Creep Prediction of an Undisturbed Sensitive Clay

Transcription

1 Creep Predition of an Undisturbed Sensitive Clay Gilberto Alexandre, Ian Martins, Paulo Santa Maria To ite this version: Gilberto Alexandre, Ian Martins, Paulo Santa Maria. Creep Predition of an Undisturbed Sensitive Clay <hal > HAL Id: hal Submitted on 6 Nov 2013 HAL is a multi-disiplinary open aess arhive for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or private researh enters. L arhive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou privés.

2 Creep Predition of an Undisturbed Sensitive Clay Alexandre, G. F., D.S. Martins, I.S.M., D.S., Assoiate Professor, COPPE/UFRJ, Rio de Janeiro, Brazil. Santa Maria, P.E.L, Ph.D., Senior Speialist Engineer, Subsea7, Rio de Janeiro, Brazil Abstrat This paper presents numerial preditions for the behavior of the sensitive marine Haney Clay subjet to undrained reep and onstant load tests. The preditions were arried-out based on the framework model proposed by Martins (1992) and in aordane with onepts developed by Terzaghi (1941), Taylor (1942, 1948), Bea (1960), Lo (1969a, 1969b), Bjerrum (1973), Finn and Snead (1973), Vaid and Campanella (1977) and others. The omplete differential equations as well as simple numerial proedures used to predit the undrained reep and onstant load tests are presented. In addition, analytial solutions are presented for the simplified differential equations of both undrained reep and onstant load tests. It is shown that satisfatory preditions were ahieved both qualitatively and quantitatively for most of the 11 undrained reep and 9 onstant load tests. Keywords Undrained reep. Constitutive equations. Soil behavior. Strain rate effets. Time effets. Visosity. Adsorbed water layer. Introdution Clays present time-dependent behavior. Experimental evidene of the this behavior based on onsolidation and shear strength harateristis have been presented by Buisman (1936), Taylor (1942, 1948), Casagrande & Wilson (1950), Murayama & Shibata (1958), Bishop and Henkel (1962), Mithell et al. (1968), Crawford (1964), Bishop and Lovenbury (1969), Bjerrum (1973), Finn and Snead (1973), Laerda (1976), Vaid and Campanella (1977), Tavenas et al (1978), Mesri et al (1981), Leroueil et al (1985), Martins (1992) and others. The study presented herein deals with the behavior of lays subjeted to shearing under undrained loadings and is applies to the tests arried-out by Vaid and Campanella (1977) on the Haney Clay. In more speifi terms, this study aims to answer the following questions regarding the undrained reep behavior of a saturated lay: Will failure our to a lay speimen subjeted to a given stress state? If the speimen fails, how long will it take to fail? And; If not, what will be the final state of strain of the speimen and how long will it take for the speimen to reah it? To answer these questions, the model developed by Martins (1992) for saturated layey soils, as modified by Alexandre (2006), was used. Previous Studies and Approahes It seems that Casagrande and Wilson (1951) were among the first investigators to study reep and the effets of rate of loading on the shear harateristis of soils. They showed that soil speimens subjeted to undrained reep loadings failed with deviatori stresses in the range of 40% to 80% of the maximum deviatori stresses of onventional tests.

3 Bishop and Lovenbury (1969) arried-out long term drained reep tests on London Clay as well as on Panone Clay that lasted for about 3.5 years. They observed the lak of seondary or steady-state reep and demonstrated the limitation of the power law or logarithmi funtions in representing strain vs time urves. Finn and Snead (1973) arried out undrained reep tests on the Haney Clay. The speimens were left with losed drainage prior to the shearing phase for 8 hours when most of the pore-pressure dependent on the seondary onsolidation developed. The investigators observed the lak of seondary (steady-state) reep. They attributed the start of failure to the minimum strain rate of reep tests and to an upper yielding strength. Also, aording to them, speimens subjeted to deviatori stresses below the upper yielding strength would not fail and speimens subjeted to deviatori stresses greater than the upper yielding strength would fail. They also proposed an equation for the upper yielding strength, reprodued below: Where: " d = " uy + K # $ 1/ n (1) " d is the maximum deviatori stress in a onstant strain rate test or the deviatori stress in a reep test; " uy is the upper yielding strength; " is the strain rate in a onstant strain rate test or the transient minimum strain rate in a reep test; and K and n are onstants; For the Haney Clay the authors found that n = 3, a value whih was onfirmed by Sherif (1965) for the lays of Seattle. Aording to Finn and Snead (1973), the idea of an upper yielding strength is also postulated by Murayama and Shibata (1961) and by Vialov and Skibitsky (1957). Bjerrum (1973), aepting that the shear strength an be represented by the parameters proposed by Hvorslev, explains a mehanism for reep mentioned in Shmertmann and Hall (1961) and proposed by Bea (1960). Aording to Bjerrum, if the applied stress in a reep test is not greater than the maximum frition resistane available, the ohesion, whih is fully mobilized for very small strains at the beginning of the test, would eventually be entirely transferred to frition with reep deformations oming to an end. On the other hand, if the applied stress is greater than the available frition, the transferene of ohesion to frition will ontinue until all the available frition is mobilized. However, beause the applied stress is greater than the available frition, the differene between the stress applied and frition will be arried by the ohesion. As the ohesion is assumed to be strain rate dependent, the strain rate will derease until all the available frition is mobilized and remain onstant thereafter. Having these onepts in mind, the reep proess involves the transferene of effetive ohesion to effetive frition. Vaid and Campanella (1977) arried out several strength tests to simulate various deformation rate histories. Tests suh as reep (onstant stress), onstant load, onstant rate of loading, onstant rate of strain and step reep were arried out on the Haney Clay. The intent was to test the hypotheses that the shear stress, q, is a funtion of the strain, ", as well as the strain rate, ". That is q = q (", ").

4 The authors were able to show that the relationship between these variables holds throughout the entire reep proess for the Haney Clay, even when the strain rate reahes a minimum and starts to inrease again. They also showed that the minimum strain rate for the Creep Test orresponds to a strain of about 2.5%, whih is lose to the strains at maximum deviatori stress of the Constant Rate of Strain Test. With regards to models for explaining reep behavior, one of the main models is the Rate Proess Theory. This theory was developed in the area of Physial Chemistry and was originally intended for assessing the speed at whih hemial reations our. Various investigators suh as Murayama and Shibata (1958), Mithell et al (1968) and Anderson and Douglas (1970) applied the Rate Proess Theory to soil mehanis with suess. It is reommended that works by Glasstone et al. (1940) as well as Mithell et al (1968) be referred to for the fundamentals of this theory. Other models for assessing reep inlude viso-elasti, viso-plasti or viso-elasto-plasti models ombined with or not with the Rate Proess Theory or the onept. A few of the models in this ategory were desribed by Murayama and Shibata (1958, 1961, 1964), Mesri et al (1981), Adahi and Okano (1974), Sekigushi (1984) and Kutter and Sathialingam (1992). As it will be seen in the following setion the model developed by Martins (1992) falls within this ategory. Basi onepts of Martins' Model (1992) Aording to Terzaghi (1941), the ontat between lay partiles an be separated into solid bonds and film bonds. In his view, both ontats are able to transmit effetive stresses and would result from the adsorbed water layers that surround the lay partiles. The solid bonds would result from the ontat between the adsorbed water layers in the immediate viinity of the lay partile, whih, aording to Terzaghi, would be in the solid state. The film bonds would result from the ontat between adsorbed water layers whih would not be in the solid state but whih would possess a higher visosity than the visosity of the free water (by free water it refers to the water that flows out of the voids between soil partiles during seepage or onsolidation). Having this piture in mind, Martins assumes as a hypothesis that the shear strength of a saturated normally onsolidated lay has two omponents; the fritional resistane and the visous resistane. The fritional resistane would develop between Terzaghi's solid bonds and it would be a funtion of the shear strain. The visous resistane would develop between Terzaghi's film bonds and it would be a funtion of the strain rate. The equation for shear strength would then be: Where: " = $ #% tg & mob # + '( e) % ( (2) #" is the normal effetive stress (taken as the differene between the normal total stress, ", and pore-pressure, u); # mob " is the mobilized effetive angle of internal frition; "( e) is the oeffiient of visosity of the adsorbed water layer surrounding the lay partiles (a funtion of void ratio for a normally onsolidated lay); and " is the strain rate. The model assumed that pore-pressure that develops in a shear test would be a funtion of the shear strains as shown by Lo (1969a, 1969b). In addition, it is assumed in the model that normalization is valid. In other words, both the fritional and visous resistanes are proportional to onsolidation pressure, σ', and are a funtion of Over Consolidation Ratio (OCR). Equation (2) is similar to the

5 equation proposed by Taylor (1948). Mehanism of Creep Modified Martins Model Alexandre (2006) modified Martins model by replaing Equation (2) by the following: Where: " d = " df (#) + K( e) $ # n (3) " d is the deviatori stress of a reep test; " df is the deviatori frition resistane (onsidered a funtion of the shear strain for normally onsolidated soils); K and n are onstants (K is also a funtion of the onsolidation pressures, #" ). Equation (3) an be seen as a generalization of the equation proposed by Finn and Snead (1973) and is onsistent with the hypothesis raised by Vaid and Campanella (1977), where the shear stress, q, is a funtion of strain, ", and the strain rate, ". The mehanis of reep an be understood with the aid of Equation (3) and the Figure below, where two reep tests are represented together with what is alled in aordane to Martins model, the basi deviatori urve for a normally onsolidated lay. " d " d 2 " d1 Senario 2 A Senario1 B " = 0 basi urve C " df max " dfa " A O " B " f ε " Figure 1 Deviatori stress x strain urves for two reep tests The basi deviatori urve is the omponent of the strength whih is independent of rate or time effets. This urve is the one that would be obtained in a test where the strain rate is zero, provided suh a test ould be arried out. In addition, a Creep Test is a strength test where the deviatori stress is held onstant throughout its duration. Senario 1 will be analyzed first. As an be seen in Figure 1, the maximum fritional deviatori stress is " df max, whih ours for a given shear strain, " f. At the beginning of Test 1, at t = 0, the deviatori stress, " d1 < " df max is applied instantaneously. Considering Equation (3) and Figure 1, the vertial distane between the urve of Test 1 and the basi urve is therefore the visous resistane. Therefore, at this point, the visous resistane is idential to the applied deviatori stress, ( ) 1 n. After some time, at Point A, the fritional " d1. At this point " assumes the value " = # d K resistane will be " dfa, relative to shear strain " A. At Point A, beause of Equation (3), the visous

6 [( ) /K] 1n. As the proess resistane will be smaller than before, assuming the value " = # d1 $# dfa ontinues, the fritional resistane is mobilized and the visous resistane is demobilized, with ontinuously dereasing strain rate, to balane Equation (3). Beause Creep Test 1 has an applied deviatori stress " df 1 < " df max, the transferene of visous resistane to fritional resistane will ontinue until the fritional resistane equals the applied deviatori stress, at Point B. At this point, the visous resistane and the strain rate are both equal to zero and the shear strain is " B. The same proess ours in Senario 2, but, beause the applied deviatori stress is now greater than the maximum fritional resistane, there will not be not enough fritional resistane to be mobilized. At Point C, where the fritional resistane is maximum, the visous resistane will be minimum and equal to " d 2 #" df max and the strain rate will be " = # d 2 $# df max point on, the soil will ontinue to reep at onstant strain rate indefinitely. [( ) /K] 1 n. From this Having in mind the two Creep Tests desribed above, reep, in the light of the model developed by Martins, would be the proess of transferene of visous resistane to fritional resistane with time, and failure in a reep test would be ahieved when " > 0 and when " # 0. Failure, in other words, would be ahieved when all the available fritional resistane is mobilized and the soil element still ontinuous to deform at a onstant strain rate or with a strain rate that inreases with time. The transferene of visous resistane to fritional resistane during reep as idealized in Martins Model is in general agreement with the reep failure riteria proposed by Bea (1960). The reep proess explained here is able to explain the so-alled primary stage of reep, where the strain rate dereases with time, and seondary stage of reep, where the strain rate remains onstant with time, but is unable to explain the tertiary reep, where the strain rate inrease with time. However, a onjeture for the inrease in time of the strain rate observed during the tertiary stage of reep is presented below, based on Figure 2. " d Visous resistane Creep test " = 0 ( basi urve) ε Figure 2 Conjeture for explaining the tertiary reep Aording to Figure 9, the fritional resistane no longer reahes a plateau in strength but instead passes through a peak. In other words, the fritional resistane, whih is initially zero for shear strain equal to zero, inreases with time, reahes a maximum value and then starts to derease again. This behavior an be observed in over-onsolidated lays, sensitive soils or whenever the pore-pressure ontinues to inrease with time after the soil have reahed the maximum shear strength. In addition to these ases, even normally onsolidated lays an experiene a peak, as long

7 as large deformations our. In this ase the maximum shear strength observed dereases until the residual strength is reahed. Referring to Figure 2 and also to Equation (3), the differene between the applied deviatori stress and the fritional resistane is the visous resistane. As the fritional resistane inreases with the development of shear strains, reahes a peak and starts to derease, the visous resistane will have the opposite behavior. That is, the visous resistane will derease, pass through a minimum and then starts to inrease again. Beause the visous resistane is proportional to the strain rate, the strain rate, as well, will derease with time, reah a minimum (at the peak strain) and then start to inrease again. This possible explanation for the tertiary reep was studied by Alexandre (2006) for the tests arried out by Vaid and Campanella (1977) on the sensitive Haney lay and will be explained in details in this paper. Another onjeture for explaining the tertiary reep is a redution in the oeffiient of visosity with the development of shear strains. If the oeffiient of visosity is shear strain dependent and dereases with the development of strains, then the strain rate will have to inrease to balane Equation (3). Finally, in this onjeture, the tertiary reep may also our by any ombination of the fators explained above. Predition of Creep It an be shown that Equations (2) and (3) are the differential equations for the reep proess. It an also be shown that these equations are non-linear and an analytial solution is very diffiult (if not impossible) to obtain. Therefore, a numerial proedure is required for prediting reep. Referring to Equation (3), with the knowledge of the fritional and visous resistanes, for a given applied deviatori stress, " d, there is a relationship between the shear strain and strain rate. This relationship an be better seen by re-arranging Equation (3) as follows: [ ] %' " = # d $# df (") )' & * (' K + ' 1 n (4) Equation (4) allows the onstrution of a plot for the relationship between the inverse of the strain rate and the shear strain for a given applied deviatori stress, as the one below: " -1 For a given deviatori stress " d Figure 3 Relationship between strain rate and strain for a given deviatori stress in a reep test. Therefore, the expression for the omputation of time is equal to the area between the urve and the ε

8 horizontal axis in Figure 3. That is: t (") = ) 0 # d" & % ( $ dt ' " *1 d" (5) Equations (3) to (5) are valid for any stage of reep, as aording to Martins model, no signifiant differene exists between what is arbitrarily onsidered primary, seondary (or steady-state) or tertiary reep. Creep is, aording to the model, the transferene that ours between visous and fritional resistanes. Although an analytial solution for Equations (2) or (3) may never be obtained, one partiular ase of interest may be solved. Referring to Equation (3) and onsidering a speified stress range, the fritional deviatori stress may be onsidered linearly proportional to the shear strains (although not neessarily implying elasti behavior), and the following differential equation an be written: " d = E #$ + K # $ n (6) Equation (6) is similar to Kelvin-Voigt's rheologial model, although the visosity funtion is nonlinear. The solution of Equation (6) and the expression of the variation of the strain rate with time are presented below. $ " = # ' d & % E ( ) * $ K ' & % E ) + (,. $ K '.& ) %# d ( -. " =,. $ K '.& ) %# d ( -. $ 1*n ' & ) % n ( $ 1*n ' & ) % n ( 1 1 $ + 1* n ' & ) E + t / 1 % n ( K 1 01 $ + 1* n ' & ) + E + t / 1 % n ( K 1 01 $ 1 ' & ) % 1*n ( $ n ' & ) % 1*n ( (7) (8) Referring to Equation (7), it an be seen that for t = 0, " = 0, and for t " # the strain is equal to "= # d E. In addition, onsidering Equation (8), it an be shown that, after applying log to both sides of the equation, an approximate linear relationship (apart from the very beginning of the reep proess) between log( ") and log( t) exists as shown on Figure 4 below. The slope of this urve is, aording to the equation, equal to "1 ( 1" n).

9 " (log sale) 1 1-n Figure 4 Relationship between the strain rate and time in a bi-log sale. Although, stritly speaking, Equations 6 to 8 an be applied only for primary reep for a stress range where E an be onsidered onstant, it an be shown that it may be onsidered as an approximation for the entire proess by onsidering different E's by parts and aording to stress range. Vaid and Campanella (1977) also arried out Constant Load Tests on the Haney. As it will be shown in the next setion, these tests an be analyzed in a similar fashion as the reep tests. For developing the numerial proedure and the analytial solution for the onstant load tests it is only neessary to aount for the derease in the applied initial deviatori stress during the test as a funtion of the shear strain. Predition of Constant Load Tests t (log sale) A Constant Load Test is similar to a Creep Test, but, instead of maintaining the deviatori stress onstant during the entire test, a deviatori load is held onstant. Due to Poisson s effet, as the deformation of the speimen progress with time under onstant loading, the ross setion of the speimen (subjeted to ompressive loads) inreases and therefore dereasing the initial deviatori stress. Referring to Equation (3) and onsidering that the Poisson ratio in an undrained test for a saturated soil is 0.5, it follows that: " d = " d 0 # ( 1$% ) = " df (%) + K # % n (9) Similarly to what was done in the numerial proedure for the Creep Tests, Equation (9) an also be re-arranged to show expliitly the relationship between strain rate and shear strain for a given initial deviatori stress, σ d0. This relationship is: [ ( ) %# df (")] & ( " = # d 0 $ 1%" ' )( K * ( +,( 1 n (10) In a similar fashion, Equation (5) an be used to alulate time required for a speimen subjeted to a given initial deviatori stress to ahieve a ertain strain, ". Referring to Equation (9) and again assuming that the fritional deviatori stress is linearly proportional to the strain (although not neessarily implying elasti behavior) the following equation an be written:

10 " d # ( 1$% ) = E # % + K # % n (11) The solution for differential equation (11) and the expression of strain rate are as follows: $ # ' $ " = d 0 & %# d 0 + E ) * K ' & ( %# d 0 + E ) + 1 (, $ 1*n ' & ) $ K '% n (. $ & %# ) + 1* n ' & ) # d 0 + E. d 0 ( % n ( K -. " =,. $.& % -. K # d 0 ' ) ( $ 1*n ' & ) % n ( 1 ( ) + t $ + 1* n ' # d & ) E % n ( K ( ) + t $ 1 ' & ) % 1*n ( / $ n ' & ) % 1*n ( / (12) (13) It an be seen from Equation (12), that for t = 0, " = 0, and for t " # the strain reahes "= # d 0 (# d 0 + E). In addition, onsidering Equation (13), it an be shown that, after applying log to both sides of the equation, that an approximate linear relationship (apart from the very beginning of the reep proess) between log " Assessment of the Parameters of the Model ( ) and log( t) exists similar to the one represented in Figure 4. Aording to Vaid and Campanella (1977), Haney Clay is believed to have been deposited in a marine environment and later subjeted to partial leahing due to surfae infiltration. It is a grey silty lay with liquid limit = 44%, plasti limit = 26%, maximum past pressure of about 3.5 kg/m2 (340 kpa) and a sensitivity from 6 to 10. Majority of the laboratory tests undertaken by Vaid and Campanella (1977) were normally onsolidated hydrostatially to 515 kpa. After onsolidation, the speimens were left resting in an undrained ondition for 12 hours under the onsolidation pressures prior to shear loading. Aording to Vaid and Campanella (1977), the pore-pressure generated during this undrained period was attributed to the arrest of seondary onsolidation. All measurements were done eletronially and all data were automatially reorded on a digital magneti tape using a high speed (10 hannels per seond) Vidar Digital Data Aquisition System. In addition, the test program was arried out in a onstant temperature environment with a maximum temperature variation of ± 0.25 o C. Aording to Vaid (2004), the tests were not arried out with internal load ells and did not use the free-ends tehnique to minimize frition between the speimens and the top ap and pedestal. Instead, an external load ell with a speially designed ontinuously air leaking seal was used. Aording to Vaid (2004), the maximum frition in the air seal on the loading ram was 10 grams, and was independent of the ell pressure. The experimental results of the Constant Rate of Strain Tests were used for deriving parameters of the Modified Martins Model (the fritional and the visous resistanes) in order to allow for the preditions of the undrained reep and onstant load tests and are reprodued below.

11 Figure 5 Constant rate of Strain Test arried out by Vaid and Campanella (1977)

12 Figures 6 and 7 Creep Tests arried out by Vaid and Campanella (1977).

13 Figures 8 and 9 Constant Load Tests arried out by Vaid and Campanella (1977) Aording to the model developed by Martins the visous resistane in a Constant Rate of Strain test is instantaneously mobilized at the beginning of the test and remains onstant thereafter. This effet an be seen both on a deviatori stress vs strain plot as well as on an effetive stress path plot.

14 The best proedure for assessing the visous resistane, if possible, is by using both plots. The assessment of the visous resistane using only the deviatori stress vs strain plots is very diffiult as it involves the assessment of deviatori stress for very small strains, and may lead to apparent disrepanies in the assessed visous resistane, if not used together with the stress path plots. As pore-pressure measurements of the tests were not available, the assessment of the visous resistanes of the Constant Rate of Strain tests ould only be arried out using the deviatori stress x strain plot. However, as this assessment led to apparent disrepanies, another proedure was developed. First, the visous resistane from the test with the highest strain rate was assessed, as it possesses the highest visous resistane. The visous resistane for the test with the strain rate of 1.1%/min, when normalized with respet to the onsolidation pressure, is about 0.2. However, beause of the reasons explained above, it ould be greater or smaller than this value. Subtrating the visous resistane from the deviatori stress urve of the test with the strain rate of 1.1%/min, the fritional resistane urve was assessed. The Figure below presents the fritional resistane for the test with a strain rate of 1.1 %/min. Figure 10 Fritional resistane urve from the Constant Rate of Strain test with strain rate = 1.1 %/min. Having assessed the fritional resistane urve from this test, the visous resistanes of the other tests were assessed by subtrating, for a given strain, the fritional deviatori resistane for that strain from the deviatori stress at the same strain. For a strain of 2.5%, the normalized fritional resistane is about Subtrating this fritional resistane from the deviatori stresses of the other tests for the same strain, the following visous resistane were obtained: " (%/min) 2V #" 1.5x x x x Table 1 Normalized visous resistanes and respetive strain rates.

15 The following plot shows the pairs of values of strain rate and normalized visous resistanes, 2V " #, as well as a power funtion of the strain rate fitting the data. Figure 11 Assessment of the visous resistane funtion. Where 2V= #" dv is the visous deviatori stress. By assessing the visous resistanes for all the other tests, the fritional resistane for eah test an be assessed. This an be done by subtrating the assessed visous resistane from the deviatori urves for eah one of the other tests. Figure 2 shows the fritional resistanes of all tests as well as a urve representing the average deviatori stress vs strain urve. Figure 12 Assessment of the normalized fritional resistane urves.

16 It is worth noting that, beause the test results are normalized in relation to the onsolidation pressures, both the assessed visous and fritional resistanes are also normalized with respet to the onsolidation pressure. This normalization, however, does not make the analysis invalid, as aording to Martins's model, the normalizing behavior is one of the hypotheses. Finally, it is also worth mentioning that the maximum fritional resistane, 0.46, assessed above, is onsistent with the upper yielding strength onept from Finn and Snead (1973). Although lose, this value differs from the upper yielding value suggested by Vaid and Campanella (1977). For Vaid and Campanella (1977), the upper yielding would be between 0.5, whih is the reep test that did not fail within 3 weeks, and 0.518, whih is the reep test that did fail. Chek of the Assessed Parameters Using Dimensional Analysis Dimensional analysis is frequently used for providing guidane for the oneption, onstrution, exeution and interpretation of physial models. In this work, the tests arried out by Vaid and Campanella (1977) an be onsidered the physial models and therefore dimensional analysis was used to assess the onsisteny of the parameters of the model. The basis of the theory will not be presented here but an be found on Vashy (1890), Bukingham (1915), Bridgman (1922), Langhaar (1965) and Carneiro (1993), By using the Theorem of Vashy-Bukingham, the dimensional matrix of the time-dependent strength problem was assembled and the following " numbers were obtained: " 1 = # % d and " 2 = t #' # df max & $ df max Therefore, if the physial understanding of the proess is orret and if the parameters assessed are onsistent, there must by a funtional relationship between " 1 and " 2. The following figure shows a plot of the two " numbers assessed above for the Constant Rate of Strain, Undrained Creep and Constant Load Tests for a strain of 2.5%. K ( * ) 1 n Figure 13 Dimensional Analysis of the Time-Dependent Strength for the Haney Clay.

17 Numerial Verifiation Equations (3) and (11) an be re-arranged to the following formats: " df # ( ) + K $ # n " d =1 (14) " df # ( ) + K $ ( ) " d 0 $ 1%# # n =1 (15) Using the re-arranged equations above, the onsisteny of the assessed parameters an be heked by using any point of the reep or onstant load tests. Tables 2 and 3 below show the results of this numerial verifiation for the reep and onstant load tests.! Time (min) (% ) d!"! dt d! (%/min)! Equation (16) Error (%) df!" E E E E E E E E E E E E E E E E E E E E E E Table 2 Numerial verifiation for the reep tests.

18 !" Time (min) (% ) d 0!"! dt d! (%/min)! Equation (17) Error (%) df!" E E E E E E E E E E E E E E E E E E E E Table 3 Numerial verifiation for the Constant Load tests. The numerial verifiation arried out in this setion an be seen as similar to the one arried out by Vaid and Campanella (1977) for heking the validity of the equation q = q (", "). Undrained Creep and Constant Load Test Preditions Figures 14 to 52 shows the strain x time and strain rate x time plots for all the reep and onstant load tests. For arrying out the preditions of all the Figures (exept Figures 32 to 35), the numerial proedures desribed in this paper were used. For integrating the areas below the inverse of the strain rate x strain urve (Figure 3) the method on the Trapezoids with a strain step of 0.05% was used. The preditions arried out for Figures 32 to 35 were made using the analytial solution of the differential equation, Equations (7) and (8) with a different E every 0.05% strain interval. All preditions made use of the assessed visous resistane funtion and three different fritional resistane urves. The fritional resistane urves used were the average urve (showed as a thik blak solid line) and two other urves (showed as thin blak dashed line with rosses) representing the upper and lower bounds of the fritional resistane data shown in Figure 12. A disussion of the numerial sensitivity of the preditions is inluded in the disussion of the results. The tests data presented in the plots were obtained by interpolation of Figures 4, 5, 6 and 7 from Vaid and Campanella (1977).

19 Figures 14 and 15 Creep test -" d = 0.638

20 Figures 16 and 17 Creep test - " d = 0.616

21 Figures 18 and 19 Creep test - " d = 0.600

22 Figures 20 and 21 Creep test - " d = 0.586

23 Figures 22 and 23 Creep test -" d = 0.572

24 Figures 24 and 25 Creep test -" d = 0.552

25 Figures 26 and 27 Creep test - " d = 0.530

26 Figures 28 and 29 Creep test - " d = 0.518

27 Figures 30 and 31 Creep test - " d = 0.500

28 Figures 32 and 33 Creep test - " d = 0.446

29 Figures 34 and 35 Creep test -" d = 0.374

30 Figures 36 and 37 Constant Load test -" d 0 = 0.630

31 Figures 38 and 39 Constant Load test -" d 0 = 0.606

32 Figures 40 and 41 Constant Load test -" d 0 = 0.592

33 Figures 42 and 43 Constant Load test -" d 0 = 0.578

34 Figures 44 and 45 Constant Load test -" d 0 = 0.558

35 Figures 46 and 47 Constant Load test -" d 0 = 0.542

36 Figures 48 and 49 Constant Load test -" d 0 = 0.532

37 Figures 50 and 51 Constant Load test -" d 0 = 0.528

38 Figure 52 Constant Load test -" d 0 = Disussion Fritional and Visous resistanes As attested by the orrelation oeffiient ( R 2 = ), the power funtion represents the visous resistane well for the range of strain rate of the laboratory tests. Also, the fritional resistanes of all the laboratory tests, apart from some small variation, are very lose to one another, supporting the hypotheses that the fritional resistane (for a given onsolidation pressure and OCR) is an exlusive funtion of the shear strain and therefore independent of the strain rate. Creep Tests In general, it an be said that the numerial preditions arried out for the Creep Tests are satisfatorily, both qualitatively and quantitatively. However, a more detailed disussion will be presented below. As mentioned in the Introdution, the three most important questions about reep behavior are: Will failure our to a lay speimen subjeted to a given stress state? If the speimen fails, how long will it take to fail? And; If not, what will be the final state of strain of the speimen and how long will it take for the speimen to reah it? To answer the first question, it is suffiient to ompare the applied deviatori stress to the maximum fritional deviatori stress. If the applied stress is greater than the maximum fritional resistane, the speimen will fail. On the other hand, if the applied stress is less than the maximum fritional resistane the speimen will not fail. Making this omparison for all the 11 laboratory tests, in aordane to the model, for reep tests with " d = 0.638, " d = 0.616, " d = 0.600, " d = 0.586, " d = 0.572, " d = 0.552, " d = 0.530, " d = and" d = 0.500, failure was expeted to our and for reep tests with " d = and " d = failure

39 was not expeted to our. Therefore, exept for test " d = 0.500, the numerial preditions are in agreement with the laboratory tests results. The reep test with " d = presents harateristis of failure and stabilization at the same time. This test appears to be stabilizing as the strain rate is ontinuously dereasing, but on the other hand, the strain after 32,000 min is about 4.6% and therefore above the peak strain whih is about %. Taking into aount these peuliar fats, the predition made for this test is in agreement with the test results in the sense that strains greater than % were expeted. Despite of this, the predition is not in agreement with the test results as this test was expeted to reah a minimum strain rate at about 10,000 to 15,000 minutes, whih did not our. As Vaid and Campanella (1977) observed that thixotropi effets were observed after a time of about 20,000 minutes in reep tests with " d = and " d = 0.374, perhaps the differene between the predition and the test results for reep test with " d = may be related to thixotropy as well. Regarding the seond question, the answers will be provided in the ontext of the definition of failure in a reep test aording to Martins model. Aording to the definition, failure will our when " > 0 and when " # 0. Therefore the onset of failure for a soil with a peak in strength is when the reep tests reah the minimum strain rate (whih is onsistent with the proposition made by Finn and Snead, 1973). For the Constant Rate of Strain tests, the peak is in the strain range of about 2.5 % to 3 % and therefore the minimum strain rates should our within this range of strain. Table 4 below presents the preditions and the laboratory tests data: Experimental Data! d!" Minimum " Time (%/min) (min) " % Preditions " Time (min) " % (%/min) ( ) Minimum ( ) x a 3.3 x x a 1.1 x x a 2.2 x x a 7.1 x Table 4 Comparison of minimum strain rate, time and strain for reep tests. Finally, the third question an be answered by omparing the strains from the fritional resistane urve for the reep deviatori stresses of tests " d = and " d = with the strains of these tests. The preditions for test " d = and " d = are between " =1.4% and " =1.6%, and for test " d = are between " = 0.5% and " = 0.7%. The experimental results for tests " d = and " d = 0.374, when they were roughly terminated between about 2 and 3 weeks, are respetively " =1.5% and " =1.0%. Regarding the time for stabilization, the omparison between predition and test results are not possible as, aording to Equation (7), the behavior is asymptoti. However, it is possible to ompare the values of the strain rates for the last experimental data point. For this data point, the strain rate for tests " d = and " d = at t =10,000min and t = 7,000min are 1.55 x 10-5 %/min and 1 x 10-5 %/min, respetively. Aording to the preditions, the strain rates range from 1.14 to 1.55 x 10-5 %/min for test with" d = and from 1.3 to 1.6 x 10-5 %/min for test

40 with " d = In relation to the variation of the strain rate with time, Figure (7), it an be seen that the urves for tests " d = and " d = 0.374, when represented in a log (de/dt) x log (t) plot, are not perfetly straight but slightly urved downwards. The examination of Equation (8) allows for an interpretation of the shape of this urve. Equation (8) was obtained onsidering that the relationship between fritional deviatori stresses and strains an be represented by a straight line. However, the stress-strain diagram of a real soil is not straight. Considering that Equation (8) an be applied by parts, in intervals in whih E an be assumed onstant, for eah interval a different Equation (8) with its respetive E modulus an be applied. The effet of the modulus E on Equation (8) is suh that, having all the other parameters the same, the urve is displaed to the right for dereasing values of the E modulus. The Figure below exemplifies this point. " (log sale) E 0 > E 1 > E 2 > E n E 0 E 1 E 2 E 3 Figure 53 Effet of E on Equation (8). t (log sale) Now onsider a fritional deviatori stress urve suh as the one represented in the Figure below and an undrained reep test with a deviatori stress " d 3. σ d " d 3 E 3 E 2 E 1 E 0 ε 1 ε 2 ε 3 ε f ε 4 ε 5 ε 6 Figure 54 Disretization of a fritional deviatori stress urve with onstant E by parts. ε

41 Aording to the onepts developed so far, it is expeted that this reep test will stabilize at a strain " = " f where the modulus is E 3. However, before reahing this strain, the speimen will reah strains between " = " 3 and " = " 2, where the modulus is E 2, and before that, strains between " = " 2 and " = " 1 where the modulus is E 1. Therefore, the strain rate x time urve of this speimen will start at a urve representing Equation (8) where the modulus is E 0, for " = 0, ross the urves relative to E 1 and E 2 and reah (asymptotially) the urve where the modulus is E 3. In the proess of rossing these urves and reahing the urve relative to E 3, the urve of the speimen subjeted to the reep stress presents itself slightly onvex. This effet is expeted to be more pronouned for reep tests with higher applied deviatori stresses (that do not fail) and for soils presenting strongly urved stressstrain urves before reahing its maximum strength value. For assessing these boundary lines, the following equation an be used: " = - /% K $ # ( /' * & $ d $ # )./ % 1+n ( ' * & n ) 1 % + 1+ n ( ' *, & n ) ( E $ # ), t ( K # ) $ % 1 ( ' * & 1+n ) (16) Equation (16) is Equation (8) normalized in relation to #". Regarding the reep test with " d = 0.446, from Figure 12, for ε = 1.55 % and ε = 0 %, E #" modulus of about E #" = 0.033% -1 and E #" = 1.65% -1 an be assessed respetively. For reep test with " d = 0.374, from Figure 12, for ε = 1.0 % and ε = 0 %, E #" modulus of about E #" = 0.084% -1 and E #" = 1.65% -1 an be assessed respetively. For these values and onsidering that K #" = min and that n = 0.174, the strain rate for a given time an be assessed using Equation (16) and the parameters mentioned above, the following strain rates were assessed for the seleted times shown in the table. " d "(%) E " # (% -1 ) Time (min) " (%/min) x x x x x x x x x x x x10-5 Table 5 Assessment of the boundary lines for reep tests with " d = and " d = The boundary lines assessed on Table 5 are shown as thik dashed lines in the figures below:

42 Figure 55 Strain rate x time boundary lines for test with " d = Figure 56 Strain rate x time boundary lines for test with " d = In addition, the onvexity of the strain rate x time urve an, of ourse, be seen on Figure 7 as well. When all the test results are ompared together it an be seen that, in general, for reep tests with a deviatori stress equal or smaller than " d = the predition deviate more from the tests results than the other tests. Looking at the strain rate, it appears that tests that presented strain rates below 1x10-3 %/min show greater deviations than the others. The effet of the strain rate an also be seen on the Constant Rate of Strain tests. The tests with onstant strain rate equal or lower than 2.8x10-3 %/min show a smaller derease in the strength with strain than the others. Therefore it is believed that " d = was affeted by the thixotropy.

43 Constant Load Tests In general, it an be said that the preditions arried out for the Constant Load Tests are also satisfatorily, both in qualitative and quantitative terms. As for the Creep tests results, a more detailed disussion will be presented below for the Constant Load tests. As the load is onstant and the ross setion area of the speimen inreases with strain, the initial deviatori stress dereases with strain as well. As pointed out before, the urrent deviatori stress is related to the initial deviatori stress by the equation " d = " d 0 # ( 1$% ). In this ontext, to answer the first question about the failure of a speimen subjet to a given deviatori stress, it is neessary to ompare the urrent deviatori stress with the fritional deviatori stress for the same strain. This omparison an be made with the help of Figure 57 below. σ d Minimum visous resistane Constant Load test " d = " d 0 # 1$% ( ) " = 0 basi urve Visous resistane Visous resistane inreasing " for minimum " ε Figure 57 - Constant Load Test. If the urrent fritional deviatori stress is greater than the fritional deviatori stress (for the same strain), the speimen will ontinue to deform. That means, if the onstant load test urve is above the fritional deviatori urve for any strain, the speimen will not stabilize. Stabilization will our only if the urrent stress urve touhes the fritional resistane urve. Undertaking this omparison for the Constant Load Tests, it is predited that no test will stabilize for the range of strains experiened by the tests. The tests results show that in fat 7 of the 9 tests fail by reahing the minimum strain rate and presenting an inrease in the strain rates afterwards. Tests " d 0 = and " d 0 = did not pass through a minimum in strain rate although they experiened large strains (greater than about 7% in both tests) as predited. In aordane to the onept that, by having a onstant load urve above the fritional urve the speimen will not stabilize, these two tests an also be onsidered to have failed. Therefore preditions and test results regarding stabilization agree. The question about the minimum strain rate and its relationship with the strains will be addressed onsidering the stress derease with the development of strains that ours in a Constant Load Test. Beause of the shape of the urrent deviatori stress funtion, the minimum visous resistane, and therefore, the minimum strain rate, will not neessarily our for the strain related to the peak strength, but for a strain somewhat greater than that. The minimum strain rate will our for the

44 strain where the visous resistane is minimum. For the values of the initial deviatori stresses of the tests arried out by Vaid and Campanella (1977) and onsidering the shape of the fritional resistane, the strain interval within whih the minimum strain rates are expeted to our are between 2.9% and 4%. Table 6 below presents the preditions and the tests results data. σ d0 /σ Experimental Data Minimum Time dε/dt (%/min) (min) ε (%) Minimum dε/dt (%/min) Preditions Time (min) ε (%) to to to x to 7.6 x to to x to 4.0 x to to x to 1.9 x to to x to 6.7 x to to x to 1.7 x to to No minimum strain rate reahed 4.1 to 6.7 x to to No minimum strain rate reahed 2.65 to 4.44 x to to 4.0 Table 6 Comparison between preditions and the Constant Load tests results for the point of minimum strain rate. Comparison with test " d 0 = was not possible as the strain rate vs time urve of this test was not presented by Vaid and Campanella (1977). Apart from this test for whih a omparison was not possible, it an be seen agreement between preditions and test results in 6 of the 8 tests. It is believed that the differene between the preditions and the tests results for Constant Load tests with " d 0 = and " d 0 = may also be attributed to the thixotropy effets as these were the only Constant Load tests that presented strain rates below 1x10-3 %/min. Conlusions Considering the results of the preditions for the sensitive undisturbed Haney Clay, the following onlusions an be made: The separation of the shear strength of the Haney Clay into the fritional and visous resistanes for explaining the undrained reep behavior of the Haney Clay, as established by Martins (19992), an be onsidered adequate; The visous resistane an be represented by a power law funtion of the strain rate for the range of strain rates observed in this study; The hypothesis of onsidering the fritional resistane as a unique funtion of the shear strain, and therefore independent of the strain rate, as established by Martins, was verified for the Haney Clay; The hypothesis of normalization, as adopted by Martins (1992) was also verified for the Haney Clay; The undrained behavior under onstant stress or onstant load for the Haney Clay an be onsidered as a unique proess, an interation between fritional and visous resistanes, and not a segmented one. The minimum strain rates in reep tests are assoiated with the peak strength strain range as the minimum visous resistane ours within this strain range. The minimum strain rates for the onstant load tests our at strains greater than the peak strength strain. However, the minimum strain rates for these tests are also assoiated with

45 the minimum visous resistane whih our for a different strain range than the reep tests. The so-alled tertiary reep, for the Haney Clay, an be onsidered as a onsequene of the derease of the fritional resistane and therefore inrease in the visous resistane of the soil; The model developed by Martins (1992) for non-sensitive, saturated, normally onsolidated lays, as modified by Alexandre (2006), was able to predit qualitatively the behavior of the sensitive Haney Clay. The model was also able to predit quantitatively the behavior of the Haney Clay for the tests not affeted by the thixotropy. Aknowledgements The authors wish to thank Professor YoginderVaid, Professor Emeritus of the University of British Columbia for the disussions and ritiism and Mrs. Mei Cheung, M.S., P. Eng., for the reading of this manusript, disussions and for the many suggestions presented. Referenes Adahi T., Okano M., (1974), A onstitutive equation for normally onsolidated lay, Soils and Foundations, 14, (4), Alexandre (2006), Contribution to the Understanding of the Undrained Creep, D.S. thesis, COPPE/UFRJ, Rio de Janeiro, Brazil (in Portuguese) Anderson, O. B. and Douglas, A. G. (1970), Bonding, Effetive Stresses and Strength of Soils, J. Soil Meh. Found. Div., Pro. Am. So. Civ. Eng. 96: Bea, R. G. 1960, An experimental study of ohesion and frition during reep in saturated lay. Master s thesis to the University of Florida, 107 pp. Bishop, A. W., and Henkel, D. J., The Measurement of Soil Properties in the Triaxial Test: London (Edward Arnold). Bishop, A.W. & Lovenbury, H.T. 1969, Creep harateristis of two undisturbed lays, Pro. 7th ICSMFE, Mexio, Vol. I, pp Bjerrum, L. 1973, Problems of soil mehanis and onstrution on soft lays., State-of-the-art- Paper to Session IV, 8th ICSM & FE, Mosow, Vol. 3, pp. 124, 134. Buisman, A. S. Keverling (1936), "Results of Long Duration Settlement Tests," Pro., Intern. Conf. on Soil Meh. and Found. Engr., Vol. 1, pp Casagrande, A. and Wilson, S.D. (1951) Effet of rate of loading on strength of lays and shales at onstant water ontent. Geotehnique, 2(3), Crawford, C.B Interpretation of the onsolidation test. Journal of the Soil Mehanis and Foundation Division, ASCE, 90: Finn, W. D., and Snead, D. E., (1973), Creep and reep rupture of an undisturbed sensitive lay, Pro. 8 th International Conferene on Soil Mehanis and Foundation Engineering. Mosow, USSR. Glasstone, S., Laidler, K. J., & Eyring, H., 1940, Theory of Rate Proesses. First Edition, New York: MGraw-Hill Book Co. Kutter, B.L., and Sathialingam, N., (1992), Elasti visoplasti modelling of the rate-dependent