Dislacement-Softening Constitutive Model fo Geosynthetic Intefaces Mohamed G. Aab & Edwad Kavazanjian, J. School of Sustainable Engineeing and the Built Envionment Aizona State Univesity, Teme, AZ, USA Patick J. Fox Deatment of Stuctual Engineeing, Univesity of Califonia-San Diego, La Jolla, CA, USA Jason D. Ross Staff Enginee, BBC&M Engineeing, Inc., Dublin, OH, USA ABSTRACT A dislacement-softening constitutive model has been develoed to model the cyclic shea load-defomation behaviou of textued geomembane/geosynthetic clay line (GMX/GCL) intefaces. The dislacement-softening fomulation is based on the assumtion that the stength eduction at the inteface can be elated to the cumulative elative shea dislacement. This model was develoed based on the esults of cyclic shea tests on GMX/GCL intefaces. The constitutive model has been imlemented in a finite-diffeence softwae ackage fo solution of geotechnical oblems. The efomance of the constitutive model comaes well with the exeimental data, showing a ogessive degadation in shea stength of the inteface with inceasing cumulative dislacement in unifom cyclic tests. Moe eseach is needed to test the efomance of the model fo the case of non-unifom cyclic loading. RÉSUMÉ Une loi de comotement avec adoucissement a été déveloée ou modélise le comotement cyclique de l inteface ente une géomembane en PEHD avec aséités et un géosynthétique bentonitique. L hyothèse de base de cette loi de comotement avec adoucissement est que la éduction de la ésistance au cisaillement à l inteface eut ête eliée au cumul du cisaillement elatif. Cette loi de comotement avec adoucissement a été déveloée au vu des ésultats d essais cycliques conduits su des intefaces ente géomembanes en PEHD avec aséités et géosynthétiques bentonitiques. La loi de comotement a été intégée dans un model numéique aux difféences finies ou la ésolution de oblèmes géomécaniques. Les ésultats obtenus avec le model numéique se comaent bien aux ésultats exéimentaux, indiquant une éduction ogessive de la ésistance au cisaillement de l inteface avec augmentation du délacement total los d essais cycliques unifomes. Des tavaux comlémentaies sont nécessaies ou évalue la efomance de la loi de comotement ou chagements cycliques non unifomes. 1 INTRODUCTION Comehensive modeling of the cyclic shea behaviou of landfill line and cove systems used in waste containment facilities equies a suitable constitutive model fo geosynthetic intefaces. Modeling of line and cove system cyclic shea behaviou can be imotant in the assessment of the long-tem efomance of waste containment facilities subjected to eathquakes o othe dynamic loading. Although a significant body of eseach has been conducted on the static shea modeling of geosynthetic intefaces (e.g., Byne 1994, Estehuizen et al. 21, Tilett and Fox 21), only vey limited wok has been ublished on modeling the shea behaviou of geosynthetic intefaces unde cyclic loads (Desai and Fishman 1991, LoGasso et al. 22, Kim et al. 25). This ae esents a constitutive model fo modeling the in-lane cyclic shea behaviou of textued geomembane/geosynthetic clay line (GMX/GCL) intefaces. This model is based uon exeimental data fom unifom cyclic tests. Comaison between model edictions and exeimental esults is good, as the model catues the ogessive degadation of the hysteesis loos deemed the most imotant facet of the obseved behaviou. The model has been imlemented in a finitediffeence ogam fo edicting the esonse of landfill line and cove systems to eathquake loading. 2 EXPERMENTAL DATA FROM DYNAMIC SHEAR TESTING OF GMX/GCL SPEICMENS The constitutive model esented heein was develoed using the esults of a seies of cyclic diect shea tests on GMX/GCL secimens descibed by Ross (29) and Ross et al. (21). These tests wee conducted using the cyclic diect shea device descibed by Fox et al. (26). GMX/GCL secimens ae sheaed between the floo of the test device and a hoizontal ullout late. Both test device floo and the ullout late ae configued to event secimen sliage and elated ogessive failue effects (Fox and Kim 28). Each GMX/GCL secimen was fee to fail at the inteface o intenally (within the GCL). Ross et al. (21) descibe the exeimental ogam emloyed to develo the constitutive model descibed heein. Testing was conducted in the geotechnical laboatoy at the Univesity of Califonia at San Diego using two geosynthetic oducts: a double non-woven (NW) needle-unched (NP) GCL with no themal bonding
Shea Stess, τ (kpa) and a 6 mil, HDPE GM with single-sided stuctued (Mico Sike ) textuing. All secimens wee 132 x 35 mm in lan dimension. Each GCL secimen was hydated unde the sheaing nomal stess using the two-stage acceleated hydation ocedue develoed by Fox and Stak (24). Ross (29) conducted twenty-nine dislacement-contolled cyclic shea tests on this GMX/GCL combination to detemine the effects of the dislacement amlitude, Δ a, on mateial esonse. The loading consisted of 25 cycles of sinusoidal dislacement with a fequency, f, equal to 1 Hz. Tests wee conducted at five nomal stess levels (13, 348, 692, 1382 and 271 kpa) and seven dislacement amlitudes (±2, 1, 15, 2, 3, 6, and 12 mm). Afte each test was comleted, the ullout late and GCL wee caefully emoved fom the shea machine. The obseved failue mode was ecoded along with any indications of localized stess, teaing, o GMX sliage. 2.1 Results Insection of the cyclic shea stess-dislacement cuves indicated that that the dislacement at which the eak shea esistance was mobilized was tyically between ±1 and ±15 mm. Theefoe, cyclic testing done with dislacement amlitudes of (±2, 1 and 15 mm) wee consideed e-eak tests and cyclic testing done with dislacement amlitudes of (±2, 3, 6 and 12 mm) wee consideed ost-eak tests (Ross 29). The diffeences between e- and ost-eak esonse wee not as clea at a nomal stess, σ n, equal to 13 kpa as they wee at the lage nomal stesses (e.g., 348, 692, and 1382 kpa). Ross (29) concluded that the 13 kpa nomal stess was too small to ceate intimate contact at the GMX/GCL inteface. In the cyclic inteface shea stength tests conducted by Ross (29), inteface failues wee consistently obseved at the thee smallest nomal stesses (13, 348, and 692 kpa). The only cyclic test in which a atial intenal failue was obseved was at σ n = 1382 kpa fo a dislacement amlitude of ±1 mm. All tests at ±15 mm and geate caused inteface failues. Figue 1 and Figue 2 show the e-eak (±1 mm) and ost-eak (±2 mm) esonse, esectively, of the GMX/GCL inteface at σ n = 1382 kpa. In the ost-eak case (Figue 2) a continuous degadation in shea stength is obseved ove the 25 cycles of the test. Howeve, most of the shea stength degadation occus within the fist five cycles of loading in the e-eak tests. The ost-eak test esults fom Ross (29) fo cyclic testing of the GMX/GCL combination at nomal stesses of 1382 kpa and 271 kpa ae shown in Figue 3 and Figue 4, esectively, fo fou dislacement amlitudes (±2, 3, 6, and 12 mm). Fo these ost-eak cyclic tests, a eak in the shea stess may be obseved in the thid quadant of the lot. Both Figue 3 and Figue 4 show a continuous degadation in shea stength with cyclic loading until the cyclic stength eaches a stable minimum value nea the end of the test (i.e. eaches a lage dislacement shea stength). Figue 1. Pe-eak shea stess vs. dislacement fo ±1 mm cyclic shea test at σ n = 1382 kpa (Ross 29). Figue 2. Post-eak shea stess vs. dislacement fo ±2 mm cyclic shea test at σ n = 1382 kpa (Ross 29). 6 4 2-2 -4-6 Δa=2mm Δa=3mm Δa=6mm Δa=12mm -14-1 -6-2 2 6 1 14 Dislacement (mm) Figue 3. Shea stess vs. dislacement fo ±2, 3, 6, and 12 mm amlitudes cyclic shea tests at σ n = 1382 kpa.
Shea Stess, τ (kpa) Shea Stess, τ (kpa) Shea Stess, τ (kpa) Shea Stess, τ (kpa) 8 6 4 2-2 -4-6 -8 Δa=2mm Δa=3mm Δa=6mm Δa=12mm -14-1 -6-2 2 6 1 14 Dislacement (mm) Figue 4. Shea stess vs. dislacement fo ±2, 3, 6, and 12 mm amlitudes cyclic shea tests at σ n = 271 kpa. 2.2 Inteetation of Test Results The ogessive degadation of the hysteesis loos in Figue 2 though Figue 4 was deemed the most imotant asect of the obseved inteface behaviou with esect to modeling the in lane shea behaviou of the GMX/GCL combination. To illustate the tyical ost eak degadation behaviou of the GMX/GCL combination modeled in this ae, the shea stess time histoy ove 25 cycles of loading with a constant dislacement amlitude of ± 12 mm unde σ n = 692 kpa is esented in Figue 5. The fist five cycles exeience most of the degadation in mobilized shea stength and afte 1 cycles the mobilized shea stength is almost constant. Based uon the shea stess time histoy esented in Figue 5, it was hyothesized that the obseved degadation in the mobilized shea stength may be elated to the cumulative elative shea dislacement. 3 2 1-1 -2-3 5 1 15 2 25 Time (sec) Figue 5. Shea stess vs. time fo dislacement amlitude ± 12 mm cyclic shea tests at σ n = 692 kpa. A lot of the absolute value of the mobilized shea stess at the eak dislacement vesus the cumulative elative shea dislacement fo ost-eak dislacement amlitudes of 2, 3, 6 and 12 mm at σ n = 692 kpa is esented in Figue 6. A distinctive tend of mobilized shea stess vesus cumulative elative shea dislacement may be obseved in the figue. As the cumulative dislacement inceases, it aeas that a stable minimum value of mobilized shea stength (i.e. a lage dislacement shea stength) is eached, afte which futhe degadation is negligible. The elationshi between mobilized shea stength and cumulative elative shea dislacement illustated in Figue 6 indicates that the eduction in GMX/GCL mobilized shea stength may easonably be exessed as a function of cumulative elative shea dislacement. Relating the eduction in mobilized inteface shea stength to the elative shea dislacement is not a new concet. Estehuizen et al. (21) develoed a hyebolic model fo inteface behaviou unde monotonic loading that elates the eduction in inteface shea stength to elative shea dislacement based on monotonic inteface diect shea test esults. 3 2 1 Δa=2mm Δa=3mm Δa=6mm Δa=12mm 2 4 6 8 1 12 Total (Cumulative) Relative shea dislacement (mm) Figue 6. Mobilized shea stength vs. cumulative elative shea dislacement fo ± 2, 3, 6 and 12 mm amlitudes cyclic shea tests at σ n = 692 kpa. The eak and lage dislacement shea-stength failue enveloes fo the in-lane shea stength of the GMX/GCL combination modeled heein subject to cyclic loading ae shown in Figue 7. 8 6 4 2 Peak Lage Dislacement τ =.2971σ n + 28.274 R² =.9923 τ =.259σ n + 24.662 R² =.9834 Moh-Coulomb 5 1 15 2 25 Nomal stess, σ n (kpa) Figue 7. Peak and lage dislacement failue enveloes fo the in-lane stength of a GMX/GCL combination.
Fiction angle (degee) s Both the eak and lage dislacement shea stength enveloes can be eesented by the Moh-Coulomb model as follows: c tan [1] n whee c is the in-lane cohesion and is the in-lane fiction angle. Table 1 esents the best fit aametes fo both the eak and lage dislacement Moh-Coulomb enveloes. 25 2 15 1 5 5 si (348kPa) 1 si (692 kpa) 2 si (1382 kpa) 3 si (269 kpa) Table 1. Peak and lage dislacement shea stength failue enveloe aametes. Moh-Coulomb Paametes Failue Enveloe c (kpa) (degees) Peak 28.2 16.5 Lage Dislacement 24.7 1.5 The eak stengths in Figue 7 and Table 1 wee mobilized at shea dislacements of 1.-15. mm. If the cohesion is assumed to be constant, the fiction angle at any stage duing degadation can be elated to the mobilized shea stength,, at the eak dislacement amlitude accoding to Eq. 1 as: 1 c tan [2] n The fiction angle evaluated using Eq. 2 and the lagedislacement cohesion fom Table 1 is lotted vesus the cumulative elative shea dislacement fo ost-eak tests at fou diffeent nomal stesses in Figue 8. The tends of the fou cuves ae quite simila ove the ange of nomal stesses fom 348 kpa to 271 kpa. At each nomal stess, the eak fiction angle (and thus eak shea esistance) develos at a elatively small cumulative shea dislacement (i.e. 1.-15. mm). Then, as cumulative shea dislacement inceases the fiction angle educes until it eaches a stable lage dislacement fiction angle. The data in Figue 8 dislay a highe vaiability at lowe nomal stesses. The data in Figue 8 also show a distinctive tend between in-lane fiction angle and cumulative elative dislacement. This tend may change slightly with nomal stess. Howeve, the hyothesis of using cumulative elative dislacement as the govening vaiable to estimate the eduction in in-lane shea stength that accomanies cyclic loading of a GMX/GCL combination is clealy suoted by the data in Figue 8. 2 4 6 8 1 12 14 Figue 8. Fiction angle vesus cumulative elative shea dislacement at nomal stesses of 348, 692, 1382 and 269 kpa fo ost-eak tests only. 3 DISPLACEMENT-SOFTENING RELATIONSHIP To model the in-lane behaviou of a GMX/GCL secimen subject to cyclic shea loading, a nonlinea dislacementsoftening model was been develoed. The model assumes that a unique elationshi exists between the inlane fiction angle and the cumulative elative shea dislacement, as suggested by the data in Figue 8. The fomulation esented heein esults in hysteesis loos eesentative of those esented in Figue 3. 3.1 Constitutive elationshi Total (Cumulative) elative shea dislacement (mm) The genealized shea stength-dislacement elationshi fo the in-lane mobilized fiction angle,, of a GMX/GCL combination is illustated in Figue 9. Key aametes in this elationshi include fo the lastic cumulative elative shea dislacement ( ), the eak fiction angle ( ), the lage dislacement fiction angle ( ), the cumulative dislacement at the eak fiction angle ( ) and cumulative dislacement at the lage dislacement fiction angle ( ). This fomulation assumes elastic behaviou at elative shea dislacements less than and lastic behaviou afte the shea dislacement exceeds. Peak Residual e Figue 9. Genealized inteface fiction angle vesus cumulative shea dislacement elationshi.
s The assumtion that the mobilized fiction angle deceases as the cumulative dislacement inceases once is exceeded and continues to decease until it eaches a stable esidual value (the lage dislacement fiction angle), as illustated in Figue 9, is the basis fo the dislacement-softening model develoed heein. This elationshi can be eesented mathematically as follows: 1 k a b ( ) [3] whee a, b and k ae model aametes and ae assumed to be constant fo a aticula GMX/GCL combination. These aametes can be defined using the coodinates (, ), (, ), and (, ) fom the cumulative shea dislacement elationshi in Figue 9. The shea stength of the secimen is equal to until the cumulative defomation exceeds. At this oint, the mobilized fiction angle stats to degade until it eaches a value of at a dislacement equal to. The ate of decay of the mobilized fiction angle is descibed by the exonent k. The elationshi between the mobilized fiction angle and the cumulative lastic shea dislacement at dislacements geate than can be witten as: 1 [4] k a d whee 1 d 1 [5] a 1 [6] 3.2 Cyclic behaviou The hysteetic behaviou fo ost-eak cyclic loading edicted by Eq. 4 is illustated in Figue 1. The model will initially behave elastically until the cumulative dislacement exceeds the dislacement at the eak fiction angle ( ), at which oint cumulative lastic shea dislacements will stat to accumulate. Once lastic shea dislacements begin to accumulate, the mobilized fiction angle (o shea stength) will follow Eq. 4 until unloading begins. Plastic shea dislacements begin to accumulate in the evese diection when the shea stess exceeds the mobilized shea stength fom any io loading cycle. This model will geneate shea hysteesis loos simila to the ost-eak behaviou esented in Figues 3 and 4. The dislacement-softening model assumes elastic behaviou fo e-eak loading, so the model does not coectly edict the e-eak hysteesis loos esented in Figue 1. K s Figue 1. Genealized shea stess vesus cumulative shea dislacement elationshi. 4 IMPLEMENTATION Fowad loading o Unloading An elasto-lastic fomulation of the new constitutive model was imlemented in FLAC 6. (Itasca, 28), a finite diffeence softwae ackage fo solving geneal stessdefomation oblems in geotechnical engineeing. Elastic behaviou was assumed fo stess conditions below failue. Elastic behaviou is govened by the unload-eload stiffness, K s. Fo ost-failue stess conditions, the new constitutive model descibed above was used to model softening and degadation of the inlane shea stength of an inteface element. A simle numeical model, illustated in Figue 11, was used to test the efomance of the new constitutive model. The single element igid block in Figue 11 eesents the ue ull out late and the five element base in Figue 11 eesents the base of the diect shea device emloyed by Fox et al. (26). Values of bulk and shea modulus eesentative of stuctual steel wee used to model both the igid block and base. Inteface elements wee used between the igid block and the base to eesent the GMX/GCL combination. The constitutive model descibed above was used to model the shea behavio of the inteface elements. Table 2 esents the aametes used fo the constitutive model in the numeical analysis. The inteface elements wee assigned an elastic shea stiffness equal to 5x1 7 Pa/m (calculated fom the test esults shown in Figues 4 and 5). A velocity time histoy was alied to the base of the mesh to model K s Revese loading
the cyclic test inut motion (25 cycles of sinusoidal dislacement with f = 1 Hz). Y Rigid Block X Nomal stess (σ n) Inteface Elements The numeical and exeimental esults unde the same nomal stess as in Figue 12 (271 kpa) but fo a 6 mm dislacement amlitude (instead of 12 mm) ae comaed in Figue 13. While the numeical model still seems a little off with esect to the initial stiffness, in this case it accuately edicts the eak shea stess uon initial loading and the shea stength at the oint of stess evesal. Howeve, the numeical model does not catue the eak in the shea stess at the beginning of evey cycle. Base Figue 11. Finite diffeence model (maco elements shown). Table 2. Model aametes values fo model veification. c Shea stength degee degee e Plastic aametes Elastic aametes k kpa mm m 26 16.5 1.5 1.6 12..52 5x1 7 K s a/m The esults calculated using the numeical model ae comaed to the exeimental esults fom Ross (29) fo a nomal stess of 271 kpa and a 12 mm dislacement amlitude in Figue 12. While thee is some disceancy with esect to the initial stiffness and eak shea stength, the numeical model aeas to accuately edict the degadation in shea stess. Figue 13. Comaison of test data to ediction using new constitutive model fo ± 6 mm dislacement amlitude at σ n = 271 kpa. In Figue 14, the shea stess time histoy calculated using the numeical model is comaed to the esults fo the test with a dislacement amlitude of 12 mm at a nomal stess of 692 kpa (the test esults esented in Figue 5). The geneal tend of the shea stess time histoy is catued well by the numeical model esults esented in Figue 14. Howeve, simila to the disceancy between the numeical model and test esults esented in Figue 13, the eak at the beginning of evey stess cycle was not catued by the numeical model. The shea stess sikes obseved in the test data at the begging of evey cycle may be elated to the intenal shea behaviou of the GCL, as oosed to the behaviou of the GMX/GCL inteface. Figue 12. Comaison of test data to ediction using new constitutive model fo ± 12 mm dislacement amlitude at σ n = 271 kpa. Figue 14. Shea stess vs. time calculated fo (± 12 mm) dislacement amlitudes cyclic shea tests at σ n = 692 kpa.
5 SUMMARY AND CONCLUSIONS An elasto-lastic constitutive model fo the in-lane shea behaviou of a GMX/GCL combination has been develoed to simulate the degadation in mobilized shea stength accomanying cyclic loading that exceeds the eak shea stength of the mateial. The model uses the Moh- Coulomb shea stength citeion fo both eak and ost-eak shea stength chaacteization. When imlemented in a finite diffeence comute ogam, the model is shown to catue the degadation of hysteesis loos (i.e. the degadation in mobilized shea stength) obseved in diect shea testing of a GMX/GCL combination. Comaison of numeical and exeimental esults indicates that the model has some shotcomings with esect to the initial shea stiffness and the shea stess immediately afte shea stess evesal. Howeve, catuing the degadation of the mobilized shea stength is consideed to be of aamount imotance in modeling the behaviou of a GMX/GCL combination subject to eathquake loading. The famewok of the model can be exanded to any geosynthetic inteface that follows the same tend of stength eduction, i.e. a stength eduction deendent uon the cumulative elative shea dislacement duing cyclic loading. A shotcoming of the model is that it is based solely uon the esults of unifom cyclic loading tests. Howeve, no data was available on the behaviou of a GMX/GCL combination subject to non-unifom cyclic loading. Additional exeimental data is needed to assess the efomance of the model unde non-unifom cyclic loading, e.g. unde loading conditions eesentative of seismic loading. Futhemoe, additional wok is needed to imove model efomance duing initial loading and immediately following stess evesal. ACKNOWLEDGEMENTS The wok in this ae is at of a joint Aizona State Univesity / Univesity of Califonia at San Diego eseach ogam titled GOALI: Collaboative Reseach: The Integity of Geosynthetic Elements of Waste Containment Baie Systems Subject to Lage Settlements o Seismic Loading. This oject is funded by the Geomechanics and Geotechnical Systems, Geoenvionmental Engineeing, and Geohazads Mitigation ogam of the National Science Foundation (NSF) Division of Civil, Mechanical, and Manufactuing Innovation unde gant numbe CMMI- 8873. The authos ae gateful fo this suot as well as suot of oject s industial atnes, Geosyntec Consultants, Inc and CETCO. Oinions and ositions exessed in this ae ae the oinions and ositions of the authos only and do not eflect the oinions and ositions of the NSF. REFERENCES Byne, R.J. 1994. Design issues with stain-softening intefaces in landfill lines. Poc. Waste Tech. 94, Chaleston, South Caolina, USA, 1-26. Desai, C.S., and Fishman, K. L. 1991. Plasticity-based constitutive model with associated testing fo joints. Int. J. Rock Mech. Min. Sci. Geomech. Abst., 28, 15 26. Estehuizen, J.J.B., Filz, G.M., and Duncan, J.M. 21. Constitutive behavio of geosynthetic intefaces. J. Geotech. and Geoenvi. Engg., ASCE, 127(1): 834-84. Fox, P.J., and Kim, R.H. 28. Effect of ogessive failue on measued shea stength of geomembane/gcl inteface. Jounal of Geotechnical and Geoenvionmental Engineeing, 134(4): 459-469. Fox, P.J., Nye, C.J., Moison, T.C., Hunte, J.G., and Olsta, J.T. 26. Lage dynamic diect shea machine fo geosynthetic clay lines. Geotechnical Testing Jounal, 29(5): 392-4. Itasca Consulting Gou, Inc. 28. FLAC Fast Lagangian analysis of continua, use s manual, www.itasca.com. Kim, J., Rieme, M., and Bay, J.D. 25. Dynamic oeties of geosynthetic intefaces. Geotechnic al Testing Jounal, 28(3): 1-9. Lo Gasso, S.A., Massimino, M.R., and Maugei, M. 22. Dynamic analysis of geosynthetic intefaces by shaking table tests. Poc. 7th Intenational Con feence on Geosynthetics, P. Delmas and J. P. Gouc, eds., Nice, 4, 1335-1338. McCatney, J.S., Zonbeg, J.G., and Swan, R.H., J. 29. Analysis of a lage database of GCL geomembane inteface shea stength esults. Jounal of Geotechnical and Geoenvionmental Engineeing, 134(2), 29-223. Ross J.D., Fox P. J. and Olsta, J. T. 21. Dynamic shea testing of a geomembane/geosynthetic clay line inteface. 9th Intenational Confeence on Geosynthetics, IGS-Bazil, 2: 649-652. Ross, J.D. 29. Static and dynamic shea stength of a geomembane/geosynthetic clay line inteface. M.S. thesis, Deatment of Civil and Envionmental Engineeing and Geodetic Science, The Ohio State Univesity, Columbus, Ohio, USA. Tilett, E.J. and Fox, P.J. 21. Shea stength of HDPE geomembane/geosynthetic clay line intefaces. Jounal of Geotechnical and Geoenvionmental Engineeing, 127(6): 543-552.