f C13 (uxlly-ldd quz) Ufd 1 M 33 M 1 M Cfd Fgu 12 Fu f ml l mdl wh vyg lvl f fm ( Cfd f ll dlm ) Fgu 14 Id fl fu fm lb C17 md h mdl ul f mul f C13 69 AUGUS 212 VOLUME 112 h Jul f h Suh Af Iu f Mg d Mllugy
7 Numl mul f h fu vlu lby-l ml dmd h fmd bw h l fu d h f h ml fu ud h mdl ul, hugh h C17 ul h l ffvly hfd uwd by h f h bddgl-d ub-hzl fug. Fu d (budy-d) h h l ld. C17 l quz ubjd 33 M ll ld d g xl ld Fu m fm vu g f h ly d Fgu 15. h m fu h C13 mdl ul ld h: A f l l fu F uhg Fu vu h l Lg h fu xdg fm h ml hgh ld. h l l fu fm f, bu d gw muh h uxl mul. h fu d fm xly h f h d mmdly uv wd h. Sl fu h, whh l uv wd v h l. Sub-vl fu d fmly f fu, m ld v h ld, h. h fu gw d bfu h ml fu. Ulmly h ml d l fu m ml h h uxl mul. h l fu z hw g vy f h h uxl. h fu whh uv v h l ml h h lby ul (Fgu 2 d Fgu 3). I dd, fug whh w dbd by Gy. h hghlghd Fgu 16. h fu uvd wh f h l. hy lly fm ud ld d f h fu, bu ubquly d by h l h fu. h f h fu Gy xm my hv b hbd by h f -xg wk h quz. h fu m lly mbl h f jug fu dbd l by Gy (1973), whh h hz ublll h m f h xvd l (Fgu 17) M 124 M Fgu 16 Fu m fm mul f C17 22 M ld, wh uvd fu hghlghd 15 M 19 M 22 M 39 M Fgu 15 Fu f mul f C17 (bxlly ldd quz) Fgu 17 Fu ml h hghlghd Fgu 16 (f Gy, 1973), bvd hydlly ldd ml g 5.8:1 dmd l g h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 112 AUGUS
8 Numl mul f h fu vlu lby-l ml Mdllg w udk l h ul. A yldl ml g l w ubjd h m ldg d h xm. h ulg mdlld fu hw Fgu 18. Bh m-lll d h fu vd h ul, dg h h uvd fu bvd h mul f C17 f h jug ym f h h ully u ud bxlly d g. h fu bvuly hbd Gy uxl d bxl xm, bly by fl h kg ub-lll h mj, by h f bddg l h ml. I h mdl, h fu u ly h fd ml, whh l h hyd ldg Gy fm xm. A mul f C17 w u wh ld ld d g, ju d by Gy. Dug h g, ld d d h mdl llwd u dy. h mum by Gy md wh mum h m h mul. h mud d mdlld md Fgu 19. Fgu 18 Id d mdlld fu f hydlly ldd yldl ml g l Wh h x f gug A, ll h mdlld gug fl h d bvd h xm d ε 2 (.. gly mv) d ε 1 (.. gly l). I m, h f ml, bu h l vlu qu dff. S h md g udmgd h ldg g, m h h h mdlld Yug mdulu hgh h h f h ml, h m -l ml bhvu w vd h ml lw. h dg fm gug A dd g wll wh h mdl ul. Cul g h ul dg kwldg f h dg f dmg dug ldg g I-VI ( Gy Fgu 3). h mum dly h h f h l l fu. h mud mxmum ld f 65 M, uggg h h l k h h mum h lvl. I h mdl, h k ly 75 M, d d h h mum f h 98 M ld ld. h ml h h l gh fu gy y f h ml lw h h mdlld. h f u fd fm h dg ly w bl k h dvlm f h k h hyl. I l bl h h mum fl h l f h, h f h l l fu; hwv, h hgh l l bh l d gu g h. All d h h C17 ml w wk d bly l ff h qud. I my b bl mh h fd qu f flu h xm by djug h ml m; hwv, h bddg l huld l b ludd h mdl. S h S (m) Gg A -6 Ald (M) S (m) Gg B -25 Ald (M) 3 2 Mdlld ε1 Mdlld ε2 Mud ε1 Mud ε2 1 5 Mdlld ε1 Mdlld ε2 Mud ε1 Mud ε2 S (m) S (m) Gg C -4 Ald (M) -15 Gg D -2 Ald (M) Mdlld ε1 Mdlld ε2 Mud ε2 Mud ε1 Mdlld ε1 Mdlld ε2 Mud ε1 Mud ε2 Fgu 19 Mud d mdlld l f mul f C AUGUS 212 VOLUME 112 h Jul f h Suh Af Iu f Mg d Mllugy
9 Numl mul f h fu vlu lby-l ml l d f h l, h mly my ukw llw uh mh b bd bl m fm. C18 mg l quz ubjd 27 M ll ld d 135 M xl ld Dug l g, w fud h h ml hwd gf dmg h ldg-u h. A l l k d xv f uhg ud bf h l w xdd fm l f 25 mm. Cl xm f d h vld h dym vh ud m f h dmg. h mdl w u llly h ldg-u h lm dmg ud by h vh. Sm dmg w ll vd, hugh h dmg w k h fm f l fu, dg v h l. h ul f h hyl mdl f h l f l kw. I my b bl h Gy ldd h m ul m dmg w vd, h by udbl kg, d h bg h mg mul. I l bl h dmg ud, d h h ml w g h h mdl mlyd h uml mul. h mum vd m gh h g f dmg (blw). h g f fu h mul d Fgu 2. h m fu f h hyl fu m (Fgu 3) ld, ludg f uhg, fu whh glly uv v h l, d dml fu xdg ll h wy v h l. Igly, h l l fu d h mul, uggg h h l fm f h l fu my hb h fm f h fu. Cm f dg fm h lby d mul d Fgu 21. h ul f gug A (bv d gh f h l) hw vy ml d: ε 1 g 4 45 mm d h dg; d ε 2 hwg h d wh mm. El ly hwd vy ml d, hwv h u mm f ε 1 d mm f ε 2. I h l ul, h u f ε 2 wh h l udmd, d hly hf f ε 1. I m h fug u h u l h muld d mud, bly du dmg h f whh ffvly h f h l. h d h h ff f fug wll ud by h mul. h mdlld vlu ufmly g h h mud, bu by l h.75 mll. h my d h, f h mdl, h muld ff w l h h f h ml. 2 1 Gg A 25 mm 33 mm S (m) S (mm) Mdl A 1 Mdl A 2 D A 1 D A 2 El A 1 El A Gg B 41 mm 45 mm S (m) mm 61 mm -4 S (mm) Mdl B 1 Mdl B 2 D B 2 D B 1 El B 1 El B 2 Fgu 2 Fu f mul f C18 (mg bxlly ldd quz) Fgu 21 Mud d mdlld l f mul f C18 h Jul f h Suh Af Iu f Mg d Mllugy VOLUME 112 AUGUS
10 Numl mul f h fu vlu lby-l ml h mul f gug B l hw ml d h xm, bu h m bl dff. h f w f h ( 25 mm ) fl h hg du h f l fu. Bh ε 1 d ε 2 gfly d (.. bm l mv) wh h u: muld ε 1 h h mud, whl muld ε 2 bm l mv h mud. h hg h l hw h h f fug h l d h dv -, bu h mgud f h l. Subqu d ml, bu ε 2 hw mud d mdlld d fm mm. h mum l wll wh h l mdl ul. I h fu mul, l l k dd ; hwv, gwg z f l dmg w, whh my u f h d h hy. Adg h f bv, h l l fu dd u h ml f h md g. A l ub-vl fu b h fu m (Fgu 3); hwv, h fu w lghly ff- d h u-f-l wh h mum. I m lkly h h C18 ml w g h md by Gy. C9 mg l d ubjd 71.1 M xl ld A gf mu f mfu-d dmg ud h f g f h ly, wh h ml w f ld blk whu h dll-hl. Fug ud mmdly wh h dll hl w xvd. A h w bvd h lby ul, bl h h ml w g h dd Gy. I my l b h m ll fm w ld. A umb f ly w hf fmd wh vu ml gh d ldg d. Aly w dud wh ld ll f z (uxl), 1 M (7.1:1) d 2 M (3.6:1). Du fug h f xv g (g h dll hl) h ml UCS w d 135 M d h l gh d 13.5 M (1 f UCS). h m fu f h lby xm (Fgu 5) w: A ub-vl l fu f lmd lgh (<9 mm lg) Mull h (< 7 mm) fu dg v h l Lg ub-vl fu dg bh d g h 4 5 mm. h l g f h ly, wh umbly h ml fld, d. A bl l wh bvd fu w bd f ld ll ld f 2 M (Fgu 22). I h, bvu fug hbd u f 31 mm, whh ml h whh mj fu d h lby. Subqu dvlm f l fu mh h xd d, hugh h mdlld fu lw uv v h l m dly. Sub-hzl fug l u bw h m x fu. Fu h w ufully muld lud: A l l fu F uhg Ex fug ud h f whh d v h l lw d v h ld g. Cmm fu f h uml ul whh w hyl xm lud: 31 mm 37 mm 47 mm 55 mm Fgu 22 Fu f mul f C9 wh 2 M ll ld (k=.28) 694 AUGUS 212 VOLUME 112 h Jul f h Suh Af Iu f Mg d Mllugy
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