A unified constitutive model for both clay and sand with hardening parameter independent on stress path

Transcription

1 vailable online at Comuters an Geotechnics () unifie constitutive moel for both clay an san with harening arameter ineenent on stress ath Y.P. Yao a, *, D.. Sun b, H. Matsuoka c a Deartment of Civil Engineering, Beihang University, Beijing, China b Deartment of Civil Engineering, Shanghai University, Shanghai 7, China c Deartment of Civil Engineering, Nagoya Institute of Technology, Nagoya 66-, Jaan Receive ugust 6; receive in revise form March 7; accete March 7 vailable online May 7 bstract unifie constitutive moel for both clay an san uner three-imensional stress conitions is erive from the moifie Cam-clay moel, by taking the following two oints into consieration. irst, a transforme stress tensor base on the SMP (satially mobilize lane) criterion is alie to the Cam-clay moel. The roose moel consistently escribes shear yieling an shear failure an combines critical state theory with the SMP criterion for clay. Seconly, a new harening arameter, which is ineenent of the stress ath, is erive in orer to evelo a unifie constitutive moel for both clay an san. It not only escribes the ilatancy for lightly to heavily ilatant san, but also reuces to the lastic volumetric strain for clay. The valiity of the harening arameter is confirme by the test results of triaxial comression an extension tests on san uner various stress aths. Only five conventional soil arameters are neee in the roose moel. Ó 7 Elsevier Lt. ll rights reserve. Keywors: ; Dilatancy; Elastolastic moel; San; Stress ath; Three-imensional stress. Introuction * Corresoning author. Tel.: aress: y-yao@6.com (Y.P. Yao). The moifie Cam-clay moel, calle the Cam-clay moel for short in this aer, which is suitable for clay (in this aer, the term clay is to be interrete as normally consoliate clay), was roose by Roscoe an Burlan []. The Cam-clay moel an many other moels have been generalize by assuming a section of the yiel surface to be circular in the -lane. The mean stress ð¼ r ii =Þ an eviator stress qð¼ ffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi = s ij s ij Þ are use as the stress arameters in the moels, where s ij ð¼ r ij ij Þ is a eviatoric stress tensor an ij is Kronecker s elta. That is to say, the criterion of the Extene Mises tye (g = q/ = const.) is aote for the shear yieling an the shear failure of clay in the Cam-clay moel. Shear yieling is cause by an increase in stress ratio g = q/, while comressive yieling is cause by an increase in the mean stress. However, as exerimental evience shows, the Extene Mises criterion grossly overestimates the strength in triaxial extension, an also results in incorrect intermeiate stress ratios in lane strain []. In contrast to the Extene Mises failure, the SMP failure criterion [], which is consiere to be a three-imensional extension of the Mohr Coulomb criterion, is a failure criterion that exlains the high quality test results for soils. transforme stress tensor has been roose by Matsuoka et al. [9] which makes the SMP criterion become circular in the transforme - lane. revise transforme stress tensor r ij for the SMP criterion is eveloe in this aer base on the transforme stress metho [9] an the work of Yao an Sun [,] in orer to take the lastic strain increment irection into account aitionally. The new transforme stress tensor r ij is alie to the Cam-clay moel. The roose moel consistently escribes the shear yieling an shear failure of soils uner three-imensional stresses, both 66-X/$ - see front matter Ó 7 Elsevier Lt. ll rights reserve. oi:.6/j.comgeo.7..

2 Y.P. Yao et al. / Comuters an Geotechnics () of which obey the SMP criterion, an combines critical state theory with the SMP criterion for clay. Traitionally, the lastic volumetric strain e v is taken as the harening arameter in the Cam-clay moel, which is not aroriate for ilatant san. To ate, a lot of harening arameters which can escribe the ilatancy of soils have been assume (e.g., [,,,,,6]), an several ilatant lasticity moels have been roose (e.g., [,6, 7,7,,6]). In this aer, a new harening arameter, that is ineenent of stress ath, is erive by consiering unifie yiel an lastic otential functions which are the same those for Cam-clay. s will be escribe later, the hysical meaning of this harening arameter is clear. The roose harening arameter not only escribes the ilatancy from lightly to heavily ilatant san, but also reuces to the lastic volumetric strain e v for clay. The valiity of the harening arameter H is confirme by triaxial comression an extension test results on san uner various stress aths. The ability of the roose moel to reict the raine behavior of normally consoliate clay an saturate san is examine along various stress aths uner triaxial comression an extension conitions. The results reicte by the roose moel agree well with the test results. Only five soil arameters are neee in the roose moel. The values of these arameters can be etermine by a loaing an unloaing isotroic comression test an a conventional triaxial comression test. In this aer, the term stress is to be interrete as effective stress. The imlementation of the moel into the finite element metho follows the methos escribe in Sheng et al. [] an Sloan et al. [].. transforme stress tensor base on the SMP criterion transforme stress tensor, by which the SMP criterion is mae circular in the transforme -lane, has been roose by Matsuoka et al. [9]. revise transforme stress tensor r ij is eveloe in this aer base on the transforme stress metho [9] an the work of Yao an Sun [,] in orer to take the lastic strain increment irection into account aitionally. The outline of the new transforme stress tensor r ij is introuce as follows: The SMP criterion can be written as I I =I ¼ const ðþ where I, I an I are the first, secon an thir stress invariants, resectively. The soli curve in ig. is the shae of the SMP criterion in the -lane. The transforme stress is euce from what makes the SMP curve in the - lane become a circle, the ot line of ig., with the center being the origin in the transforme -lane (-lane). When = const, we assume that r is equal to r, where r is the stress raius in the -lane of the transforme stress sace an r is the stress raius in the -lane of the orinary stress sace when h =. rom Eq. (), r can be exresse by Matsuoka et al. [9] θ ( ) Circle SMP ( ) ( ) rffiffiffi rffiffi r ¼ r ¼ q I ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðþ ði I I Þ=ðI I 9I Þ The generalize eviatoric stress in the transforme stress sace qð¼ ffiffiffiffiffiffiffi = rþ can be written as q ¼ q I ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðþ ði I I Þ=ðI I 9I Þ To realize the transformation from r ij to r ij, the following equations shoul be mae: >< ¼ q ¼ q ðþ >: h ¼ f ðhþ where is the mean transforme stress; h an h are Loe s stress angles in the transforme stress sace an the orinary stress sace resectively, which are one-to-one corresonence with intermeiate rincial stress arameters b an b (=(r r )/(r r )) in the transforme stress sace an the orinary stress sace resectively; f (h) is a function of h which also can be exresse by b. So, Eq. () can be rewritten as >< ¼ q ¼ q ðþ >: b ¼ f ðbþ where b ¼ r r r r >< ¼ ðr þ r þ r Þ¼ r ii qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ ffiffi ðr r Þ þðr r Þ þðr r Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >: ¼ ðr ij ij Þðr ij ij Þ θ r = r ig.. SMP criterion in -lane (soli curve) an transforme -lane (broken circle). f (h)orf (b) is a function to transform the flow irection of the lastic strain increment in the orinary stress sace into the flow irection in the transforme stress sace. By ð6þ

3 Y.P. Yao et al. / Comuters an Geotechnics () f (h) orf (b), the eviation of the lastic strain increment irection from the stress irection in the orinary -lane can be taken into consieration, although the irections of stress an lastic strain increments in transforme - lane are coincient. reasonable lastic otential surface for frictional materials shoul be between the curve-triangle shae (I I /I = const) an the circular shae (q/ = const) in the -lane. So, the flow irection of the lastic strain increment is also between the normal to the triangular shae of SMP criterion an that to the circular shae, as shown ig.. The following f (b) [] has the above characteristics ffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffi r þ r r r f ðbþ ¼ffiffiffiffiffi ffiffiffiffiffi b ¼ ffiffiffiffiffi ffiffiffiffiffi ð7þ þ r r r So, the transforme stress tensors uner the general stress state can be obtaine as r r ij ¼ ij þ q q s ðrs ij s ij Þ an s ¼ r s ii = 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q s ¼ ð=þðr s ij s ij Þðr s ij s ij Þ >= r s ij ¼ðI s r ik þ I s ikþðr kj þ I s kjþ I s ¼ ffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffi r þ r þ r I s ¼ ffiffiffiffiffiffiffiffiffi r r þ ffiffiffiffiffiffiffiffiffi r r þ ffiffiffiffiffiffiffiffiffi r r I s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi >; r r r When the stress tensor r ij is given, the transforme stress tensor r ij can be calculate from Eq. (). rom the above erivation, it can be shown that the shae of the SMP criterion becomes a cone with the axis being the sace iagonal r ¼ r ¼ r in the transforme rincial stress sace. Its cross-section is a circle with the center being the origin O in the -lane (see ig. ). Noting the similarity in the shaes of the Extene Mises criterion in the rincial stress sace an the SMP criterion in the transforme rincial stress sace, we can revise existing elastolastic moels such as the Cam-clay moel by using the transforme stress tensor r ij base on the SMP criterion.. Basic metho for constructing a unifie harening arameter ineenent of stress ath Harening of materials is ue to the occurrence of lastic strain. Therefore, a arameter reresenting the egree of harening must een on the lastic strain. Harening arameters are internal variables that are use to inicate the rate of lastic eformation, an shoul be a function of lastic volumetric strain an lastic shear strain. or a single yiel surface moel, the harening arameter shoul have the following roerties: () its increments shoul be the same from one oint on a yiel surface to ifferent oints on another yiel surface, an () its increment shoul ðþ ð9þ also be same from one oint to another oint along ifferent stress aths. Here, the harening arameter is ifferent with the harening moulus which obviously changes from oint to oint. However, the lastic strain is usually eenent on the stress ath. or examle, the lastic volumetric strain an lastic shear strain for san are all eenent on stress ath (the test results in the following section can confirm this characteristic). So, the lastic strain increments instea of the total lastic strains shoul be use to construct a unifie harening arameter for various kins of soils. That is to say, the harening arameter shoul be a function of the lastic volumetric strain increment e v an the lastic shear strain increment e. It is assume that the lastic volumetric strain increment an lastic shear strain increment can be relate through the stress-ilatancy equation e v =e ¼ f ðgþ. or examle, the stress-ilatancy equation in the original Cam-clay moel is e v =e ¼ M g (stress ratio g = q/; M is the value of g at the critical state), so that f (g) =M g. Therefore, we can choose either the lastic volumetric strain increment e v or the lastic shear strain increment e as a basic arameter to construct the harening arameter. However, lastic shear strain oes not occur uring isotroic comression, which means that it is imossible to escribe harening uner isotroic comression by use of the lastic shear strain increment. Therefore, we have to choose the lastic volumetric strain increment e v to construct the harening arameter. Realizing that the increment of the harening arameter shoul be ineenent of the stress ath, we assume there exists a stress function R(g) such that the integral R e v is ineenent of stress RðgÞ ath. Thus, this integral can be use as a general harening arameter Z H ¼ e v RðgÞ ðþ In the Cam-clay moel, R(g) =. suitable R(g) for both clay an san will be introuce in the next section.. unifie harening arameter for both clay an san basic metho for constructing a unifie harening arameter ineenent of stress ath has been introuce in the receing section. We will now erive a new harening arameter for both clay an san in this section. The moifie Cam-clay moel is consiere to be one of the best basic elastolastic moels for clay, which is better than the original one [9]. In the moifie Cam-clay moel, the yiel an lastic otential functions are assume to be of the same form as follows: f ¼ g ¼ ln þ ln þ q M Z e v ¼ ðþ c where is the initial mean stress, e v ð¼ e iiþ is the lastic volumetric strain increment an c is written as c ¼ k j ðþ þ e

4 Y.P. Yao et al. / Comuters an Geotechnics () with k being the comression inex, j being the swelling inex, an e the initial voi ratio at =. It is reasonable that Eq. () is also chosen as the lastic otential function for ilatant san, because this equation reicts that the lastic volumetric strain increment is ositive before the characteristic state (g = M) [7] an negative after the characteristic state (see ig. ). To aot an associate flow rule in new moel, a yiel function similar to Eq. () is also assume for san. However, the lastic volumetric strain cannot be use as the harening arameter for san because it is eenent on the stress ath an oes not increase monotonically with loaing. In this aer, a new harening arameter H is erive to escribe the harening behavior of clay an san. The yiel an lastic otential functions for san are written as f ¼ g ¼ ln þ ln þ q M H ¼ ðþ Nova an Woo [,] have chosen a combination of the lastic volumetric strain an lastic eviator strain as a harening arameter. In the resent moel, the harening arameter is consiere to be a combination of the stress tensor r ij an lastic strain increment tensor e ij, i.e., a lastic work tye of harening arameter. Thus, the following harening arameter is assume by using the lastic strain invariants Z Z H ¼ H ¼ c ðr ij Þe v þ c ðr ij Þe ðþ where c (r ij ) an c (r ij ) are the functions of the stress tensor, resectively, an e ¼ ffiffiffiffiffiffiffi = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe ij e v ij =Þðe ij e v ij =ÞÞ is the lastic eviator strain increment. In the moifie Cam-clay moel, the following stress-ilatancy equation is aote: e v e ¼ M g g Substituting Eq. () into () gives Z H ¼ c ðr ij Þe v þc g ðr ij Þ M g e v Z ¼ cðr ij Þe v ðþ ð6þ where c(r ij ) is a function of the stress tensor. Note that c(r ij ) in Eq. (6) is also equal to /R(g) in Eq. (). ig. shows the constant mean stress ath (ath B) an the isotroic comression stress ath (ath C) along which the harening arameter changes from H to H. The rocess for fining the harening arameter H is () to erive a general equation of H along ath B an () to etermine an exlicit equation of H along ath C... long the constant mean stress ath B fter substituting Eq. (6) into Eq. (), the total ifferential form of the yiel function is exresse as f ¼ of of of þ q þ o oq oh H ¼ of of þ o oq q cðr ijþk of o ¼ ð7þ So the roortionality constant K can be written as K ¼ of o cðr ij Þ þ of oq q of o ðþ Base on Eq. (), the following two ifferential equations can be obtaine: of o ¼ M g ð9þ M þ g of oq ¼ g ðþ M þ g By substituting Eqs. (9) an () into Eq. (), the lastic eviator strain increment along a constant mean stress ath is as follows: e ¼ K of oq ¼ g cðr ij Þ M g q ðþ ig. shows the results of triaxial comression tests on clay an san (ata from []). It can be seen from ig. a that the shaes of the curves g e for clay an san are alike. The stress ratios (q/) at failure for clay an san are M an M f resectively. In fact, M f is not constant uring shearing. If the variation of M f is consiere uring shearing, more comlex behavior (e.g., the softening) of soils can be escribe []. In this aer, M f is assume to be constant for the sake of simlicity. The arameters M an M f are similar to those of Nova an Woo []. q, v η= M f η=m v q η= M f H η= M B (,q) H, v (,) C ( x,) ig.. Direction of lastic strain increment vectors. ig.. Stress aths for eriving harening arameter H.

5 Y.P. Yao et al. / Comuters an Geotechnics () San M f M Substituting Eq. (6) into Eq. (6) gives Z Z M f g H ¼ H ¼ q M g e v ð7þ η (M f - η )/ η Therefore, comare with the equation of the lastic eviator strain increment for clay in the Cam-clay moel when the mean stress is constant (Eq. ()), the equation of the lastic eviator strain increment for san is assume to be Eq. () when the mean stress is constant e ¼ c g q ðfor clayþ ðþ M g e ¼ q g q ðfor sanþ ðþ M f g where q is a constant. Eqs. () an () can also be written in the linear forms M g g ¼ c g e ðfor clayþ ðþ M f g ¼ q g g e ðfor sanþ ðþ The valiity of Eqs. () an () is confirme by the triaxial comression results in ig. b. So, the assume form of Eq. () is aroriate. It is worth noting that M f = M for clay an the elastic eviator strain is very small uner a constant mean stress in ig. b. By combining Eqs. () an (), c(r ij ) in Eq. () is exresse as M f g cðr ij Þ¼ q M g 6 Com. =96 kpa 6 Com. =96 kpa San η / ig.. Triaxial comression test results for clay an san arrange in (a) g e ; (b) ðm f g Þ=g g=e. ð6þ.. long the isotroic comression stress ath C When g = (ath C), Eq. (7) becomes Z Z M f H ¼ H ¼ q M e v ðþ Moreover, uner the isotroic comression conition (g = q/ = ) an e v ¼ c lnð= Þ. In aition, Eq. () becomes H = ln(/ ) when g =. So, the following equation can be obtaine from the above two equations: Z Z e H ¼ H ¼ v ð9þ c Letting Eq. () equal Eq. (9) gives M f q ¼ c ðþ M inally, we can obtain the following equation of the new harening arameter for san by substituting Eq. () into Eq. (7): Z Z M M f g H ¼ H ¼ c M f M g e v ðþ Comaring Eq. () with Eq. () gives M f M g RðgÞ ¼c ðþ M M f g If M = M f, Eq. () becomes H ¼ R H ¼ R e v =c, which is the same as the harening arameter for clay in the Cam-clay moel. H in Eq. () is a unifie one for both clay an san.. unifie elastolastic moel for both clay an san In the roose moel, the equations of the yiel locus an lastic otential remain the same as in Cam-clay, but the transforme stress tensor r ij (base on the SMP criterion) an the new harening arameter are aote to moel the mechanical behavior of clay an san uner three-imensional stresses. The total strain increment is given by the summation of the elastic comonent an the lastic comonent as usual: e ij ¼ e e ij þ e ij ðþ Here, the elastic comonent is given by the following equation e e ij ¼ þ m E r ij m E r mm ij ðþ where m is Poisson s ratio, an the elastic moulus E is exresse as

6 Y.P. Yao et al. / Comuters an Geotechnics () E ¼ ð mþð þ e Þ ðþ j The lastic comonent is given by assuming the flow rule not in r ij sace but in r ij sace. e ij ¼ K og ð6þ or ij where the lastic otential function g (or the yiel function f), the harening arameter H, the roortionality constant K an the ifferential equation og=or ij are given, resectively, as follows: f ¼ g ¼ ln þ ln þ q H ¼ ð7þ M Z Z M M f g H ¼ H ¼ c M f M g e v ðþ M f M g q K ¼ c þ M M f g M q q ð9þ og M q ¼ or ij M þ q ij þ ðr ij ij Þ ðþ In the above equations, the eviator stress q an the transforme stress ratio g in r ij sace, Mðg at critical state) an M f ðg at eak) are written resectively as follows: g ¼ q= ðþ rffiffiffiffiffiffiffiffiffiffiffiffi q ¼ s ijs ij ðþ M ¼ 6 sin / c ðþ sin / c M f ¼ 6 sin / ðþ sin / where ð¼ Þ is the initial mean stress, an (/ c,/) are the internal friction angles at the characteristic state an shear failure, resectively. The main features of the roose moel for both clay an san are resente as follows... or san The stress-ilatancy equation of the roose moel is exresse as follows: e v e ¼ M q ðþ q Eq. () can be rawn as ig. a in the q= e v =e lane an as ig. b in the q= e v =e lane uner triaxial comression an extension conitions. It is seen from these two figures that the unique relationshi between q= an e v =e can exlain the ifference in the q= e v =e relations between triaxial comression an extension. Uner triaxial comression an extension stress conitions, the yiel function in Eq. (7) is lotte in ig. 6, where CL an L show the characteristic state line an the shear failure line, resectively. r a an r r are the transforme stresses corresoning to r a an r r resectively, in which r a an r r are the axial an raial stresses in triaxial stress conitions. It is seen from ig. 6 that although the yiel curves in triaxial comression an extension are symmetrical with resect to the -axis in the ðr a r r Þ lane (see ig. 6a), the yiel curves are not symmetrical with resect to -axis, an the value of q in triaxial extension is smaller than that in triaxial comression in the (r a r r ) lane at the same (see ig. 6b). This tren is similar to the test results from various kins of soils (e.g., []). How the roose moel escribes the ilatancy of soil is exlaine as follows. rom Eq. (), we obtain e v ¼ c M f M g H M ð6þ g M f Since H is always larger than or equal to, the following conclusions can be obtaine from Eq. (6): () g ¼ (isotroic comression conition): e v ¼ c H. () 6 g < M (negative ilatancy conition): e v >. () g ¼ M (characteristic state conition): e v ¼. () M < g 6 M f (ositive ilatancy conition): e v <. s inicate above, the ilatancy of clay an san are reasonably escribe by the new harening arameter H. The valiity of H an the other quantities use usually as harening arameters will be checke in the various stress aths as follows. ig. 7 shows the stress aths in triaxial tests on Toyoura san (ata from []) in terms of the relation between mean stress an eviator stress q. The values of the rincial stress ratios are the same (r /r = ) at oints an in ig. 7. We will check the stress ath eenency of four quantities, the lastic volumetric strain e v, the lastic eviator strain e, the lastic work W an the roose harening arameter H, in four kins of triaxial comression tests (aths: DE, BC, G an BE) an three kins of triaxial extension tests (aths: D, C an ). igs. show the variations of those quantities along the seven kins of stress aths uner triaxial comression an extension conitions. It is obvious from igs. an 9 that the lastic volumetric strain e v an the lastic eviator strain e are unsuitable for the harening arameter for san because these quantities een on the stress aths at the same stress state. rom ig. the lastic work W is almost ineenent of the stress aths only in triaxial comression or triaxial extension, but its value at the stress state is ifferent from that at although the stress invariants ( an q) of the stress states an are the same. The other roblem inuce by the lastic work harening arameter is that san will always haren even though it is at a eak or failure state. Hence, the lastic work is also not goo for the harening arameter in general stress states. However, ig. shows that the values of the

7 6 Y.P. Yao et al. / Comuters an Geotechnics () q Com. an Ext. M f Com. Ext. q M f M M - v / - v / ig.. Stress-ilatancy relationshis of roose moel uner triaxial comression an extension conitions exresse in (a) q= e v =e an (b) q= e v =e. - r a L L q(kpa) CL CL Com. D B C D' a- r ig. 6. Yiel curves of roose moel uner triaxial comression an extension conitions exresse in (a) ðr a r r Þ lane an (b) (r a r r ) lane. Ext. - ' 6 (kpa) ig. 7. Stress aths of triaxial tests for examining new harening arameter. roose harening arameter H are uniquely etermine at the same stress state, regarless of the stress ath in triaxial comression an extension an the revious stress E G L L CL CL v v history. So, the roose harening arameter H is a state quantity, an we emloy it as a new harening arameter for san... or clay Com. D 6 (kpa) -. Ext. D' ' B '. C '. 6 s mentione before, if M = M f, the new harening arameter H becomes the lastic volumetric strain, which is the same as the harening arameter for normally consoliate clay in the Cam-clay moel. Therefore, in this case the ifference between the roose moel an the Cam-clay moel is only the stress tensor use. Let us iscuss the critical state conitions in three-imensional stresses in etail. The critical state conitions of the roose moel in three-imensional stresses can be exresse as follows: E B G (kpa) ig.. Relation between lastic volumetric strain e v an mean rincial stress. C

8 Y.P. Yao et al. / Comuters an Geotechnics () 7 Com. D E E G B C 6 (kpa) Ext. B C 6 (kpa) ig. 9. Relation between lastic eviator strain e stress. D' ' ' an mean rincial H H Com. E D G B C 6 (kpa) Ext. ' D' B C 6 (kpa) ig.. Relation between new harening arameter H an mean rincial stress. W (kpa) Com. E same as the critical state conitions of the Cam-clay moel in triaxial comression, so we might say Eq. (7) is the extension form for the critical state conitions of the Cam-clay moel uner three-imensional stresses. W (kpa) B C 6 (kpa) Ext. 9 g cs ¼ q cs = cs ¼ M e v =e ¼ >= e v ¼ c ½lnð cs = Þþln Š >; e! D D' B C 6 (kpa) ig.. Relation between lastic work W an mean rincial stress. ð7þ where the suffix cs means the critical state. When Eq. (7) is satisfie, soil will be continuously istorte. Eq. (7) is the G ' 6. Preictions versus exerimental results series of triaxial comression an triaxial extension tests on normally consoliate ujinomori clay an saturate Toyoura san have been comlete by Nakai an Matsuoka [] an Nakai []. ig. shows the initial yiel surfaces from Eq. (7) an the test stress aths conucte in triaxial comression an extension for clay an san. The test ata are use to examine the caability of the roose moel in reicting raine behavior of clay an san. The values of soil arameters use in the moel are M = M f =., k/( + e ) =., j/( + e )=. an m =. for ujinomori clay, an M =.9, M f =.66, k/( + e ) =., j/( + e ) =. an m =. for Toyoura san, resectively. The above soil arameters are etermine from isotroic comression tests an conventional triaxial comression tests. 6.. Path = const ig. comares the reicte an observe results for the raine behavior of ujinomori clay an Toyoura san uner triaxial comression an extension conitions when

9 Y.P. Yao et al. / Comuters an Geotechnics () a- r η=m =c =c =c a- r η =M f η San =M =c =c =c = 96 kpa. It can be seen from this figure that the reictions (soli lines) of the roose moel agree well with the observe test results (marke s) for clay an san at constant mean stress uner triaxial comression an extension conitions. 6.. Path r = const η=m =c =c =c η= M f η=m =c =c =c ig.. Stress aths in triaxial comression an extension tests for (a) clay an (b) san. ig. comares the reicte an observe test results for the raine behavior of ujinomori clay an Toyoura san uner triaxial comression an extension conitions when r = 96 kpa. It can be seen from this figure that the reictions (soli lines) of the roose moel agree well with the observe test results (marke s) for clay / Comression =96 kpa - - i V / San Comression =96 kpa - - v Extension =96 kpa / San Extension =96 kpa / - - i - - V (a) or clay (b) or san v ig.. Comarison between reicte an test results uner triaxial comression an extension conitions when = constant for clay an san. / / San Comression Comression =96 kpa =96 kpa - - i - - V v / Extension =96 kpa - - i V (a) or clay / San Extension =96 kpa - - (b) or san v ig.. Comarison between reicte an test results uner triaxial comression an extension conitions when r = constant for clay an san.

10 Y.P. Yao et al. / Comuters an Geotechnics () 9 / Comression =96 kpa - - V / Extension =96 kpa - - V (a) or clay / San Comression =96 kpa - - / San Extension =96 kpa - - (b) or san v v ig.. Comarison between reicte an test results uner triaxial comression an extension conitions when r = constant for clay an san. q(kpa) an san with increasing mean stress uner triaxial comression an extension conitions. 6.. Path r = const B(, ) E(,) D 6 (kpa) ig. 6. Stress aths incluing constant an ecrease in stress ratio in triaxial comression for san. ig. comares the reicte an observe test results for the raine behavior of ujinomori clay an Toyoura C y (kpa) 6 C B R = ' C R = R = 6 x(kpa) ig.. Stress aths uner lane strain conition. san uner triaxial comression an extension conitions when r = 96 kpa. Since the most of the stress ath in triaxial comression for clay is within the initial yiel surface (ig. a), the reicte strain is a little smaller than the ' B comression BC / - comression DE / v v ig. 7. Comarison between reicte an test results along stress aths BC an DE in triaxial comression for san.

11 Y.P. Yao et al. / Comuters an Geotechnics () 6 y x Path B x = 9kPa - - x y v (a) Stress-strain behavior - - x y - v - (a) Stress-strain behavior Path C ( ) x + y =7kPa y x 6 Path B Path C z / x z / x y / x (b) Intermeiate rincial stress ig. 9. Comarison between reicte an measure (a) stress strain behavior an (b) intermeiate rincial stress uner lane strain ath B without change in rincial stress irections (ata after []). y / x (b) Intermeiate rincial stress ig.. Comarison between reicte an measure (a) stress strain behavior an (b) intermeiate rincial stress uner lane strain ath C without change in rincial stress irections (ata after []). test results (ig. a). But, it can be seen from ig. a an b that the reicte results (soli lines) agree well with the test results (marke ) for clay uner triaxial extension conition an for san uner triaxial comression an extension conitions. 6.. Stress aths incluing constant an ecrease in stress ratio in triaxial comression for san ig. 6 shows two secial stress aths, which contain a constant an a ecrease in stress ratio uner triaxial comression, for tests on a san. The values of the san arameters use in the moel are M =., M f =.6, k/ ( + e ) =.6, j/( + e ) =.9 an m =., resectively. The above soil arameters were etermine by an isotroic comression test an a conventional triaxial comression test. ig. 7 shows the reicte an observe test results for the raine stress strain behavior along the stress aths BC an DE, resectively. It can be seen from this figure that the results (soli lines) reicte by the roose moel agree well with the test results (marke s) in those secial stress aths. 6.. Stress aths in lane strain for clay In orer to valiate the resent moel uner lane strain conition, the moel is use to reict the results of the lane strain tests on ujinomori clay []. ig. shows the stress aths teste in the lane strain tests. The tests are conucte along four kins of stress aths (B, C, B an C ) uner lane strain conition from K -consoliation state (oint : r y = 96 kpa, r x = r z = 9 kpa). The moel arameters use in the reictions are the same as the above for ujinomori clay. igs. 9 an show the comarison between reicte an measure stress strain behavior an intermeiate rincial stress along aths B an ath C, resectively, without change in rincial stress irections. igs. an show the comarison between reicte an measure stress strain behavior an intermeiate rin- z / y y x 6 Path B y =96kPa Path B 6 x y v (a) Stress-strain behavior x / y (b) Intermeiate rincial stress ig.. Comarison between reicte an measure (a) stress strain behavior an (b) intermeiate rincial stress uner lane strain ath B with change in rincial stress irections (ata after []).

12 Y.P. Yao et al. / Comuters an Geotechnics () z / y cial stress along aths B an ath C, resectively, with change in rincial stress irections. It is seen from igs. 9 that the resent moel gives relatively goo reictions of the measure stress strain behavior an the measure intermeiate rincial stress values. Therefore, it can be seen from the above comarisons that the roose moel can reasonably escribe the stress strain characteristics of clay an san uner threeimensional stresses, as well as the ilatancy of san along various stress aths. 7. Conclusions Path C ( x + y ) - - y x v (a) Stress-strain behavior Path C x y x/ y (b) Intermeiate rincial stress ig.. Comarison between reicte an measure (a) stress strain behavior an (b) intermeiate rincial stress uner lane strain ath C with change in rincial stress irections (ata after []). () new harening arameter is erive by consiering unifie yiel an lastic otential functions for both clay an san. It not only escribes the ilatancy for lightly to heavily ilatant san, but also reuces to the lastic volumetric strain for clay. The valiity of the harening arameter is confirme by triaxial comression an extension tests on san along various stress aths. () n elastolastic moel is roose by alying the transforme stress tensor r ij (base on the SMP criterion) an the new harening arameter H to the Cam-clay moel. The roose moel can reasonably escribe the stress strain behavior of clay an san uner three-imensional stress states. () The five soil arameters (k, j, M, M f an m) in the roose moel can be etermine by a loaing an unloaing isotroic comression test an a conventional triaxial comression test. 6 cknowlegements This stuy was suorte by the National Natural Science ounation of China, NSC (No. 67 an No. 79). The authors thank Prof. S.W. Sloan an Dr. D.C. Sheng at the University of Newcastle, ustralia, for their hels in imroving the quality of the aer. The authors are also areciative of the contributions mae by D.C. Lu an W. Hou at Beihang University, China, who mae art of the reictions in this aer. References [] nanarajah. granular material moel base on associate flow rule alie to monotonic loaing behavior. Soils oun 99;(): 9. [] Baret JP. bouning surface moel for sans. SCE J Eng Mech 96;(EM):9 7. [] Dafalias Y. Bouning surface lasticity. I: Theory. SCE J Eng Mech 96;(EM): 9. [] Hashiguchi K, Ueno M. Elasto-lastic constitutive laws of glanular materials. In: Proceeings of secial session 9 of 9th international conference on soil mechanics an founations engineering, Tokyo, [] Lae PV. Elasto-lastic stress strain theory for cohesionless soils with curve yiel surface. Int J Solis Struct 977;:9. [6] Lae PV, Prabucki MJ. Softening an reshearing effects in san. Soils oun 99;():9. [7] Luong MP. Stress strain asects of cohesionless soils uner cyclic an transient loaing. In: International symosium on soils uner cyclic an transient loaing, 9... [] Matsuoka H, Nakai T. Stress-eformation an strength characteristics of soil uner three ifference rincial stresses. Proc JSCE 97;:9 7. [9] Matsuoka H, Yao YP, Sun D. The Cam-clay moels revise by the SMP criterion. Soils oun 999;9(): [] Miura N, Murata H, Yasufuku N. Stress strain characteristics of san in a article-crushing region. Soils oun 9;():77 9. [] Nakai T. n isotroic harening elastolastic moel for san consiering the stress ath eenency in three-imensional stresses. Soils oun 99;9():9 7. [] Nakai T, Matsuoka H. generalize elastolastic constitutive moel for clay in three-imensional stresses. Soils oun 96;6(): 9. [] Nakai T, Tsuzuki K, Yamamoto M, Hishia T. nalysis of lane strain tests on normally consoliate clay by an elastolastic constitutive moel. In: Proceeings of the st Jaan national conference on soil mechanics an founation engineering, vol., [in Jaanese]. [] Nova R, Woo DM. n exerimental rogramme to efine the yiel function for san. Soils oun 97;():77 6. [] Nova R, Woo DM. constitutive moel for san in triaxial comression. Int J Numer nal Methos Geomech 979;: 7. [6] Pastor M, Zienkiewicz OC, Leung KH. simle moel for transient loaing in earthquake analysis. Part II: Non-associative moel for sans. Int J Numer nal Methos Geomech 9;9:77 9. [7] Prevost JH. simle lasticity theory for frictional cohesionless soils. Soil Dyn Earthquake Eng 9;():9 7. [] Roscoe KH, Burlan JB. On the generalise stress strain behavior of wet clay. In: Heyman J, Leckie, eitors. Engineering lasticity. Cambrige, Englan: Cambrige University Press; [9] Roscoe KH, Schofiel N, Thurairajah. Yieling of clay in states wetter than critical. Geotechnique 96;:.

13 Y.P. Yao et al. / Comuters an Geotechnics () [] Sheng D, Sloan SW, Yu HS. sects of finite element imlementation of critical state moels. Comut Mech ;6(): 96. [] Sloan SW, bbo J, Sheng D. Refine exlicit integration of elastolastic moels with automatic error control. Eng Comut ;( ):. [] Wroth CP, Houlsby GT. Soil mechanics-roerty characterization an analysis roceures. In: Proceeings of th international conference on soil mechanics an founations engineering, San rancisco, vol., 9... [] Yao YP, Matsuoka H, Sun D. unifie elastolastic moel of san eenent on stress level an voi ratio. In: Proceeings of n international symosium on re-failure eformation characteristics of geomaterials, Setember 7 9, Torino, Italy, [] Yao YP, Sun D. lication of Lae s criterion to Cam-clay moel. J Eng Mech SCE ;6(): 9. [] Yao YP, Sun D. lication of Lae s criterion to Cam-clay moel (Closure). J Eng Mech SCE ;7(6):6. [6] Yu HS. CSM: unifie state arameter moel for clay an san. Int J Numer nal Methos Geomech 99;:6. [7] Zienkiewicz OC, Leung KH, Pastor M. simle moel for transient loaing in earthquake analysis. Part I: Basic moel. Int J Numer nal Methos Geomech 9;9: 76.