Strength and Deformation Behavior: F (E, C S)

 

CHAPTER 11

Strength and Deformation  Behavior

11.1

INTRODUC INTR ODUCTION TION

All asp aspects ects of soi soill sta stabili bility—bea ty—bearin ring g cap capacit acity y, slo slope pe stability, the supporting capacity of deep foundations, and penetration resistance, to name a few—depend on soil so il st stre reng ngth th.. Th Thee st stre ress–d ss–defo eforma rmati tion on an and d st stre ress– ss– deform def ormatio ation–time n–time beh behav avior ior of soi soils ls are imp import ortant ant in any problem where ground movements are of interest. Most Mo st re rela latio tions nshi hips ps fo forr th thee cha chara ract cter eriza izatio tion n of th thee stress–deformation and strength properties of soils are empirical empiric al and based on pheno phenomenolog menological ical descri descriptions ptions of soil behavior. The Mohr–Coulomb equation is by far the most widely used for strength. It states that

 ff 

   c       ff  tan      

 ff 

   c       ff   tan      

 

(11.1)  

(11.2)

where    ff  is shear stress at failure on the failure plane, c  is a cohesion intercept,   ff  is the normal stress on the failure plane, and    is a friction angle. Equation (11.1) applies for   ff  defined as a total stress, and  c  and    are referred to as total stress parameters. Equation (11.2) applies app lies for  ff  defi defined ned as an ef effec fectiv tivee str stress ess,, and c   and     are effective stress parameters. As the shear resistance of soil originates mainly from actions at interpart pa rticl iclee co cont ntact acts, s, the se secon cond d eq equa uatio tion n is th thee mo more re fundamental.

In reality, the shearing resistance of a soil depends on many factors, and a complete equation might be of  the form Shearing resistance      F (e,  c ,    ,    ,   C ,   H ,   T ,   , ˙ , S ) (11.3) in which   e   is the void ratio,   C   is the composition,   H  is the stress history,  T  is   is the temperature,   is the strain, ˙ is the strain rate, and   S   is the structure. All param eters in these equations may not be independent, and thee fu th func nctio tiona nall fo form rmss of all of th them em ar aree no nott kn know own. n.  c Consequently, the shear resistance values (including  and    ) are determined using specified test type (i.e., direct shear, triaxial compression, simple shear), drainagee co ag cond nditi ition ons, s, ra rate te of lo load adin ing, g, ra rang ngee of con confin finin ing g pressures, and stress history. As a result, different friction angles and cohesion values have been defined, includin clu ding g par paramet ameters ers for tot total al str stress ess,, eff effect ectiv ivee str stress ess,, drai dr aine ned, d, un undr drai aine ned, d, pe peak ak st stre reng ngth th,, an and d re resi sidu dual al streng str ength. th. The she shear ar res resist istanc ancee va values lues app applic licable able in practice depend on factors such as whether or not the problem is one of loading or unloading, whether or not shortsho rt-ter term m or lon long-t g-term erm sta stabil bility ity is of int intere erest, st, and stress orientations. Emphasis in this chapter is on the fundamental factorss con tor contro trollin lling g the str streng ength th and str stress–de ess–defor formati mation on behavior of soils. Following a review of the general characteristi charac teristics cs of streng strength th and deformation, some re369

 

370

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  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

lationship lations hipss amo among ng fab fabric, ric, str struct ucture ure,, and str streng ength th are examined. The fundamentals of bonding, friction, particulate behavior, and cohesion are treated in some detail in order to relate them to soil strength properties. Micromechanical interactions of particles in an assemblage and the relationships between interparticle friction and macroscopic friction angle are examined from discrete particle simulations. Typical values of strength parameters are listed. The concept of yielding is introduced, and the deformation behavior in both the preyield (including small strain stiffness) stiffness) and post-y post-yield ield regions is summarized. Time-dependent deformations and aging effects are discussed separately in Chapter 12. The details of strength determination by means of  laboratory and in situ tests and the detailed constitutive modeling of soil deformation and strength for use in numerical analyses are outside the scope of this book.

11.2 GENE GENERAL RAL CHAR CHARA ACTER CTERISTIC ISTICS S OF OF STRENGTH AND DEFORMATION Strength

1. In the absence of chemical cementatio cementation n between grains, the strength (stress state at failure or the ulti ul timat matee st stre ress ss st state ate)) of sa sand nd and cl clay ay is approximated by a linear relationship with stress:  ff 

    ff   tan    

 

(11.4)

or

Shear Stress τ or Stress Ratio τ/σ

( 1ff 

    3f f )      ( 1ff      3f f )sin      

  (11.5)

where whe re the pri primes mes des design ignate ate eff effecti ective ve str stress esses es  1ff an and   d  3ff  ar aree th thee ma majo jorr an and d mi mino norr pr prin inci cipa pall effective stresses at failure, respectively. 2. The basic contributions contributions to soil strength strength are frictional resistance between soil particles in contact and int intern ernal al kin kinema ematic tic con constr strain aints ts of soi soill particles associated with changes in the soil fabric. The mag magnitu nitude de of the these se con contri tribu bution tionss depends pen ds on the eff effecti ective ve str stress ess and the vo volum lumee change tendencies of the soil. For such materials the stress–stra stress–strain in cur curve ve fro from m a she sheari aring ng tes testt is typically of the form shown in Fig. 11.1a. The maximum or peak strength of a soil (point   b) may be greater than the critical state strength, in which the soil deforms under sustained loading at constant volume (point  c ). For some soils, the particles align along a localiz localized ed failure plane afterr lar afte large ge she shear ar str strain ain or she shear ar dis displa placeme cement, nt, and the strength decreases even further to the residual residu al streng strength th (poin (pointt   d ). The corres correspondin ponding g three failure envelopes can be defined as shown in Fig. 11.1b, with peak, critical, and residual friction angles (or states) as indicated. 3. Peak failure failure envelopes envelopes are usually curved curved in the manner shown in Fig. 4.16 and schematically in Fig. 11.1b. This behavior is caused by dilatancy supp su ppre ress ssio ion n an and d gr grai ain n cr crus ushi hing ng at hi high gher er stresses. Curved failure envelopes are also observed for many clays at residual state. When

Secant Peak Strength Envelope

Peak

b Shear  τ Stress c

At Large Strains

Critical state Strength Envelope

Tangent Peak Strength Envelope Peak Strength

φ

peak

d

φcritical state

b, c

Critical State b

Residual Strength Envelope

c

Residual

φ residual d

d

a

a

Strain

Dense or Overconsolidated

(a)

a Normal effective stress σ Loose or Normally Consolidated

(b)

Figure 11.1   Peak, critical, and residual strength and and associated friction angle: angle: (a) a typical

stress–strain curve and (b) stress states.

 

GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION

expressed expres sed in ter terms ms of the she shear ar str streng ength th nor nor-malized by the effective normal stress as a function tio n of ef effect fectiv ivee nor normal mal str stress ess,, cur curves ves of the type shown in Fig. 11.2 for two clays are obtained.

4. The peak strength of cohesionless cohesionless soils is influinfluenced en ced mo most st by de dens nsity ity,, ef effe fect ctiv ivee co confi nfini ning ng pres pr essu sure res, s, tes testt ty type pe,, an and d sa samp mple le pr prep epar arat atio ion n methods. For dense sand, the   secant   peak friction angle (poin (pointt  b  in Fig. 11.1b) consists in part

Figure 11.2   Variat ariation ion of resid residual ual strength strength with stres stresss level (after Bishop Bishop et al., 1971): (a)

Brown London clay and (b) Weald clay.

  371

 

372

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

of internal rolling and sliding friction between grains gra ins and in par partt of int interlo erlockin cking g of par particl ticles es (Tayl (T aylor or,, 194 1948). 8). The int interlo erlocki cking ng nece necessi ssitate tatess either eit her vo volum lumee exp expans ansion ion (di (dilat latanc ancy) y) or gra grain in fra ract ctur uree an and/ d/ or cr crus ushi hin ng if th ther eree is to be deformation. For loose sand, the peak friction angle (point  b  in Fig. 11.1b) normally coincides with the critica critical-state l-state friction angle (point   c), and there is no peak in the stress–strain curve. 5. The peak strength strength of saturated saturated clay is influenced influenced most by overconsolidation ratio, drainage conditions, dition s, effec effective tive confin confining ing press pressures, ures, origin original al structure, struc ture, disturbance disturbance (which causes a change in effective stress and a loss of cementation), and creep or deformation rate effects. Overconsolidated clays usually have higher peak strength at a giv given en effec effective tive stress than normal normally ly consol consoliidated clays, as shown in Fig. 11.3. The differences in strength result from both the different stress histories and the different water contents at peak. For comparisons at the same water content but different effective stress, as for points  A   and   A, the Hvorslev strength parameters   ce and       are obt obtain ained ed (Hv (Hvors orsle lev v, 193 1937, 7, 196 1960). 0). e Further details are given in Section 11.9. 6. Dur During ing critical critical sta state te def deform ormatio ation n a soi soill is completely destructured. As illustrated in Fig. 11.1b, the critical state friction angle values are independent of stress history and original structure; for a given set of testing conditions the shearing

  w

   t   n   e   e   t   n   o    i    t   o   a   C    R  r   e    d   t    i   o   a

Normally Consolidated Virgin Compression

 A

   V   W

 e ff 

 A

Rebound Overconsolidated

     τ

τ

  s   s   e   r    t    S   r   a   e    h    S

resistance depends only on composition and effective stress. The basic concept of the critical state is that under sustained uniform shearing at failur fai lure, e, the there re exi exists sts a uni unique que com combin binatio ation n of  void vo id rat ratio io   e, me mean an pr pres essu sure re   p, an and d de devi viato atorr 1 stress   q. The cri critica ticall sta states tes of rec recons onstitu tituted ted Weald clay and Toyoura sand are shown in Fig. 11.4. The critical state line on the  p  – q  plane is linear,2 whereas that on an   e-ln   p   (or   e-log   p) plane tends to be linear for clays and nonlinear for sands. 7. At failure, dense sands and heavily heavily overconsoloverconsolidated clays have a greater volume after drained shear or a higher effective stress after undrained shear than at the start of deformation. This is due to its dilative tendency upon shearing. At failure, loose sands and normally consolidated to moderately overconsolidated clays (OCR up to about 4) have a smaller volume after drained shear or a lower effective stress after undrained shear than they had initially. This is due to its contractive tendency upon shearing. 8. Under further further deformation, platy clay particles begin to align along the failure plane and the shear resistance may further decrease from the critical state condition. The angle of shear resistance at this condition is called the residual friction angle, as illustrated in Fig. 11.1 b. The post po stpe peak ak sh shea earin ring g di disp spla lacem cemen entt re requ quir ired ed to cause a reduction in friction angle from the critical state value to the residual value varies with the soil type, normal stress on the shear plane, and test conditions. For example, for shale mylonite3 in contact with smooth steel or other polished hard surfaces, a shearing displacement of  only 1 or 2 mm is sufficient to give residual strength.4 For soil against soil, a slip along the

σ ff 

1

In three-dimensional stress space       (   x,   y,   z,     xy,     yz,     zx ) or the eq the equi uiv val alen entt pr prin inci cipa pall st stre ress sses es ( 1,   2,   3), th thee me mean an ef effe fect ctiive stress  p , and the deviator stress  q  is defined as

σ e

 p     (   x      y      z ) / 3       (   1       2       3) / 3

Peak Strength Envelope

φcrit

 (  x      y)2    (  y      z)2    (  z      x)2    6  2xy    6  2yz    6  2zx 

Overconsolidated

 A

 A

φ e

Hvorslev Envelope

c  e

Normally Consolidated

0

σ ff  Normal Effective Stress σ 

Effect ct of ove overcons rconsolida olidation tion on effe effectiv ctivee stres stresss Figure 11.3   Effe strength envelope.

q     (1 /  2) 2)

  (1

/  2) 2) ( 1      2)2    (  2      3)2    (  3      1)2

For tri For triaxi axial al com compre pressi ssion on con condit dition ion ( 1      2      3),  p      (  1    2 2)/3,  q      1      2 2 The critical state failure slope on   p –q  plane is related to friction angle   , as described in Section 11.10. 3 A rock that has undergone differential movements at high temperature and pressure in which the mineral grains are crushed against one another. The rock shows a series of lamination planes. 4 D. U. Deere, personal communication (1974).

 

GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION

4

500 Critical State Line

   )   a    P 400    k    (   q

   )   a    P    M3    (

Critical State Line

  q

  s   s 300   e   r    t    S   r 200   o    t   a    i   v   e    D 100

0

  373

Overconsolidated Normally Consolidated 0

100

  s   s   e   r    t 2    S   r   o    t   a    i   v 1    D   e

0

200 300 400 500 600 Mean Pressure p Pressure p (kPa)

0

(a-1) p versus (a-1) p versus q  q

Critical State Line

Initial State

0.95 Isotropic Normal Compression Line

0.6

4

(b-1) p versus q (b-1) p

Critical State Line

0.7

1 2 3 Mean Pressure p Pressure p (MPa)

0.90   e

  e

  o    i    t   a 0.85   r    d    i   o    V 0.80

  o    i    t   a   r 0.5    d    i   o    V 0.4

Overconsolidated 0.3

0.75

Normally Consolidated 100

200

300 4 40 00 5 50 00

Mean Pressure p Pressure p (kPa) (a-2) e (a-2)  e versus  versus ln p (a)

0.02

0.05 0.1

0.5

1

5

Mean Pressure p Pressure p (MPa) (b-2) e (b-2)  e versus  versus log p (b)

Figure 11.4   Criti Critical cal states states of clay and sand: sand: ( a) Critical state of Weald clay obtained by

drained triax drained triaxial ial comp compress ression ion test testss of norm normally ally cons consolida olidated ted () and ove overcons rconsolida olidated ted ( ●) specimens: (a-1)   q– p  plane and (a-2)  e –ln   p  plane (after Roscoe et al., 1958). ( b) Critical state of Toyoura sand obtained by undrained triaxial compression tests of loose and dense specimens consolidated initially at different effective stresses, (b-1)  q – p  plane and (b-2)  e – log   p   plane (after Verdugo and Ishihara, 1996).

shear plane of several tens of millimeters may be req requir uired, ed, as sho shown wn by Fig Fig.. 11. 11.5. 5. Ho Howe wever ver,, signifi sig nifican cantt sof softeni tening ng can be cau caused sed by str strain ain localiza loca lizatio tion n and de devel velopm opment ent of she shear ar ban bands, ds, especially for dense samples under low confinement. 9. Streng Strength th anisotropy anisotropy may result from both stress stress and fabric anisotropy. In the absence of chemical cementation, the differences in the strength

of two samples of the same soil at the same void ratio but with different fabrics are accountable in ter terms ms of dif differ ferent ent eff effect ectiv ivee str stress esses es as dis dis-cussed in Chapter 8. 10. Undra Undrained ined strength in triaxial compressio compression n may differ significantly from the strength in triaxial extension. However, the influence of type of test (triaxial (triax ial compre compression ssion versus extension) on the effective stress parameters  c   and     is relatively

 

374

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

Figure 11.5   Dev Develop elopment ment of resid residual ual strength with incre increasin asing g shea shearr displ displacem acement ent (after

Bishop et al., 1971).

small. Effective stress friction angles measured in pla plane ne str strain ain are typ typical ically ly abo about ut 10 per percen centt greater than those determined by triaxial compression. 11. A change in temperature temperature causes either either a change in void ratio or a change in effective stress (or a combination of both) in saturated clay, as discussed in Chapter 10. Thus, a change in tempera pe ratu ture re ca can n ca caus usee a st stre reng ngth th in incr crea ease se or a streng str ength th dec decrea rease, se, dep depend ending ing on the circ circumumstances, as illustrated by Fig. 11.6. For the tests on ka kaoli olini nite te sh show own n in Fi Fig. g. 11 11.6 .6,, all sa samp mple less were prepared by isotropic triaxial consolidation drainagee allow allowed, ed, at 75F. Then, with no further drainag temperatures were increased to the values indicated, cat ed, and the samples samples wer weree tes tested ted in unc uncononfined fine d comp compres ressio sion. n. Sub Substa stanti ntial al red reduct uction ionss in strength accompanied the increases in temperature. Stress–Strain Behavior

1. Stress–strain Stress–strain behavior ranges from very brittle for som somee qui quick ck cla clays, ys, ceme cemente nted d soi soils, ls, hea heavily vily overconsolidated clays, and dense sands to ductile for insensitive and remolded clays and loose sands, as illustrated by Fig. 11.7. An increase in

Figure 11.6   Effe Effect ct of temp temperatu erature re on undra undrained ined strength strength of 

kaolinite in unco kaolinite unconfine nfined d comp compress ression ion (afte (afterr Sheri Sheriff and Burrous, 1969).

 

GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION

  375

(a) Typical Strain Ranges in the Field Retaining Walls

   E

Foundations

  r   o  

Tunnels

   G Linear Elastic

  s   s   e   n    f    f    i    t    S

Nonlinear Elastic Preyield Plastic Full Plastic

10-4

10-3

10-2

Figure 11.7   Types of stress–strain behavior. behavior.

10-1

100

101

Strain % Dynamic Methods Local Gauges

confining pressure causes an increase in the deform fo rmati ation on mo modu dulu luss as we well ll as an in incr creas easee in strength, as shown by Fig. 11.8. 2. Str Stress–st ess–strai rain n rel relatio ationsh nships ips are usu usuall ally y non nonlinlinear; soil stiffness (often expressed in terms of  tangent or secant modulus) generally decreases with increasing shear strain or stress level up to peak failure stress. Figure 11.9 shows a typical stiffne stif fness ss deg degrad radatio ation n cur curve, ve, in ter terms ms of she shear ar modulus   G  and Young’s modulus  E , along with typical strain lev levels els dev developed eloped in geotech geotechnical nical construction (Mair, 1993) and as associated with different diff erent laboratory testing techniq techniques ues used to measure the stiffness (Atkinson, 2000). For example, Fig. 11.10 shows the stiffness degradation of sands and clay subjected to increase in shear strain. As illustrated in Fig. 11.9, the stiffnesss deg nes degrad radatio ation n cur curve ve can be sep separat arated ed into

Conventional Soil Testing (b) Typical Strain Ranges for Laboratory Tests

Figure re 11.9   Stif Stiffnes fnesss degr degradat adation ion curv curve: e: stif stiffnes fnesss plott plotted ed Figu

against logarithm of strains. Also shown are ( a) the strain levels lev els obse observed rved durin during g cons construct truction ion of typi typical cal geot geotechn echnical ical structures (after Mair, 1993) and (b) the strain levels that can be measured by various techniques (after Atkinson, 2000).

four zones: (1) linear elastic zone, (2) nonlinear elastic zone, (3) pre-yield plastic zone, and (4) full plastic zone. 3. In the line linear ar elastic elastic zon zone, e, soil particles particles do not slide relative to each other under a small stress increment, and the stiffness is at its maximum. The soi soill sti stiff ffnes nesss dep depend endss on con contact tact inte interac rac-tions, particle packing arrangement, and elastic stiffness of the solids. Low strain stiffness values can be determined using elastic wave velocity measurements, measurements, resona resonant nt column testing, or local strain transducer measurements. The magnitudes of the small strain shear modulus (Gmax) and Young’s modulus ( E max) depend on applied confini con fining ng pre pressu ssure re and the pack packing ing con conditi ditions ons of soil particles. The following empirical equations tio ns are often emp employ loyed ed to exp expres resss thes thesee dependencies:

Figure Figu re 11.8   Eff Effect ect of con confini fining ng pre pressu ssure re on the con consol solii-

dated-drained stress–strain behavior of soils.

nG

 

(11.6)

n E 

 

(11.7)

Gmax  

    AG F G   (e) p

 E i(max)  

    AE F E  (e) i

where   F G(e) and   F  E (e) ar aree fu func ncti tion onss of vo void id ratio,   p   is the mean effecti effective ve con confini fining ng pre presssure, su re,  i is th thee eff effec ecti tive ve st stre ress ss in th thee   i  direction, and the other parameters are material constants.

 

376

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

TC

   ) 140   a    P    M    (  120

Confining Pressure

PSC

Toyoura Sand

78.4 kPa

Ticino Sand

49 kPa

   G

  s   u100    l   u    d   o 80    M   r   a 60   e    h 40    t    S   n   a   c 20   e    S

10-4

10-3

10-2

10-1

100

   )   a    P120    M    (     G100   s   u    l   u    d 80   o    M   r 60   a   e    h 40    S    t   n   a   c   e 20    S

Confining Pressures

σc = 400 kPa σc = 200 kPa

σc = 100 kPa σc = 30 kPa

10-5

10-4

10-3

10-2

10-1

100

Shear Strain (%)

Shear Strain (%) (a)

(b)

Figure 11.10   Stiffness degradation degradation curve at different confining confining pressures: (a) Toyoura and

Ticino sand Ticino sandss (TC: triax triaxial ial comp compress ression ion tests tests,, PSC: plain stra strain in comp compress ression ion tests) (after Tatsuoka et al., 1997) and (b) reconstituted Kaolin clay (after Soga et al., 1996).

Figure 11.11 shows to examples of thedata. fitting of  the above equations experimental 4. The stiffness stiffness begins to decrease decrease from the linear elasti ela sticc val value ue as the app applie lied d str strains ains or str stress esses es increa inc rease, se, and the def deform ormati ation on mov moves es int into o the nonlinear nonlin ear elasti elasticc zone. However, However, a comple complete te cycle of loading, unloading, and reloading within thiss zon thi zonee sho shows ws ful fulll rec recov overy ery of str strains ains.. The strain at the onset of the nonlinear elastic zone ranges from less than 5  104 percent for non-

104   a    P    M   x   a 103   m

At each vertical effective stress, horizontal effective stress σh (kPa) was varied between 98 kPa and

Undisturbed Remolded Remolded with CaCO3 nG = 0.13

   G  ,

  s   u 2    l   u 10    d   o    M   r 1   a 10   e    h    S 100 100

plastic soils at low pressure conditions to greater than 5 confining 102 percent at high confining pressure or in soils with high plasticity (Santamarina et al., 2001). 5. Irr Irreco ecover verabl ablee str strains ains de devel velop op in the pre pre-yi -yield eld plastic zone. The initiation of plastic strains can be determined by examining the onset of permanent volumetric strain in drained conditions or residual excess pore pressures in undrained conditi con ditions ons aft after er unl unload oading ing.. Ava vaila ilable ble exp experi eri--

nG = 0.65 nG = 0.63

101 102 103 104 Confining pressure, p pressure, p (kPa) (a)

   l   u   s 500   u   )    d   a450   o   P    M    M   s   (    '   e 400   g   )   n   (    E   u   F   o   /350    Y   a    l   a   m   c   v 300    i    t   r   E   e    V 250

196 kPa

n E = 0.49

     x

100

150 200

250

300

Vertical Effective Stress,σv (kPa) (b)

Figure 11.11   Smal Smalll stra strain in stif stiffnes fnesss vers versus us confining confining pres pressure sure:: (a) Shear modulus   Gmax   of 

cemented silty sand measured by resonant column tests (from Stokoe et al. 1995) and (b) vertical Young’s modulus of sands measured by triaxial tests (after Tatsuoka and Kohata, 1995).

 

GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION

mental data suggest that the strain level that initiates plastic strains ranges between 7      103 and 7      102 percent, with the lower limit for uncemen unc emented ted nor normall mally y con consol solida idated ted san sands ds and the upper limit for high plasticity clays and cemented sands. 6. A dis distin tincti ctive ve kin kink k in the str stress–st ess–strai rain n rel relatio ationnship defines yielding, beyond which full plastic strains str ains are gen generat erated. ed. A loc locus us of str stress ess states that initiate yielding defines the yield envelope. Typi ypical cal yiel yield d en envel velope opess for san sand d and nat natura urall clay are shown in Fig. 11.12. The yield envelope expands, shrinks, and rotates as plastic strains develop. It is usually considered that expansion is related to plastic volumetric strains; the surface fa ce ex expa pand ndss wh when en th thee so soil il co comp mpre ress sses es an and d shrinks when the soil dilates. The two inner envelopes shown in Fig. 11.12b  define the boundariess bet arie betwee ween n lin linear ear elas elastic, tic, non nonline linear ar elas elastic, tic, and an d pr pree-yi yiel eld d zo zone nes. s. Wh When en th thee st stre ress ss st stat atee moves in the pre-yield zone, the inner envelopes envelopes move with the stress state. This multienvelope concept allows modeling of complex deformations observed for different stress paths (Mroz, 1967; 196 7; Pre ´vost, 197 ´vost, 1977; 7; Daf Dafalia aliass and Her Herrma rman, n, 1982; Atkinson et al., 1990; Jardine, 1992). 7. Plas Plastic tic ir irre reco cove vera rabl blee sh shea earr de defo form rmat atio ions ns of  satu sa tura rated ted so soil ilss ar aree ac acco comp mpan anied ied by vo volu lume me

changes when drainage is allowed or changes in poree wat por water er pre pressu ssure re and ef effect fectiv ivee str stress ess whe when n drainage is prevented. The general nature of this behavior is shown in Figs. 11.13a   and 11.13b for dra draine ined d and und undrain rained ed con conditi ditions ons,, res respec pec-tive ti vely ly.. Th Thee vo volu lume me an and d po pore re wa water ter pr pres essu sure re changes depend on interactions between fabric and stress state and the ease with which shear defo de form rmat atio ions ns ca can n de deve velo lop p wi with thou outt ov over eral alll changes in volume or transfer of normal stress from the soil structure to the pore water. 8. The The st stre ress–s ss–str train ain re relat latio ion n of cl clay ayss de depe pend ndss largely lar gely on ov overc ercons onsoli olidat dation ion rat ratio, io, ef effec fectiv tivee confini con fining ng pre pressu ssures res,, and dra drainag inagee con conditi ditions ons.. Figure 11.14 sho shows ws triaxia triaxiall compres compression sion behavior of clay specimens that are first normally consolid so lidat ated ed an and d th then en is isotr otrop opica ically lly un unlo load aded ed to different overconsolidation ratios before shearing. The specimens are consolidated at the same confi co nfini ning ng pr pres essu sure re  p0, bu butt ha have ve di diff ffer eren entt vo void id ratios rat ios due to the dif differ ferent ent str stress ess history history (Fi (Fig. g. 11.14a). Draine Drained d tests on normal normally ly conso consolidated lidated clays and lightly overconsolid overconsolidated ated clays show ductile behavior with volume contraction (Fig. 11.14b). Heavily overconsolidated clays exhibit a stif stifff res respon ponse se init initiall ially y unt until il the str stress ess state reac re aches hes th thee yi yiel eld d en enve velo lope pe gi givin ving g th thee pe peak  ak  strength and volume dilation. The state of the

Yield State Pre-yield State

Initial Condition q = σa-σr MPa Failure Line 0.8

Yield State Stress Path

q = σa-σr

Initial State Surrounded by Linear Elastic Boundary

MPa 0.6

Yield Envelope

Yield Envelope

0.6

0.4 Preyield Boundary

0.4 0.2 0.2

Linear Elastic Boundary

MPa 0.0

0.2

0.4

0.6

-0.2 -0.4

0.8

1.0  p = (σa + 2σr)/3

0.0

0.2  

MPa 0.4

0.6 p = (σa + 2σr)/3

-0.2 Failure Line

(a)

(b)

Figure 11.12   Yie Yield ld envelopes: envelopes: (a) Aoi sand (Yasufuku et al., 1991) and ( b) Bothkennar clay

(from Smith et al., 1992).

  377

 

378

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

Same Initial Confining Pressure

Same Initial Confining Pressure

Dense Soil Metastable Fabric

Deviator Stress

Critical State

Dense Soil Deviator Stress

Cavitation

Critical State Loose Soil

Loose Soil

Critical State

Metastable Fabric Axial or Deviator Strain

Axial or Deviator Strain

Dense Soil

Dense Soil +∆ V   / V  V0 

-∆ u

0

0

Cavitation

Loose soil

Loose Soil +∆ u

-∆ V   / V  V0  Metastable Fabric

Metastable Fabric (b)

(a)

Figure 11.13   Volume and pore pressure pressure changes during shear: shear: ( a) drained conditions and

(b) undrained conditions.

Initial State Failure at Critical State (D: Drained, Drained, U: Und Undrained) rained)

Void Ratio

Deviator Stress

3 Heavily Overconsolidated 2 Lightly Overconsolidated

Deviator Stress

U3 2 Lightly Overconsolidated U2

D Critical State

Virgin Compression Line

3 Heavily Overconsolidated

1 Normally Consolidated U1

1 Normally Consolidated

1 Normally consolidated

U1 Axial or Deviatoric Strain

Axial or Deviatoric Strain 2 Lightly Overconsolidated U2 D U3

+∆ V   / V  V0 

3 Heavily Overconsolidated

-∆ u

3 Heavily Overconsolidated

3 Heavily Overconsolidated Critical State Line  p0

-∆ V   / V  V0 

2 Lightly Overconsolidated

2 Lightly Overconsolidated +∆ u 1 Normally Consolidated

log p 1 Normally Consolidated (a)

(b)

(c)

Figure 11.14   Stress–strain relationship of normally consolidated, consolidated, lightly overconsolidated, overconsolidated,

and heavily overconsolidated clays: (a) void ratio versus mean effective stress, ( b) drained tests, and (c) undrained tests.

 

FABRIC, STRUCTURE, AND STRENGTH

soil then progressively moves toward the critical state exhibi exhibiting ting soften softening ing beha behavior vior.. Undra Undrained ined sheari she aring ng of nor normall mally y con consol solida idated ted and lig lightly htly overconsolidated clays generates positive excess pore pressures, whereas shear of heavily overconsol con solida idated ted cla clays ys gen genera erates tes neg negati ative ve exc excess ess pore pressures (Fig. 11.14c). 9. The mag magni nitu tude dess of po pore re pr pres essu sure re th that at ar aree de de-veloped in undrained loading depend on initial consolidation consol idation stres stresses, ses, overc overconsoli onsolidation dation ratio, density, and soil fabric. Figure 11.15 shows the undrained undrai ned effe effectiv ctivee stres stresss paths of anisot anisotropiropically and isotro isotropically pically consolidated consolidated specim specimens ens (Ladd and Varally arallyay ay,, 1965). The diff difference erence in undrained shear strength is primarily due to different excess pore pressure dev developmen elopmentt associated with the change in soil fabric. At large strains, the stress paths correspond to the same friction frictio n angle. 10. A temperature temperature increase causes causes a decrease in undrained modulus; that is, a softening of the soil. As an ex exam ample ple,, ini initia tiall st stra rain in as a fu func nctio tion n of  stre st ress ss is sh sho own in Fi Fig. g. 11 11.1 .16 6 fo forr Os Osak akaa cla clay y

  379

Figure 11.16   Effect of temperature on the stiffness of Osaka

clay in undrained triaxial compression (Murayama, 1969). Failure Line in Triaxial Compression

MPa)) (MPa 0.3

tested in undrained triaxial compression at different fer ent tem temper peratur atures. es. Inc Increas reasee in temp tempera eratur turee causes consolidation consolidation under drained conditi conditions ons and softening under undrained conditions.

σr/σa = 0.54 0.2      σ   r                            

     σ

  + 0.1

11.3

  a

FABRI ABRIC, C, STRUCTU STRUCTURE, RE, AND AND STRENGTH STRENGTH

                           

     σ

  =   q

  s 0.0   s   e   r    t    S   r   o -0.1    t   a    i   v   e    D

0.1

0.2

0.3

0.4

(MPa)

 Mean Pressure Pressure  p = (σa + 2σr )/3

-0.2

σr/σa = 1.84 -0.3 Initial

At Failure

Failure Line in Triaxial Extension

Anisotropically Consolidated σr/σa = 0.54 Isotropically Consolidated Anisotropically Consolidated σr/σa = 1.84

Figure 11.15   Undrained effective effective stress paths of anisotropanisotrop-

ically and isotr ically isotropic opically ally cons consolida olidated ted spec specimen imenss (afte (afterr Ladd and Varallyay, 1965).

Fabric Changes During Shear of Cohesionless Materials

The deformation of sands, gravels, and rockfills is influenced by the initial fabric, as discussed and illustrated in Chapter 8. As an illustration, fabric changes associated with the sliding and rolling of grains during triaxial compression were determined using a uniform sand composed of rounded to subrounded grains with sizes in the range of 0.84 to 1.19 mm and a mean axial length ratio of 1.45 (Oda, 1972, 1972a, 1972b, 1972c). Samples were prepared to a void ratio of 0.64 by tamping and by tapping the side of the forming mold. A delayed setting water–resin solution was used as the pore po re flu fluid. id. Sa Samp mples les pr prep epar ared ed by ea each ch met metho hod d we were re tested tes ted to suc succes cessi sivel vely y hig higher her str strains ains.. The res resin in was then allowed to set, and thin sections were prepared. The differences in initial fabrics gave the markedly different stress–strain and volu volumetric metric strain curves shown in Fi Fig. g. 11 11.1 .17, 7, wh where ere th thee pl plun ungi ging ng me metho thod d re refe fers rs to

 

380

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

Figure 11.17   Stres Stress–strai s–strain n and volumetric volumetric strain relationsh relationships ips for sand at a void ratio of 

0.64 but with different initial fabrics (after Oda, 1972a). ( a) Sample saturated with water and (b) sample saturated with water–resin solution.

tamping. There is similarity between these curves and those for Monterey No. 0 sand shown in Fig. 8.23. A statistical analysis of the changes in particle orientation with increase in axial strain showed: 1. For samples samples prepared by tapping, tapping, the initial fabric tended toward some preferred orientation of  long axes parallel to the horizontal plane, and the intensity of orientation increased slightly during deformation. 2. For samples samples prepared by tamping, tamping, there was very weak preferred orientation in the vertical direction initially, but this disappeared with deformation. Shear deformations break down particle and aggregate assemblages. Shear planes or zones did not appear until after peak stress had been reached; however, the distr dis trib ibut utio ion n of no norm rmal alss to th thee in inte terp rpar artic ticle le co cont ntac actt planes   E ( ) (a me meas asur uree of   fabric anisot anisotro ropy py) did change with strain, as may be seen in Fig. 11.18. This figure shows different initial distributions for samples prepared by the two methods and a concentration of  contact plane normals within 50   of the vertical as deformation progresses. Thus, the fabric tended toward great gr eater er an anis isot otro ropy py in ea each ch cas casee in te term rmss of co cont ntac actt plane orientations. There was little additional change in   E ( ) after the peak stress had been reached, which implies impl ies that par particl ticlee rear rearran rangem gement ent was pro proceed ceeding ing without significant change in the overall fabric.

As the stress state approaches failure, a direct shearinduce ind uced d fab fabric ric for forms ms that is gen genera erally lly com compos posed ed of  region reg ionss of hom homoge ogeneo neous us fab fabric ric sep separa arated ted by dis discon con-tinuit tin uitie ies. s. No di disc scon onti tinu nuiti ities es de deve velo lop p be befo fore re pe peak  ak  strength is reached, although there is some particle rotation in the direction of motion. Near-perfect preferred orientation develops during yield after peak strength is reached rea ched,, bu butt lar large ge def deform ormatio ations ns may be req requir uired ed to reach this state. Compaction Versus Overconsolidation of Sand

Specimens at the same void ratio and stress state before shearing, but having different fabrics, can exhibit different stress–strain behavior. For example, consider a ca case se in wh which ich on onee sp spec ecime imen n is ov over erco cons nsol olida idate ted, d, wherea whe reass the oth other er is com compac pacted. ted. The two specimens specimens are prepared in such a way that the initial void ratio is the same for a given initial isotropic confining pressure. Coop (1990) performed undrained triaxial compression tests of carbonate sand specimens that were either overconsolidated or compacted, as illustrated in Fig.. 11. Fig 11.19 19a. The und undrain rained ed str stress ess paths and str stress– ess– strain curves for the two specimens are shown in Figs. 11.19b  and 11.19c, respectively. The overconsolidated sample was initially stiffer than the compacted specimen. The difference can be attributed to (i) different soil fabrics developed by different stress paths prior to shearing and (ii) different degrees of particle crushing prior to shearin shearing g (i.e., some breakage has occurred dur-

 

FABRIC, STRUCTURE, AND STRENGTH

  381

Figure 11.18   Dist Distribu ribution tion of inter interparti particle cle contact contact normals as a function of axia axiall stra strain in for

sand samples prepared in two ways (after Oda, 1972a): (a) specimens prepared by tapping and (b) specimens prepared by tamping.

ing the precon preconsolid solidation ation stage for the over overconsol consolidated idated specimen). Therefore, overconsolidation and compaction tio n pr prod oduce uced d ma mater teria ials ls wi with th di diff ffer erent ent me mech chani anica call properties. However, at large deformations, both spec-

much lower peak strength for the sample prepared by kneading compaction. The recov recoverable erable deformation deformation of compact compacted ed kaolini kaolinite te with wit h floc floccul culent ent str struct ucture ure ran ranges ges bet betwee ween n 60 and 90

imen imens s exh exhibit ibited ed sim similar ilar str streng engths ths bec becaus ausee the ini initial tial fabrics were destroyed.

per percen cent, t,structures whereas whe reas is theonly recover reco y order of sam samples with dis-dis persed ofvery the ofples 15 to 30 percent of the total deformation, as may be seen in Fig. 11.21. This illustrates the much greater ability of the braced-box type of fabric that remains after static compaction to withst withstand and stres stresss withou withoutt perman permanent ent defor defor-mation mat ion tha than n is pos possib sible le wit with h the bro broken ken-do -down wn fab fabric ric associated with kneading compaction. Different macrofabric features can affect the deformation behavior as illustrated in Fig. 11.22 for the undraine dra ined d tria triaxial xial com compre pressi ssion on tes testin ting g of Bot Bothke hkenna nnarr clay, Scotland (Paul et al., 1992; Clayton et al., 1992). Samples with mottled facies, in which the bedding feature tu ress ha had d bee been n di disr srup upted ted an and d mix mixed ed by bu burr rrow owin ing g mollusks mollus ks and worms (bioturbation), (bioturbation), gave the stiff stiffest est response, respo nse, whereas sample sampless with distinct laminat laminated ed features showed the softest response.

Effect of Clay Structure on Deformations

The high sensitivity of quick clays illustrates the principle that floccul flocculated, ated, open microf microfabrics abrics are more rigid but more unstable than deflocculated fabrics. Similar behavior may be observed in compacted fine-grained soils, and the results of a series of tests on structuresensit sen sitiv ivee kao kaolini linite te are illu illustr strati ative ve of the dif differ ferenc ences es (Mitchell (Mitch ell and McCon McConnell, nell, 1965). Compac Compaction tion conditions and stress–strain curves for samples of kaolinite comp co mpact acted ed us usin ing g kn knea eadi ding ng an and d st stat atic ic me meth thod odss ar aree shown in Fig. 11.20. The high shear strain associated with wi th kn knea eadi ding ng co comp mpac actio tion n we wett of op opti timu mum m br break eakss down flocculated structures, and this accounts for the

 

382

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR 1.0 Overconsolidated 0.8

2

Normal Compression Line

   ) 0.6   a    P    M    (   q 0.4 Compacted 0.2

Overconsolidated Sample

Void 1.5 Ratio

0

0

0.2 0.4  p (MPa)

0.6

(b) q (MPa)

Compacted Sample

1.0

1

Compacted 0.75

0.1 1 Mean Pressure p

(MPa)

0.5

Overconsolidated

0.25

(a) 0

0

4

8 12 16 Axial strain ε a (%)

20

( c)

overconsolidated specimen specimen Figure 11.19   Undrained response of compacted specimen and overconsolidated of carbonate sand: (a) stress path before shearing, (b) undrained stress paths during shearing, and (c) stress–strain relationships (after Coop, 1990).

If slip planes develop at failure, platy and elongated particles align with their long axes in the direction of  slip. By then, the basal planes of the platy clay particles are enclosed between two highly oriented bands of particles on opposite sides of the shear plane. The

clay can also be accounted for in terms of differences in ef effect fectiv ivee str stress ess,, pro provid vided ed par partt of the und undistu isturbe rbed d strength does not result from cementation. Remolding breaks down the structure and causes a transfer of effective stress to the pore water.

dominant of displacementt sh men shea earmechanism r zo zone ne is ba basa salldeformation plan pl anee sl slip ip,, in and an dthethe over ov erall all thickness of the shear zone is on the order of 50   m. Fabrics associated with shear planes and zones have been studied using thin sections and the polarizing microscope and by using the electron microscope (Morgenstern and Tchalenko, 1967a, b and c; Tchalenko, 1968; McKyes and Yong, 1971). The residual strength associated with these fabrics is treated in more detail in Section 11.11.

An example of of this is shown in Fig. 11.23, shows the results incremental loading triaxialwhich compression tests on two samples of undisturbed and remold mo lded ed Sa San n Fr Fran ancis cisco co Ba Bay y mu mud. d. In th thes esee te test sts, s, th thee undist und isturb urbed ed sam sample ple was firs firstt bro brough ughtt to equ equilib ilibriu rium m under und er an iso isotro tropic pic con consol solida idation tion pre pressu ssure re of 80 kPa kPa.. After undrained loading to failure, the triaxial cell was disassembled, and the sample was remolded in place. The apparatus was reasse reassembled, mbled, and pore press pressure ure was measured. Thus, the effective stress at the start of compression of the remolded clay at the same water content te nt as the or orig igina inall un undi dist stur urbe bed d cla clay y wa wass kn know own. n. Stress–strain and pore pressure–strain curves for two sample sam pless are sho shown wn in Fig Figs. s. 11. 11.23 23a   and 11. 11.23 23b, and stress paths for test 1 are shown in Fig. 11.23c. Differences in strength that result from fabric differences caused by thixotropic hardening or by different compaction methods can be explained in the same

Structure, Effective Stresses, and Strength

The effective stress strength parameters such as  c   and     are are iso isotrop tropic ic pro proper perties ties,, wit with h ani anisot sotrop ropy y in undrained dra ined str streng ength th exp explain lainabl ablee in ter terms ms of exc excess ess por poree pres pr essu sures res de deve velo lope ped d du duri ring ng sh shear ear.. Th Thee un undr drain ained ed streng str ength th los losss ass associ ociated ated wit with h rem remold olding ing und undist isturb urbed ed

 

  383

FRICTION BETWEEN SOLID SURFACES

Figure 11.21   Ratio of recoverable recoverable to total strain for samples

of kaolinite with different structure.

Stress-Strain Relationships

Stress Paths

0.6

0.6

  o   a

  o   a

                           

     σ

                           

     σ

0.4    2    /    )   r

   2    /    )   r

                           

     σ

                           

     σ

  –

  a

  –

Facies

                           

   ( 0.2      σ

  a

                           

Mottled Bedded Laminated

0.0

Figure 11.20   Stress–strain behavior of of kaolinite compacted compacted

by two methods.

0.4

     σ

   (

0.0 0

2

4

Facies

0.2

Mottled Bedded Laminated 0.4

0.6

0.8



Axial Strain (%)



1.0 

(σ a + σ r)/2σ ao

Figure 11.22   Effe Effect ct of macr macrofab ofabric ric on undra undrained ined response response

way. Thus, in the absence of chemical or mineralogical changes, different strengths in two samples of the same soil so il at th thee sa same me vo void id ra ratio tio can be ac acco coun unte ted d fo forr in terms of different effective stress.

11.4

FRICTION FRIC TION BETWEE BETWEEN N SOLID SURF SURFA ACES

The fri fricti ction on ang angle le use used d in equ equatio ations ns suc such h as (11 (11.1) .1),, (11.2) (11 .2),, (11 (11.4) .4),, and (11 (11.5) .5) con contain tainss res resist istanc ancee con contritributio bu tions ns fr from om se seve vera rall so sour urces ces,, in inclu cludin ding g sl slid idin ing g of  grains gra ins in con contact tact,, res resista istance nce to vo volum lumee cha change nge (di (dilalatancy),, grain rearrangement, tancy) rearrangement, and grain crushing. crushing. The

of Both Bothkenn kennar ar clay in Scotl Scotland and (after Hight and Lerou Leroueil, eil, 2003).

true friction coefficient is shown in Fig. 11.24 and is represented repres ented by  

T     tan      N 

 

(11.8)

where   N  is   is the normal load on the shear surface,   T   is the shear force, and    , the intergrain sliding friction angle, is a compositional property that is determined by the type of soil minerals.

 

384

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

Figure 11.23   (a)Bay and mud. (b) Effect of remolding undrained strengthtests and pore water pressure in San Francisco ( c) Stress paths foron triaxial compression on undisturbed and

remolded samples of San Francisco Bay mud.

Basic ‘‘Laws’’ of Friction

Two law lawss of fri frictio ction n are rec recogn ognized ized,, beg beginn inning ing wit with h Leonardo da Vinci in about 1500. They were restated by Amontons in 1699 and are frequently referred to as  Amontons’ laws . They are: 1. The frictional frictional force is directly proportiona proportionall to the normal force, as illustrated by Eq. (11.8) and Fig. 11.24. 2. The fri frictio ctional nal res resista istance nce bet betwee ween n two bodies is inde in depe pend nden entt of th thee si size ze of th thee bo bodi dies es.. In Fi Fig. g.

 T  is 11.24, the value of  T   is the same for a given value of   N   regardless of the size of the sliding block.

Although Althou gh the these se prin principl ciples es of fri frictio ctional nal res resist istanc ancee have ha ve lon long g been kno known, wn, sui suitab table le exp explan lanatio ations ns came much later. It was at one time thought that interlocking betwee bet ween n irr irregu egular lar sur surfac faces es cou could ld acco account unt for the behavior. On this basis,     would be given by the tangent of the average inclination of surface irregularities on the sli sliding ding plane. This cann cannot ot be the cas case, e, ho howe wever ver,, because such an explanation would require that      de-

 

FRICTION BETWEEN SOLID SURFACES

  385

Figure 11.23 Continued )) (Continued 

contacting surfaces. He observed that the actual area of contact is very small because of surface irregularities, and thus the cohesive forces must be large. The foundation for the present understanding of the mobiliz mob ilizatio ation n of fri frictio ction n betw between een sur surfac faces es in con contact tact was laid by Terzaghi (1920). He hypothesized that the normal load   N   acting between two bodies in contact causes yielding at   asperities,   which are local ‘‘hills’ ‘hills’’’ on the surface, where the actual interbody solid contact develops. The actual contact area   Ac   is given by  Ac  

 N 

 

(11.9)

  y

where     y   is the yie yield ld st stre reng ngth th of th thee mat mater eria ial. l. Th Thee shearing strength of the material in the yielded zone is assumed to have a value    m. The maximum shearing force that can be resisted by the contact is then T      Ac  m

Figure 11.24   Coefficie Coefficient nt of friction for surfaces in contact.

 

(11.10)

The coefficient of friction is given by   T  /   /  N   N , crease as surfaces become smoother and be zero for perfec per fectly tly smo smooth oth sur surfac faces. es. In fac fact, t, the coe coeffi fficie cient nt of  friction can be constant over a range of surface roughness ne ss.. Ha Hard rdy y (1 (193 936) 6) su sugg gges este ted d in inst stea ead d th that at st stat atic ic fric fr ictio tion n or orig igina inate tess fr from om co cohe hesi sive ve fo forc rces es bet betwe ween en

 

T  N



Ac  m Ac  y



 m  y

 

(11.11)

This co This conc ncep eptt of fr frict ictio iona nall re resi sist stan ance ce wa wass su subs bseequently further developed by Bowden and Tabor (1950,

 

386

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

1964). The Terzagh erzaghi–Bowden i–Bowden and Tabor hypoth hypothesis, esis, commonly referred to as the  adhesion theory of friction,  is the basis for most modern studies of friction. Two char charact acteris eristics tics of sur surfac faces es pla play y ke key y rol roles es in the adhesion theory of friction: roughness and surface adsorption. Surface Roughness

The surfaces of most solids are rough on a molecular scale, with succes successions sions of asperi asperities ties and depres depressions sions ranging from 10 nm to over 100 nm in height. The slopes of the nanoscale asperities are rather flat, with individual angles ranging from about 120   to 175   as shown in Fig. 11.25. The average slope of asperities on met metal al su surf rface acess is an in incl clud uded ed an angl glee of 15 150 0; on rough quartz it may be over 175  (Bromwell, 1966). When two surfaces are brought together, contact is established at the asperities, and the actual contact area is only a small fraction of the total surface area. Quartz surfaces polished to mirror smoothness may consist of peaks and valleys with an average height of  about 500 nm. The asperities on rougher quartz surfaces may be about 10 times higher (Lambe and Whitman, 1969). Even these surfaces are probably smoother than most soil particles composed of bulky minerals. The actual surface texture of sand particles depends on geolog geo logic ic his histor tory y as wel welll as min minera eralog logy y, as sho shown wn in Fig. 2.12. Thee cl Th clea eava vage ge fa face cess of mi mica ca fla flake kess are am amon ong g th thee smoothest naturally occurring mineral surfaces. Even in mica, however, there is some waviness due to rotation of tetrahe tetrahedra dra in the silica layer, and surfaces usually contain steps ranging in height from 1 to 100 nm, reflecting different numbers of unit layers across the particle. Thus, large areas of solid contact between grains are not probable in soils. Solid-to-solid contact is through asperities, and the corresponding interparticle contact stress str esses es are hig high. h. The mol molecul ecular ar str structu ucture re and composition in the contacting asperities determine the magnitude of    m  in Eq. (11.11).

Surface Adsorption

Because of unsatisfied force fields at the surfaces of  solids, the surface structure may differ from that in the interior, and material may be adsorbed from adjacent phases. Even ‘‘clean’’ surfaces, prepared by fracture of  a solid or by evacuation at high temperature, are rapidly idl y con contami taminat nated ed whe when n ree reexpo xposed sed to nor normal mal atm atmosospheric conditions. According to the kinetic theory of gases, the time for adsorption of a monolayer   t m   is given by

t m 



1  SZ  SZ 

 

(11.12)

where      is the area occupied per molecule,   S   is the fraction of molecules striking the surface that stick to it, and   Z  is   is the number of molecules per second striking a square centimeter of surface. For a value of   S  equal to 1, which is reasonable for a high-energy surface, the relation relationship ship between   t m   and gas pressure is shown in Fig. 11.26. The conclusion to be drawn from this figure is that adsorbed layers are present on the surface of soil particles in the terrestrial environment,

Figure 11.26   Mono Monolaye layerr formation formation time as a funct function ion of atFigure 11.25   Contact between between two smooth surfaces. surfaces.

mospheric pressure.

 

FRICTION BETWEEN SOLID SURFACES

and contacts through asperities involve adsorbed material, unless it is extruded under the high pressure.5 Adhesion Theory of Friction

The basis for the adhesion theory of friction is in Eq. (11.10), that is, the tangential force that causes sliding depends on the solid contact area and the shear streng str ength th of the contact. contact. Pla Plastic stic and / or elas elastic tic defordeformations determine the contact area at asperities. Plastic Plast ic Junc Junctions tions   If asp asperit erities ies yield and und underg ergo o plastic def plastic deform ormatio ation, n, then the con contact tact are areaa is pro propor por-tional to the normal load on the asperity as shown by Eq. (11.9). Because surfaces are not clean, but are covered by adsorbed films, actual solid contact may develop vel op onl only y ov over er a fra fracti ction on      of the contact contact area as shown in Fig. 11.27. If the contaminant film strength is    c, the strength of the contact will be

T      Ac [  m 

   (1       ) c  ]

(11.13)

Equation (11.13) cannot be applied in practice because      and    c   are unknown. However, it does provide a possible explanation for why measured values of fric-

cles. The asperities, caused by surface waviness, are more regular but not as high as those for the bulky minerals. Thus, it can be postulated that for a given number of contacts per particle, the load per asperity decreases with decreasing particle size and, for particles of the same sa me si size ze,, is le less ss fo forr pla platy ty mi mine nera rals ls th than an fo forr bu bulk lky y minerals. Because    should  should increase as the normal load per asperity increases, and it is reasonable to assume thatthe thesolid adsorbed film(  strength is less than the strength of material c       m), it follows that the true friction angle ( ) is less for small and platy particles than for large and bulky particles. In the event that two platy particles are in face-to-face contact and the surfacee wa fac wavin viness ess is ins insuf uffici ficient ent to cau cause se dir direct ect sol solidid-totosolid contact, shear will be through the adsorbed films, and the effective value of     will   will be zero, again giving a lower value of    . In reality, the behavior of plastic junctions is more comp co mple lex. x. Un Under der co comb mbin ined ed co comp mpre ress ssio ion n an and d sh shear ear stress str esses, es, def deform ormatio ation n fol follo lows ws the vo von n Mis Mises–Hen es–Henky ky criterion, which, for two dimensions, is 2

tion angle for bulky minerals such as quartz and feldspar are greater than values for the clay minerals and othe ot herr pl platy aty mi mine neral ralss su such ch as mic mica, a, ev even en th thou ough gh th thee surface structure is similar for all the silicate minerals. The small particle size of clays means that the load per particle, for a given effective stress, will be small relative to that in silts and sands composed of the bulky minerals. The surfaces of platy silt and sand size particles are smoother than those of bulky mineral parti-

2

2

       3   

     y

sorbed surface films.

5

Conditions may be different on the Moon, where ultrahigh vacuum exists. This vacuum exists. vacuum prod produces uces cleaner surfaces. surfaces. In the absence of  suitable adsorbate, clean surfaces can reduce their surface energy by cohering This could for thegradation. higher cohesion of with lunarlike soilssurfaces. than terrestrial soils account of comparable

 

(11.14)

For asperities loaded initially to           y, the application cat ion of a she shear ar str stress ess requires requires that      become become less than     y. The only way that this can happen is for the contact area to increase. Continued increase in     leads   leads to continued increase in contact area. This phenomenon is called   junction gr growth owth   and is res respon ponsib sible le for cold col d wel weldin ding g in som somee mat materia erials ls (Bo (Bowde wden n and Tabo aborr, 1964). If the shear strength of the junction equals that of the bulk sol solid, id, then gro gross ss sei seizur zuree occ occurs urs.. For the case where the ratio of junction strength to bulk material strength is less than 0.9, the amount of junction growth is small. This is the probable situation in soils. Elastic Junctions   The contact contact area between between par partiticles cl es of a pe perf rfect ectly ly ela elast stic ic mat mater erial ial is no nott de defin fined ed in terms of plastic yield. For two smooth spheres in contact, application of the Hertz theory leads to d      (  NR)1 / 3

Figure Figu re 11.2 11.27 7   Plas Plastic tic junction between between aspe asperitie ritiess with ad-

  387

(11.15)

where   d   is th thee di diam amete eterr of a pla plane ne ci circ rcul ular ar ar area ea of  contact;      is a functio function n of geometr geometry y, Poiss Poisson’ on’ss ratio, 6 and Young’s modulus ; and  R  is the sphere radius. The contact area is

6

For a sphere in contact with a plane surface       12(1      2) /  E   E .

 

388

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

   Ac     (  NR)2 / 3 4

( 11. 16)

If the shear strength of the contact is    i, then T       i A  A c

 

(11.17)

and   1 / 3     T      i     (  R)2 / 3 N     N    4

(11.18)

According to these relationships, the friction coefficient for two elastic asperities in contact should decrease with increasing load. Nonetheless, the adhesion theory would still apply to the strength of the junction, with the frictional force proportional to the area of real contact. If it is assumed that the number of contacting asperities in a soil mass is independent of particle size and effective stress, then the influences of particle size and effective stress on the frictional resistance of a soil with asperities deforming elastically may be analyzed. For uniform spheres arranged in a regular packing, the gros gr osss ar area ea co cove vere red d by on onee sp sphe here re alo along ng a po pote tent ntial ial 2 plane of sliding is 4 R . The normal load per contacting asperity, assuming one asperity per contact, is  N      4 R2   

 

(11.19)

Using Eq. (11.16), the area per contact becomes    Ac     (4  R3   )2 / 3 4

( 11. 20)

and the total contact area per unit gross area is ( Ac ) T     

 

1    2   R   (4 )2 / 3 4 R2   4



 

  (4 )2 / 3 16

( 11. 21)

The tot total al she sheari aring ng res resist istance ance of       is equal to the contact area times    i, so   i       (4 )2 / 3   16    

    iK ( )  

1 / 3

( 11. 22)

where   K       (4 (4 )2 / 3 / 16. On this basis, the coefficient of frictio friction n should decrease with increasing    , but it should be independent of sphere radius (particle size). Data have been obtained that both support and contradict these predictions. A 50-fold variation in the normal load on assemblages of quartz particles in contact with a quartz block was found to have no effect on

frictional resistance (Rowe, 1962). The residual friction angles of quartz, feldspar, and calcite are independent of normal stress as shown in Fig. 11.28. On the other hand, a decreasing friction angle with increasing normal load up to some limiting value of  normal stress is evident for mica and the clay minerals in Fig. 11.28 and has been found also for several clays and an d cl clay ay sh shale aless (B (Bis isho hop p et al al., ., 19 1971 71), ), fo forr di diamo amond nd (Bowde (Bo wden n and Tabo aborr, 196 1964), 4), and for sol solid id lub lubrica ricants nts such as graphite molybdenum disulfide (Campbell, 1969). 1969) . Additi Additional onaland data for clay mineral minerals s show that fric1 / 3 tional resistance varies as ( ) as predicted by Eq. (11.22) up to a normal stress of the order of 200 kPa (30 psi), that is, the friction angle decreases with increasing normal stress (Chattopadhyay, 1972). There are at least two possible explanations of the normal stress independence of the frictional resistance of quartz, feldspar, and calcite: 1. As the load per particle increases, increases, the number of  asperities in contact increases proportionally, and the deformation of each asperity remains essentially constant. constant. In this case, the assumption of one asperity per contact for the development of Eq. (11.22) is not valid. Some theoretical considerations of multiple asperities in contact are available (Johnson, 1985). They show that the area of  contact is approximately proportional to the applied load and hence the coefficient of friction is constant with load. 2. As the load per asperity increases increases,, the value of     in Eq. (11.13) increases, reflecting a greater proportion of solid contact relative to adsorbed film contact con tact.. Thu Thus, s, the av avera erage ge str streng ength th per con contact tact increases more than proportionally with the load, while whi le the contact contact area increases increases less tha than n pro pro-portio por tionall nally y, wit with h the net res result ult bein being g an ess essenentially constant frictional resistance. Quartz is a hard, brittle material that can exhibit both elastic and plastic deformation. A normal pressure of  11 GPa (1,500,000 psi) is required to produce plastic deformation, deform ation, and brittle failure usually occurs before plastic deformation. deformation. Plasti Plasticc deform deformations ations are evid evidently ently restricted restri cted to small, highly confined asperities, asperities, and elastic deformations control at least part of the behavior (Bromw (Br omwell, ell, 196 1965). 5). Eit Either her of the pre previou viouss two explanation nat ionss migh mightt be app applic licable able,, dep depend ending ing on det details ails of  surface texture on a microscale and characteristics of  the adsorbed films. With the exception of some data for quartz, there appear app earss to be litt little le inf inform ormatio ation n con concer cernin ning g pos possib sible le variations of the true friction angle with particle size. Rowe Ro we (19 (1962) 62) found that the value of       for assem assem-blages of quartz particles on a flat quartz surface de-

 

FRICTIONAL BEHAVIOR OF MINERALS

  389

Figure 11.28   Variat ariation ion in frict friction ion angl anglee with normal stress for diff differen erentt mine minerals rals (after

Kenney, 1967).

creased from 31  for coarse silt to 22  for coarse sand. This is an apparent contradiction to the independence of particle size on frictional resistance predicted by Eq. (11.22). On the other hand, the assumption of one asperity per contact may not have been valid for all particle sizes, and additionally, particle surface textures on a microscale could have been size dependent. Furthermore, there could have been different amounts of particle tic le re rearr arrang angeme ement nt an and d ro rolli lling ng in the tes tests ts on th thee different size fractions. Sliding Friction

The frictional resistance, once sliding has been initiated, may be equal to or less than the resistance that had to be overcome to initiate movement; that is, the coefficient of sliding friction can be less than the coeffic ef ficien ientt of st stat atic ic fr fric ictio tion. n. A hi high gher er va valu luee of st stati aticc frictio fri ction n than sli slidin ding g fri frictio ction n is exp explain lainabl ablee by time time-de depen penden dentt  motion, bond bo nd fo form rmat atio ions ns at as aspe peri rity ty or junc ju nctio tions ns.. Stick–slip wherein      varies more less erratically as two surfaces in contact are displaced, appears common to all friction measurements measurements of minerals involving single contacts (Procter and Barton, 1974). Stick–slip is not observed during shear of assemblages of large numbers of particles because the slip of individual contacts is masked by the behavior of the mass as a whole. However, it may be an important mechanism of energy dissipation for cyclic loading at very small strains when particles are not moving relative to each other.

11.5

FRICTION FRIC TIONAL AL BEHA BEHAVIOR OF MI MINERA NERALS LS Evaluation Evalu ation of the true coeffi coefficient cient of frictio friction n    and friction angle       is difficult because it is very difficult to

do tes tests ts on two very small par particl ticles es that are sli slidin ding g relative to each other, and test results for particle assemblages sembla ges are influen influenced ced by partic particle le rearran rearrangement gements, s, volume volu me changes changes,, surfa surface ce prepar preparation ation factors, and the like. Some values are available, however, and they are presented and discussed in this section. Nonclay Minerals

Values of the true friction angle     for several minerals are listed in Table 11.1, along with the type of test and conditions used for their determination. A pronounced antilubricating effect of water is evident for polished surfaces of the bulky minerals quartz, feldspar, and calcite. This apparently results from a disruptive effect of  water on adsorbed films that may have acted as a lubricant for dry surfaces. Evidence for this is shown in Fig. 11.29, where it may be seen that the presence of  wate wa terr had no ef effe fect ct on th thee fr frict ictio iona nall res resis istan tance ce of  quartz surfaces that had been chemically cleaned prior to th thee me meas asur ureme ement nt of th thee fr fric ictio tion n co coef effic ficie ient nt.. Th Thee samples tested by Horn and Deere (1962) in Table 11.1 had not been chemically cleaned. An app apparen arentt ant antilub ilubric ricatio ation n eff effect ect by wat water er migh mightt also arise from attack of the silica surface (quartz and feldsp fel dspar) ar) or car carbon bonate ate sur surfac facee (ca (calcit lcite) e) and the for for-mation of silica and carbonate cement at interparticle contacts. Many sand deposits exhibit ‘‘aging’’ effects wherein their strength and stiffness increase noticeably within wit hin per period iodss of wee weeks ks to mon months ths aft after er dep deposi osition tion,, disturbance, or densification, as described, for example, by Mitchell and Solymar (1984), Mitchell (1986), Mesr Me srii et al al.. (1 (199 990) 0),, an and d Sc Schme hmert rtma mann nn (1 (199 991) 1).. In In-creases in penetration resistance of up to 100 percent have been measured in some cases. The relative importance of chemical factors, such as precipitation at

 

390

11

Table 11.1 11.1 Mineral Quar Qu artz tz

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

Values of Frictio Friction n Angle Angle ( ) Between Mineral Surfaces Type of Test Block Bloc k over pa part rtic icle le set in mortar

Conditions

 

    (deg)

Dry

6

Moist Wat ater er sa satu tura rate ted d Wat ater er sa satu tura rate ted d

24.5 24.5 24 .5 21.7 21 .7

Dry Wat ater er satu satura rate ted d Wat ater er sa satu tura rate ted d

Comments

Reference

Dried over CaCl2  before testing

Tschebotarioff and Welch (1948)

Normal Norm al lo load ad pe perr pa part rtic icle le increasing from 1 to 100 g Polished surfaces

Hafiz (1950)

Quar Qu artz tz

Threee fix Thre fixed ed par arti ticl cles es over block 

Quartz

Block on block

Quartz Quartz

Particles on polished block  Block on block

Quartz

Particle – particle

Saturated

7.4 24.2 24.2 22–31 22– 31     decreasing with increasing particle size 0 – 45 Depends on roughness and cleanliness 26 Single-point contact

Feldspar

Particle – plane Particle – plane Block on block

Saturated Dry Dry Wat ater er sa satu tura rate ted d Water saturated

22.2 17.4 6.8 37.6 37 .6 37

Polished surfaces

Horn and Deere (1962)

25 – 500 sieve

Lee (1966)

Saturated

28.9

Single-point contact

Dry Wat ater er sa satu tura rate ted d Dry

8.0 34.2 34 .2 23.3

Polished surfaces

Procter and Barton (1974) Horn and Deere (1962)

Oven dry

Horn and Deere (1962)

Dry Saturated Dry

16.7 13.0 17.2

Air equilibrated

Dry Saturated Dry

14.0 8.5 17.2

Air equilibrated

Dry Saturated Dry

14.6 7.4 27.9

Air equilibrated

Dry Saturated

19.3 12.4

Air equilibrated

Feld Fe ldsp spar ar Feldspar

Free Fr ee pa part rtic icle less on fla flatt surface Particle – plane

Calcite

Block on block

Muscov Mus covite ite

Along cle Along cleav avage age faces

Phlogo Phl ogopit pitee

Biotite

Chlo Ch lori rite te

Along cle Along cleav avage age faces

Along cleavage faces

Along Alon g cl clea eav vag agee faces

Variable

interparticle contacts, change interparticle changess in surfa surface ce charact characteriseristics,, and mec tics mechan hanical ical fac factor tors, s, suc such h as time time-de -depen penden dentt stress redistribution and particle reorientations, in causing the observed behavior is not known. Further details of aging effects are given in Chapter 12.

Oven dry

Oven dry

Oven dry

Horn and Deere (1962) Rowe (1962) Bromwell (1966) Procter and Barton (1974)

Horn and Deere (1962)

Horn and Deere (1962)

Horn and Deere (1962)

As surface roughness increases, increases, the appare apparent nt antilubric lub ricatin ating g ef effect fect of wa water ter dec decreas reases. es. This is sho shown wn in Fi Fig. g. 11 11.2 .29 9 fo forr qu quar artz tz su surf rfac aces es th that at ha had d no nott be been en cleaned clea ned.. Che Chemic mically ally cle cleaned aned qua quartz rtz sur surfac faces, es, whi which ch give the same value of friction when both dry and wet,

 

FRICTIONAL BEHAVIOR OF MINERALS

  391

Figure 11.29   Frict Friction ion of quart quartzz (data from Brom Bromwell well,, 1966 and Dickey Dickey, 1966).

show a loss in frictional resistance with increasing surface roughness. Evidently, increased roughness makes it easier for asperities to break through surface films, resu re sulti lting ng in an in incr crea ease se in      [Eq. [Eq. (11 (11.13 .13)) and Fig Fig.. 11.27]. The decrease in friction with increased roughness is not readily explainable. One possibility is that the clea cleanin ning g pro proces cesss was not ef effec fectiv tivee on the rough surfaces. For soils in nat nature ure,, the surfaces surfaces of bu bulky lky mineral mineral particles are most probably rough relative to the scale in Fig. 11.29, and they will not be chemically clean. Thus, values of         0.5 and          26  are reasonable for quartz, both wet and dry. On the other hand, water apparently acts as a lubricant in sheet minerals, as shown by the values for muscovite, phlogopite, biotite, and chlorite in Table 11.1. This is because in air the adsorbed film is thin, and surface ions are not fully hydrated. Thus, the adsorbed layer is not easily disrupted. Observations have shown that the sur surfac faces es of the she sheet et min minera erals ls are scratched scratched when tested in air (Horn and Deere, 1962). When the surfaces of the layer silicates are wetted, the mobility of the surface films is increased because of their increased thickness and because of greater surface ion hydrat hyd ration ion and dis dissoc sociat iation ion.. Thu Thus, s, the va value luess of     listed in Table 11.1 for the sheet minerals under saturated conditions (7 –13 ) are probably appropriate for sheet mineral particles in soils.

Clay Minerals

Few, if any, directly measured values of        for the clay minerals are available. However, because their surface structures are similar to those of the layer silicates discussed cus sed pre previo viousl usly y, app approx roximat imately ely the sam samee va values lues would be anticipated, and the ranges of residual friction angles measured for highly plastic clays and clay mineral min eralss sup suppor portt thi this. s. In ver very y acti active ve col colloid loidal al pur puree clays, such as montmorillonite, even lower friction angles have been measured. Residual values as low as 4 for sodium montmorillonite are indicated by the data in The Fig. 11.28. effect ef fectiv ivee str stress ess fai failur luree en envel velope opess for calc calcium ium and sodium montmo montmorilloni rillonite te are different, different, as shown by Fig. 11.30, and the friction angles are stress dependent. For each material the effective stress failure envelope was the same in drained and undrained triaxial compression press ion and unaf unaffected fected by electro electrolyte lyte concen concentration tration over the range investigated, which was 0.001   N  to   to 0.1  N . The water content at any effective stress was independent of electrolyte concentration for calcium montmorillo mor illonit nite, e, bu butt var varied ied in the man manner ner sho shown wn in Fig Fig.. 11.31 for sodium montmorillonite. This consolidation consolidation behavior is consis consistent tent with that described in Chapter 10. Interlayer expansion in calcium montmorillonite is restricted to a   c-axis spacing of 1.9 nm, leading to formation of domains or layer aggregates of several unit layers. The interlayer spac-

 

392

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

Figure 11.30   Effective stress stress failure diagrams for calcium and sodium sodium montmorillonite (af(af-

ter Mesri and Olson, 1970).

the same time there is little increase in shear strength because the shearing strength of water and solutions is essenti ess entially ally ind indepe epende ndent nt of hyd hydros rostati taticc pre pressu ssure. re. The small friction angle that is observed for sodium montmorillo mor illonit nitee at low eff effect ectiv ivee str stress esses es can be asc ascrib ribed ed mainly to the few interparticle contacts that resist particle rearrangement. Resistance from this source evident de ntly ly ap appr proa oach ches es a co cons nsta tant nt val alue ue at th thee hi high gher er effective stresses, as evidenced by the nearly horizontal

Figure 11.31   Shea Shearr and consolidatio consolidation n beha behavior vior of sodiu sodium m

montmorillonite (after Mesri and Olson, 1970).

ing of sodium montmorillonite is sensitive to doublelaye la yerr re repu puls lsio ions ns,, wh whic ich, h, in tu turn rn,, de depe pend nd on th thee electrolyte concentration. The influence of the electrolyte concentrat concentration ion on the beh behavi avior or of sod sodium ium mon monttmorillonite is to change the water content, but not the strength, at any effective consolidation pressure. This suggests that the strength generating mechanism is independent of the system chemistry. The platelets of sodium montmorillonite act as thin films held apart by high repulsive forces that carry the effective stress. For this case, if it is assumed that there is essentially no intergranular contact, then Eq. (7.29) becomes  i             A      u0 

 R    0

( 11. 23)

Since         u0  is the conventionally defined effective stress    , and assuming negligible long-range long-range attractions, Eq. (11.23) becomes        R

 

(11.24)

This accounts for the increase in consolidation pressure required to decrease the water content, while at

failur failure e env envelope elope at 50 values valu of aver average ageaseffe effectiv ctiveeinstress greater than about psies(350 kPa), shown Fig. 11.30. 11. 30. The vis viscou couss res resist istanc ancee of the pore flui fluid d may contribute a small proportion of the strength at all effective stresses. An hypothesis of friction between fine-grained particles in the absence of interparticle contacts is given by Sa Sant ntam amar arin inaa et al. (2 (200 001) 1) us usin ing g th thee co conc ncep eptt of  ‘‘electrical’’ surface roughness as shown in Fig. 11.32. Consider two clay surfaces with interparticle fluid as shown in Fig. 11.32b. The clay surfaces have a number of di disc scre rete te ch char arge ges, s, so a se serie riess of po pote tent ntial ial en ener ergy gy wells exists along the clay surfaces. Two cases can be considered: 1. When the particle separation separation is less than several several nanometers, there are multiple wells of minimum energy ener gy betw between een nearby surfaces surfaces and a for force ce is required to overcome the energy barrier between the wells when the particles move relative to each other. Shearing involves interaction of the molecules of the interparticle fluid. Due to the multiple energ energy y wells wells,, the interparticle fluid molecules go thr throug ough h suc succes cessi sive ve sol solidl idlike ike pin pinned ned sta states tes.. This sti stick–sli ck–slip p mot motion ion con contri tribu butes tes to fri frictio ctional nal resistance and energy dissipation. 2. When the particle separation separation is more than several several nanometers, the two clay surfaces interact only by the hyd hydrod rodyna ynamic mic vis viscou couss ef effec fects ts of the interparticle fluid, and the frictional force may be estimated using fluid dynamics.

 

PHYSICAL INTERA INTERACTIONS CTIONS AMONG PART ARTICLES ICLES

  393

disorder of particles, (i.e., local spatial fluctuations of  coordination number, and positions of neighboring particles) produce packing constraints and disorder. This leads to inhomogeneous but structured force distributions tio ns wit within hin the gra granul nular ar sys system. tem. Def Deform ormatio ation n is associa so ciate ted d wi with th bu buck ckli ling ng of th thes esee fo forc rcee ch chai ains ns,, an and d energy is dissipated by sliding at the clusters of particles between the force chains. Discrete particle numerical simulations, such as the discrete (distinct) element method Strack, 1979 19 79)) an and d th thee co cont ntact act dy dyna namic micss(Cundall metho met hod dand (Mor (M orea eau, u, 1994), offer physical insights into particle interactions and an d lo load ad tr tran ansf sfer erss th that at ar aree dif diffic ficult ult to de dedu duce ce fr from om physica phy sicall exp experi erimen ments. ts. Typi ypical cal inp inputs uts for the sim simulaulations are particle packing conditions and interparticle contact characteristics such as the interparticle friction angle   . Complete details of these numerical methods are bey beyond ond the sco scope pe of thi thiss boo book; k; add additio itional nal inforinformatio ma tion n ca can n be fo foun und d in Od Odaa an and d Iw Iwas ashi hita ta (1 (199 999) 9).. Howe Ho weve verr, so some me of th thee mai main n fin findin dings gs ar aree us usef eful ul fo forr developing an improved understanding of how stresses are carried through discrete particle systems such as soils and how these distributions influence the deformation and strength properties. Figure 11.32   Concept of ‘‘electrical’’ ‘‘electrical’’ surface roughness roughness ac-

cording to Santamarina et al., (2001): (a) electrical roughness and (b) conceptual picture of friction in fine-grained particles.

The aggregation of clay plates in calcium montmorillonite produces particle groups that behave more like equidimensional particles than platy particles. There is more physical interf interference erence and more intergrain contact than in sodium montmorillonite since the water content range for the strength data shown in Fig. 11.30 was only about 50 to 97 percent, whereas it was about 125 to 450 percent for the sodium montmorillonite. At a consolidation pressure of about 500 kPa, the slope of  the failure envelope for calcium montmorillonite was about 10, which is in the middle of the range for nonclay sheet minerals (Table 11.1).

11.6 PHYS PHYSICAL ICAL INTE INTERA RACTIO CTIONS NS AMON AMONG G PARTICLES

Continuum mechanics assumes that applied forces are transmitted uniformly through a homogenized granular system. In reality, however, the interparticle force distributions are strongly inhomogeneous, as discussed in Chapter 7, and the applied load is transferred through a net networ work k of inte interpa rpartic rticle le for force ce cha chains ins.. The gen generi ericc

Strong Force Networks and Weak Clusters

Examples of the computed normal contact force distrib tr ibut utio ion n in a gr gran anul ular ar sy syst stem em ar aree sh show own n in Fi Figs gs.. 11.33a   for for an is isot otro ropi pica cally lly loa loade ded d co cond nditi ition on an and d 11.33b   for a biax biaxial ial loa loaded ded con condit dition ion (Th (Thorn ornton ton and Barnes, 1986). The thickness of the lines in the figure is proportional to the magnitude of the contact force. The external loads are transmitted through a network  of int interp erparti article cle con contact tact for forces ces rep repres resente ented d by thic thicker ker lines. This is called the  strong force network  and   and is the key ke y micr microsc oscopi opicc fea featur turee of load tra transf nsfer er thr throug ough h the granular system. The scale of statistical homogeneity in a two-dimensional particle assembly is found to be a few tens of particle diameters (Radjai et al., 1996). Forces averaged over this distance could therefore be expected to give a stress that is representative of the macroscopic stress state. The particles not forming a part of the strong force network are floating like a fluid with small loads at the interparticle contacts. This can be called the  weak cluster,  which has a width of 3 to 10 particle diameters. Both normal and tangential forces exist at interparticle contacts. Figure 11.34 shows the probability distributions (P N   and   PT ) of normal contact forces   N   and tangential contact forces   T   for a given biaxial loading condition. The horizontal axis is the forces normalized by their mean force value ( N    or   T ), which depend on particle size distribution (Radjai et al., 1996). The individual normal contact forces can be as great

 

394

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

tric compression of a dense granular assembly (Thornton, 2000). The strong force network carries most of  the whole deviator load as shown in Fig. 11.36 and is the load-bearing part of the structure. For particles in the strong force networks, the tangential contact forces are much smaller than the interparticle frictional resistance tan ce beca because use of the large normal contact contact for forces ces.. In contrast, the numerical analysis results show that the tangential contact forces in the weak clusters are close to the interparticle resistance. Hence, the frictional resistance is frictional almost fully mobilized between particles in the weak clusters, and the particl particles es are perhaps behaving like a viscous fluid. Buckling, Sliding, and Rolling

forcee di forc disstr trib ibut utiion onss of a tw twoodimensional disk particle assembly: (a) isotropic stress condition and (b) biaxial stress condition with maximum load in the vertical direction (after Thornton and Barnes, 1986). Figur Fig uree

11.33 11 .33   Norm Normaal

as si six x ti time mess th thee me mean an no norm rmal al co cont ntac actt fo forc rce, e, but approx app roximat imately ely 60 per percent cent of con contac tacts ts carr carry y nor normal mal conta co ntact ct fo forc rces es be belo low w th thee me mean an (i. (i.e., e., we weak ak clu clust ster er particles). When normal contact forces are larger than their mean, the distribution law of forces can be approximated proxim ated by an expon exponentially entially decreasing decreasing functi function; on; Radja Ra djaii et al. (1 (199 996) 6) sh sho ow tha thatt   P N (      N  /   /  N )    1.4(1 ) ke fits the computed data well for both two-and threethr ee-di dime mens nsio iona nall si simu mula latio tions ns.. Th Thee ex expo pone nent nt wa wass found to change very slightly with the coefficient of  interparticle friction and to be independent of particle size distributions. Simulations show that applied deviator load is transferred exclusively by the normal contact forces in the stro st rong ng fo forc rcee ne netw twor orks ks,, an and d th thee co contr ntrib ibut utio ion n by th thee weak clusters is negligible. This is illustrated in Fig. 11.35, which shows that the normal contact forces contribute greater than the tangential contact forces to the development of the deviator stress during axisymme-

As particles begin to move relative to each other during shear, particles in the strong force network do not slide, but columns of particles buckle (Cundall and Strack, 1979). 197 9). Par Particl ticles es in the str strong ong for force ce netw network ork col collaps lapsee upon up on bu buck cklin ling, g, an and d ne new w fo forc rcee ch chai ains ns ar aree fo form rmed ed.. Hence, the spatial distributions of the strong force network are neither static nor persistent features. At a given time of biaxial compression loading, particle sliding is occurring at almost 10 percent of the contactss (Kuh contact (Kuhn, n, 1999) and approx approximately imately 96 percen percentt of the sliding particles are in the weak clusters (Radjai et al., 1996). Over 90 percent of the energy dissipation occurs at just a small percentage of the contacts (Kuhn, 1999). This small number of sliding particles is associated with the ability of particles to roll rather than to slide. Particle rotations reduce contact sliding and dissipa si patio tion n ra rate te in the gr gran anul ular ar sy syst stem em.. If all pa part rtic icles les coul co uld d ro roll ll up upon on on onee an anot othe herr, a gr gran anul ular ar as asse semb mbly ly 7 would deform withou withoutt energ energy y dissip dissipation. ation. However, thiss is not pos thi possib sible le ow owing ing to res restri trictio ctions ns on par particl ticlee rotations. It is impossible for all particles to move by rotation, and sliding at some contacts is inevitable due to the random position of particles (Radjai and Roux, 1995).8 Some frictional energy dissipation can therefore be considered a consequence of disorder of particle positi positions. ons. As deformation progresses, the number of particles in the strong force network decreases, with fewer particles sharing the increased loads (Kuhn, 1999). Figure

7

This assumes that the particles are rigid and rolling with a singlepoint contact. In reality, particles deform and exhibit rolling resistance. Iwashita and Oda (1998) state that the incorporation of rolling resistanc resis tancee is nece necessary ssary in disc discrete rete part particle icle simulations simulations to gene generate rate realistic localized shear bands. 8

For instance, a chain loop one of ansliding odd number Particle rotationconsider will involve at least contact.of particles.

 

PHYSICAL INTERAC INTERACTIONS TIONS AMONG PARTIC ARTICLES LES

  395

Figure 11.34   Probability distributions distributions of interparticle contact contact forces: (a) normal forces and

(b) tange tangential ntial forces. The dist distribu ributions tions were obta obtained ined for cont contact act dyna dynamic mic simulations simulations of  500, 1024, 1200 and 4025 particles. The effect of number of particles in the simulation on probability distribution appears to be small (after Radjai et al., 1996).

Figure 11.35   Contributions of normal and tangential contact

Figure Figu re 11.3 11.36 6   Con Contri tribu butio tions ns of str strong ong and wea weak k con contac tactt

forces to the evolution of the deviator stress during axisymmetric met ric com compre press ssion ion of a den dense se gra granul nular ar as assem sembly bly (af (after ter Thornton, 2000).

forces to the evolution of the deviator stress during axisymmetric met ric com compre pressi ssion on of a den dense se gra granul nular ar ass assemb embly ly (af (after ter Thornton, 2000).

11.37 shows the spatial distribution of residual deformation, mati on, in whi which ch the com comput puted ed def deform ormatio ation n of eac each h particle is subtracted from the average overall deformation (Williams and Rege, 1997). A group of interlocked particles that instantaneously moves as a rigid body in a circular manner can be observed. The outer

boundary of the group shows large residual deformation, whereas the center shows very small residual deformation. The rotating group of interlocked particles, which can be considered as a weak cluster, becomes more apparent as applied strains increase toward failure. The bands of large residual deformation [termed

 

396

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

Stress Ratio q/p

A B C

Contact Plane Normals in Initial State: More in Vertical Direction Same in All Directions More in Horizontal Direction

-8

-6

-4

Triaxial Compression 1.5 1.0 0.5

-2

Triaxial Extension

2 -0.5

4

6 8 10 Axial Strain (%)

-1.0 (a)

Fabric Anisotropy A Anisotropy A

A B C

Contact Plane Normals in Initial State: More in Vertical Direction Same in All Directions More in Horizontal Direction

-8

-6

-4

-2

0.4 0.3 0.2 0.1 2 -0.1 -0.2

4

6 8 10 Axial Strain (%)

-0.3 -0.4 -0.5

Figure 11.37   Spatial distribution distribution of residual residual deformation ob-

served in an elliptic particle assembly at an axial strain level of (a) 1.1%, (b) 3.3%, (c) 5.5%, (d ) 7.7%, (e) 9.8 9.8%, %, an and d (ƒ) 12.0% (after Williams and Rege, 1997).

microbands  by Kuhn (1999)] are where particle translations and rotations are intense as part of the strong force for ce net networ work. k. Kuh Kuhn n (19 (1999) 99) rep report ortss tha thatt the their ir thic thickknesses are 1.5 D50  to 2.5 D50  in the early stages of shearing in g an and d in incr crea ease se to be betw twee een n 1. 1.5 5 D50   an and 4 D50   as

deformation proceeds. This microband slip zone may eventually become a localized shear band. Fabric Anisotropy

The abi ability lity of a gra granul nular ar ass assemb emblage lage of par particl ticles es to carry deviatoric loads is attributed to its capability to develop deve lop anisot anisotropy ropy in contact orientations. orientations. An initial isotropic packing of particles develops an anisotropic contact network during compression compression loading loading.. This is because beca use new con contac tacts ts for form m in the direction direction of compression loading and contacts that orient along the direction perpendicular to loading direction are lost. The ini initial tial state of con contact tact anisotrop anisotropy y (or fabric) fabric) plays an important role in the subsequent deformation as illustrated in Fig. 11.18. Figure 11.38 shows results

(b)

Figure 11.38   Disc Discrete rete element element simu simulatio lations ns of drain drained ed tri-

axial compression and extension tests of particle assemblies prepared at different initial contact fabrics: (a) stress–strain relationships and (b) evolution of fabric anisotropy parameter  A  (after Yimsiri, 2001).

of discrete particle simulations of particle assemblies prepared at different states of initial contact anisotropy under an isotropic stress condition (Yimsiri, 2001). The initial void ratios are similar (e      0.75 to 0.76) and 0 both drained triaxial compression and extension tests weree sim wer simulat ulated. ed. Alt Althou hough gh all spe specime cimens ns are init initiall ially y isotro iso tropic pically ally load loaded, ed, the dir directi ectiona onall dis distri tribu bution tionss of  contact forces are different due to different orientations of contact plane normals (sample A: more in the vertical direction; sample B: similar in all directions; sample C: more in the horizontal direction). As shown in Fig. 11.38a, both samples A and C showed stiffer response when the compression loading was applied in the preferred direction of contact forces, but softer response when the loading was perpendicular to the preferr fe rred ed di dire recti ction on of co cont ntac actt fo forc rces es.. Th Thee re resp spon onse se of  sample B, which had an isotropic fabric, was in between the two. Dilation was most intensive when the contact con tact for forces ces wer weree ori oriente ented d pre prefer ferenti entially ally in the di-

 

PHYSICAL INTERA INTERACTIONS CTIONS AMONG PART ARTICLES ICLES

rection of applied compression; and experimental data pres pr esen ente ted d by Kon onis ishi hi et al al.. (1 (198 982) 2) sh show owss a si simil milar ar trend. Figure 11.38b   shows the development of fabric anisotro iso tropy py wit with h inc increas reasing ing str strain ain.. The deg degree ree of fab fabric ric anisotropy is expressed by a fabric anisotropy parameter   A; the value of   A   increases with more vertically oriented contact plane normals and is negative when there are more horizontally oriented contact plane nor-

  397

forces categorized by their magnitudes when the specimen is under a biaxial compression loading condition (Radjai, 1999). The direction of contact anisotropy of  the weak clusters ( N  /   /  N   less than 1) is orthogonal to the direction of compression loading, whereas that of the strong force network ( N  /   /  N    more than 2) is parallel. Figure 11.40 shows an example of fabric evolution with strains in biaxial loading (Thornton and Antony, 1998). The fabric anisotropy is separated into

9

mals. The fabricand parameter parame ter gradually changes with increasing strains reaches a steady-state value as the specimens fail. The final steady-state value is independent of the initial fabric, indicating that the inherent anisotropy is destroyed by the shearing process. The final fabric anisotropy after triaxial extension is larger than that after triaxial compression because the additional tion al con confine finemen mentt by a lar larger ger inte interme rmedia diate te str stress ess in the ext extens ension ion tests cre created ated a hig higher her degree degree of fab fabric ric anisotropy. Close examination of the contact force distribution for the strong force network and weak clusters gives interesting interes ting micros microscopic copic featur features. es. Figure 11.39 show showss  A  determined for the subgroups of contact the values of  A

0.1    A

  r   e    t   e   m 0.05   a   r   a    P   y   p 0.0   o   r    t   o   s    i   n    A-0.05   c    i   r    b   a    F -0.1

1

2

3

4

5

that in th that thee st stro rong ng fo forc rcee ne netw twor orks ks ( N  /   /  N    of mo more re than 1) and that in the weak clusters ( N  /  N  less than 1). Again the directional evolution of the fabric in the weak clusters is opposite to the direction of loading. Ther Th eref efor ore, e, th thee st stab abil ility ity of th thee st stro rong ng fo forc rcee ch chai ains ns aligned in the vertical loading direction is obtained by the lateral forces in the surrounding weak clusters.

Changes in Number of Contacts and Microscopic Voids

At the beginning of biaxial loading of a dense granular assembly, more contacts are created from the increase in the hydrostatic stress, and the local voids become smaller. As the axial stress increases, however, the local voids tend to elongate in the direction of loading as shown in Fig. 11.41. Consequently particle contacts are lost. As loading progr progresses esses,, vertic vertically ally elongated local voids become more apparent, leading to dilation in

6  /  N 

Figure 11.3 Figure 11.39 9   Fabr Fabric ic anis anisotrop otropy y param parameter eter   A   for dif differen ferentt

levels of contact force when the specimen is under biaxial compression loading conditions (after Radjai et al., 1996).

9

The density of contact plane normals  E ( ) with direction    is fitted with the following expression (Radjai, 1999): c  E ( )      {1     A  cos 2(      c )}  

where  c  is the total number of contacts,   c  is the direction for which the maximum   E   is reach reached, ed, and the magnitude magnitude of   A   indicates the amplitude of anisotropy. When the directional distribution of contact

Figure 11.40   Evolution of the fabric anisotropy anisotropy parameters parameters

forces force s is inde independ pendent ent of    , the system has an isotropic fabric and  A      0.

der biaxial ial comp compress ression ion load loading ing cond condition itionss (afte (afterr Thorn Thornton ton and biax Antony, 1998).

of strong forces and weak clusters when the specimen is un-

 

398

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

Figure 11.41   Simulated spatial distribution distribution of local microvoids under under biaxial loading (after

Iwashita and Oda, 2000): (a)    11      1.1% (before failure), ( b) 11     4.4% (after failure), and ( d )    11      5.5% (after failure).

terms of ove terms overal ralll sam sample ple vol volume ume (Iw (Iwash ashita ita and Oda Oda,, 2000). Void red reduct uction ion is par partly tly ass associ ociated ated wit with h par partic ticle le breakag bre akage. e. Thu Thus, s, the there re is a nee need d to inc incorp orpora orate te gra grain in crushing in discrete particle simulations to model the contractive behavior of soils (Cheng et al., 2003). Normal contact forces in the strong force network are quite high, and, therefore, particle asperities, and even particles themselves, are likely to break, causing the force chains to collapse. Local voids tend to change size even after the applied pli ed st stre ress ss re reac ache hess th thee fa fail ilur uree st stre ress ss st stat atee (K (Kuh uhn, n, 1999). This suggests that the degrees of shearing required for the stresses and void ratio to reach the criticall st ica state ate ar aree di diff ffer eren ent. t. Nu Nume meric rical al si simu mula latio tions ns by Thornt Tho rnton on (20 (2000) 00) show that at leas leastt 50 per percen centt axi axial al

   11      2.2%

(at failure), ( c)

strain is required to reach the critical state void ratio. Practical Practi cal implication of this is discu discussed ssed further in Section 11.7. Macroscopic Friction Angle Versus Interparticle Friction Angle

Discrete particle simulations show that an increase in the interpa interparticle rticle friction angle     results in an increase in shear modulus and shear strength, in higher rates of  dilation, and in greater fabric anisotropy. Figure 11.42 shows sho ws the effe effect ct of assume assumed d interp interparticle article friction angle    on the mobilized macroscopic friction angle of the particle assembly (Thornton, Yimsiri, 2001). The macroscopic friction angle is2000; larger than the interparticle friction angle if the interparticle friction angle is smaller than 20. As the interparticle friction becomes

 

PHYSICAL INTERAC INTERACTIONS TIONS AMONG PARTIC ARTICLES LES

  399

50

40    )   s   e   e   r   g   e    ( 30    d   e    l   g   n    A   n   o    i    t 20   c    i   r    F   c    i   p   o   c   s 10   o   r   c   a    M

Drained (Thornton, 2000) Drained Triaxial Compression (Yimsiri, 2001) Undrained Triaxial Compression (Yimsiri, 2001) Drained Triaxial Extension (Yimsiri (Yimsiri,, 2001) Undrained Triaxial Extension (Yimsiri, 2001) Experiment (Skinner, 1969)

0 0

10

20

30

40

50

60

70

80

90

Interparticle Friction Angle (degrees)

Relations tionships hips betw between een inter interparti particle cle fricti friction on angle and macr macrosco oscopic pic friction Figure 11.42   Rela angle from disc angle discrete rete elem element ent simu simulatio lations. ns. The macr macrosco oscopic pic frict friction ion angle was dete determine rmined d from simulations of drained and undrained triaxial compression (TC) and extension (TE) tests. The experimental data by Skinner (1969) is also presented (after Thornton, 2000, and Yimsiri, 2001).

more than 20, the contribution of increasing interparticle friction to the macroscopic friction angle becomes relatively small; the macroscopic friction angle ranges between betw een 30   and 40, whe when n the inte interpa rpartic rticle le fri frictio ction n 10 angle increases from 30  to 90. The non nonpro propor portio tional nal rela relatio tionsh nship ip betw between een macr macrooscopic friction angle of the particle assembly and interparticle friction angle results because deviatoric load is ca carr rrie ied d by th thee st stro rong ng fo forc rcee ne netw twor orks ks of no norm rmal al forces and not by tangential forces, whose magnitude is governed by interparticle friction angle. Increase in interparticle friction results in a decrease in the percentage of sliding contacts (Thornton, 2000). The interp te rpar arti ticl clee fr fric icti tion on th ther eref efor oree ac acts ts as a ki kine nema mati ticc constraint of the strong force network and not as the direct source of macroscopic resistance to shear. If the interpa inte rpartic rticle le fri frictio ction n wer weree zer zero, o, str strong ong for force ce chai chains ns could not develop, and the particle assembly will be-

have like a fluid. Increased friction at the contacts increa cr ease sess th thee st stab abil ility ity of th thee sy syst stem em an and d re redu duce cess th thee number of contacts required to achieve a stable conditio di tion. n. As lon long g as th thee st stro rong ng fo forc rcee ne netw twor ork k ca can n be formed for med,, ho howe wever ver,, the mag magnitu nitude de of the inte interpa rpartic rticle le friction becomes of secondary importance. The abo above ve find finding ingss fro from m dis discre crete te par particl ticlee sim simulaulations are partially supported by the experimental data given by Skinner (1969), which are also shown in Fig. 11.42. He performed shear box tests on spherical particles with different coefficients of interparticle friction angle. ang le. The tes tested ted mate materia rials ls inc includ luded ed glas glasss bal balloti lotini, ni, steel ball bearings, and lead shot. Use of glass ballotini was particularly attractive since the coefficient of interparticle friction increases by a factor of between 3.5 and 30 merely by flooding the dry sample. Skinner’s data shown in Fig. 11.42 indicate that the macroscopic frictio fri ction n ang angle le is nea nearly rly ind indepe epende ndent nt of inte interpa rparti rticle cle friction frictio n angle. Effects of Particle Shape and Angularity

10

Reference to Table 11.1 shows that actually measured values of    for geomaterials are all less than 45. Thus, numerical simulations done assuming larger values of     appear to give unrealistic results.

A lower porosity and a larger coordination number are achieved for ellipsoidal particles compared to spherical

 

400

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

particles (Lin and Ng, 1997). Hence, a denser packing can be achieved for ellipsoidal particles. Ellipsoid particles rotate less than spherical particles. An assembly of el ellip lipso soid id pa part rtic icles les gi give vess lar large gerr va valu lues es of sh shear ear streng str ength th and ini initial tial modulus modulus than an ass assemb emblage lage of  spherical particles, primarily because of the larger coordina ord ination tion num number ber for elli ellipso psoida idall par particl ticles. es. Sim Simila ilarr findings result for two-dimensional particle assemblies. Circul Cir cular ar di disk skss gi give ve th thee hi high ghes estt di dila latio tion n fo forr a gi give ven n

Deviator Stress

The basic concept of the critical state is that, under sustained uniform shearing, there exists a unique relations lati onship hipss amon among g voi void d rat ratio io   ecs   (or specifi specificc volu volume me vcs  1  ecs), mean eff effect ectiv ivee pres pressur suree  pcs, and and de deviat viator or stress   qcs   as shown in Fig. 11.43. An example of the critical state of clay was shown in Fig. 11.4a. The critical state of clay can be expressed as qcs  v

   Mpcs

 

(11.25)

     1      e           

cs

cs

  ln   p

cs

cs

 

(11.26)

where   qcs   is the the de devi viat ator or str stres esss at fa fail ilur ure, e,  pcs is the the mean effective stress at failure, and   M   is the critical

 

σr

σr

Mean Pressure p Critical State Line

 M (triaxial extension)

(a) Specific Volume v

Compression Lines of Constant Stress Ratio q/p

Γ 

11.7 CRIT CRITICAL ICAL ST STATE: A USEFUL USEFUL REFERENCE REFERENCE CONDITION

Clays

 M  (triaxial  (triaxial compression)

σa

stress and the or lowest coordination number pared par ed ratio to elli ellipti ptical cal diamon dia mond d sha shapes pes (W (Willi illiams amscomand Rege, 1997). An assembly of rounded particles exhibits gre greater ater sof softeni tening ng beh behav avior ior wit with h fab fabric ric ani anisot sotrop ropy y change with strain, whereas an assembly of elongated particl par ticles es req requir uires es mor moree she shearin aring g to mod modify ify its init initial ial fabr fa bric ic an anis isot otro ropy py to th thee cr crit itic ical al st stat atee co cond ndit itio ion n (Nouguier-Lehon et al., 2003).

After large shear-induced volume change, a soil under a given effective confining stress will arrive ultimately at a un uniq ique ue fin final al wa wate terr co cont nten entt or vo void id ra rati tio o th that at is independent of its initial state. At this stage, the interlocking achieved by densifi densification cation or overc overconsoli onsolidation dation is gone in the case of dense soils, the metastable structure of loose soils has collapsed, and the soil is fully destructured. A well-defined strength value is reached at this state, and this is often referred to as the  critical state strength. Under undrained conditions, the critical state is reached when the pore pressure and the effective stress remain constant during continued deformation. The critical state can be considered a fundamental state, and it can be used as a reference state to explain the effect of overconsolidation ratios, relative density, and different stress paths on strength properties of soils (Schofield and Wroth, 1968).

Critical State Line

q = σa – σr

λ  λ cs

Isotropic Compression Line

Critical State Line ln p

1

(b) Figure 11.43   Criti Critical cal state concep concept: t: ( a)   p–q   plane and (b)

v–ln   p   plane.

state stress ratio. The critical state on the void ratio (or specifi spe cificc vo volume lume)–mean )–mean pre pressu ssure re pla plane ne is defi defined ned by two materia materiall parame parameters: ters:   cs, th thee cr criti itical cal st state ate co commpression index and   , the specific volume intercept at unit pressure ( p      1). The compression lines under constant stress ratios are often parallel to each other, as shown in Fig. 11.43b. Parameter   M  in   in Equation (11.25) defines the critical state stress ratio at failure and is similar to       for the Mohr–C Mo hr–Cou oulo lomb mb fa failu ilure re lin line. e. Ho Howe weve verr, Eq Equa uatio tion n (11.25 (11 .25)) inc includ ludes es the ef effec fectt of int interm ermedia ediate te pri princi ncipal pal stress  2 bec ecaause    p     1     2        3, wher whereas eas the Mohr–Coulomb Mohr–Coulom b failur failuree criteri criterion on of Eq. (11.4) or (11.5 (11.5)) does not take the intermediate effective stress into account. In triaxial conditions,  a       r       r an  d  r      r       a for compres compression sion and extens extension, ion, respec respective tively ly 11 (see Fig. 11.43). Hence, Eqs. (11.4) and (11.25) can be related to each other for these two cases as follows:

11

 a is the axia axiall effecti effective   ve stress stress and and  r  is the radia radiall effecti effective ve stress stress..

 

CRITICAL STATE: A USEFUL REFERENCE CONDITION

 M   

6 sin    crit   3      sin    crit

for tria triaxia xiall com compre pressi ssion on

(11.27) (11.2 7)

 M   

6 sin    crit   3      sin    crit

forr tr fo tria iaxi xial al ex exte tens nsio ion n

(11. (1 1.28 28))

These equ These equatio ations ns ind indicat icatee tha thatt the cor correl relatio ation n between   M   and  crit is not un uniq iqu ue bu but dep depeends on on th the stress conditions. Neither is a fundamental property of  the soil, as discussed further in Section 11.12. Nonetheless, both are widely used in engineering practice, and,, if int and interp erprete reted d pro proper perly ly,, the they y can pro provid videe use useful ful and simple phenomenological representations of complex behavior. The drained and undrained critical state strengths are illustr illu strate ated d in Fig Figs. s. 11. 11.44 44a   and 11. 11.44 44b   for normal normally ly consolidated conso lidated clay and heavi heavily ly overc overconsol onsolidated idated clay clay,, respective respec tively ly.. The initial mean pressu pressure–void re–void ratio state of the normally consolidated clay is above the critical state line, whereas that of the heavily overconsolidated clay is below the critical state line. When the initial state sta te of the soil is nor normall mally y con consol solidat idated ed at   A   (Fig. 11.44a), the critical state is   B   for undrai undrained ned loadin loading g

Critical State Line

Deviator Stress q Stress  q

 M 

and   C  for   for drained triaxial compression. Hence, the deviator stress at critical state is smaller for the undrained case than for the drained case. On the other hand, when the initial state of the soi soill is ove overco rconso nsolida lidated ted from  D   (Fig. 11.44b), the critical state becomes   E  for undrained loading and  F  for   for drained triaxial compression. The deviator stress at critical state is smaller for the drained case compared to the undrained case. It is important to note that the soil state needs to satisfy both state equations [Eqs. (11.25) and to be at critical state. For example, point  G  in(11.26)] Fig. 11.44 b  satisfies  pcs and  q cs, but not  e cs; therefore, it is not at the critical state. Conver Con verting ting the vo void id rati ratio o in Eq. (11.26) (11.26) to wat water er content, a normalized critical state line can be written using the liquidity index (see Fig. 11.45).

LIcs 



wcs     wPL   ln( pPL  /  p)    wLL     wPL   ln( pP L  /  p  pL L  )

Deviator Stress q Stress  q

Undrained Strength

 E

Drained Strength

 F 

G 3 3

1  A

 D

1 Mean pressure p pressure p

 D

Specific Volume v

Specific Volume v

Γ 

Γ   B

Critical State Line  M 

Drained Peak Strength

C

 B

 A

Mean pressure p pressure p

 F  Isotropic Compression Line

G

 E

 D

 D

Isotropic Compression Line

C

λ 

λ cs

ln ln p  p (a)

λ 

λ cs

Critical State Line 1

(11.29)

where   wcs   is the water content at critical state when the effective mean pressure is   p.  pLL a nd pPL are the

Drained Strength

Undrained Strength

  401

Critical State Line 1

ln p ln p (b)

response using the critical critical state concept: concept: Figure 11.44   Drained and undrained stress–strain response (a) normally consolidated clay and ( b) overconsolidated clay.

 

402

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

Water Content

Critical State Line

Liquidity Index

Isotropic Compression Line

Liquid Limit wLL

Critical State Line

LI = 1 (ln(  p p), w ), w))

(ln(p’), LI)

LI

w

LIeq-1

wcs

LICS

Plastic Limit wPL

LI = 0

ln(  p pLL)

pPL) ln(  p p) ln(  p

ln(  p pLL)

Mean pressure

(a)

ln(  p pPL) Mean p) ln(  p Pressure (b )

critical state line: line: (a) water content versus mean pressure Figure 11.45   Normalization of the critical and (b) liquidity index versus mean pressure.

mean effective pressure at liquid limit (wLL) and plastic limit (wPL), respectively;  pL  L    1.5 to  6 kPa and pPL    150 to 600 kPa are expected expected considerin considering g the undrained shear strengths at liquid and plastic limits are in the ranges   suLL     1 to 3 kPa and   suPL     100 to 300 kPa, respectively12 (see Fig. 8.48). Using Eq. (11.29), a relative state in relation to the critical state for a given effective mean pressure (i.e., above or below the critical state line) can be defined as (see Fig. 11.45) LIeq 

   LI      LIcs     1      LI   

 p   ) log( p /  p    LL  pLL  ) log( pPL  /  p

(11.30)

where LIeq  is the equivalent liquidity index defined by Schofield (1980). When LIeq     1 (i.e., LI      LIcs) and 

q / p46 gi  M  , thethe claystr has thencritical Figure 11.46 11. gives ves stress essreached ratio whe when plastic plas ticstate. failure fai lure (or fracture) initiates at a given water content. When LIeq   1 (the state is above the critical state line), and the soil in a plastic state exhibits uniform contractive behavior. When LIeq     1 (the state is below the critical state line), and the soil in a plastic state exhibits localized cali zed dil dilatan atantt rup ruptur ture, e, or pos possib sibly ly fra fractur cture, e, if the stress ratio reaches the tensile limit (q / p      3 for triaxial compression and   1.5 for triaxial extension; see Fig. 11.46b). Hence, the critical state line can be used as a ref refere erence nce to char characte acterize rize pos possib sible le soi soill beh behavi avior or under plastic deformation.

Sands

The critical state strength and relative density of sand can be expressed as qcs   D R,    R,cs cs

   Mpcs 

 

ema x    ecs   1    ema x    emin   ln(  c  /  p)  

(11.31) (11.32)

where  e cs  is the void ratio at critical state,  e max  and  e min are the maximum and minimum void ratios, and    c   is the cru crushi shing ng str streng ength th of the par partic ticles. les.13 The cri critica ticall state line based on Eq. (11.32) is plotted in Fig. 11.47. The plotted critical state lines are nonlinear in the   e – ln   p  plane in contrast to the linear relationship found for clay clays. s. Thi Thiss non nonlin linear earity ity is con confirm firmed ed by exp experi eri-mental data as shown in Fig. 11.4b. At high confining pressure, when the effective mean pressu pre ssure re beco becomes mes lar larger ger tha than n the cru crushi shing ng str streng ength, th, many man y par particl ticles es beg begin in to bre break ak and the line liness bec become ome more or less linear in the   e– ln ln   p  plane, similar to the

13

Equation Equa tion (11.32) is deri derived ved from Eq. 11.3 11.36 6 prop proposed osed by Bolt Bolton on (1986) with   I  R     0 (zero dilation). dilation). Bolt Bolton’ on’ss equa equation tion is disc discussed ussed further furth er in Sect Section ion 11.8. Othe Otherr math mathemati ematical cal expressions expressions to fit the experimentally determined critical state line are possible. For example, Li et al. (1999) propose the following equation for the critical state line (ecs  versus  p ): ecs 

  e 0      s

 p  p

  

 

a

12

A revie review w by Sha Sharma rma and Bora (20 (2003) 03) gi give vess av avera erage ge va valu lues es of  suLL   1.7 kPa and   suPL     170 kPa.

where   e0  is the void ratio at   p      0,   pa   is atmospheric pressure, and s  and    are material constants.

 

CRITICAL STATE: A USEFUL REFERENCE CONDITION

Fracture

Ductile Plastic and Contractive

Dilatant Rupture

q

Tensile Fracture

q/p

  403

Triaxial Compression

3  M TC

3

 M TC

 M TE

Triaxial Compression 1

0.5 Triaxial Extension

1.0

LIeq

 p 2

-1.5

 M TE

3

Fracture

Dilatant Rupture (a)

Tensile Fracture

Ductile Plastic and Contractive

Triaxial Extension

(b)

Figure 11.46   Plast Plastic ic state of clay in rela relation tion to normalized normalized liquidity liquidity index: (a) stress ratio

when plastic state initiates for a given LIeq  and (b) definition of stress ratios used in (a) (after Schofield, 1980).

 D R,  R,cs cs =

 emax –  ecs  emax –  emin

 emax 0

 e max 0

    r

0. 0.2 2

    r

     D

  y 0. 4    t    i 0.4   s   n   e    D 0.6 0. 6   e   v    i    t   a    l 0. 8   e 0.8    R

1

1

 e

min

1. 1.2 2 0 .0 0 1

0.2 0. 2

     D

  y    t    i 0. 4   s 0.4   n   e    D 0.6 6   e 0.   v    i    t   a    l   e 0. 0.8 8    R

 e

1 In (σc/p)

=

min

1.2 1. 2 0 .0 1

0.1

0

1

 p /   σ c

(a)

 

0.1

0.2

0 .3

0 .4

0 .5

p /  

σc

(b)

Figure 11.47   Criti Critical cal state line derived derived from Eq. (11.32): (11.32): ( a)   e–log   p   plane and (b)   e– p

plane.

behavi beha vior or of cla clays ys.. Co Coop op an and d Lee (1 (199 993) 3) fo foun und d th that at there is a unique relationship between the amount of  particle breakage that occurred on shearing to a critical

would not be sufficient to cause further breakage. Coop et al. (2004) performed ring shear tests (see Section 11.11) on a carbonate sand to find a shear strain re-

state implies and the value of the normal effective This that sand at mean the critical state wouldstress. reach a stable grading at which the particle contact stresses

quir qu ired ede togra reac re ach h the e tr true criti cri tical cal state ate (i.e. (i .e., cons co nsta tant nt par particl ticle gradin ding). g).th They The yuefou found nd thatst particl par ticle e , bre breakag akage e cont co ntin inue uess to ve very ry la larrge st stra rain ins, s, fa farr be beyo yond nd th thos osee

 

404

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

reached in triaxial tests. Figure 11.48a  shows the volumetric strains measured for a selection of their tests, which were carried out at vertical stress levels in the range of 650 to 860 kPa. A constant volumetric strain is reached at a shear strain of around 2000 percent. For specime spe cimens ns at low lower er str stress ess lev levels els,, mor moree she shear ar str strain ainss (20,000 percent or more) were required. Similar findings were made for quartz sand (Luzzani and Coop, 2002). 200 2). Fig Figure ure 11. 11.48 48b   shows shows th thee de degr gree ee of pa part rtic icle le bre breakag e unt with wit shear ar strains str ainsisinqu the logari log thmic ic rdin scale. sca le. Theeakage Th amoun amo t hofshe brea br eaka kage ge quan anti tifie fied darithm by Hard Ha in’ ’s (1985) relative breakage parameter   Br  defined in Fig. 10.14. 10. 14. At ver very y lar large ge str strain ains, s, the va value lue of   Br   finally stabilizes. stabili zes. The strain required for stabili stabilization zation depends on applied stress level. Interestingly, less shear strain was needed for the mobilized friction angle to reach the steady-state value (Fig. 11.48c) than for attainment of the constant volume condition, (Fig. 11.48 a). The critical state friction angle was unaffected by the particle breakage. In summary, the critical state concept is very useful to characterize the strength and deformation properties of soils when it is used as a reference state. That is, a soil has a tendency to contract upon shearing when its state is above the critical state line, whereas it has a tendency to dilate when it is below the critical state line.. Vari line arious ous nor normali malized zed sta state te par parame ameter terss hav havee bee been n proposed propos ed to charact characterize erize the diff difference erence between the actual state and the critical state line, as illustrated in Fig. 11.49. These parameters have been used to characterize the stiffness and strength properties of soils. Some of them are introduced later in this chapter.

11.8

STRENGTH STRE NGTH PARA ARAMETE METERS RS FOR FOR SANDS SANDS

Many fac Many factor torss and phe phenom nomena ena act tog togeth ether er to det deterermined the shearing resistance of sands, for example, mineralogy mineral ogy,, grain size, grain shape, grain size distribution, (relative) density, stress state, type of tests and stress path, drainage, and the like [see Eq. (11.3)]. In this section, the ways in which these factors have become understood and have been quantified over the last several sev eral decades are summar summarized. ized. Several correl correlations ations are gi given ven to pro provid videe an ov overv erview iew and ref refere erence nce for typical typ ical va values lues and ran ranges ges of str streng ength th par paramet ameters ers for sands san ds and the infl influen uences ces of va vario rious us fac factor torss on the these se 14 parameters.

Early Studies

The important role of volume change during shear, especially peciall y dilatan dilatancy cy,, was recogn recognized ized by Taylor (1948). From direct shear box testing of dense sand specimens, he calculated the work at peak shear stress state and showed that the energy input is dissipated by the friction using the following equation:  peak    dx          n   dy        n   dx 

 

(11.33)

where   peak   is is the the appl applied ied sh shea earr stre stress ss at at peak peak,,  n is th thee confining normal (effective) stress on the shear plane, dx  is   is the incremental horizontal displacement at peak, dy  is the incremental vertical displacement (expansion positive) at peak stress, and    is the friction coefficient. The ene energy rgy dis dissip sipated ated by fri frictio ction n (th (thee com compon ponent ent in the right-hand side) is equal to the sum of the work  done do ne by sh shear earin ing g (fi (firs rstt co comp mpon onen entt in th thee le leftft-ha hand nd side) and that needed to increase the volume (the second component in the left-hand side). The latter component is referred to as dilatancy. Rearranging Eq. (11.33),  peak       tan    m            

dy   dx 



(11.34)

Thus, the peak shear stress ratio or the peak mobilized fric fr ictio tion n ang angle le  m co cons nsis ists ts of bo both th in inter terloc locki king ng (dy/dx ) and sliding friction between grains ( ). This equation that th at re relat lates es st stre ress ss to di dilat latio ion n is ca calle lled d th thee st stre ress– ss– dilatancy rule, and it is an important relationship for characterizing the plastic deformation of soils, as further discussed in Section 11.20. Rowe (1962) recognized that the mobilized friction angl an glee  m mu must st tak takee into into acc accou ount nt par parti ticl clee rear rearra rang ngeements as well as the sliding resistance at contacts and dilation. crushing, increases in id importance tan ce as Particle confini con fining ng pressu pre ssure rewhich increas inc reases es and vo void ratio rat io decreases, should also be added to these components. The general interrelationships among the strength contribut tri buting ing fac factor torss and por porosi osity ty can be rep repres resent ented ed as show sh own n in Fig Fig.. 11.5 11.50. 0. In In this this figu figure re,,   f  is the the fri fricti ction on angle corrected for the work of dilation. It is influenced by particle packing arrangements and the number of  sliding contacts. The denser the packing, the more important is dilation. As the void ratio increases, the mobilized bil ized fri frictio ction n ang angle le dec decrea reases ses.. The crit critical ical state is defin de fined ed as th thee co cond ndit ition ion wh when en th ther eree is no vo volu lume me change by shearing [i.e., (dy/dx )     0 in Eq. (11.34)]. Thee co Th corr rres espo pond ndin ing g mo mobi bili lize zed d fr fric icti tion on an angl glee  m is      crit  .

14

A number of additional useful correlations are given by Kulhawy and Mayne (1990).

The ‘‘true to friction’’ in thesliding. figure is associated with the resistance interparticle

 

STRENGTH PARAMETERS FOR SANDS

Shear Strain (%) 50,000

0

100,000

   )    % 0    (   n    i   a   r    t   s   c    i   r    t   e20   m   u    l   o    V

150,000

 

RS3 RS5 RS7 RS8

(a)

RS13 RS15

40 Luzzani & Coop, 805 kPa 650-930 kPa

1.0

248-386 kPa 60-97 kPa

0.8

  e   g   a    k   a 0.6   e   r    B

RS7 RS8

  e   v    i    t 0.4   a    l   e    R

(b)

0.2

?

0.0

800 kPa unsheared

10

100

1000

10,000

100,000 1,000,000

Shear Strain 50    )   s   e   e   r   g 40   e    d    (   e    l   g   n 30    A   n   o    i    t   c 20    i   r    F    d   e   z    i    l    i 10    b   o    M 0

RS3 RS7 RS8

(c)

RS9 RS10 RS15

1

10

100

1000

10,000

100,000

Shear Strain (%)

Figure 11.48   Ring shear shear test results of carbonate carbonate sand: sand: ( a) volumetric strain versus shear

strain, (b) the degree of parti particle cle breakage breakage with shea shearr strai strains, ns, and ( c) mobilized friction angle versus shear strains (after Coop et al., 2004).

  405

 

406

11

  STRENGTH AND DEFORMATION DEFORMATION BEHAVIOR

1. State parameter (Been and Jefferies, 1985)

Void ratio  e

 = e –  e c Ψ = e

Critical state  pc , e c) line ( p

Loose Sand

 ecD

Loose sand Ψ = e  = e L –  e cL  (>0)

( p L, e L )

2. State index (Ishihara et al., 1998)

Ψ 0 Dense sand

 ecL

Dense sand Ψ  = e  = e D – ecD  (1) Dense sand Is  = (e 0 – e cD)/(e0 – e0) (1)  I  p = p Dense sand I  sand I  p = p  = p D/pcD (