Analysis of the frictional slip between a layer and a half-space

Analysis o he ic ional slip be ween a laye and a hal -space R. Abascal, J. Dommguez Escuela Supe io de Ingenie os Indus ials, Uni e sidad de Se illa, A . Reina Me cedes, s/n, 41012-Se illa, Spain ABSTRACT The nume ical analysis o a boundless elas ic laye on an elas ic hal - space wi h di e en ma e ial p ope ies unde he e ec s o an uni o m su ace p essu e and a cyclic angen ial su ace o ce is p esen ed. F ic ional con ac condi ions a e assumed. The s udy is ocussed on he e alua ion o he maximum ampli ude o he angen ial load which p oduces localized slip be ween he wo egions du ing he i s load cycle bu no he subsequen ones. The mo e simple limi o which no slip exis e en o he i s cycle is also es ablished INTRODUCTION The s udy o he slip and sepa a ion ha may ake place be ween wo su aces in con ac when hey a e unde angen ial cyclic loading condi ions is e y impo an o p e en he ailu e known as " e ing". This ailu e mechanism is ini ia ed by localized slip be ween wo su aces om which some pa icles a e de ached. These pa icles ac as an ab asi e in subsequen load cycles and de e io a e he ma e ial apidly. The damage mechanism may be combined wi h co osion in he case o me als[l]. The slip be ween he wo su aces can be a oided by in oducing a no mal p essu e be ween hem. I his p essu e is enough he p e en slip du ing he i s load cycle hen he sys em beha es linea ly and slip will no ake place du ing subsequen cycles o he same ampli ude. This welded con ac p oblem can be sol ed easily. The limi o which he i s slip akes place can be ob ained om he welded con ac model. I co esponds o he angen ial load o which he shea ac ion a a poin becomes equal o he no mal p essu e imes he ic ion coe icien . The e a e s ill highe alues o he angen ial load o which he e is pa ial slip du ing he i s load cycle bu no in he subsequen ones. The e o e no e ing will ake place o his T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X 210 Bounda y Elemen s c 0 ' 0 0 0 0 0 0 , 0 11111. c, ,, ** *: 1,1 1 1 1 , Q ^F kHoi Su eca Figu e 1. Elas ic laye on elas ic hal -space unde angen ial load Q and no mal load PO load. The slip o he i s cycle lea es esidual s esses which p e en he wo su aces o slip du ing he nex cycles. The e alua ion o he highe alue o he load o which he sys em has his kind o beha io is impo an since i is he ac ual limi o he loads which do no p oduce e ing. The s udy equi es a mo e complica ed model and a nume ical solu ion app oach as shown below. The p oblem analyzed in his wo k e e s o a semi-in ini e domain consis ing o a boundless ho izon al elas ic laye wi h dep h "a" on an elas ic hal -space. The su ace o he laye is subjec o a uni o m cons an in ime p essu e p<, and o a concen a ed angen ial load o ampli ude Q which has a cyclic ime dependence. The a ia ion o he load wi h ime is assumed o be su icien ly slow so ha ine ial e ec s a e negligible. The e o e, a quasi- s a ic analysis is ca ied ou . The ini e egion wi h he bounda y condi ions shown in Figu e 1 is used o he s udy. The p oblem a hand emains linea o small alues o he load Q. The exp essions o he ac ions along he in e ace in such case can be ound o ins ance in Re .[2]. The s udy o a simila p oblem wi h wo ma e ials o iden ical p ope ies and a single loading p ocess which p oduces i s , slip a a poin hen, one o wo slip zones and inally, sepa a ion, was done analy ically be Schmuese , Comminou and Dundu s [3]. Comminou and Ba be [4] ex ended his analysis o he case o cyclic loading and assuming a ic ion coe icien JJL = 0.5. These au ho s iden i ied a alue X, o he T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X Bounda y Elemen s 211 pa ame e X = Q/(po a) such ha o loads X < X, he p oblem emains linea , and a load ange X, < X < X] o which he e is slip bu only in he i s load cycle. Loads co esponding o X > X? p oduce mul iple slip and/o sepa a ion which may become gene al along he in e ace du ing he subsequen load cycles. The pu pose o his pape is o ca y ou a s udy o hese anges o he load o he mo e gene al case when he laye and he hal -space ha e di e en ma e ial p ope ies. The s udy is done nume ically by means o he Bounda y Elemen Me hod (BEM) which is e y well sui ed o his kind o p oblems. A pa ame ic s udy is done using as a pa ame e o de ine he ela i e s i ness o he wo ma e ials, he squa e oo o he a io be ween he shea modula RCs = (Gj/GJ^, he Poisson's a ion being he same o bo h ma e ials. NUMERICAL APPROACH The analysis o he p oblem in Figu e 1 s a s by disc e izing he bounda ies in o cons an elemen s and compu ing he usual BEM sys em o equa ions o he bounda y nodes o each one o he wo sub egions whe e u* and;/ a e he displacemen and ac ion ec o s espec i ely, along he bounda y o he "z" sub egion. The coupling be ween he wo egions is done by means o he compa ibili y and equilib ium condi ions along he in e ace co esponding o one o he ollowing si ua ions: bonding, sepa a ion o sliding wi h ic ion. A Coulomb ype ic ion is assumed. Sliding: P = p u? + w»* = 0 Bonding: P?* I < I M P? I .< . ui + «/ = 0 (3) T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X 212 Bounda y Elemen s Sepa a ion: P?* - 0 P?' = 0 5 > 0 W whe e he supe index indica es he egion o which he a iable e e s and he subindexes s and n s and o angen ial and no mal di ec ion o he in e ace, espec i ely; /x is he ic ion coe icien and 5 he dis ance along he no mal di ec ion be ween wo poin s which could be in con ac . The sign o he shea ac ion du ing he sliding is de e mined by he condi ion o nega i e wo k (dissipa ion o ene gy) du ing he p ocess. In he h ee cases abo e ou equa ions can be w i en o each couple o poin s on he con ac su aces. Those equa ions plus he wo BEM equa ions o each domain de e mine he alues o he wo displacemen componen s and wo ac ion componen s o each one o he wo poin s o a couple which a e o may be, in con ac . The solu ion o he sys em o equa ions equi es o an i e a i e p ocess wi hin each load s ep. This p ocess o a load s ep N s a s by w i ing he B.E. equa ions (1) o each bounda y node as whe e x^ and /„ a e he bounda y unknown ec o and he known ec o , espec i ely. The la e is ob ained om he p oduc o he known bounda y alues and he co esponding columns o he H and G ma ix. The sys em ma ix A con ains mo e columns han ows as bo h ac ions and displacemen s a e unknown along he in e ace. The con ac equa ions o he nodes on he in e ace a e w i en as CN *N = ° (6) The abo e sys em o equa ions (5) plus (6) allows o he solu ion o he p oblem o he load s ep N p o ided ha he con ac condi ions o all he poin s on he in e ace ( ep esen ed by €„) emain cons an du ing he load s ep. Since hese condi ions will, in gene al, change du ing he load s ep, each load s ep mus include se e al i e a ions wi h di e en con ac T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X Bounda y Elemen s 213 condi ions CN*. Fo each load s ep one wo ks wi h inc emen s o he a iables wi h espec o he p e ious s ep » -/«- 0 = A . (7) The inc emen s o he a iables a e subdi ided in o M possible sub- inc emen s and w i en as A *„ = £ A ^* (8) whe e each sub-inc emen co esponds o di e en con ac condi ions Gj. The i e a ion p ocess o load s ep N begins by sol ing he sys em (7) assuming he same con ac condi ions o he end o he p e ious s ep and applying he comple e load inc emen A/^. A scale ac o /?„' is compu ed om he solu ion o his sys em such ha i ep esen s he pa o he load which can be applied wi hou change in he con ac condi ions. This load can be w i en as and A 4 (9) whe e A * is he solu ion o (7) wi h A/*. Nex he con ac condi ions a e modi ied and he load (1 - #/) A/%, applied. The p ocess con inues up o he poin when /?/ > 1. Then he compu a ion o load s ep N is inished. A simila i e a i e p ocess was applied by he au ho s o dynamic p oblems in Re s. [6] and [7]. LAYER ON A HALF-SPACE The ollowing alues o he pa ame e s ha e been used o he model in Figu e 1: a = 1 m; L = 10 m ; H = 10 m; Poisson's a io u, = ^ = T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X 214 Bounda y Elemen s 1/3 ; po=l Nw/n ; ic ion coe icien p = 0.5 and a shea modulus o he hal -space 62 = 1 Nw/n . The pa ame ic s udy is done by changing he shea modulus o he laye ma e ial and he load ampli ude Q. The load is assumed o be applied in 100 s eps ollowing he a ia ion shown in Figu e 2. • I 32 41 64 LaW Sup Figu e 2. Tangen ial load a ia ion Cons an Bounda y Elemen s a e used o he s udy. The disc e iza ion consis s o 50 equal elemen s on each side o he con ac in e ace and on he laye ee su ace, 3 equal elemen s on each la e al bounda y o he laye , 5 on each la e al bounda y o he hal -space and 10 equal elemen s o he bo om o he model. In o de o es his model which ex ends o a ini e egion a ound he load o ep esen an in ini e domain and includes a cons an Bounda y Elemen app oxima ion, a p oblem wi h known analy ical solu ion is sol ed. Such solu ion was ob ained by Comninou and Ba be [5] o he case o wo iden ical ma e ial p ope ies o he laye and he hal -space. Figu e 3 shows a compa ison be ween he analy ical and he p esen nume ical esul s o he maximum load o he second load cycle (poin C o Figu e 2) wi h a load ampli ude X = 2.353. Slip be ween he laye and he hal -space akes place o his alue o X du ing he i s cycle bu no in he subsequen ones. The no mal and shea ac ions along he in e ace e sus he ho izon al dis ance x o he poin load a e shown in Figu e 3. The no mal ac ions ha e a smoo h a ia ion and a e almos he same as in he linea case. On he o he hand, he shea ac ions show a clea di e ence wi h hose o he linea case because o he esidual ac ions exis ing in he zones whe e slip T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X Bounda y Elemen s 215 ook place in he p e ious load cycle. These esidual ac ions a oid he sliding in he second and subsequen load cycles. The esul s p esen ed in Figu e 3 show a e y good ag eemen be ween he analy ical and he p esen nume ical solu ion. - No mal Theo . * * Lo.d • No mal B.E.M. « {?Sl£ —Shea Theo . ^ « Rd - l • Shea B.EJ*. * . ,.oj -4-3-2' -1 Figu e 3. Resul s o wo iden ical ma e ials 20 18 16 14 I*" O 10 Zone 1 0 Oj 1 lj 2 2J 3 3J 4 4j Ra« Figu e 4. F e ing zones T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X 216 0.4 0 3 02 4 « •0.4 Bounda y Elemen s -5 -4 3 -2 -1012345 «/* 0.4 035 O " & 025 g 02 I 0.15 3 0.1 * 0.05 0 4X05 LoadC #-03 X- X - RC -0.25 -RCs-0.50 RCm-l -RCs-2 - RC«-4 -5-4-3-2-1012345 0.4 035 0.15 0.1 0.05 0 •0.05 Lo dA P-O5 X • X - RCs-025 RCs-OJO -RCs-2 -RCm-4 5 -4 3 -2 -1012345 %/* OJ5 03 025 02 0.15 0.1 0.05 0 -0.05 5 -4 -3 -2 -1012345 %/a Figu e 5. T ac ioo5 dis ibu ion along he in e ace Once he B.E. model and' he i e a i e app oach ha e been alida ed he pa ame ic s udy o a ange o he s i ness a io RCs = (GJG^ going om 0 o 4.5 is ca ied ou . Fo each s i ness a io wo limi ing alues o he load pa ame e s X = Q/PO a a e de e mined. The i s one, X, is he limi o he loads which do no p oduce non-linea e ec s on he con ac in e ace. This limi de ine he linea zone (zone 1) in Figu e 4. The second limi X? de ines he load zone (zone 2) o which he e is slip along he in e ace T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X Bounda y Elemen s 217 du ing he i s load cycle bu he esidual ac ions a oid slip in he subsequen load cycles. Values o X > X% (zone 3) co espond o loads which p oduce slip in all he load cycles and he e o e he e ing p ocess can no be con olled. The alues o X, and X% ob ained analy ically by Comninou and Ba be [5] o he case o RCs = 1 a e shown as do s in he igu e. Figu e 5 shows some o he ac ion dis ibu ions which appea along he in e lace du ing he loading p ocess. Only in one case a e he no mal ac ions ep esen ed (Figu e 5a) since hose ac ions show e y li le di e ence wi h he linea ones o zone 2. The shea ac ions ep esen ed in Figu e 5b co espond o he second cycle (Poin C) and X = X,. The e o e, no slip has aken place and he cu es in his igu e ep esen he shea ac ion dis ibu ion o a linea p oblem. Figu es 5c and 5d show he shea ac ion dis ibu ion o X = X2 du ing he i s and second load cycles (Poin s A and C, espec i ely). In he o me case, he e ha e been slip only in he nega i e pa o he x-axis and esidual ac ions ha e appea ed. I can be seen in Figu e 5d ha he esidual ac ions co esponding o poin A emain and some mo e appea along he posi i e pa o he x-axis which ha e been p oduced du ing he nega i e pa o he i s load cycle (poin B). 2.0 1.5 1.0 7« 3- O'O & -OJ -1.0 -1J -zo - RCs-0.25 RCs-0.50 -RCs-1 -RCs-2 -•RCs -4 LoadC -0.5 A - -5-4-3-2-1012345 x/a Figu e 6. Rela i e angen ial displacemen s along he in e ace T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X