ANALYQS OF CEAY' CONSOLIDATtON BY RATE PROCESS THEORY ThuEs far the Degree- lof M. S. MICHIGAfl STATE UMVERS! Robes“? 3. Neukirchmr 3965 THESE LIBRARY Michigan State University ABSTRACT ANALYSIS OF CLAY CONSOLIDATION BY RATE'PROCESS THEORY by Robert J. Neukirchner The consolidation of clay is composed of two processes; the first is drainage of excess pore water to reduce pore pressures and the second involves the deformation of the clay particle structure. In this thesis the deformation phenomenon is considered to occur through the breaking and reforming of bonds at the clay particle contacts and is assumed to be gov- erned by the rate process theory. A rheological model is proposed which represents the behavior of the clay structure under load. The model para- meters are related to properties of the clay structure and the parameters are used to evaluate the deformation charactertistics of the clay structure. ANALYSIS OF CLAY CONSOLIDATION BY RATE PROCESS THEORY By Robert J. Neukirchner A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil Engineering 1965 ACKNOWLEDGMENTS The writer is indebted to his major professor, Dr. T. H. Wu, Professor of Civil Engineering, for his everpresent aid, guidance and encouragement throughout the writer's graduate program and during the preparation of this thesis. Thanks also to Dr. A. K. Loh and J. R. Adams for their helpful suggestions and criticisms, and to Professor D. Resendiz of the University of Mexico for supplying the time-consolidation data on the Mexico City clays. Thanks are due also to the Civil Engineering Department of Michigan State University for the financial assistance which made the completion of these studies possible. CHAPTER II III IV VI APPENDIX TABLE OF CONTENTS INTRODUCTION ........................ THEORY ............................... EXPERIMENTAL PROGRAM ............. CALCULATION OF MODEL PARAMETERS TEST RESULTS 1. Theoretical Curves vs. Test Curves . . . . 2. Variation of Model Parameters . . . CONCLUSION .......................... BIBLIOGRAPHY ....................... TEST DATA — TABULAR FORM .......... EXPERIMENTAL TIME- DE FORMATION CURVES CALCULATED PARAMETERS ........... VARIATION OF OLB AND FOR _kL_ INCREASING STRESS - GRAPHICAL SAM PL E CALCULATIONS ............... PAGE 11 22 29 34 37 39 56 61 LIST OF FIGURES FIGURES PAGE 1 Proposed Rheological Model .............. 4 2 Loading Conditions for General Model ..... 7 3 Index Properties of Clays Used ........... 10 4 Graphical Solutions for Case II ............ l4 5 Relation of U* to U** Curves ............. 15 6 Construction of U* vs. t Curve ........... 23 7 Experimental vs. Theoretical Curves - 1 . . . 24 8 Experimental vs. Theoretical Curves - Z . . . 25 9 Experimental vs. Theoretical Curves - 3 . . . Z6 10 Experimental vs. Theoretical Curves - 4. . . 27 11 Variation of Model Parameters with Stress . 31 12 Variation of Model Parameters with Stress . 32 13 Classification of Clay Deformation Curves. . 35 k1,k2 K,L, M,N tP U* U** Uo. U00 zm NOTATION Slope parameter Constant Coefficient of consolidation Boltzmann's constant Planck's constant rheological model parameters constants with respect to time applied pressure increment total effective stress on the model effective stress on the flowing bonds gas constant absolute temperature time time for 100% primary consolidation to occur determined according to log time method axial deformation percent total axial deformation dimensionless strain used in Case I solution initial and final values of axial deformation time parameter rate process theory parameters vi deformations of model (shear strain) octahedral shear strain principal strains initial and final values of strain, 6| average distance between interparticle bonds distance between adjacent planes of slip number of flowing particle contacts per unit area principal stresses octahedral normal stress shear stress shear stress in flowing bonds octahedral shear stres s CHAPTER I INTRODUCTION The reduction in volume with time under a constant applied load is called consolidation. Consolidation of clay has customarily been considered to consist of two distinct parts; a primary and a secondary part. With refer- ence to primary consolidation, Terzaghi and Peck (1948) say, "The gradual decrease of the. water content at constant load is known as consolidation. " The rate of primary consolidation is considered to be governed by the vel- ocity of the flow of water from the material under a hydraulic gradient. The "secondary time effect" is described by the same authors as " . . . due to the gradual adjustment of the soil structure to stress, com- bined with the resistance offered by the viscosity of the adsorbed layers to a slippage between grains. " The primary phase is considered to be adequately described by the flow of water from the voids under a hydraulic gradient and was solved by Terzaghi's one-dimensional consolidation theory. Secondary consolidation, however, cannot be attributed to the hydraulic phenomenon, and may con- tinue for long periods after the excess hydrostatic pressure has approached zero. In an effort to determine the nature of this phenomenon, several authors (Altschaffel, Gibson and Lo, Leonards, Tan, Schiffmann and others) have proposed rheological models designed to describe clay consolidation. In recent studies on clay deformation in general, the rate process theory of Eyring et a1 (1948) has been used to describe the mechanism of slip at interparticle contacts. The result of these investigations seems to give a satisfactory insight into the nature of deformation of clay. Christensen and Wu (1964) and Muryama and Shi-bata (1964) analyzed creep and stress relaxation in clay by the rate process theory. They also proposed a rheological model consisting of a parallel arrangement of a spring, and a spring and a dashpot to approximate the deformation of the clay structure under creep. The results of this analysis agreed well with test data. Considering that consolidation of clay also involves contact slips and is a special case of clay deformation, the author has used the model of Christensen and Wu to describe this phenomenon. The solution is generalized to take into account the stress changes in the system during primary consolidation. This approach is considered because it seems reasonable to expect that one mechanism rather than two distinct processes should be used to represent clay consolidation. In addition, the rate process theory offers an opportunity to study the mechanism of clay deformation. CHAPTER II THEORY It is commonly held that clay particles are thin, plate-like structures with a high surface area to volume ratio . Tan (1953) (also Rosenquist, 1959, and Mitchell, 1964,) has proposed that these particles form edge to face con- tacts resulting in a continuous clay structure. The nature of these contacts and the bonding forces at the contacts are still the subject of much disagreement. However, deformation of the clay structure under an applied load is considered to be the result of slip at these particle contacts due to local stresses. On the molecular level this involves a continuous breaking and reforming of bonds at the contacts (Christensen and Wu, 1964, and Mitchell, 1964) and is considered to be a rate process. It should be mentioned that a wide range in bond strengths may be encountered in deformation. Some contacts break under the small- est stress while others remain intact .until shear failure of the soil occurs. The deformational properties of the particle structure of a clay can be approximately represented by the rheological model shown in figure 1. The spring k.2 represents the behavior of the non-flowing bonds in the clay structure under stress :5 . These bonds are assumed to behave elastically under stress. The right hand side of the model accounts for the effects of the flowing bonds. The bonds at any particular contact are con- sidered to have a specific bond strength. When the applied stress exceeds this strength flow occurs at the contact. Once flow is initiated the bond may be broken or reestablished depending on the contact displacement. Bonds are distinct from particle contacts as there may be many bonds at a single contact. The flow is assumed to obey the rate process theory. Figure 1. Proposed rheological model Considering the behavior of the model under an applied stress T, the combined resistance of k1 and k2 account for the initial deformation which represents the elastic response of the structure. As deformation proceeds stress is transferred from the weaker contacts, which flow, to the stronger, non-flowing contacts. In the model, as the dashpot flows, the stress is transferred to the left half of the model until ultimately all the stress resides in spring k2. Then the flow ceases. Experiments show that this model satisfactorily represents the major deformational characteristics observed in clays. Under an applied stress 1 T the stress in the left—hand side of the model is T—Trr- 1.27 (1) where 7; is the stress in the right side of the model and 7is the total shear strain. In the right half, the stress is Tr: k1 (7-7,) (2) If we consider the viscous flow in the dashpot to behave according to the rate process theory we have (according to Glasstone, Laidler, and Eyring, 1941) fig—Bsinha’frd +__ (111' 1 (3) k1 w r A he 6 0‘: ZUkT )8: 2 fih —k—T exp(— AF/RT) I and k = Botlzmann's constant; T =. absolute temperature; h = Planck's constant; R = gas constant; AF = Activation energy; A = average jump distance; A1 = distance between planes of slip; ~ U = number of flowing contacts From eq. (1) 7, 7=§r=lzlg (it) _ (4) Combining eqs. (3) and (4) we get k1+k2 371. 1 klkz T - = k—z % "/8 SinhOCTr (5) dt which is the differential equation governing the deformation of our model. As proposed, the model represents the behavior of the clay structure under load but does not take into account the presence of the pore water pres- sure. In a clay-water system the load on the soil skeleton is the effective stress, which in a consolidation test increases with decreasing pore pressure. If we consider the stress under a given axial stress increment p0 p0: pw + E (6) in which pW is pore water pressure and S is effective axial stress. From the Terzaghi one—dimensional consolidation theory we have at t=0 , pW : p0 at t=tp , pW = 0 where tp represents the time for 100% primary consolidation as determined graphically from a time-consolidation curve. Rewriting (6) we have 5 = 130 - pw (6a) which implies _ at t=0 , p=0 at t=tp , p-=pO The change in pW and E are given by the theory of consolidation (Terzaghi, 1923). Graphically this can be shown by figure 2. From this it is seen that 3 represents the loading condition on the . soil structure (the model) and we note that for Otp, p is constant. We may apply the condition fi- = O to eq. (5), which yields dt k1+ kZ d Tr or ma 1.) _—-k—11-<2_ : -/B sinhCX Tr (8) d O( t) k1+k2 ON. on; Writing sinhCXZ: e -6 and substituting this into (8) Z we get: M _ Q chr -otp; Tzconst. ;—-—=O The deformation is given by eq. (12). CHAPTER III EXPERIMENTAL PROGRAM The purpose of the experimental program is to compare the observed behavior of the soils tested with the theoretical predictions. Further, eval- uation of the significance of the model parameters will be in order if test data and the model behavior are in agreement. Clays Tested - Time-deformation curves were obtained on three different types of clay. The author conducted tests on two types of glacial lake clays. The specimens consist of remold'ed samples of a glacial clay from a site approximately 15 miles south of Sault Ste. Marie, Michigan, and undisturbed and remol’ded samples of a glacial clay from Marine City, Michigan. Data on the Mexico City clay were provided by Professor D. Resendiz of the University of Mexico. The index properties of the glacial clays tested are shown in fig. 4. Testing Procedure - All samples were tested with standard consoli- dation test equipment. The samples are two and one—half inches in diameter with an original height of l. 00 inches. 10 Clay PL P1 Clay LL Fraction % Marine City #1 46. 8 24.1 23.7 - Marine City #2 41.4 21.7 19.7 68 Sault Ste. Marie 60.5 23.6 36.2 60 Figure 3. Index properties of clays used Loading increments varied throughout the testing sequence and are given with other pertinent information for each test in Appendix A. Wherever possible each load increment was maintained until deformations stopped. As can be seen from the test data, the tests extended over. long periods and many had to be stopped before the final deformation was reached because of time considerations. The shortest time duration of a load increment was 4 days, while several tests lasted for more than 100 days and one was terminated after 253 days. On such long-term tests, the effects of temperature variations and other minor disturbances are readily apparent; however these effects are, in most cases, short term and do not seriously affect the long-term behav- ior of the clays. Test results and calculations of parameters are shown in Appendix C. Time-consolidation curves for each load increment are shown graphically in Appendix B. CHAPTER IV CALCULATION OF MODEL PARAMETERS In order that deformation equations (5) and (12) for the proposed model may be used to analyze the results of the experimental program we must first adopt a consistent and convenient method for calculating our model parameters k1,k2,06 and B . Let us define Tand Tin eqs. (5) and (12) as the effective shear stress and shear strain on octahedral plane in our stress system. Toct : '31: \/(Ei—Uz)2+(©}—O’3) 2+(G/3“ av (13) and UJN 70“: = _ \/(g]:' €2fii" (62" €3)Z+ (63" £1) 2 (14) where 01 2 3 are major, intermediate and minor principal stresses and ’ i . 61, 2, 3 are the major, intermediate and minor principal strains. In a consolidation test we have the conditions that (_ _ O’2=O’3 (15) ( 62:53 =0 (16) \ Putting (15) into (13) gives us: Toct = 71 \/(c’)’1 -5’3)Z+(c“)’343",)2 <17) The stress 0; is indeterminate for a consolidation test but a good a s sumption is that 63 = k 5" : kopo (18) 12 where k0 is the at rest coefficient of earth pressure k can be evaluated by the equation k0: 1- sinfi where¢ represents the angle of internal friction in terms of effective stress (19) (Bishop, 1958) For the clays studied, an average value of¢ 13 30° therefore (20) k0: l - sin 300: 0.5 Subst1tuting (20) and (18) into( (13) our expression becomes Toct= —\/2( (21) where po 13 the 1ncrement in consolidation pressure applied At t>tp, the entire load is carried by the effective stress and 137-130 Now if we turn to our equation for octahedral shear stra1n and put (16) into eq. (14) we get You: 2 (5 51121316): 2_\/_§__€| (22) Let us now consider the two conditions for the deformat1on of the Begmning with Case 11, the condition T: 0 g1ves us 130(7) (12) t + tanh lexp k1+k2 model separately. kl+kz 1 ‘ 1 7: Ff+arz 1n tanh —(X/8 Substituting (21) and (22) into (12) we have 6 = 3 . 1 4p + —— g1n1:a.nh_1_0(/Bk1k2 t+t h_1 -O(k1Po k +k an exp M 23 1 2 3f? (kl‘l'kz ( ) 2 fiakz 2 13 For purposes of analyzing the test results, eq. (23) would be much more valuable if it were written in a dimensionless form. Consider the following: at t=0 (6111— 214—3-9— (24) where (€11i is the elastic response (instantaneous) to the applied stress. at t=o<> (€11 ._.., _P_o_ (25) N We now propose a dimensionless strain U** such that U**:U£ (26) (61106 (€111 Making use of conditions (24) and (25), we may write eq. (23) in the form: U... 1.21/2 (14:122.) .1. La 4.2 p0 k1 a 1n tanh 2 kl+k2t + tanh'l exptakl PO 3(/?(k1+k2) (27) 01‘ 1 U**=1+ X In tanh [ Z(t)+tanh'l exp(—A)] (28) “ \ where _CXk1 Po 3J7 (k1+k2) 5 Z(t)=__ 098 k1R2 (29) k1+kzt / We can now make use of the fact that eq. (28) expresses U** as a function of Z(t) and can plot solutions of the equation in the form of U** vs. Z(t) for different values of A. The solutions in this form are shown in figure 4. 14 OH H ommO H3 maofigom HMUEQGHO ~-oH EN .v .2:de MIOH L 7198 Tags - Si is E _ < - mu: H 3.9 15 To relate the curves of figure 4 to the consolidation data we define a parameter where U represents axial deformation. . ‘ dz" A graph showmg U* vs. Z(t) for dt = 0 (case 11) would appear as shown in figure 5. l. | ' I U* ' I I | I I I U. | ) 1 1 I 0 : a It} ltuz i fiL log time Figure 5. Relation of U* to U** curves From figure 5, U* is related to U** by the equation U** = M (31) 1 - Ui where Uizglk. p0 k2 (63100 4(k1+k2) po - k1+kz (31a) 4k2 As shown, the slope of the straight portion of the curve is given by slope m* where 111* _ Ug- Uf‘ log t2 - log t1 16 The corresponding slope of the U** curve in figure 4 is given by m** where ~ *2: _‘** U201 ‘ UI m _1og Z(t)2 — log Z(t)1 For the same time differences we have the relationship that rn'*=(m**) _ m** (k1+k2) 1 1-Ui T (321 Therefore for a given curve of U* vs t we can evaluate the slope m'* , calculate m** by eq. (32) and then obtain a value of A which gives the best fit to m** from figure 4. We note that the initial conditions of eq. (28) differ from that in the consolidation test. However, we restrict the use of eq. (28) through (32) to the range t>tp. Hence, the initial conditions do not effect the eqs. in this range. All the calculations require that we have a quantity ( 61100' This value of ultimate deformation, (6" )oo , may be obtained by letting each test run until an equilibrium is reached. However, waiting for the deforma- tions to stop in a consolidation test is often impractical as the time required may be extremely long. We can however compute the value of ultimate deformation, or at least an estimate of this value from the following relationships. The variation of the strain rate with increasing strain is d6 dé dt d - dt d6 (33) Now we take eq. (25) and express it as €1=K +1. ln tanh (Mt +N) (34) 17 Taking the first derivative we get d 3%; : 2LM csch 2(Mt +N) (35) Differentiating again yields —_Lgt : -4LM2 csch 2(Mt+N) 36 tanh 2(Mt +N) ( ) from which d 'B—E : -ZM ctnh 2 (Mt +N) (37) We see that if (Mt+N) -- 00 as t -—-00 , then ctnh 2(Mt+ N)--l. Therefore, for large values of t dé _ 56 _-2M (38) Thus an arithmetic plot of €vs. E for large t has a constant slope of -2M. The intercept of this curve at é=0 gives (Q) as the1ultimate oo deformation. Note also that equation (38) gives us another equation for 098 in that dé k k _: _2 = __ 2 d€ M “18151.3; (39) With ( 6" 100 known, the value of k2 can be calculated from eq. (25). To evaluate k1 we must turn our attention to case I. For this case the deformation of the model for t < tp is governed by the general equation Mail- 1 EL: . . klkz. dt — 12-; dt —l['3 sth(T,. (5) To be consistent with our analysis of case 11 we consider 7;. and T as stresses on the octahedral plane. Using (21) and letting 0" :5 we may rewrite (5) as 18 1.1.1., 95:1. -_1._ (:13..- s... 414 3 3 2k1k2 dt 3J—Z-k2 dt 18 3 [2' 1 01‘ as, k. as (3..., s o. - dt k1+k2 dt k1+k2 3\/'2' where E is the axial effective stress on the model and 51 is the axial effective stress on the flow component. If we write (40) in the form of finite differences we have A13 k AS _ 1 : 1 - (3’8 sinh 0‘ pl (.41) At k1+k2 At 3‘12. where G - 3 \FZ-k1k2 The initial condition for this case is 51:0 at t=0. For these conditions eq. (41) becomes A31 h. .AE At = k1+k2 At (42) which is valid as the limit when At-u-O OI‘ lim A151 _ k1 AS At-.O At -k1+k2 At t = o (43) If we approximate (43) by limiting t to small values we have eq. (42) 01' AR) k1 A? " k1+kZ (441 Using eq. (21) and (22) we rewrite eq. (1) as - 1 _ — 61*213 w-pp (4m 19 From eq. (30) and (25) we have U*:_§_1_ (30) P (25) (ence ,( 61); 41:2 Combining (25) and (45) with (30) we have I5 51 5 '5 U>k = ' - : _:l' (46) P0 P0 P0 P Differentiating with respect to _p_ gives us po dU* l d'fSI/d (1) dis]. p : ' Po = 1 - _— (47) d(-) — d? P o dp/ d (.2) p0 If we write this in difference form it becomes AU“ -1 A51 A r/po) A}; (48) which becomes upon substituting (44) Au* k1 k2 _ = 1 - = —— (49) A p /Po k1+k2 k1+k2 We know that the variation of p/po for O l< . U* __ p p1 P0 P0 (46) We can now plot a theoretical curve for the deformation of our model for t < tp. In fact this solution can be extended over the entire range of t if desired. However it is cumbersome to apply and its use is not warranted as we already have a convenient solution for t > tp in eq. (28) . All the model parameters have now been determined and a step by step procedure for calculating k1,k2,O( and [B is given as follows. For a complete set of sample calculations see Appendix D. 1 - Plot test data as U*= U/Uoo vs. t . 2 - Evaluate(€l)ooand get k2 from eq. (25) . 3 - Determine Cv from experimental time-deformation curve and plot p/p0 vs. t . 21 Evaluate kZ/(k1+ k2) from eq. (49) and consequently k1 . Calculate m* and m**, then determine the value of A (figure 5) which best fits m**. Calculate 098 by eqs. (29) and (39) . Use eq. (29) to calculate (Xandfl . CHAPTER V TEST RESULTS 1. Comparison of Theoretical Curves with Test Data In the previous section steps are given for calculating the model parameters k1 , k2 , (X , and 18 Having these parameters along with the value of A, CV and Z(t)/t , we can construct a curve of U* vs. t and com- pare it with the experimental curve. The construction is carried out in two steps corresponding to the two cases given for the model deformation, however the finate difference solution used in case I would be extended over the whole test range. This process is cumbersome and not necessary. For ttp can be gotten with the aid of eq. (28) and figure 4. Ui is computed by eq. (31a). A solution of eq. (28) from figure 4 can be plotted for known values of Z(t)/t, A and the ratio of m**/m*. Of course we only use the part of the curve for t>tp. Figure 6 gives a graphical picture of these solutions. Figures 7, 8, 9 and 10 give graphical comparisons of the actual and computed results. It is seen that in each case the agreement of the pre- dicted curves and the experimental data is very good over the entire test range. The curves shown are typical of several types of consolidation curves . 23 1.0 solution of eq. (28) for t>tp U* P" . . . finite difference solution for t Hmucocfiuomxm Amounfiev . 633. God .N. oudmfm OH 000 .2: 000 A: _ 054 pfiom t t wok/HAD woumgonO 930.20 1 3mm “mop. _ 3 N62? SIN u 24 N83? 8 .m n a L Avonnsumwpca a - >30 36 @552 N . mob/H50 3036.893. .m> Amazogwuomxm 25 Amen—55c: ) 09:8 000 .2: coo .oH oooH oo~ .w 0.23th Ga 654 pfiom . o I 95.90 woumgoamu mmHofiO .. 4“..qu «mom. 90 ~82? mmgumd ~83? om.~ u a b 6630505 4 - .36 3.82 .Bm Scam *D 26 goo .ooH m . mm>HSU Hmofimuooafi. .m> Hmucoafinomxm 000 .OH 003 Am 3538; I math 03 .a onswmh 05.4 view .. .. ®>HSU 633.9030 moHUHHU u 3mm ummH 0 N83? om.o Had Naimx om; Hm H.o . .70 *D no.0 INSO 68:32an E - >20 36 8382 o v . mm>HSU H‘mufiouooflH .m> Hmudvafiuomxm .oH ohswfih 7 2 Amoudcfiev n 05TH. 000 .03 000 .3 oooH 02 OH H H. _ _ a _ . . 0 rl O 0 1| 0 9:4 mfiom . a I . 9/de wwumgoflmo o mmflofio u mudQ ummH n30 I o O o I O O I o O O o 0 ~82? Sauna Negwx oo .H H m _ _ F _ _ _ gunman—muggy Hm - >20 36 03on H.o 04 28 has been completed, the viscous flow of the particles is still very small and in some cases negligible. However most of the viscous flow occurs after tp and very large secondary consolidation may take place before an equilibrium is reached. Our model is still able to reproduce faithfully the experimental data. Z9 2. Variation of Model Parameters As there is good agreement between theoretical and experimental data, we now look at the model parameters individually and study their variation with the loading conditions. A summary of the calculated para- meters is given in Appendix C, while the variation of ' kl/(k1+k2), a and )8 with stress is shown graphically in Appendix D. The spring constants k1 and k2 represent the elastic response of the clay structure while k2 represents the effect of stress on non-flowing (elastic) contacts. In the remolded samples of Marine City and Sault Ste. Marie clays, k2 remains fairly constant at low stress levels but increases at high stresses. On the undisturbed samples of the same clay, it again remains constant at low stresses but increases rapidly at stresses that exceed the preconsolidation pressure, pC. In the undisturbed Mexico City clays, k2 decreases with increasing stress until the preconsolidation pressure is reached after which it increases markedly. The increase in k2 at stresses exceeding pC is what we would expect because in this stress range the clay is compressed to progressively smaller void ratios. Consequently the structure becomes more rigid and the elastic bonds are able to carry larger stresses. The ratio of kl/(kl-l-kz) represents the portion of the applied stress that acts on the flowing contacts. We would expect that, for undisturbed samples, at high stress levels more contacts are broken and flow. This should result in an increasing trend in the ratio of kl/(k1+k2). Such a prediction agrees with the data shown in Appendix B. 30 In all cases except one, the undisturbed samples showed increases in k1/(kl+ kg) with increasing stress. Figures 11 and 12 show variation of kl/(k1+k2) as well as (X and ’8 with increasing stress for two samples. For the remolded clays, the ratio varies little over the entire stress range and has a low value. This indicates that under each new load increment the stress on the flowing bonds is a constant proportion of the total stress. The parameter (X , as defined by rate process theory, is an indicator of interparticle movements. It is directly proportional to A , the average distance between bonds (or the average "jump" distance) and inversely pro- portional to the number of flowing contacts, U . As the total stress increases, the stress on the flowing contacts increases: we would expect the number of flowing contacts and flowing bonds to increase directly with the stress. The data shows that for all tests Ot decreases with increasing stress as we predicted it should. It might be noted that O( would also decrease if A. decreased at high stresses, however Arepresents an average value of all jump distances and is unlikely to decrease significantly. The parameter )8 also represents viscous movement of bonds and is defined as B = 2%? exp(-AF/RT) l where A1 is the distance between planes of slip and AF represents the activation energy required to initiate bond "jumps". Marine City Clay - la 1. 0 .. (remolded) .75».- k1 k1+k2 . 50.? .25" O o 0.5 1.0 1T5 2.0 10 4b (x10'8) S'Ol' 1500 + J 10 00 l J 500 " o L 0 0.5 1.0 1.5 2.0 po - kg/crn2 Figure 11. Variation of Model Parameters with Stress 31 32 Mexico City Clay - B5 (undisturbed) 10 .- .75 ‘- k1+k2 .50 1" .25 ‘- 15 L to 11.1 to 3.82 (x10-7) 800 1" 600 < 400 -- 200 4%- 0 0.5 1.0 1.5 pO - kg/cm2 Figure 12. Variation of Model Parameters with Stress 33 A. If the ratio of A1 remains constant over the entire stress range fl would depend only on the activation energy. In this case we would expect fl to decrease with increasing stress, for at high stresses the bonds with low strengths have been broken and those remaining have progressively higher strength and greater activation energy. In a consolidation test, however, the strains are essentially in one direction. Final strains can approach 10 to 12% of the original height of the sample‘fwhich could bring about a decrease in A. , the distance between the planes of slip. A decrease in Alwould increase the ratio of A1 and would cause an increase in B. The undisturbed and remolded Marine City clays have high intial ’8 values which generally decrease with increasing stress, indicating a con- stant increase in A F. However, from the data we see that the Mexico City clays have an initially large value of B which decreases with increasing stress but increases rapidly at stress levels around the preconsolidation pressure. Such behavior is also present in the remolded Sault Ste. Marie clay. It would be well to note here that both of these clays have very high initial void ratios. The decrease in B for low stresses in the above clays can be explained by an increase in AF. As the preconsolidation pressure is reached, however, large compressions and large structural changes take place under additional stresses. This would suggest that Amay increase, as the average jumps would be longer, and Al decreases due to large compression in the axial direction. Both of these factors would increase B . At stresses beyond pc it is probable that 3 will again decrease because AF will steadily increase. CHAPTER VI CONCLUSION The analysis in the previous section indicates that the model adequately describes the deformational characteristics of a variety of clays. It has been shown that there is good agreement between the calculated curves and the experimental data. The main factors governing the shape of the curves produced by our deforming model are the slope constant, A, the ratio of kl/(kl-i- k2), the coefficient of compressibility, CV , and the ratio of Z(t)/t. A further look tells us that the value we pick for A depends on the ratio of m*/in which in turn is a function of kl/(k‘1 + k2). By combining these factors in various proportions we can duplicate a number of typical deformation curves for clays. Figure 13 shows two different systems for classifying types of con- solidation curves; one by K. Y. Lo (1961) and the other by G. A. Leonards (1961). Girault (1960) applied Leonard's classification to Mexico City clay and showed their relation to the loading ratio used in testing. Wehave already shown that these curves can be duplicated with the model of figure 1. To duplicate curves of Type (1) (requires that we have a small k1/,(k1+ k2) ratio, a fairly small value for CV and a large ratio of Z(t)/t. Lo's type II curve is merely an extension of Type I and can be pro- duced with a value of k1/(kl-1-k2) between 0.3 and O. 6, a fairly small value of CV and a large ratio of Z(t)/t. 0.1 l 10 100 1000 10, 000 \II a) Leonard's Classification of Types of Clay Deformation Curves 10 100 1000 10, 000 \ III(b) . \ b) K. Y. Lo's Classification of Types of Clay Figure 13. Deformation Curves Classification of Clay Deformation Curves 36 Similarly, Leonard's Type II curve can be duplicated with a value of kl/,(k1+k2) around 0.15. Lo's Type III(b) curve is similar to Leonard's Type III curve and neither shows any primary stage. A curve of this type can be approximated by using a very large value of kl/(k1+ k2), a large value for Cv and a small ratio of Z(t)/t. Curves of this type occur most frequently in very compres- sible clays subjected to equal loading increments as can be seen from the tests on Mexico City clay. Thus we conclude that the proposed model represents an effective way of representing the consolidation characteristics of a wide range of clay types, including both remolded and undisturbed samples. The fact that the calculated curves compare very well with the actual deformation data indicates that this is alsoan effective way to calculate the rheological parameters of the model. These have been shown to behave con- sistently with the material parameters as proposed by the rate-theory analysis. The variation of these material parameters gives us an important insight into the mechanics of clay deformation. The rheological model also gives us a solution for the entire consol- idation process, which would seem to be more convenient and reasonable than an arbitrary separation of the phenomenon into primary and secondary stages. Also, while the model gives good agreement with a wide variety of deformation curves, it is itself simple and the equations governing its behavior offer relatively simple solutions. BIBLIOGRAPHY Bishop, A. W. , "Test Requirements for Measuring the Coefficient of Earth Pressure at Rest, " Brussels Conference on Earth Pressure Prob- lems, Vol. 1, 1958, pp. 2-14. Christensen and Wu, "Analysis of Clay Deformation as a Rate Process, " Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 90, No. SM6, Nov., 1964, pp. 125-157. Eyring, H. and Halsey, G. , "The Mechanical Properties of Textiles-‘ The Simple Non-Newtonian Model, " High Polymer Physics - A Sympo- sium, 1948, pp. 61-116. Gibson, R. E. and Lo, K. Y. , "A Theory of Consolidation for Soils Exhibiting Secondary Compression, " Norwegian Geotechnical Institute Publication No. 41, (also Acta Polytechnica Scandinavia, 296/191, c110), 1961 Girault, Pablo, "A Study on the Consolidation of Mexico City Clay, " Ph.D. Thesis, Purdue University, 1960. Glasstone, S. , Laidler, K. , and Eyring, H. ,"The Theory of Rate Processes, " McGraw-Hill Publishing Company, Inc. , New York, New York, 1941. Hildebrand, F. B. , "Introduction to Numerical Analysis, " McGraw-Hill New York, New York, 1956. Leonards, G. A. and Altschaffel, A. G., "Cornpressibility of a Clay," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 90, No. 5M5. Sept. 1964, pp. 133-155. Leonards, G. A. and Girault, P. , "A Study of the One-Dimensional Con- solidation Test, " Proceedings of the Fifth International Conference on Soil Mechanics and Foundation Engineering, Vol. 1, Paris, 1961, pp. 213—224. Lo, K. Y. , ”Secondary Compression of Clays, " Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 87, No. 5M4, August, 1961, pp. 61-87. Mitchel, J. K. , "Shearing Resistance of Soils as a Rate Process, " Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 90, 5M4, 1964. 38 Murayama and Shibata, "Flow and Stress Relaxation of Clays, " Rheology and Soil Mechanics Symposium of the International Union of Theo- retical and Applied Mechanics, April, 1964. Rosenquist, 1. Th. , "Physical-Chemical Properties of Soils: Soil-Water Systems, " Journal of the Soil Mechanics and FoundationsDivision, ASCE, Vol. 85, No. SMZ, Proc. Paper 2000, April, 1959, pp. 31-53. Schiffmann, R. L., Ladd, C. C., Chen, A. TTF., "The Secondary Consolidation of Clay, " Symposium on Rheology and Soil Mechanics of the International Union of Theoretical and Applied Mechanics, April, 1964. Tan, T. K. , (Discussion), Proceedings of the Third International Conference on Soil Mechanics and Foundation Engineering, Vol. 3, 1953, p. 129'. Terzaghi, K. and Peck, R. B. , "Soil Mechanics in Engineering Practice, " John Wiley and Sons, Inc. , New York, New York, 1948. Zeevaert, L. , "Consolidation of Mexico City Volcanic Clay, " ASTM, Spec. Tech. Pub’. 232, 1957, pp. 18-32. APPENDIX A TEST DATA - TABULAR FO'RM TEST DATA 1. Clay - Marine City (undisturbed) Sample 1 Load Inc. Duration of CV (kg/cmz) Test (days) (inZ/min) 0 - 0.625 5 3. 38x10-2 0.625- 1.25 5 1.77x10-2 1.25 -2.50 8 2.325.10-2 2.50 -5.00 234 2.56x10'2 5.00 -1000 114 2.44x10'2 2. Clay - Marine City (remolded) Sample 2a Load Inc. Duration of CV (kg/cmz) Test (days) (inZ/min) 0 -o.031 4 3.025.10-Z 0.031 -0.0625 8 9.2 5610‘3 0. 0625-0.125 53 6.0 x10-4 0.125 -0.250 81 5.0 x10-4 0. 250 .-0.500 69 8.5-.xio-4 0. 500 -1 .00 139 1.78x10-3 1 .00 -2.00 117 2.6 x10-3 3. Clay - Marine City (undisturbed) Sample 2b Load Inc. Duration of Cv (kg/cmz) Test (days) (inzlmin) 0.500- 1.00 20 4.255.10-2 1 .00 - 2.00 48 4.88x10‘2 2.00 - 4.00 185 2.92.;10-2 4.00 - 8.00 156 1.5 x10-2 8.00 -16.00 253 0.69x10-2 5.30 7.60 7.60 9.90 9.90 .90 .60 .60 .60 .60 .60 44.81.560.800 5.30 7.60 9.90 12.0 9.90 (mo-5) 2.66 4.26 5.35 2.24 (mo-5) 2.77 0.05 . 9O. 0 .075 0431 (x10-5) 156 11 5. 4 O 0 O .O 02 . 10 .953 .760 .726 0981 (x10‘7) 795 49 .6 O .068 9192 ) (x10'5 260 10. 3 2. 1 0 O 8 .89 64 . 37 . 105 . 301 0% (x10-7) 885 52 .4 6.56 7.64 2.24 3.59 4.96 1. 15 TEST DATA (cont'd) 4. Clay - Sault Ste. Marie (remolded) Sample 1 Load Inc. Duration of Cv (kg/cmz) Test (days) (inz/min) 0 - 0.156 6 2.45:.10'2 0.312 - 0.625 73 1.36x10-3 0.625- 1.25 172 .802x10-3 1 .25 - 2.50 114 .982x10'3 2.50 - 5.00 27 1. 21x10-3 5. Clay - Mexico City (undisturbed) Sample A5 Load Inc. Duration of CV (kg/cmz) Test (days) (cmZ/mi'n) 0 - 0.25 4 0.770 0.25 - 0.50 40 0.705 0.50 — 1.00 132 0.458 1.00 - 2.00 125 0.0026 2.00 - 4.00 31 0.0108 4.00 - 8.00 69 0.0078 6. Clay - Mexico City (undisturbed) Sample Bl Load Inc. Duration of CV (kg/cmz) Test (days) (cmZ/min) 0 - 0.25 11 0.467 0.25 - 0.50 56 0.348 0.50 - 1.00 108 0. 167 1.00 - 1..50 90 0.0475 1.50 - 2.00 132 0.128 2.00 - 2.50 180 0.0653 6.20 4.60 4.60 5.30 6.20 4.60 .90 .90 .22 .60 . 90 erhwaQ .60 .60 .20 .30 .60 #momqp (xlo-é) 53.7 18.8 12.9 5. 16 26.0 (x10'5) 105 1.06 0.529 7.64 100 5.20 41 (x10-6) 33.8 19.3 11.6 6.78 8.96 0982 (x10'5) 147 0.358 0. 157 3.36 26.8 3. l3 (x10‘5> 256 1.55 0.106 .06 .29 .404 ONI—I 7. HHHOOOO CD wNHooo Sample B5 Load Inc. (kg/cmz) - 0.25 .25 - 0,50 .50 - 0.75 .75 - 1.00 .00 - 1.25 .25 - 1.50 .50 - 1.75 Sample C5 Load Inc. (kg/cmz) - 0.25 .25 - 0.50 .50 - 1.50 .50 - 2.50 .50 - 3.50 .50 - 4.50 Duration of Te st (days) 4 35 70 167 90 132 183 Duration of Test (day s) 33 7O 76 97 57 10 TEST DATA (cont‘d) 0000000 000000 C (cmZ/min) Clay - Mexico City (undisturbed) V .192 .196 .115 .065 .057 .047 .046 Clay - Mexico City (undisturbed) Cv (cmZ/min) .625 .315 .125 . 185 .162 .171 mexomxo H t—IUJUTNKJKIUJ .90 .60 .60 .30 .22 .00 .90 .30 .90 .20 .60 .00 .45 .95 .55 .70 .84 .34 (x10‘5) vhp—iNNp—ad .10 .59 .07 .08 .17 .77 42 O<fl2 (x10-6) 733 10.8 0.305 9.45 (3410' ) 5.85 0.730 5.00 15.2 1.68 2.04 APPENDIX B EXPERIMENTAL TIME- DE FORMATION CURVES 44 Amen—555V u 053. 000 .2: 000 .2 003 03 OH H H. a d _ q d . .. NEo\mx 5 oHSmmOHm CofimmuSOmdoo .233 36065 mnongdz 6qu r T o .3 . mm In 4 o 1 .\. . 1 I om N ‘<\ I mNQ . O u - «fl . 1 e 4 o 1 L N O L\_. \\ .‘M o o o .. \ 4 o o - _ _ _ 666833683 7 .88 36 65.82 OJ .2 45 000 .00.. 000 .0.” 000a $39550 . 083. 00a 0H Nanimx E onsmmoum GowudEHOmnoo H.300 363ch wnmngz 3qu 6020503 em - .36 56 65.32 04 *D 46 000 .02 000 .0a $3339: 7 08E. 003 00a 0H Nagwx a: ouammmum u cowumwfiomaoo H83 36305 mumnfisz 6qu 66663265; £ - .86 36 6532 004 04 *D 47 Amougflfiv 1 mafia. 000 .03 000 .3 000a 00H 0H H0 . q r T NEo\mM a: oudmmoum r Gown—3030330 H33 300203 muonasz "on—oz .. I v .0 I .. m.0 F l O .O 1 1 >0 4 F <‘\‘\ I w .o 4“ e 4 I 1. .. 0 .0 \.\ . u L _ _ _ O .H 663686... A - .36 332 .8m :58 *D 48 36.35535: .. 05.53. 000 .003 000 .OH 0003 OCH OH H 3.0 d _ _ 3 _ . . O NEU\MV3 53 05558on 5036020980 3.30» 0303053 "1.503552 8qu *D _ _ _ 0 .3 goofing—335$ m... - >20 .36 63x62 1‘1. ’11.” V >JHWV NJLI‘N‘”: 49 Ammusfiav . 08TH. 000 .03 ooo .3 003 . CS 3 ad . . _ _ .1|.|| o Naimx GM ondmmmnn . ., Gofimufiowaoo H33 . .\ I @339: whoagz 6qu X\\.\.x : H . o . . . O \ - um .o .. o 1 m .o I «rem .N L w o 1 L m o T o I o .o \\. 4 cm .o . ‘ mm .o 1 N. . o r a X 4 L w .0 ‘ \ 4 o o I 4 o o 1 ® . O 4 éulb L, . r _ o g Aconuamflgv a - >20 36 8332 *D O 5 Amen—555V u 053. ooo “o3 ooo .3 . oooH 03 .3 . H H. o‘ J q . - p - :IIl“ .illHHw Oa‘llllu‘r O >\\.....>|\IJM.‘ .. . .\\.\ n O u r . o v T N . OJ. 1 m .o a» . o g m . o o . o p . o m .o magnum 5 madmmonm nomumcSOmnou H33 0 . C 03055 939.532 8qu p P rV h n b o . H Avoaufiummwndv mm u >MHU >§U 00332 *D 51 33.95:: u 05TH 000 .2: 000 .3 Good OCH 3 H are NEo\m& E ondmmonm r Gowudvfio‘mdoo H38 030va muonfigz nouoz - . — _ P 0.." Avonugnwwgv m0 .. >NHU knumD’ouflnmE APPENDIX C CALCULATED PARAMETERS TEST RESULTS 1. Clay - Marine City (undisturbed) Sample 1 Load Inc. k1 k2 k1 (kg/cmz) (kg/cmz) (kg/cmz) k1+k2 O - 0.625 3.55 26.0 0.12. 0.625-1.25 3.76 13.3 0.22 1.25 - 2.50 5.81 21.6 0.24 2.50 - 5.00 21.6 26.4 0.45 5.00 -10.00 39.1 44.0 0.47 2. Clay - Marine City (remolded) Sample 2a Load Inc. k1 k2 k1 (kg/cmz) (kg/cmz) (kg/cmz) E1+k2 0 - .031 1.17 1.05 0.53 .031 - .0625 7.12 2.50 0.74 .0625- .125 0.46 1.37 0.36 . 125 - .250 0.68 1.74 0.28 .250 - .500 2.05 3.64 0.36 .500—1.00 2.84 5.06 0.36 1.00 -2.00 5.26 8.60 0.38 3. Clay - Marine City (undisturbed) Sannple 2b Load Inc. k1 kg 1‘1 (kg/cmz) (kg/cmz) (kg/cmz) k1+k2 .500- 1.00 8.5 24.2 0.26 1.00 - 2.00 11.2 16.85 0.40 2.00 - 4.00 23.2 21.4 0.52 4.00 - 8.00 10.3 17.25 0.375 8.00 - 16.00 38.9 31.8 0.5.5 0% (cmZ/kg-min) 8.20 x10‘5 3.573 2. 77 .050 .068 9‘52 . (cm /kg-m1n) 2.60 x10"5 10.8 3.89 2.64 1.37 0.105 O. 301 0% (cmZ/kg-min) 88. 5 0. 359 0.496 0. 115 (X (HM/kg) 300 235 107 75. 20.4 0( (cmZ/kg) 1060 350 587 559 21:7 179 5 84.6 (mg W?) 172. 80. 36. 28. 3‘. '\l \J‘ UT, [\J 53 ,8 10‘7(min'1) 2. 1. 73 52 2.59 .066 .333 B 10-8(mm-1) 246 12. .63 .73 .31 .59 .56 6 WOCPD-P 10-*(mjn-~=) (_ OO‘tt-a 3.4 {a o o 7 1 <3 (3‘ .‘ NKIQUTUJ ’T O TEST RESULTS (cont'd) 4. Clay - Sault Ste. Marie (remolded) Sample 1 Load Inc. (kg/cmz) 0 - 0.156 0. 312 - 0.625 0.625- 1.25 1.25 - 2.50 2.50 - 5.00 5. Clay - Mexico City (undisturbed) Sample A5 Load Inc. (kg/cmz) 0 - 0.25 0.25 - 0.50 0.50 - 1.00 1.00 - 2.00 2.00 - 4.00 4.00 - 8.00 6. Clay - Mexico City (undisturbed) Sample B1 @L'oad Inc. (kg/cmz) O - 0.25 0.25 - 0.50 0.50 - 1.00 1.00 - 1.50 1.50 - 2.00 2.00 - 2.50 k1 2.22 1.87 2.47 3.48 4.36 k1 1.89 7.45 7.62 8.70 31. 2 1.9.3 k2 (kg/cmz) (kg/cmz) 3.18 3.18 5.25 7.39 14.6 k1 k2 (kg/cmz) (kg/cmz) 1.38 11.1 11.3 9.25 (8.65 5.30 0.72 1.34 0.275 3.16 1.15 5.26 k2 (kg/6.4.2) (kg/cmz) 9.2 7.45 3.92 1.30 1.08 1.23 k1 0982 k1+ k2 (cmzz/kg-min) .40 .37 .32 .32 .23 00000 k1 33.8 5110-6 19.3 11.6 6.78 8.96 0982 k1 + kg (cm2 /kg-min) .11 .55 .62 .35 .08 .18 000000 k1 .17 .50 .66 .87 000000 .94 . 968 14.7 x10-5 0.358 0.157 3.36 26.8 3.13 01,82 k1 +kz (cmZ/kg-min) 256 x10-5 1.55 0.106 1.06 2.29 0.404 (cmZ/kg) 421 168 97.5 56. O 45. 7 (cmZ/kg) 710 305 120 52.7 122 23. 0 (cmZ/kg) 459 25.8 97. 60. 46. 41. Lurkv-Pm 54 10‘8(min- ) 8.03 11.5 11.9 12.1 19.6 10' 8(min-1) 207 1 . 17 1 . 31 63.8 220 136 TEST RESULTS (cont‘d) 7. Clay - Mexico City (undisturbed) Sample B5 Load Inc. k1 k2 k1 (kg/cmz) (kg/cmz) (kg/cmz) kl+k2 0 - 0.25 1.04 9.40 0.10 0.25 - 0.50 5.23 8.70 0.375 0.50 - 0.75 7.90 5.27 0.60 0.75- 1.00 10.50 4.50 0.70 1.00- 1.25 14.9 2.63 0.85 1.25-1.50 23.2 1.55 0.938 1.50- 1.75 22.6 0.82 0.965 8. Clay - Mexico City (undisturbed) Sample C5 Load Inzc. k1 k2 k1 (kg/cm ) (kg/cmz) (kg/cmz) 144 kg 0 - 0.25 2.07 4.40 0.32 0.25- 0.50 5.12 3.14. 0.62 0.50- 1.50 3.66 2.54 0.59 1.50- 2.50 11.9 1.32 0.90 2.50 - 3.50 21.4 2.38 0.90 3.50- 4.50 32.7 3.52 0.90 "‘ 0([31 2 (cmZ/kg—min) 733 10 2 4 3 9 6 >110-6 .8 76 553g 18 45 72 O( 2 (cmZ/ké—min) Nt—‘NOOU‘I . 85 1-110'5 .730 .50 .08* .68 .04 (cmZ/kg) 662 344- 215 291 106 58. 5 17.6 (cmZ/kg) 525 145 71 29.1 16.9 4. 92 55 10-8(min-l) 111 3.14 1.28 1.56 3.00 15.9 38.2 10'7(min"1) 1.11 0.503 0.705 7.15 9.95 41.5 APPENDIX D k1 k FOR INCREASING STRESS - GRAPHICAL l VARIATION OF (X. ['3 AND n..1 .I..s.v .fin—yv vfi-hI-n;~)~ A0ur0-I-Iilnul'.-V \ Ahioifiv~.\v-~rivhy nu. ut¢.-. NEo\mx u mmmnum H308 H - .66 6:62 .65 :28 W00 03v 0..m 0.N 04 0 u u u T , j _ J A; q . \\ x \\ S u h h. n \ _ $602050: §u\wx n mmvuum H308 N 0 .N m .H 0 .H m .0 o + n o _.I fir Com 04 11. 0.0a : «0:2 I. 0.3 «0-0 h V ( 0.N .fi 0 .0N 63 3 0 w u 0 .1 0mm . 0mm -1 00m x 00 .. om... com n.oooH o, 0 o mN. .. mm. 0m . H g. cm ._ If .+ N 4A.. x J 2. 2. m»... 8030803 m. - .3 h: 22.3.. 04 N G .0 . E t o; ~25ng . mmohum H.308 58 Eo\wx . mmonum .2308 N m: m 0 m2 2 m o Ai u 4 u "I .F n J u 1.! O w \. N W . “A903 r m.~ m. w w Q «92¢ \\. o m. m \ I o.m v. w 000 on m 4m 3 V .4 1. 1 a u .4 0 u m 04 .r u u \ 0 “ \.. m a \ 1 0mm \1 00H m 8 m- 8 m. 1 00m .100N “ u L w u v w u o u w u o . mm .-m~. . om. +3. Nu?! mmHm 1.91 F. . ms. :2. Avonnsumwvgv 369933050 0 - .66 .36 65.32 r o; 6N - >66 36 66:62 -4 o; 59 Nagmx . mmouum 1309 ¢ N db 10.? wood l 1 00m .003 m< >30 >30 03on 1 .mm. . om . 3+; rmh. NEo\mM . mmonum H.309 +- $61 Iqu- )- up . 0mm . 00m 1mm. m0 - >30 .30 00332 - 0m . N61; 3 0. mp. NEo\mx - mmmhm H308 Eo\wx u mmouum 1300.. N N m4 H .m.o 0 m.~ N m4 H . m0 0 "i .. L. 0 o .11 _.I|Il .. OH 061 -- 6N. .- 3 Q .- om ~92 .. 04 .- om SH 3 + .1 . o . . r|1|l _1 u o ,1. com 1.. 06¢ -dmN 00 00 000 1. com Lroom .1 o o 11 mN o Iva . -1 om . :3. ~61; Nib. le Iml .. m5. :mn 6696366ch mm - >30 30 09.382 Lr o; 636360.50 Hm - >60 .30 02on Ho; APPENDIX E SAMPLE CALCULATIONS 62 SAMPLE CALCULATIONS Clay - Marine City (undisturbed) Sample 1 Pressure - 2. 50 kg/cm2 Increment - 1. 25 kg/cm2 See figure Ea for plot of test data. Figures Eb and c are used to evaluate U00 . We found U0 by plotting U vs. t (for small t) arithmetically and projecting the resulting curve back to its zero intercept. CV is calculated to be 2. 32 5:10"2 in2 /min. Elbe vs. t is plotted in figure Ed. Take several small increments of At and plot AU*/A('p-/.po) vs. t. Extending this line to its intersection with t=0 yields k2/(k1+k2)=‘ 0.76 (figure E1). 1.0 0.76 AU* M 23(1) 0.5 p o : 1 % ‘ 0 . 0.1 0.2 0.3 0.4 0.5 Time - (minutes) Figure E1. Determination of kz/(k1+k2) 1.25 k =—-E-9— : -—-———— = 21.6 kg/cm2 2 4(6)0° 4x0.0144 k = 21.6x0.24 : 5.81 kg/cm2 10.75 mka.5‘85~o =7.5/cyc1e log 1000- log100'" m**=m*(kl+k2 :7'5 = 31.3/cyc1e k1 .24 1.0 (a) 63 ! E 3 1 l . m* o a // J o 6 / U* . l , o 4 ' / i 0.2// o o .1 1 10 100 1000 10, 000 100,000 Time - (minutes) 150 (b) 150 (C) . / -2M=-2. 50x10-4(min-1) 140 3" 140 .1. U 130 .. 5 .. (5:10-41. U 130 (ado-4) 120 120 .1 110 . ~ : : t + 110 4. i i ‘r T‘ o 2.5 5 7.5 10 12.5 o 2 4 6 8 10 Time - (minutes: x103) dU/dt (x10-7 in/min) 1.0 (d ii —\ .75 p0 p — .50 p0 / 25 /rf}_ (fromfinite difference ' po solution) * r- o 0.1 1 10 100 1000 Time - (minute 5) Figure E therefore A =7. 60 (from figure 4) Z(t)_1.6x10'2 _ -5 “7““??? - 6-4’40 ._ Z(t) ' . 0982 " 2““. (4%): 2(6. 4x10 5) (. 218)= -2. 77x10” 5 Note: 0651, 2 refer to the two methods for calculating this quantity. 66 The slope of the ? curve in figure Ec gives us a value of -2M=2.50x10"4 k k O(,8,=-2M J—i— = 2.5ox10‘4 (.218)=5.35x10-5 k1+k2 k k a=3\/—2_ (A) ..__._..__. 1+: 2 =107 cmZ/kg (X o- 5 )9: -—&§3=2 771317 = 2.59x10-7 min-1 The calculations above and the figures in Appendix A indicate that the two methods described previously for calculating afi differ by a factor of two. Because of the approximations and estimates made in each of the methods, agreement is considered good if the ratio of the smaller to the larger of these corresponding values is less than 3. In cases where the ratio exceeds 3, the author has used 0982 for calculations, having more confidence in the reliability of this method. 64 lllllllllli I