Japanese Getechnical Sciety Special Publicatin The 15th Asian Reginal Cnference n Sil Mechanics and Getechnical Engineering Effects f time and rate n the stress-strain-strength behavir f sils Jian-Hua Yin i) i) Department f Civil and Envirnmental Engineering, The Hng Kng Plytechnic University, Hng Kng, China ABSTRACT This paper presents main data frm bth edmeter tests and triaxial tests n Hng Kng Marine Clay (HKMC) and a mixture f bentnite-silica. The edmeter tests include (i) multi-staged lading tests with unlading/relading and enugh time fr creep and swelling in 1D straining and (ii) step-changed cnstant rate f strain cmpressin tests with unlading/relading in 1D straining as well. The triaxial tests include K -cnslidated undrained cmpressin r extensin tests with cnstant (r stepped changed) strain rates. This paper will discuss the time effects and rate effects with a special attentin t the nn-linear and elastic visc-plastic behavir. Based n the test data presented abve, this paper presents a brief review f the wrks f elastic visc-plastic (EVP) mdelling f the time-dependent stress-strain behaviur f sils in ne-dimensinal straining (1D) and in 3D stress state. A few imprtant cncepts and their physical meanings are explained. The 1D EVP mdel is briefly reviewed with a cmparisn with the classic Maxwell s rhelgical mdel. It is fund that Yin and Graham s 1D EVP mdel is an extensin f Maxwell s rhelgical mdel fr cnsidering the nnlinear behaviur f sils. The recent extensin f the EVP mdelling framewrk t cnsider the swelling f a saturated sil is intrduced. A 3D EVP mdel is als intrduced and discussed. A nnlinear creep functin prpsed by the authr is presented. This functin has been used in refined 1D and 3D EVP mdels. It is cncluded that the time and rate effects f clayey sils shall be cnsidered in a suitable cnstitutive mdel. Keywrds: clay, sil, creep, time effects, rate effects, elastic, visc-plastic, stress-strain 1. INTRODUCTION The stress-strain behaviur f many sils is time-dependent r rate-dependent. Creep and swelling are tw special aspects f the time-dependence. Sme clayey sils exhibit mre r less bth creep and swelling. In this paper, creep means viscus cmpressin under a cnstant lad in an edmeter cnditin; while, swelling means viscus expansin under a cnstant lad in an edmeter cnditin and is a reverse behaviur t creep, all fr saturated sils. Clayey sils cntaining clay mineral mntmrillnite shw strng creep and swelling characteristics. Bth creep and swelling characteristics have a significant influence n the defrmatin and failure f getechnical structures. Stress relaxatin is anther time-dependent phenmenn in which the effective stress is decreasing with time when strain is keep cnstant. The rate effects include the strain rate and stress-rate effects in which the effective stress increases with the rate. This paper presents main data frm bth edmeter tests and triaxial tests n Hng Kng Marine Clay (HKMC) and a mixture f bentnite-silica, discuss the effects f time and rate, and elastic visc-plastic mdelling.. OBSERVATIONS FROM LABORATORY TESTS.1 Oedmeter tests n HKMC Fig.1 shws the curves f lg(time) and vertical strain frm a multi-stage lading (MSL) edmeter test n undisturbed Hng Kng Marine Clay (HKMC) (Yin and Cheng 6). It is seen frm the figure that the HKMC exhibits creep strain after the End f the Primary Cnslidatin (EOP). Fig. shws curves f effective vertical stress and vertical strain a multi-stage CRS (Cnstant Rate f Strain) edmeter test and MSL test n the same HKMC (Yin and Cheng 6). The CRS rates are.18%/hurs,.18%/hurs, and 1.8%/hurs. The strain rate effects are seen clearly. The larger the strain rate, the higher the effective vertical stress. http://di.rg/1.38/jgssp.hkg-1 44
Time in Lg Scale (min).1 1 1 1 1 1 1 1kPa 5kPa 4 5kPa 6 8 1kPa kpa 1 1 Creep 14 16 18 Fig.1. Results frm a multi-stage lading edmeter test 4 6 8 1 1 14 16 18.18 %/hr Frm Oedmeter Test.18 %/hr 1.8 %/hr.1 1 1 1 1 Effective Vertical Stress (Lg Scale), σ (kpa) Fig.. Results frm a multi-stage CRS edmeter test. Triaxial tests n HKMC Fig.3 shws results frm fur multi-stage CRS K -cnslidated cmpressin triaxial tests (deviatr stress q vs axial strain in tp and prewater pressure u vs axial strain in the bttm) (Yin and Cheng 6). The CRS rates are.%/hurs, %/hurs, and %/hurs. It is seen clearly that the strain rates affect curves f stress-strain and the prewater pressure and strain. The larger the strain, the higher deviatr stress and larger the excess prewater pressure. Fig.4 shws results frm multi-stage CRS K -cnslidated extensin triaxial tests (q vs axial strain in tp and prewater pressure u vs axial strain in the bttm) (Yin and Cheng 6). The extensin test was dne that the cnfining stress was kept cnstant while the vertical stress was reduced at a step-changed strain rate. The strain rate effects are seen clearly..3 Oedmeter tests n a mixture f bentnite-silica Fig.5 shws curves f lng(time) and vertical strains frm a multi-stage lading and unlading/relading edmeter test n a mixture f bentnite and 7% f silica (Yin and Tng 11). It is seen that the mixture shws bth creep and swelling. Deviatr Stress, q (kpa) Pre Water Pressure, u (kpa) 5 45 4 35 3 5 C4 C15b Test schedule fr all cmpressin tests +%/hr +%/hr +.%/hr Stage1 Stage Stage3 Stage4 +%/hr +.%/hr +%/hr -%/hr (U) +%/hr (R) 15 C15a 1 Sample N.: BH-4- Plastic limit: 6 Liquid limit: 49 5 C5 Plastic index: 3 Clay 7%/Silt 43%/Sand 3% 4 6 8 1 1 14 16 18 5 15 1 5 C15b C5 C4 C15a +%/hr +%/hr +.%/hr 4 6 8 1 1 14 16 18 Fig.3. Results frm multi-stage CRS K -cnslidated cmpressin triaxial tests (q vs axial strain in tp and prewater pressure u vs axial strain in the bttm) Deviatr Stress, q (kpa) Pre Water Pressure, u (kpa) 4 3 Test schedule fr all Stage1 Stage Stage3 Stage4 -%/hr -.%/hr -%/hr +%/hr (U) extensin tests -%/hr (R) Sample N.: BH-4- Plastic limit: 6 1 Liquid limit: 49 E5 Plastic index: 3 Clay 7%/Silt 43%/Sand 3% -4 - - -18-16 -14-1 -1-8 -6-4 - -1 - -3-4 3 5 15 1 5-1 -15 -.%/hr -%/hr -%/hr -.%/hr -%/hr -%/hr -4 - - -18-16 -14-1 -1-8 -6-4 - -5 Fig.4. Results frm multi-stage CRS K -cnslidated extensin triaxial tests (q vs axial strain in tp and prewater pressure u vs axial strain in the bttm) E5 E15 E15 E4 E4 441
Fig.5. Results frm a multi-stage lading and unlading/relading edmeter test (time and strain curves) Vertical strain ε (%) _ (a) Time (lg scale) (min).1.1 1 1 1 1 1 Vertical strain (%)_ 1 3 4 5 1 1 1 1 1 3 4 1kPa 5kPa 5kPa 1kPa kpa 4kPa 8kPa (b) Time (lg scale) (min).1.1 1 1 1 1 1 1 Vertical strain (%)_ Vertical strain (%)_ 3 4 4 34 44 Effective vertical stress σ (kpa) Lps Creep Swelling 5kPa 5kPa 1kPa kpa 1kPa 5kPa (c) Time (lg scale) (min).1.1 1 1 1 1 14 5kPa 5kPa 1kPa 1kPa kpa 4kPa Fig.6 shws the curve f effective vertical stress in lgarithmic scale and vertical strain frm data in Fig.5. Large unlading and relading lps are bserved. This is caused by significant swelling f the bentnite. 3. ELASTIC VISCO-PLASTIC MODELLING Cnstitutive mdelling f the viscus stress-strain behavir f sils is very imprtant and practically needed, nt matter simple and cmprehensive mdels. This is because (a) analysis, design and cnstructin f mst getechnical structures need t cnsider the time and strain effects and nn-linear and plastic defrmatins, and (ii) a cnstitutive relatin is a mathematic expressin f the physical stress-strain behavir including the time and strain effects, nn-linearity, plastic strains etc. with in-depth understandings and lgical expressin f these physical phenmena. The item (ii) is nt an easy task. 3.1 An elastic visc-plastic mdel and extensin f the classic Maxwell s linear rhelgical mdel Based n the cncepts ( equivalent time, reference time line, etc.) and understandings f the time-dependent behavir, Yin and Graham (1989, 1994) derived rigrusly a cnstitutive equatin f a ne-dimensinal Elastic Visc-Plastic (1D EVP) mdel: κ σ V σ ε = (1) V σ where λ / + exp[ ( ε ε ) ]( ) r Vt σ r ε and ε are vertical strain rate and strain; σ and σ are vertical effective stress rate and stress; κ/v is a cnstant related t elastic cmpressin (see Fig.6); λ/v is a cnstant related t the reference time r line. The meaning f σ and ε r are the define a pint where the reference time line passes. As explained befre /V and t (in units f time) are tw cnstants related t creep f the sil. Mre details can be fund in Yin and Graham (1989, 1994). In (1), the elastic strain rate is visc-plastic strain rate is: e κ σ ε = V σ ; the 5 Stress-strain (7% sand) V σ ε = exp[ ( ε ε r ) ]( () Vt σ λ / ) r Fig.6. Results frm a multi-stage lading and unlading/relading edmeter test (stress and strain curve) It is seen frm Eqn.(1) that the visc-plastic strain rate (the creep strain rate) is a functin f the stress-strain state ( σ, ε ) nly, nthing t d with hw t reach this state. It is well knwn that the classic Maxwell s rhelgical mdel can be expressed as: 44
σ σ ε = where ε is the ttal strain rate (i = 1,,3; j = 1,,3); + (3) (11) E η e ε and ε are the elastic and visc-plastic strain rate; where E is the elastic mdulus and η is a viscus cnstant. Maxwell s rhelgical mdel is derived frm a σ is effective stress; the mean effective stress p is series cnnectin f a linear spring and a linear dashpt. defined as p = σ kk / 3 ; s is the deviatr stress rate; In this apprach, the ttal strain rate is divided int elastic strain rate and a visc-plastic strain rate. It is nted that the deviatr stress s is defined as s = σ δσ kk / 3, Maxwell s rhelgical mdel is nt a visc-elastic mdel since the strain due t the dashpt is nt recverable. where δ = if i j, δ = 1 if i = j; G is the elastic Maxwell s rhelgical mdel is, in fact, a linear elastic shear mdulus; κ / V (V is specific vlume), / V, and linear visc-plastic mdel. Cmparing Eqn.(3) with r Eqn. (1), it is fund that Yin and Graham s 1D EVP (Yin t, λ / V, p m and ε vm are mdel parameters. and Graham 1989, 1994) is a nnlinear elastic and The F in Eqn.(5) is a functin describing the nnlinear visc-plastic mdel, that is, a nnlinear visc-plastic flw surface (Yin and Graham 1999): rhelgical mdel. Yin and Graham s 1D EVP can be cnsidered t be an extensin f the classic Maxwell s q linear rhelgical mdel (Yin 1). F = p + p p = m (6) M 3. An elastic visc-plastic mdel cnsidering bth creep and swelling The swelling here is the time-dependent expansin f the skeletn f a sil, in ppsite t creep (cmpressin). Fr saturated sil with bth creep and swelling, Yin and Tng (11) prpsed the fllwing cnstitutive mdel, based n Yin and Graham (1989, 1994): c κ σ 1 ε = + exp[ V σ V t s V s 1 exp[ + t rs ( ε ε ) c rc ( ε ε ) V σ ] s rs σ V σ ] c σ λ s λ c rc Eqn. (4) is a general cnstitutive mdel fr the time-dependent stress-strain behaviur f sils exhibiting bth creep and swelling in 1D straining. This mdel is valid fr all lading cnditins such as cnstant rate f strain (CRS) lading, relaxatin, unlading, relading etc. This new mdel may be called ne-dimensinal Elastic Visc-Plastic mdel cnsidering swelling, namely 1D EVPS mdel. The explanatin f all parameters in (4) can be fund in Yin and Tng (11). 3.3 3D elastic visc-plastic mdels Using the cncept f equivalent time, the Mdified-Cam Clay mdel, and the apprach in 1D EVP mdel (Yin and Graham 1989, 1994), Yin and Graham (1999) derived a three-dimensinal Elastic Visc-Plastic (3D EVP) mdel: ε = ε + ε e exp ε Vt = r vm 1 G κ p s + δ + 3V p λ p + ln V p m m V 1 ε vm p p m F σ (5) where M is the slpe f the Critical State strength envelpe in the q-p plane; and q is the generalied deviatr stress 3 s s. In Eqn.(5) and Eqn.(6), pm is the mean effective stress at which the flw surface in Eqn.(6) intercepts the p axis in the q-p plane. The sub-index m represents the mean stress r vlume strain under istrpic stressing cnditins, that is, p = p m with q =. Fr example, ε vm in Eqn.(5) is the ttal vlumetric strain under istrpic stressing. (4) Yin and Graham (1999) shw that the rate (14) f ε vm can be expressed: κ r λ p V m ε = + vm p m exp εvm + ln εvm (7) Vpm Vt V pm Eqns.(5), (6) and (7) are differential equatins f a 3D EVP mdel fr describing time-dependent stress-strain behaviur f clays. The prpsed mdel has been verified using data frm a number f sils. It is nted that the 3D EVP mdel in Eqn.(5) uses the natural lgarithmic functins fr creep, instant time line, and the reference time line withut limit. T vercme the limitatins, Yin et. al. () extended the 3D EVP mdel t describe the time-dependent behaviur f nrmally and vercnslidated clays using nnlinear functins (see Sectin 4 belw) fr creep. The validatin f the mdel in Eqn.(5) can be fund in Yin and Graham (1999). 4. A NON-LINEAR FUNCTION FOR CREEP PF SOILS IN 1D STRAINING A semi-lgarithmic functin using e = e Cαe lg t where C α e is the c-called the cefficient f secndary cnslidatin has a prblem: when the time t is very big r infinite, the vid rati e becmes 443
negative. A semi-lgarithmic functin using c ε = ε + [ C /(1 + e )]lg( σ / σ ) r e = e C c lg( σ / σ ) has the same prblem ( Cc is the cmpressin index; e is the vid rati crrespnding stress). Yin (1999, 11) suggested a new functin fr fitting nn-linear creep behaviur f sils. In rder t help readers t understand, we use the cmmn lg-functin and parameters t express this nn-linear functin: ε C V = C 1 + V αe αe tp ε l t + t lg t t + t lg t where ε is creep strain (nt including the initial strain befre creep) and t is the duratin f lading and ε l is the creep limit. In Eqn.(8) C α e and t stand fr parameters at time t = and can be determined by fitting Eqn.(8) t creep test data. It is nted that Eqn.(8) is valid fr time t=. When the time t is t + t infinite, lg, ε = ε l. It is nted that if t t + t we treat lg as a new variable x, Eqn.(8) t x x becmes ε = = which V / Cαe + (1/ ε l ) x a + bx is the well-knwn hyperblic functin ( a = V / Cαe and b = 1/ ε l ). It is nted that expnential functin c t ( ε = ε 1 + c1e ) and the hyperblic functin in terms t f time t ( ε = ) are nt suitable fr the nnlinear a + bt creep in 1D straining. We can use the same type f nn-linear functin t fit the relatin between the vertical effective stress and vertical strain (Yin, Zhu and Graham ). Fr example fr the reference time lime (similar t the nrmal cnslidatin line): Ccr σ lg V σ r ε r = (9) Ccr σ 1 + lg Vε σ If rl σ r (8) r σ =, we have ε =. When the stress σ ε = ε rl which is the limit. We can have the same functin type in Eqn.(9) fr the limit time line in Figure 5: ll l σ σ l Ccl σ lg V σ ε l = (1) Ccl 1 + lg Vε If σ σ l σ =, we have ε =. When the stress ε = ε which is the limit. ll 5. CONCLUSIONS Frm the previus presentatin and discussin, the fllwing cnclusins are made: (a) Mst sils, such as Hng Kng Marine Clay (HKMC) and a mixture f bentnite and sand, exhibit varius viscus phenmena, such as creep, swelling, rate effects, nnlinearity, and plastic strains. (b) All these time effects, rate effects, nnlinearity, and plastic strains shall be cnsidered a cnstitutive mdel. (c) The statement f the magnitude f a creep strain rate at a stress-strain state pint is unique, independent f the lading path t reach this pint can be seen in the 1D EVP mdel (Yin and Graham 1989, 1994). (d) The ne-dimensinal Elastic Visc-Plastic (1D EVP) (Yin and Graham 1989, 1994) is rigrusly derived. This 1D EVP mdel is an extensin f Maxwell s linear rhelgical mdel fr cnsidering the nnlinear behaviur f sils. (e) The 1D EVP mdel has been extended t cnsider bth creep and swelling f a mixture f bentnite and sand. (f) The 3D EVP mdel is rigrusly derived based n the 1D EVP mdel apprach and the mdified Cam-Clay mdel fr the time-dependent stress-strain behaviur f clayey sils. Further imprvements f this mdel have been dne and are still needed. (g) The nnlinear functins prpsed by the authr are gd fr fitting the creep cmpressin (with time) and the cmpressin under high stress f mst sft sils in 1D straining. ACKNOWLEDGEMENTS The wrk in this paper is supprted by a research grant (prject N. 517844) frm Natinal Natural Science Fundatin f China (NSFC), a grant (PlyU 15196/14E) frm Research Grants Cuncil (RGC) f Hng Kng Special Administrative Regin Gvernment f China, PlyU Shenhen Research Institute in Mainland, China, and The Hng Kng 444
Plytechnic University, Hng Kng, China. REFERENCES 1) Yin, J-H. 1999, Nn-linear creep f sils in edmeter tests, Getechnique, 49, N.5, pp.699-77. ) Yin, J-H (11). Frm cnstitutive mdeling t develpment f labratry testing and ptical fiber sensr mnitring technlgies, Chinese J f Getechnical Engineering, 33(1), 1~15. (14th Huang Wen-Xi Lecture in China). 3) Yin, J-H. (1). Review f elastic visc-plastic mdeling f the time-dependent stress-strain behaviur f sils and its extensins and applicatins. Cnstitutive Mdelling f Gematerials Advances and New Applicatins, Springer Series in Gemechanics and Geengineering, Yang, Q., Zhang, JM, Zheng, H. and Ya YP (eds.), 149~158. 4) Yin, J-H and Cheng, CM, (6). Cmparisn f Strain-rate Dependent Stress-Strain Behaviur frm K-cnslidated Cmpressin and Extensin Tests n Natural Hng Kng Marine Depsits. Marine Geresurces and Getechnlgy, Vl.4, N., pp119-147. 5) Yin, J. H. and Graham, J. 1989. Viscus elastic plastic mdelling f ne dimensinal time dependent bahavir f clays. Canadian Getechnical Jurnal 6, pp.199 9. 6) Yin, J. H., and Graham, J. 1994, Equivalent times and ne dimensinal elastic visc plastic mdelling f time dependent stress strain behaviur f clays., Canadian Getech. Jurnal 31, 4 5. 7) Yin, J.-H. and Graham, J., 1999, Elastic visc-plastic mdelling f the time-dependent stress-strain behaviur f sils, Canadian Getech. Jurnal, 36, 736-745. 8) Yin, JH and Tng, F. (11). Cnstitutive Mdelling f the Time-dependent Stress-strain Behaviur f Saturated Sils Exhibiting bth Creep and Swelling. Canadian Get J. 48, 187~1885. 9) Yin J-H., Zhu J-G. and Graham, J (): A new elastic visc-plastic mdel fr time-dependent behaviur f nrmally and vercnslidated clays: thery and verificatin. Canadian Getechnical Jurnal 39, 157-173. 445