THREE-DIMENSIONAL MODELING OF STRESS-STRAIN RELATIONSHIP OF SAND SUBJECT TO LARGE CYCLIC LOADING

Transcription

1 Per No. TMSNA THREE-DIMENSIONAL MODELING OF STRESS-STRAIN RELATIONSHIP OF SAND SUBJECT TO LARGE CYCLIC LOADING Tutomu NAMIKAWA, Junichi KOSEKI,Lddu Indik Nlin DE SILVA 3 ABSTRACT In order to redict ground behvior ubjected to cclic loding uch tht induced b erthquke, it i necer to ccurtel evlute the deformtion chrcteritic of oil ubjected to cclic loding nd develo model which exree thee deformtion chrcteritic roritel. Thi er rovide imle three-dimenionl elto-ltic model for nd ubjected to lrge cclic loding. It i n extenion of the work on one-dimenionl modeling of the cclic loding tre-trin reltionhi of nd (De Silv 008), in which erie of lrge mlitude cclic torionl her tet were imlemented on ecimen of nd. The concet of the model with infinite number of neting urfce (INS model) (Mroz et l. 978) w emloed to introduce the Ming rule, which h been often ued in the onedimenionl modeling of the cclic tre-trin reltionhi, into the three-dimenionl modeling. In the concet of the INS model, the ctive loding nd tre reverl urfce re defined to exre the cclic deformtion behvior. In the rooed model, the effect of cclic hrdening behvior which were oberved in the torionl her tet reult were tken into ccount b uing the reltionhi between the mobilized friction ngle nd the ccumulted devitor ltic trin. Moreover, the tre-diltnc reltionhi w defined b the reltionhi between the rtio of the ltic volumetric trin to the ltic devitor trin nd the ize of the ctive loding urfce. The rooed model w lied to imulte tre-trin reltionhi of dene nd ubjected to drined lrge cclic loding. Keword: Snd, Cclic loding, Elto-ltic model, Torionl her INTRODUCTION Soil re ubjected to lrge cclic loding during lrge erthquke. Therefore, in order to deign tructure with dequte fet nd to nlze the erformnce of the exiting tructure in region in which lrge erthquke i redicted to occur, it i necer to reonbl redict the ground deformtion under the lrge cclic loding condition. Recent dvnce in comuter technolog hve rendered oibilit nd feibilit to the numericl nli of ground deformtion during n erthquke. In uch nli, the ccurte rediction of the ground deformtion require n elborte modeling of oil tre-trin reltionhi under cclic loding condition. De Silv (008) erformed erie of cclic torionl her tet on Toour nd. The tet reult reveled everl chrcteritic of the cclic behvior of nd ubjected to lrge mlitude cclic loding. On the bi of thee exerimentl reult, the one-dimenionl model uing the extended Ming rule, which h been often ued in one-dimenionl modeling of the cclic tre-trin reltionhi, w rooed to imulte the nd behvior under the cclic loding condition. Aocite Profeor, Kobe Cit College of Technolog, Jn, e-mil: nmikw@kobe-koen.c.j Profeor, Intitute of Indutril Science, Univerit of Toko, Jn 3 Lecturer, Dertment of Civil Engineering, Univerit of Mortuw, Sri Lnk

2 Thi er rovide three-dimenionl elto-ltic model for the tre-trin reltionhi of nd ubjected to lrge cclic loding. It i n extenion of the work on one-dimenionl modeling rooed b De Silv (008). In thi tud, the concet of the model with infinite number of neting urfce (INS model) (Mroz et l. 978) w emloed to introduce the Ming rule into the three-dimenionl modeling. In the concet of the INS model, the ctive loding nd tre reverl urfce re defined to exre the cclic deformtion behvior. The effect of cclic hrdening behvior which w oberved in the torionl her tet reult w lo tken into ccount b uing the reltionhi between the mobilized friction ngle nd the ccumulted devitor ltic trin. Moreover, the tre-diltnc reltionhi w defined b the reltionhi between the rtio of the ltic volume trin to the ltic devitor trin nd the ize of the ctive loding urfce. The rooed model w lied to imulte tre-trin reltionhi of nd ubjected to lrge cclic loding. BASIC THEORY OF MODEL Bic Aumtion The model with n infinite number of neting urfce (INS model) (Mroz et l. 978), which i bed on the elto-ltic theor, w emloed to extend one-dimenionl model in which the Ming rule i introduced. Yield urfce, reverl urfce nd ctive loding urfce re defined to exre the cclic loding behvior in the INS model (Figure ). The urfce decribing the intntneou ltic reone i trnlted with the tre oint. The tre oint remin in contct with the ield urfce during the rimr loding roce. When the tre increment directed into the interior of the domin encloed b the ield urfce occur, the ctive loding urfce imilr to the ield urfce i defined nd the tre oint remin in contct with the ctive loding urfce. Then the revere oint become the imilrit center of thee two urfce. During the ubequent revere loding roce, the intntneou ltic reone i decribed b the ltic theor lied to the ctive loding urfce. Moreover, when the tre increment directed into the interior of the domin encloed b the ctive loding urfce occur, the new ctive loding urfce i defined nd the reviou ctive loding urfce become the reverl urfce. Then the new ctive loding urfce contct with the reverl urfce t the revere oint, which i the imilrit center of thee two urfce. Yield urfce σ Revere oint Reverl urfce Active loding urfce Current tre oint Revere oint σ 3 σ : Stre th Figure. Model with n infinite number of urfce (INS model)

3 When the tre on the ctive loding urfce reche the reverl urfce, the reverl urfce become the ctive loding urfce gin nd the reviou ctive loding urfce i eliminted. In rticulr, when the tre reche the ield urfce gin, the ltic reone i decribed b the ltic theor lied to the ield urfce. B uing the INS model, we cn exre the cclic loding reone ccording to the Ming rule in the three-dimenionl tre ce. During the rimr loding roce, the keleton curve could be decribed b the hrdening rule in the ltic theor lied to the ield urfce. After the tre revere, the hterei curve tht follow the Ming rule could be decribed b the hrdening rule in the ltic theor lied to the ctive loding urfce. Generl Formultion It i firt otulted tht the totl trin increment & i um of the eltic comonent comonent &.Thu e & nd ltic e & & & () The eltic trin increment i linerl relted to the effective tre increment σ& & e Ekl & σ kl () in which E kl i the mtrix of eltic contnt. B uing the theor of lticit, the ltic trin increment i defined with ltic otentil Q Q & Λ (3) in which Λ i the ltic multil determined b conitenc condition. B emloing hrdening rule nd the conitenc condition in Eq. (3), we obtin the following exreion for the ltic trin increment: F & σ kl kl Q H & (4) in which F i the loding function nd H i the ltic modulu. F correond to the ield function f during the rimr loding roce nd the ctive loding function f during the revere loding roce. Yield Function The ield function f i elected of the following form : f σ 0 σ k (5) in which σ i the effective devitor tre, i the effective men tre nd k i the internl vrible for the ield function. Emloing the non-ocited flow rule, the derivtive of the ltic otentil function i given b

4 Q (6) The ltic modulu H in Eq. (4) could be given b the conitenc condition nd the hrdening rule. Active Loding Function The ctive loding function i elected of the following form : f ( α )( σ α ) k 0 σ (7) in which α i the center of the ctive loding urfce nd k i the internl vrible for the ctive loding function. Emloing the non-ocited flow rule, the derivtive of the ltic otentil function i given b Q (8) The ltic modulu H in Eq. (4) could be given b the conitenc condition nd the hrdening rule. MODELING OF SAND RESPONSE SUBJECT TO LARGE CYCLIC LOADING Generl Herbolic Eqution Modeling of Primr Stre-trin Reltion Ttuok nd Shibu (99) rooed the following the generl herbolic eqution (GHE) to imulte the monotonic loding tre-trin reltion of mn te of geomteril. τ τ mx x C x C, x γ γ ref (9) in which τ i the her tre, τ mx i the her trength, γ i the her trin, γ ref i the reference her trin, C nd C re the vrible coefficient. C nd C vr with the trin level, given C C { C () 0 C ( ) } { C () 0 C ( ) } { C () 0 C ( ) } { C () 0 C ( ) } π co α x π co β x (0) in which C (0), C ( ), C (0), C ( ), α nd β re the mteril rmeter. For the cclic loding, the GHE could be ued to decribe the keleton curve in the Ming rule. The GHE w doted in the hrdening rule in the ltic theor lied to the ield urfce. Emloing the devitor ltic trin invrint, Eq. (9) i rewritten for the hrdening rule

5 in which x k, x, x C C () i the devitor ltic trin occurring during the rimr loding roce. The ubtitution of thi hrdening rule nd the conitenc condition in Eq. (4) ield the following exreion for the ltic modulu for the rimr loding roce. H k k, () Figure how the comrion between the rimr tre-trin reltionhi of drined torionl her tet on turted Toour nd t reltive denit Dr of bout 78% nd tht imulted b the rooed model. The rmeter ued in the imultion re hown in Tble. The imulted tre-trin reltion gree brodl with the exerimentl reult, indicting tht the hrdening rule emloing the GHE could decribe dequtel the rimr tre-trin reltion of nd. Tble. Mteril rmeter Eltic Modulu Poion rtio C (0) C ( ) C (0) C ( ) 95 MN/m α β S S ult A B In order to follow the Ming rule, Eq. (9) i rewritten for the hrdening rule in the ltic theor lied to the ctive loding urfce during the revere loding roce k x, x, x (3) C C in which i the devitor ltic trin occurring fter the tre revere. 0.7 Sher tre τ / Men tre Toour nd Dr 78% 00 kp Exeriment (SAT) Simultion 0..0 Pltic her trin γ (%) Figure. Meured nd imulted tre-trin reltionhi on rimr loding roce

6 The ubtitution of thi hrdening rule nd the conitenc condition in Eq. (4) ield the following exreion for the ltic modulu for the revere loding roce. H k α kl αkl k k, (4) Hrdening Reone with Lrge Cclic Loding De Silv (008) erformed erie of drined cclic torionl her tet on turted Toour nd. A ticl tre-trin reltion obtined from the cclic loding tet with contnt lrge tre mlitude i hown in Figure 3. Thi reult how tht the trin mlitude decree with the cclic loding. Thi exerimentl evidence indicte tht the tiffne of nd incree with the cclic loding. Therefore uch cclic hrdening behvior due to the liction of cclic loding hould be conidered in modeling the cclic behvior of nd. In order to decribe the cclic hrdening behvior of nd, De Silv (008) introduced the hrdening fctor denoting the effect of the mobilized friction ngle in Eq. (9). B emloing the hrdening fctor, the hrdening rule exreed Eq. () nd Eq. (3) re rewritten k, k C SC C SC (5) in which S i vrible coefficient for the effect of the cclic hrdening. S i defined b herbolic function of the totl ltic trin tht i ccumulted u to the current turning oint & the following eqution th ccle 4th ccle t ccle Sher tre τ (kn/m ) dτ > 0 3rd ccle Sher trin γ (%) nd ccle dτ < 0 Toour nd Dr 78% σ z σ r σ θ 00 kp Figure 3. Cclic hrdening behvior due to contnt tre mlitude cclic loding (tet SAT) (De Silv 008)

7 & S (6) & S S in which S ult i the mximum vlue of S fter ling infinite number of ccle nd S i the mteril rmeter. Diltnc Behvior with Lrge Cclic Loding In the drined cclic torionl her tet, the diltnc behvior w oberved with the lrge cclic loding. A ticl tre-diltnc reltionhi i hown in Figure 4. It cn be een tht tht the diltnc rtio vrie linerl with the tre rtio τ/. Since γ under the torionl her tet condition, thi tre-diltnc reltionhi could be rewritten Figure 5. In thi figure, τ rev i the her tre t the reverl loding oint. Thi reult indicte tht the diltnc rtio vrie lmot linerl with the modified tre rtio. The bove liner reltionhi between the diltnc rtio nd the tre rtio i emloed to decribe the tre-diltnc behvior of nd. The tre-diltnc reltionhi on the ield urfce tke the form ult vol k A & B & (7) in which vol i the ltic volumetric trin nd A, B re the mteril rmeter. The tre-diltnc reltionhi on the ctive loding urfce tke the form vol k kr A & B (8) & in which k r i the internl vrible for the reverl urfce. k nd k r in Eq. (8) correond to Sher tre τ / Men tre t ccle nd ccle 3rd ccle 0. 4th ccle 5th ccle dτ < 0 dτ > d vol / d γ Figure 4. Stre-diltnc reltionhi under contnt tre mlitude cclic loding (tet SAT)

8 ( τ - τ rev - τ rev )/ ( τ τ τ ) 0. 6( & & ) rev rev vol & & vol Figure 5. Arrnged tre-diltnc reltionhi (tet SAT) τ τ rev nd τ rev in Figure 5, reectivel. B uing Eq. (7) nd Eq. (8), the ltic volumetric trin increment cn be clculted form the ltic devitor trin increment. NUMERICAL EXAMPLE The drined cclic torionl her tet on Toour nd erformed b De Silv (008) were imulted b the rooed model. Two tet, SAT nd SAT were elected in the imultion. In thee tet, the ecimen were ubjected to iotroic comreion u to 400 kp nd iotroic unloding to 00 kp followed b drined lrge cclic torionl her loding. The initil reltive denit of ecimen w round 78%. The rmeter ued in the imultion re hown in Tble. The vlue of C (0), C ( ), C (0), C ( ), α nd β were rovided from the keleton curve imultion hown in Figure. The vlue of S nd S ult were determined b tril nd error. The vlue of A nd B were determined bed on the diltnc reltionhi hown in Figure 5. Comrion between the meured nd imulted her tre-trin reltionhi for tet SAT re mde in Figure 6. The imultion reult without the cclic hrdening effect i comred with the exerimentl reult in Figure 6(). Since the model without the cclic hrdening effect imulte the tre-trin reltionhi ocited with the Ming rule, tht model cn not decribe the trin mlitude reduction oberved in the exeriment reult. Converel, Figure 6(b) how tht the reult imulted b the rooed model with the cclic hrdening effect gree reonbl with the exerimentl reult. Thee reult indicte tht the conidertion of the cclic hrdening effect i needed to model roerl the nd behvior ubjected to lrge cclic loding. Comrion between the meured nd imulted her tre-trin reltionhi for tet SAT re mde in Figure 7. In tet SAT, the tre mlitude w controlled to incree with the number of loding ccle. Like the imultion for tet SAT, the model without the cclic hrdening effect overetimted the trin mlitude. Converel, Figure 7(b) how tht the model with the cclic hrdening effect could decribe the reduction of the trin mlitude under the cclic loding. However, there till i ome difference between the exerimentl nd numericl reult. More imrovement of the model will be necer to obtin more tifctor imultion. Comrion between the meured nd imulted ltic volumetric trin for tet SAT nd SAT re mde in Figure 8. Deite emloing the imlified liner diltnc reltionhi, the rooed

9 model decribe reonbl the feture of the vrition of ltic volumetric trin occurring under the lrge cclic loding. Sher tre τ / Men tre Exeriment Simultion t to 5th ccle Sher trin γ (%) () Simultion without cclic hrdening effect Sher tre τ / Men tre Exeriment Simultion 3rd ccle t ccle nd ccle 4th nd 5th ccle Sher trin γ (%) (b) Simultion with cclic hrdening effect Figure 6. Comrion between meured nd imulted tre-trin reltion ( tet SAT) Sher tre τ / Men tre Exeriment Simultion Initil tte Sher trin γ (%) () Simultion without cclic hrdening effect Sher tre τ / Men tre Sher trin γ (%) (b) Simultion with cclic hrdening effect Figure 7. Comrion between meured nd imulted tre-trin reltion (tet SAT) Exeriment Simultion Initil tte

10 Volumetric trin vol (%) th ccle 4th ccle 3rd ccle nd ccle t ccle Exeriment Simultion Volumetric trin vol (%) Exeriment - Simultion Initil tte Sher treτ / Men tre Sher treτ / Men tre () tet SAT () tet SAT Figure 8. Comrion between meured nd imulted volumetric trin CONCLUSIONS A three-dimenionl elto-ltic model for the tre-trin reltionhi of nd ubjected to cclic loding i rooed. The cclic hrdening effect oberved in the tre-trin reltionhi of nd ubjected to lrge cclic loding i introduced into the rooed model. Moreover, in thi model, imlified liner tre-diltnc reltionhi i defined bed on the exerimentl reult. Simultion of drined cclic torionl her tet on dene Toour nd demontrted tht the rooed model cn roerl exre the behvior of nd ubjected to the lrge cclic loding. The decree in the trin mlitude with the cclic loding under contnt tre mlitude could be imulted b the rooed model with the cclic hrdening effect. Moreover, the rooed model could cture the bic feture of the volumetric chnge roduced b the cclic loding. REFERENCES De Silv, L.I.N. (008). Deformtion chrcteritic of nd ubjected to cclic drined nd undrined torionl loding nd their modelling. Ph.D. thei, The univerit of Toko. Mroz, Z., Norri, V.A. nd Zienkiewicz, O.C. (978). An niotroic hrdening model for oil nd it liction to cclic loding. Int. Jour. Numer. Anl. Meth. Geomech., Vol.,. 03. Ttuok, F. nd Shibu, S. (99). Deformtion chrcteritic of oil nd rock from field nd lbortor tet, Kenote Lecture (Seion ). Proc., 9th Ain Regionl Conf. on SMFE.,