The effects of couple stresses on dislocation strain energy

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1 Intenatona Jouna of Sods and Stuctues 4 (3) The effects of coupe stesses on dsocaton stan enegy V.A. Lubada * Depatment of Mechanca and Aeospace Engneeng, Unvesty of Cafona San Dego, La Joa, CA 993-4, USA Receved 4 Ap 3; eceved n evsed fom 4 Ap 3 Abstact The coespondence theoem whch eates the soutons of dspacement bounday vaue pobems n cassca and coupe stess eastcty s fomuated and apped to deve the soutons fo edge and scew dsocatons n an nfnte medum. The effects of coupe stesses on the dsocaton stan enegy s evauated fo both types of dsocatons. It s shown that wthn a adus of nfuence of each dsocaton n a metac cysta wth dsocaton densty of cm, the stan enegy contbuton fom coupe stesses (excudng the coe enegy) s about 5% n the case of an edge dsocaton, and about % n the case of a scew dsocaton. It s then shown that coupe stesses make age effect on the tota wok of tactons actng on the dsocaton coe suface. Ó 3Eseve Scence Ltd. A ghts eseved. Keywods: Dsocatons; Coupe stesses; Stan enegy; Coespondence theoem; Dsocaton coe; Hoow dsocaton. Intoducton In a mcopoa contnuum the defomaton s descbed by the dspacement vecto and an ndependent otaton vecto. The otaton vecto specfes the oentaton of a tad of decto vectos attached to each matea patce. A patce (matea eement) can expeence a mcootaton wthout undegong a macodspacement. An nfntesma suface eement tansmts a foce and a coupe vecto, whch gve se to non-symmetc stess and coupe stess tensos. The fome s eated to a non-symmetc stan tenso, and the atte to a non-symmetc cuvatue tenso, defned as the gadent of the otaton vecto. Ths type of the contnuum mechancs was ognay ntoduced by Vogt (887) and the bothes Cosseat (99). The fundamentas of the theoy wee futhe deveoped n the sxtes, most notaby by G unthe (958), Go (96), Aeo and Kuvshnsk (96), Mndn (964), and Engen and Suhub (964). In a smpfed mcopoa theoy, the so-caed coupe stess theoy (Toupn, 96; Mndn and Testen, 96), the otaton vecto s not ndependent of the dspacement vecto, but eated to t n the same way as n cassca contnuum mechancs. The physca atonae fo the extenson of the cassca to mcopoa o coupe stess theoy was that the cassca theoy was not abe to pedct the sze effect expementay obseved n pobems whch had a * Te.: ; fax: E-ma addess: vubada@ucsd.edu (V.A. Lubada) /3/$ - see font matte Ó 3Eseve Scence Ltd. A ghts eseved. do:.6/s-7683(3)8-

2 388 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) geometc ength scae compaabe to mateaõs mcostuctua ength, such as the gan sze n a poycystane o ganua aggegate. Fo exampe, the appaent stength of some mateas wth stess concentatos such as hoes and notches s hghe fo smae gan sze; fo a gven voume facton of dspesed had patces, the stengthenng of metas s geate fo smae patces; the bendng and tosona stengths ae hghe fo vey thn beams and wes. An extensve st of efeences to mcopoa and coupe stess eastcty can be found n evew atces by Dhawa and Sngh (987) and Jasuk and Ostoja- Stazewsk (995). The eseach n coupe stess and eated non-oca and stan-gadent theoes of matea esponse (both eastc and pastc) has ntensfed dung the ast decade, agey because of an nceasng nteest to descbe the defomaton mechansms and manufactung of mco- and nanostuctued mateas and devces, as we as neastc ocazaton and nstabty phenomena (Feck and Hutchnson, 997; De Bost and Van de Gessen, 998). Thee has been a sgnfcant amount of eseach devoted to dsocaton theoy n coupe stess, mcopoa and non-oca eastcty. The epesentatve efeences ncude K one (963), Msßcu (965), Teodosu (965), Anthony (97), Knes and Semea (97), J.P. Nowack (974, 978), W. Nowack (986), Engen (977a,b, 983), Mnagawa (977, 979), Hseh et a. (98), and Gutkn and Afants (996). In ths pape we deve the soutons fo edge and scew dsocatons n an nfnte medum by usng the coespondence theoem of coupe stess eastcty, whch eates the soutons of dspacement bounday vaue pobems n cassca and coupe stess eastcty. The basc equatons of coupe stess eastcty ae summazed n Secton, wth an accent gven to dspacement fomuaton n Secton 3. Both compessbe and ncompessbe eastc mateas ae consdeed. The coespondence theoem of coupe stess eastcty fo the pobems wth pescbed dspacement bounday condtons s fomuated n Secton 4. The pane stan and ant-pane stan equatons of coupe stess eastcty ae sted n Secton 5. The coespondence theoem s apped n Sectons 6 and 8 to deve the soutons fo edge and scew dsocatons n an nfnte medum. The souton fo the edge dsocaton n a hoow cynde s deved n Secton 7. The contbuton fom coupe stesses to dsocaton stan enegy s evauated and dscussed fo both types of dsocatons. It s shown that wthn a adus of nfuence of each dsocaton n a metac cysta wth the dsocaton densty of cm, the stan enegy contbuton fom coupe stesses (excudng the coe enegy) s about 5% n the case of an edge dsocaton, and about % n the case of a scew dsocaton. It s then shown that coupe stesses make age effect on the tota wok of tactons actng on the dsocaton coe suface. Concudng emaks ae gven n Secton 9.. Basc equatons of coupe stess eastcty In a mcopoa contnuum the defomaton s descbed by the dspacement vecto and an ndependent otaton vecto. In the coupe stess theoy, the otaton vecto u s not ndependent of the dspacement vecto u but subject to the constant u ¼ e jkx jk ¼ e jku k;j ; x j ¼ e jk u k ; ðþ as n cassca contnuum mechancs. The skew-symmetc atenatng tenso s e jk, and x j ae the ectangua components of the nfntesma otaton tenso. The atte s eated to the dspacement gadent and the symmetc stan tenso by u j; ¼ j þ x j, whee j ¼ ðu j; þ u ;j Þ; x j ¼ ðu j; u ;j Þ: ðþ The comma desgnates the pata dffeentaton wth espect to Catesan coodnates x.

3 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) A suface eement ds tansmts a foce vecto T ds and a coupe vecto M ds. The suface foces ae n equbum wth the non-symmetc Cauchy stess t j, and the suface coupes ae n equbum wth the non-symmetc coupe stess m j, such that T ¼ n j t j ; M ¼ n j m j ; ð3þ whee n j ae the components of the unt vecto othogona to the suface eement unde consdeaton. In the absence of body foces and body coupes, the dffeenta equatons of equbum ae t j;j ¼ ; m j;j þ e jk t jk ¼ : ð4þ By decomposng the stess tenso nto ts symmetc and antsymmetc pat t j ¼ j þ s j ð j ¼ j ; s j ¼ s j Þ ð5þ fom the moment equbum equaton t eady foows that the antsymmetc pat can be detemned as s j ¼ e jkm k; : ð6þ If the gadent of the coupe stess vanshes at some pont, the stess tenso s symmetc at that pont. The ate of stan enegy pe unt voume s _W ¼ j _ j þ m j _j j ; whee j j ¼ u j; ð7þ ð8þ s a non-symmetc cuvatue tenso. In vew of the dentty x j;k ¼ k;j kj;, the cuvatue tenso can aso be expessed as j j ¼ e jk k; : ð9þ These ae the compatbty equatons fo cuvatue and stan feds. In addton, thee s an dentty j j;k ¼ j kj; ð¼ u j;k Þ, whch defnes the compatbty equatons fo cuvatue components. The compatbty equatons fo stan components ae the usua Sant VenantÕs compatbty equatons. Snce j s symmetc and e jk s skew-symmetc, fom Eq. (9) t foows that the cuvatue tenso n coupe stess theoy s a devatoc tenso (j kk ¼ ). Assumng that the eastc stan enegy s a functon of the stan and cuvatue tensos, W ¼ W ð j ; j j Þ, the dffeentaton and the compason wth Eq. (7) estabshes the consttutve eatons of coupe stess eastcty j ¼ ow o j ; m j ¼ ow oj j : In the case of sotopc matea wth the quadatc stan enegy, ðþ W ¼ k kk þ k k þ aj k j k þ bj k j k ; ðþ whee, k, a, and b ae the Lame-type constants of sotopc coupe stess eastcty. The stess and coupe stess tensos ae n ths case j ¼ j þ k kk d j ; m j ¼ 4aj j þ 4bj j : ðþ By the postve-defnteness of the stan enegy, t foows that a þ b >, and a b >. Thus, a s postve, but not necessay b. Snce the cuvatue tenso s devatoc, fom the second Eq. () t foows that the

4 38 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) coupe stess s aso a devatoc tenso (m kk ¼ ). In some pobems the cuvatue tenso may be symmetc, and then the coupe stess s aso symmetc, egadess of the ato a=b. If the dspacement components ae pescbed at a pont of the boundng suface of the body, the noma component of the otaton vecto at that pont cannot be pescbed ndependenty. Ths mpes (e.g., Mndn and Testen, 96; Kote, 964; Geman, 973) that at any pont of a smooth bounday we can specfy thee educed stess tactons T ¼ n j t j e jkn j ðn p m pq n q Þ ;k ; and two tangenta coupe stess tactons M ¼ n j m j ðn j m jk n k Þn : ð3þ ð4þ 3. Dspacement equatons of equbum The coupe stess gadent can be expessed fom Eqs. (9) and () as m k; ¼ ae kpq u p;q ; ð5þ ndependenty of the matea paamete b. The substtuton nto Eq. (6) gves an expesson fo the antsymmetc pat of the stess tenso s j ¼ ax j;kk ¼ a x j ; ð6þ whch s aso ndependent of b. The Lapacan opeato s ¼ o =ox k ox k. Consequenty, by addng () and (6) the tota stess tenso s t j ¼ j þ k kk d j a x j : ð7þ Incopoatng ths nto the foce equbum equatons (4), we obtan the equbum equatons n tems of dspacement components u 4 u þ o ð$ uþ þ ð$ uþ ¼ ; ð8þ ox m whee $ u ¼ u k;k, the bhamonc opeato s 4 ¼,and ¼ a ; þ k ¼ m : ð9þ The Posson coeffcent s denoted by m. Upon appyng to Eq. (8) the pata devatve o=ox, thee foows kk ¼. Thus, the voumetc stan s govened by the same equaton as n cassca eastcty wthout coupe stesses. The substtuton nto Eq. (8) yeds the fna fom of the dspacement equatons of equbum u 4 u þ o ð$ uþ ¼: ðþ m ox Thee components of dspacement and ony two tangenta components of otaton may be specfed on the bounday. Atenatvey, thee educed stess tactons and two tangenta coupe stess tactons may be specfed on a smooth bounday.

5 The genea souton of Eq. () can be cast n the fom (Mndn and Testen, 96) u ¼ U o o ð$ UÞ ½u þ x ð ÞUŠ; ðþ ox 4ð mþ ox whee the scaa potenta u and the vecto potenta U ae soutons of the Lapacan and Hemhotz pata dffeenta equatons u ¼ ; ðu U Þ¼: ðþ The genea souton of the atte equaton can be obtaned by obsevng that U U ¼ U must be a hamonc functon, satsfyng the Lapace equaton U expessed as U ¼ U U V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) þ U, whee ð3þ ¼. Thus, the genea souton can be U ¼ : ð4þ 3.. Incompessbe mateas Fo ncompessbe eastc mateas ( kk ¼ ), the stess esponse s t j ¼ j a x j pd j ; ð5þ whee p ¼ pðx ; x ; x 3 Þ s the pessue fed, ndetemnate by the consttutve anayss. The coespondng dspacement equatons of equbum ae u 4 u ¼ op : ð6þ ox The genea souton can be expessed as u ¼ u þ u, whee u and u satsfy the non-homogeneous pata dffeenta equatons u ¼ op ox ; ð7þ u u ¼ op ox : ð8þ 4. The coespondence theoem of coupe stess eastcty Fo equbum pobems of coupe stess eastcty wth pescbed dspacement bounday condtons, and wth no body foces o body coupes pesent, we state Theoem. If u ¼ ^u s a souton of the Nave equatons of eastcty wthout coupe stesses, ^u þ o ð$ ^uþ ¼; m ox then ^u s aso a souton of dffeenta equatons () fo coupe stess eastcty. ð9þ

6 38 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) Poof. It suffces to pove that ^u s a bhamonc functon. By appyng the Lapacan opeato to Eq. (9), we obtan 4^u þ o ð$ ^uþ ¼: ð3þ m ox Snce $ ^u s a hamonc functon, as can be vefed fom Eq. (9) by appyng the pata devatves o=ox, Eq. (3) educes to 4^u ¼ : ð3þ Ths shows that ^u s a bhamonc functon, whch competes the poof. The coespondence theoem fo coupe stess eastcty fomuated hee shoud be compaed wth a eated pncpe of assocaton by Stenbeg and Muk (967), and a theoem of coespondence n non-oca eastcty by Engen (977a,b). We now pove that the stess tenso n coupe stess eastcty wth pescbed dspacement bounday condtons and body foces o body coupes s a symmetc tenso. Fom Eq. (9) t eady foows by pata dffeentaton that the otaton components ae hamonc functons ( x j ¼, u ¼ ), and substtuton nto Eq. (6) gves s j ¼. In genea, the coupe stess tenso s st non-symmetc, athough n the case of ant-pane stan wth pescbed dspacement bounday condtons t becomes a symmetc tenso (see Secton 5.3). 5. Pane pobems of coupe stess eastcty 5.. Pane stan In pane-stan eastcty the dspacement components ae u ¼ u ðx ; x Þ, u ¼ u ðx ; x Þ, and u 3 ¼. The non-vanshng stan, otaton, and cuvatue components ae ¼ ou ; ¼ ou ; ¼ ou þ ou ; ð3þ ox ox ox ox u 3 ¼ x ¼ ou ox ou ox ; ð33þ j 3 ¼ ou 3 ; ox j 3 ¼ ou 3 : ox ð34þ The stess stan eatons ae ¼ðþkÞ þ k ; ¼ðþkÞ þ k ; ð35þ ¼ ; s ¼ a u 3 : ð36þ The noma stess 33 ¼ kð þ Þ. The coupe stess cuvatue eatons ae m 3 ¼ 4aj 3 ; m 3 ¼ 4bj 3 ; m 3 ¼ 4aj 3 ; m 3 ¼ 4bj 3 : ð37þ The eastc stan enegy pe unt voume s W ¼ þ ð þ mþ ð þ m 33 Þ þ 8a ðm 3 þ m 3 Þ: ð38þ

7 Eqs. (3) (38) can be easy ewtten n tems of poa coodnate components. Fo exampe, we have ¼ ou o ; hh ¼ ou h þ u ; h ¼ ou þ ou h oh oh o u h ; ð39þ u 3 ¼ x h ¼ oðu h Þ o ou ; j 3 ¼ ou 3 oh o ; j h3 ¼ Mndn s stess functons The ectangua components of stess and coupe stess tensos can be expessed n tems of the functons U and W as t ¼ o U o W ; ox ox ox V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) t ¼ o U þ o W ; ox ox ox t ¼ o U o W ; t ¼ o U þ o W ; ð4þ ox ox ox ox ox ox m 3 ¼ ow ; m 3 ¼ ow ; ð43þ ox ox whee the functons U and W satsfy the pata dffeenta equatons 4 U ¼ ; W 4 W ¼ : ð44þ The cuvatue stan compatbty equatons eque that the functons U and W be eated by o ðw WÞ¼ ð mþ o ð UÞ; ox ox o ðw WÞ¼ð mþ o ð UÞ: ð46þ ox ox The souton of the equaton fo W n (44) can be expessed as W ¼ W þ W, whee W ¼ ; W W ¼ : ð47þ Thus, Eqs. (45) and (46) can be ewtten as ow ¼ ð mþ o ð UÞ; ð48þ ox ox ou 3 oh : ð4þ ð4þ ð45þ ow ¼ ð mþ o ð UÞ: ox ox The countepats of Eqs. (4) (43), and Eqs. (48) and (49) n poa coodnates ae t ¼ ou o þ o U oh o W ooh þ ow oh ; t hh ¼ o U o þ o W ooh ow oh ; t h ¼ o U ooh þ ou oh ow o o W oh ; ð49þ ð5þ ð5þ ð5þ

8 384 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) and t h ¼ o U ooh þ ou oh þ o W o ; m 3 ¼ ow o ; m h3 ¼ ow oh ; o o ðw WÞ¼ ð mþ o oh ð UÞ; o oh ðw WÞ¼ð mþ o o ð UÞ: ð53þ ð54þ ð55þ ð56þ 5.. Ant-pane stan Fo the ant-pane stan pobems, the dspacements ae u ¼ u ¼, u 3 ¼ wðx ; x Þ. The non-vanshng stan, otaton, and cuvatue components ae 3 ¼ 3 ¼ u ¼ x 3 ¼ ow ; 3 ¼ 3 ¼ ow ; ox ox ow ; u ox ¼ x 3 ¼ ow ; ox ð57þ ð58þ j ¼ j ¼ It eady foows that o w ox ox ; j ¼ o w ; j ox ¼ o w : ð59þ ox t 3 ¼ o ox ðw wþ; t 3 ¼ o ox ðw wþ; t 3 ¼ o ox ðw þ wþ; t 3 ¼ o ox ðw þ wþ: ð6þ ð6þ The coupe stesses ae eated to the cuvatue components by m ¼ 4ða þ bþj ; m ¼ 4ða þ bþj ; ð6þ m ¼ 4aj þ 4bj ; m ¼ 4aj þ 4bj : ð63þ Snce dspacement fed s sotopc, the dspacement equatons of equbum () educe to a snge equaton w 4 w ¼ : ð64þ The genea souton can be expessed as w ¼ w þ w, whee w and w ae the soutons of the pata dffeenta equatons w ¼ ; w w ¼ : ð65þ

9 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) The non-zeo stan, otaton and cuvatue components n poa coodnates ae and h3 ¼ 3h ¼ u ¼ x h3 ¼ j ¼ ou o ¼ o o ow oh ; 3 ¼ 3 ¼ ow o ; ow oh ; u h ¼ x 3 ¼ ow o ; ow oh ; j h ¼ ou h o ¼ o w o ; ð66þ ð67þ ð68þ j h ¼ j hh ¼ ou oh u h ¼ o w oh þ ow o ; ou h oh þ u ¼ o o ow oh The coupe stesses ae eated to the cuvatue components by ð69þ : ð7þ m ¼ m hh ¼ 4ða þ bþj ; ð7þ m h ¼ 4aj h þ 4bj h ; m h ¼ 4aj h þ 4bj h ; ð7þ wth the nvese eatons j h ¼ 4ða b Þ ðam h bm h Þ; j h ¼ 4ða b Þ ðam h bm h Þ: ð73þ The eastc stan enegy pe unt voume s W ¼ ð 3 þ h3 Þþ m þ 4ða þ bþ ða bþ ½aðm h þ m h Þ bm hm h Š : ð74þ 5.3. The coespondence theoem fo ant-pane stan Fo ant-pane stan pobems wth pescbed dspacement bounday condtons, the coespondence theoem of coupe stess eastcty eads: If w ¼ w s a souton of dffeenta equaton of eastcty wthout coupe stesses w ¼, then w s aso a souton of dffeenta equatons (64) fo coupe stess eastcty. The poof s smpe. Snce w s a hamonc functon, t s aso a bhamonc functon, satsfyng Eq. (64). Fo pescbed dspacement bounday condtons, the functon w specfes the dspacement fed n both non-poa and coupe stess eastcty. The stess and coupe stess tensos n ant-pane stan pobems of coupe stess eastcty, n the case of pescbed dspacement bounday condtons, ae symmetc tensos. Indeed, snce the dspacement fed s a hamonc functon, the antsymmetc stess components n Eqs. (6) and (6) vansh,.e., s 3 ¼ s 3 ¼. Thus, the tota stess tenso s a symmetc tenso. Fom Eq. (59) t futhe foows that the cuvatue tenso s a symmetc tenso (j ¼ j ). Ths mpes fom Eq. (63) that the coupe stess tenso s aso symmetc (m ¼ m ), egadess of the ato a=b.

10 386 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) Edge dsocaton n coupe stess eastcty Fo the edge dsocaton n an nfnte medum, the ony bounday condton s the dspacement dscontnuty b, fo exampe mposed aong the pane x > andx ¼. Thus, by the coespondence theoem the dspacement fed s as n cassca eastcty,.e. (e.g., Hth and Lothe, 968), u ¼ b x p tan x u ¼ b p The stesses ae 4ð mþ ð b ¼ pð mþ ; ð75þ x x þ ð mþ x þ x mþ n x þ x þ x x b x þ x : ð76þ x ð3x þ x Þ ðx þ x Þ ; ð77þ ¼ ¼ b x ðx x Þ pð mþ ðx þ ; x Þ b x ðx x Þ pð mþ ðx þ ; x Þ ð78þ ð79þ 33 ¼ mb x pð mþ x þ : ð8þ x The otaton and cuvatue components ae u 3 ¼ b x p x þ ; ð8þ x j 3 ¼ b x x p ðx þ ; x Þ j 3 ¼ b x x p ðx þ : x Þ ð8þ The coespondng coupe stesses ae m 3 ¼ ab p m 3 ¼ ab p x x ðx þ x Þ ; x x ðx þ x Þ ; m 3 ¼ bb p m 3 ¼ bb p x x ðx þ x Þ ; x x ðx þ x Þ : ð83þ ð84þ In poa coodnates, the dspacements ae u ¼ b p h cos h þ 4ð mþ ð mþn sn h ; b ð85þ u h ¼ b p h sn h þ 4ð mþ þð mþn cos h ; b ð86þ

11 and the stesses b ¼ hh ¼ pð mþ h ¼ b pð mþ sn h ; ð87þ cos h ; ð88þ 33 ¼ mb sn h : ð89þ pð mþ The otaton and cuvatue components ae u 3 ¼ b p cos h ; j 3 ¼ b cos h ; j p h3 ¼ b sn h ; ð9þ p wth the coespondng coupe stesses m 3 ¼ ab p m h3 ¼ ab p cos h ; m 3 ¼ bb cos h ; ð9þ p sn h ; m 3h ¼ bb sn h : ð9þ p The stess components decay wth a dstance fom the cente of dsocaton as, whe the coupe stesses decay as. These aso specfy the odes of the snguates at the dsocaton coe when!. The dspacement and otaton feds fo an edge dsocaton n poa eastcty, wthout the constant (), can be found n Nowack (986). An anayss aowng fo a smooth tanston of the dspacement fed fom zeo vaue at the cente of the dsocaton coe to the vaue b aong the cut used to ceate the dsocaton has been done n non-oca eastcty by Engen (977a). Mnagawa (977, 979) deved the stess and coupe stess feds poduced by dscnatons and ccua dsocatons n a mcopoa eastc contnuum. 6.. Stan enegy V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) s The stan enegy (pe unt ength n x 3 decton) stoed wthn a cynde bounded by the ad and R E ¼ Z R Z p W d dh; whee the specfc stan enegy (pe unt voume) s W ¼ ½ h þð mþ Šþ 8a ðm 3 þ m h3 Þ¼ b 8p ð mþ ð m sn hþþ ab p : 4 Upon the substtuton nto Eq. (93) and ntegaton, thee foows b E ¼ 4pð mþ n R þ ab : ð95þ p R The second tem on the ght-hand sde s the stan enegy contbuton fom the coupe stesses. The pesence of ths tem s assocated wth the wok done by the coupe stesses on the sufaces ¼ and ¼ R. Ths can be seen by wtng an atenatve expesson fo the stan enegy, ð93þ ð94þ

12 388 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) Z R E ¼ h ð; Þbd þ Z p M 3 u 3 Rdh Z p M 3 u 3 dh; ð96þ wth M 3 ¼ m 3 gven by Eq. (9), and u 3 gven n Eq. (9). The wok of the tactons and h on the dspacements u and u h ove the suface ¼ R cances the wok of the tactons and h ove the suface ¼. These tems ae thus not expcty ncuded n Eq. (96). The second tem n Eq. (95) s the stan enegy contbuton due to ast two wok tems n Eq. (96). Fo exampe, n a metac cysta wth the dsocaton densty q ¼ cm, the adus of nfuence of each dsocaton (defned as the aveage dstance between dsocatons) s of the ode of q = ¼ nm (Meyes and Chawa, 999). Fo an FCC cystap wth the attce paamete a ¼ 4 A and the Buges vecto aong the cosed packed decton b ¼ a= ffff, the adus R p can be appoxmatey taken as R ¼ b. By choosng the matea ength to be the attce paamete ( ¼ ffff b), the coupe stess moduus s a ¼ b, and by seectng ¼ b and m ¼ =3, the stan enegy contbuton fom coupe stesses n Eq. (95) s 4.5% of the stan enegy wthout coupe stesses. The cacuatons ae senstve to seected vaue of the dsocaton coe adus, and age the vaue of smae the effect of coupe stesses n the egon beyond. Fo exampe, the stan enegy contbuton fom coupe stesses n the egon between ¼ 3b and R ¼ b deceases to 7%, and n the egon between ¼ 4b and R ¼ b to 4.%. Ths pecentage nceases by the decease of the adus R, and the stan enegy contbuton fom coupe stesses n the egon between and R ¼ 5b s.7%,.5% and 6.6% n the case of ¼ b,3band 4b, espectvey. Such sma vaues of R may be appopate fo pobems wth extemey hgh dsocaton denstes, as ase n ocazed o non-ocazed egons of sevee pastc defomaton (shea bands, wea, we dawng, hgh-pessue toson, equa channe angua pessng), o n the pastcay defomed aye behnd the shock font (Meyes et a., 3). In ths context, t shoud be noted that an ncease of dsocaton densty fom to cm esuts n the decease of R by the facto of moe than 3. It shoud aso be ponted out that the stan enegy contbuton fom coupe stesses s key to be oweed by ncuson of the mcopoa effects. The coespondng cacuatons have not been pefomed n ths pape, but t s known that the effect of coupe stesses on stess concentaton s ess ponounced f the mcootatons ae assumed to be ndependent of the dspacement fed (Kaon and Aman, 967; Cown, 97; Lakes, 985; Engen, 999). 6.. Wok of dsocaton coe tactons If an edge dsocaton s nea the fee suface o the nteface, the contbuton fom the tactons on the dsocaton coe suface appeas n the fna expesson fo the dsocaton stan enegy (e.g., Feund, 994; Lubada, 997, 998). If the dsocaton s at the dstance fom the fee suface much geate than the dsocaton coe adus, t s common pactce to evauate the contbuton fom the tactons on the dsocaton coe suface (eft afte emovng the matea of the dsocaton coe) by subjectng the coe suface to tactons of the dsocaton n an nfnte medum, aong wth the coespondng dspacement. Ths s Z p E ¼ ½ ð ; hþu ð ; hþþ h ð ; hþu h ð ; hþš dh m 3 ð ; hþu 3 ð ; hþ dh: ð97þ The fst ntega n Eq. (97) depends on cut used to ceate the dsocaton (Lubada, 998). If the dsocaton s ceated by the dspacement dscontnuty aong the cut at an ange u (Fg. ), the evauaton of the above ntegas gves E ¼ b cos u þ a b : ð98þ 8pð mþ ð mþ p Z p

13 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) b ϕ σ σ θ m 3 Fg.. A matea of the dsocaton coe s emoved and ts effect on the emanng matea epesented by the ndcated stess and coupe stess tactons ove the suface ¼. The sp dscontnuty of amount b s mposed aong the cut at an ange u. Usng a ¼, ths can be ewtten as ( E ¼ b cos u ) : ð99þ p 4ð mþ ð mþ Fo exampe, n the case of the dspacement dscontnuty aong the hozonta cut ðu ¼ Þ, the enegy becomes " # E ¼ E 8ð mþ ; ðþ m whee E m b ¼ ðþ ð mþ 6p p s the enegy contbuton wthout the coupe stess effects. Fo exampe, f m ¼ =3and ¼ ffff b (the attce paamete), " E ¼ E 64 # b : ðþ 3 The extaodnay effect of the coupe stesses on the ato E =E vs. =b s shown by a sod cuve n Fg.. If s equa to b, 3b and 4b, the coespondng ato E =E s equa to )4.33, ).37 and ).33, espectvey. If the dspacement dscontnuty s mposed aong the vetca cut ðu ¼ p=þ, the enegy becomes " # E ¼ E 8ð mþ þ ; ð3þ 3 m whee E ¼ 3 m ð mþ b 6p ð4þ

14 38 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) Fg.. The ato of the coe suface eneges wth and wthout coupe stess effects vs. the coe adus ove the Buges vecto ato n the case when the edge dsocaton s ceated by the dspacement dscontnuty aong the hozonta cut (sod ne) and vetca cut (dashed ne). In the fome case E s defned by Eq. (), and n the atte case by Eq. (4). p s the enegy contbuton wthout the coupe stess effects. By settng m ¼ =3and ¼ ffff b, we obtan " E ¼ E þ # b : ð5þ The effect of coupe stesses on the ato E =E vs. =b s n ths case shown by a dashed cuve n Fg.. If s equa to b, 3b and 4b, the coespondng ato E =E s equa to 3.6,.7 and.65, espectvey. Athough the stan enegy assocated wth the tactons on the suface of the dsocaton coe s cut dependent, the tota stan enegy due to dsocaton n an nfnte o sem-nfnte medum s not cut dependent. Ths was pevousy dscussed n the context of cassca eastcty by Lubada (997), and an anaogous dscusson appes fo coupe stess eastcty. 7. Edge dsocaton n a hoow cynde The so-caed hoow dsocaton aong the axs of a ccua cynde wth nne adus and the oute adus R s shown n Fg. 3. Both sufaces of the cynde ae equed to be stess and coupe stess fee. The dspacement dscontnuty of amount b s mposed aong the hozonta cut fom to R. The souton s deved fom the nfnte body souton by supeposng an addtona souton that cances the stesses and coupe stesses ove the nne and oute suface assocated wth the souton fo an edge dsocaton n an nfnte medum. Thus, we eque that the supeposed souton satsfes the bounday condtons t ðr; hþ ¼ b sn h pð mþ R ; t b sn h ð ; hþ ¼ ; ð6þ pð mþ b cos h t h ðr; hþ ¼ pð mþ R ; t b hð ; hþ ¼ pð mþ cos h ; ð7þ

15 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) b R Fg. 3. A sp dscontnuty of amount b s mposed aong the hozonta cut fom the nne adus to the oute adus R. The nne and oute suface of the cynde ae fee fom stesses and coupe stesses. m 3 ðr; hþ ¼ ab cos h ; m p R 3 ð ; hþ ¼ ab p cos h : ð8þ Ths can be accompshed by usng the foowng stuctue of the Mndn stess functons: U ¼ A 3 þ B sn h; W ¼ A þ B þ CI þ DK cos h: Fom the condtons (55) and (56) t eady foows that ð9þ ðþ A ¼ 6ð mþ A ; B ¼ : ðþ The stess and coupe stess components of the supeposed souton ae accodngy t ¼ A B þ C 3 I D K sn h; t h ¼ A B þ C 3 I D K cos h; m 3 ¼ 6ð mþ A C h I þ I þ D h K þ K cos h: ð4þ The expessons fo the devatves of the modfed Besse functons wth espect to = ae used (Watson, 995, p. 79). Afte a engthy but staghtfowad devaton t foows that A ¼ b pð mþ R ab Ra a p R Rb b Ra a ; ðþ ð3þ ð5þ

16 38 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) B ¼ R b Ra a ab a b a b ; ð6þ pð mþ Ra a p R Ra a C ¼ 6ð mþ 3 c c A ab c p R c ; D ¼ 6ð mþ 3 d c A þ ab d p R c : The ntoduced paametes ae a ¼ R 3 6ð mþ 3 c ðc d 3 d c 3 Þ; ð7þ ð8þ ð9þ a ¼ 3 6ð mþ 3 c ðc d 4 d c 4 Þ; ðþ b ¼ c ðc d 3 d c 3 Þ; b ¼ c ðc d 4 d c 4 Þ; ðþ whee c ¼ K R R þ K h K h c ¼ K c 3 ¼ R K R þ K þ R ; c 4 ¼ K and smay d ¼ I R R þ I þ h I h d ¼ I d 3 ¼ R I R þ I R ; d 4 ¼ I K þ K ; ðþ R þ K R ; ð3þ ; ð4þ I The paamete c s defned by c ¼ 4 K R R h þ K I R þ I ; ð5þ þ I R ; ð6þ : ð7þ þ I h 4 K þ K I R þ I R : ð8þ The effect of coupe stesses on the eastc stan enegy as a functon of the ato =b and gven R can be evauated smay as n pevous secton. If R!, we obtan the souton fo an edge dsocaton wth a stess fee hoow coe n an nfnte medum, pevousy consdeed by Knes and Semea (97). In ths case A ¼ A ¼ B ¼ D ¼, and

17 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) B ¼ b ab K 4pð mþ p K þ K ; ð9þ D ¼ 4ab p K þ K : ð3þ If coupe stesses ae negected, Eqs. (5) (8) yed b A ¼ ; B 4pð mþ R þ ¼ R A ; A ¼ C ¼ D ¼ : ð3þ The coespondng stesses ae b ¼ pð mþ h ¼ b pð mþ b hh ¼ pð mþ R þ R þ R þ þ R sn h; 3 þ R cos h; 3 3 R sn h; 3 ð3þ ð33þ ð34þ n ageement wth the souton fo the Votea edge dsocaton fom cassca eastcty (Love, 944). The esuts shoud aso be compaed wth those pesented by Hth and Lothe (968, p. 77), who emove the tactons on the nne and oute suface of the cynde ony to wthn the fst ode tems n =R. 8. Scew dsocaton n coupe stess eastcty The dspacement fed fo a scew dsocaton wth mposed dspacement dscontnuty b aong the pane x > and x ¼ s as n cassca eastcty w ¼ b p tan x x ¼ b p h: ð35þ Ths foows by a coespondence theoem snce ony dspacement bounday condtons ae pescbed. The stesses and coupe stesses assocated wth (35) ae 3 ¼ b p x x þ x ; 3 ¼ b p x x þ ; ð36þ x ða þ bþb x m ¼ m ¼ x p ðx þ ; x Þ ð37þ ða þ bþb x x m ¼ m ¼ p ðx þ : ð38þ x Þ The components of the stess and coupe stess tensos aong the poa dectons ae h3 ¼ b p ; 3 ¼ ; ð39þ

18 384 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) ða þ bþb m ¼ m hh ¼ p ; m h ¼ m h ¼ : ð4þ The stess components decay wth a dstance fom the cente of dsocaton as, whe the coupe stesses decay as. These aso specfy the odes of the snguates at the dsocaton coe when!. The dspacement and otaton feds fo a scew dsocaton n mcopoa eastcty can be found n Nowack (986), athough hs statement (at the bottom of page 37) that n cassca theoy of dsocatons we have u ¼ u ¼ s appaenty a mspnt, snce the otaton components fom Eq. (35) do not vansh but ae equa to u ¼ ow ¼ b ox 4p x x þ ; u ¼ x ow ¼ b ox 4p x x þ : ð4þ x The anayss aowng fo a smooth tanston of the dspacement fed fom zeo vaue at the cente of the dsocaton coe to the vaue b aong the cut used to ceate the dsocaton has been done n non-oca eastcty by Engen (977b, 983), and n gadent eastcty by Gutkn and Afants (996). 8.. Stan enegy The stan enegy (pe unt ength n x 3 decton) stoed wthn a cynde bounded by the ad and R s Z R E ¼ W p d; ð4þ whee the specfc stan enegy (pe unt voume) s W ¼ h3 þ 4ða þ bþ m ¼ b ða þ bþb þ 8p 4p : ð43þ 4 Upon the substtuton nto Eq. (4) and ntegaton, thee foows E ¼ b 4p n R ða þ bþb þ : ð44þ 4p R The second tem on the ght-hand sde s the stan enegy contbuton fom the coupe stesses. The pesence of ths tem s assocated wth the wok done by the coupe stesses on the sufaces ¼ and ¼ R. Ths can be seen by wtng an atenatve expesson fo the stan enegy, Z R Z p Z p E ¼ h3 ð; Þbd þ M u R dh M u dh; ð45þ wth M ¼ m gven by Eq. (4), and u ¼ x hz ¼ ow oh ¼ b 4p ð46þ beng the component of the otaton vecto. Snce 3 ¼, thee s no wok of stess tacton on the dspacement w ove the sufaces ¼ and ¼ R. The second tem n Eq. (44) s the stan enegy contbuton due to ast two wok tems n Eq. (45). Fo exampe, f we set R ¼ b, ¼ b, and a þ b ¼ b, the enegy contbuton fom coupe stesses n Eq. (44) s.9% of the stan enegy wthout coupe stesses. In the cassca eastcty a cyndca suface aound the scew dsocaton at ts cente s stess fee. On the othe hand, the souton deved n ths secton s chaactezed by the pesence of the constant coupe stess m aong that suface. Howeve, snce m n Eq. (4) does not depend on h, the educed tacton t 3 vanshes on the cyndca suface ¼ const.

19 V.A. Lubada / Intenatona Jouna of Sods and Stuctues 4 (3) Concusons We have deved n ths pape the soutons fo edge and scew dsocatons n an nfnte medum by usng the coespondence theoem of coupe stess eastcty, whch eates the soutons of dspacement bounday vaue pobems n cassca and coupe stess eastcty. The contbuton fom coupe stesses to dsocaton stan enegy s evauated and dscussed fo both types of dsocatons. It s shown that wthn a adus of nfuence of each dsocaton n a metac cysta wth the dsocaton densty of cm, the stan enegy contbuton fom coupe stesses s about 5% n the case of an edge dsocaton, and about % n the case of a scew dsocaton (excudng the enegy of the dsocaton coe of adus ¼ b). Ths contbuton deceases wth an nceasng sze of the dsocaton coe. Fo exampe, the stan enegy contbuton fom coupe stesses fo an edge dsocaton n the egon between ¼ 3b and R ¼ b deceases to 7%, and n the egon between ¼ 4b and R ¼ b to 4.%. Ths pecentage nceases by the decease of the adus R, and the stan enegy contbuton fom coupe stesses n the egon between and R ¼ 5b s.7%,.5% and 6.6% n the case of ¼ b; 3b and 4b, espectvey. Such sma vaues of R may be appopate fo pobems wth extemey hgh dsocaton denstes wthn the ocazed o nonocazed egons of sevee pastc defomaton. It s then shown that coupe stesses make age effect on the tota wok of tactons actng on the dsocaton coe suface. The souton fo the edge dsocaton n a hoow cynde (Votea dsocaton) n the pesence of coupe stesses s aso deved. The extenson of the pesent wok s n pogess to ncopoate the effect of coupe stesses on the nteacton foces between dsocatons on paae and ntesectng sp panes, and the dsocaton nteactons wth staght and cuved fee sufaces o gd boundaes. Fo exampe, fo an edge dsocaton nea the stess-fee staght bounday the mage dsocaton cances both the noma and coupe stess components at the bounday, so that ony the shea stess component has to be emoved by supeposton of an auxay pobem to acheve the stess-fee bounday condton. A study of the oganzed dsocaton stuctues n coupe stess eastcty, such as that pesented by Lubada et a. (993) and Lubada and Kous (996a,b) n the case of cassca eastcty, may aso be of nteest. In addton, the ncopoaton of coupe stesses n the anayss of stan eaxaton n thn fms (Feund, 994; Lubada, 998) s wothwhe. Snce coupe stesses sgnfcanty affect dsocaton stan eneges, they may aso have a sgnfcant effect on the ctca fm thckness and the condtons fo the fomaton of nteface dsocaton aays. The soutons fo egenstan and nhomogenety pobems fo ccua ncusons n ant-pane stan coupe stess eastcty ae pesented n the accompanyng pape (Lubada, 3). Acknowedgement Reseach suppot fom the Montenegn Academy of Scences and Ats s kndy acknowedged. Refeences Aeo, E.L., Kuvshnsk, E.V., 96. Fundamenta equatons of the theoy of eastc meda wth otatonay nteactng patces. Fz. Tved. Tea, , Tansated n Sovet Physcs Sod State, 7 8 (96). Anthony, K.H., 97. De theoe de dsokatonen. Ach. Raton. Mech. Ana. 39, Cosseat, E., Cosseat, F., 99. Theoe des Cops Defomabes. Hemann, Pas. Cown, S.C., 97. An ncoect nequaty n mcopoa eastcty. Z. Angew Math. Phys., De Bost, R., Van de Gessen, E. (Eds.), 998. Matea Instabtes n Sods. John Wey, Chcheste. Dhawa, R.S., Sngh, A., 987. Mcopoa themoeastcty. In: Hetnask, R.B. (Ed.), In Thema Stesses II. Eseve Scence, Amstedam, pp Engen, A.C., 977a. Edge dsocaton n nonoca eastcty. Int. J. Engng. Sc. 5,

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