Time dependent mechancial behvior of loose sand di prisco imposimato 1996

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MECHANICS O F COHESIVE-FRICTIONAL MATERIALS, VOL. 1, 45-73 (1 996)

Time dependent mechanical behaviour of loose sandsClaudio di Prisco and Silvia Imposimato

Dipartimento di Ingegneria Smtturale. Milan Universiy of Technologv (Politecnico). Milan, Italy

SUMMARYThe time dependency of loose sands mechanical behaviour has been experimentally analysed and theoreticallyinterpreted. A series of load controlled triaxial tests, by imposing finite load increments, has been performed. Thesingle load increments are followed by variable time periods, in o rder to cany out many classical creep tests.According to the authors, the considered time dependency is due to the internal fabric rearrangement of thegranular assembly, i.e. to the plastic strain development with time.This mechanical peculiarity is theoretically interpreted by means of an elastoviscoplastic constitutive model.This is a very simple extension of a previous incremental elastoplastic constitutive model and appears to becapable to reproduce experimental data quite well.Finally, the importance of the considered time dependency is underlined, both by considering numericalsolutions and in analysing unstable natural and experimental phenomena.KEY WORDS: oose sand; load controlled triaxial tests; creep tests; time dependency delayed mechanical response; elastoviscoplasticity

1. INTRODUCTIONThemechanical behaviour of granular materials is com monly assumed to be time independe nt; but, inthe experimental reality, the mechanical response of such materials is rapid but naturally cannot beimmediate. When a material is loaded, a stress wave passes throug h the continuum. The time of thewave propagation depe nds on the material mechanical properties; if the load increment induces a microstructural rearrangement within the granular assembly, at the microscale the stress distribution change swith time and the micro-structural fabric continuously modifies. The two effects are strictly linked andit is difficult to define which is the cause and w hich is the co nsequen ce.This progressive reconstruction of the internal fabric with increasing stresses is mostly due to thesliding along unstable contacts between grain particles and partly due to the rotation of grain

The final micro-structural configuration is reached passing through various intermediate notequilibrated configurations; the final result is not unique but statistically reached. The time periodnecessary to obtain the final configuration (several minutes) is the required amount for themicrostresses to reach a final equilibrated distribution with the external applied m acrostresses. Theinternal fabric rearrangement is caused by the micro accelerations, which develop to balance theexternally imposed macroload increm ent. When a new equilibrated microconfiguration is reached, thekinetical process may be assumed to be con cluded. This physical time may be interpreted as the timeperiod in which plastic strains take place. For t h ~ season the term deluyedplusticity will be introduced.In this pa rticular micromechanical interpretation of the phenom enon is implicit the assumption oftwo different responses: the elastic one and the plastic one. The elastic response is linked to the c urrentCCC 1082-501 0/96/0 10045-290 996 by John Wiley & Sons, Ltd. Received 4 September 1995Revised 28 September 1995


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46 C. DI PRISCO AN D S . IMPOSIMATOinternal microstructure and, in comparison with the plastic one, may be assumed to be quasi-instantaneous.Moreover, by taking into account the considered sand specimens, it will be shown that theconsolidation process also may be assumed to be qu asi-instantaneous , because the pore waterdissipates in a few seconds; it will be experimentally shown that the time scales, characterising themicrostructural rearrangement and the consolidation processes, are different and the phenomena maybe considered uncoupled.From a practical point of view, in comparison with fine soils, the time dependency of sandsmechanical behaviour, experimentally observed, is not quantitatively relevant: the time periodnecessary to reach the final configuration is much shorter. Nevertheless, when the timescale is small(some minutes), for instance when dynam ical and unstable phenomena are taken into account, the non-instantaneous material mechanical response may play a very important role.Moreover, if a delayed plastic relationship is considered, and a com putational code, for instance afinite element method (FE M) code, is used, this time dependency may be introduced as a powerful toolto regularise numerical solutions; recently, for instance, many have introduced elasto-viscoplastic constitutive models to avoid numerical instabilities during finite element analyses ofplastic flow localisation in b oundary value problems.The aim of this paper consists of show ing that the time dependency of the constitutive relationship,also by dea ling of granular materials, is physically meaningful and experimentally observable.Therefore, it has been decided to experiment and to interpret the results by means of a time-dependent constitutive relationship. Th e exp eriments consist of a load controlled drained triaxial testseries, characterised by finite stress increments, on saturated and dry loose sand specimens.The experimental programme and the proposed theoretical approach are strictly linked; theconstitutive m odel is derived from the observations and the idea to perform many particular stresspaths is derived from the theoretical structure of the model.

2. STATE OF THE ARTFrom an experimental point of view, the time dependency of sand mechanical behavior has beenalready analysed by some authors. For instance, Mitchell and S01ymar5 have published theexperimental results of an extensive in situ test programme, which aimed to analyse the mechanicalresponse of an in situ sand layer after blasting and vibrocompaction operations. They measured atdifferent depths, by means of CPT tests, the influence of the time on the penetration resistance; thephenomenon has been monitored during a period of some months and the authors concluded that thesand penetration resistance increases in the time.According to M itchell and S olymar, the observed time dependen cy must be considered in evaluatingthe results of laboratory tests on reconstituted spe cimens, in the assessment of ground improvementusing d eep densification and in the estimation of liquefaction potential. Moreover, they concluded thatthe most probable caus e of the ob served phenomena seem s to involve the form ation of silica acid gelfilms on particle surfaces.Recently, some interesting viscous phenom ena of sands have also been investigated by means of atriaxial tests series: with regard to high confining pressures. Both calcareous and silica sands havebeen clearly shown to exhibit a viscous mechanical behaviour, i.e. also in this case the time factorseems to play an important role.

When the confining pressure is sufficiently high, the grains are subjected to severe stressconcentrations, which may cause local grain ruptures, phenomenon exhaustively analysed ingeotechnical literature for m any years. T his may be related to time-mechanical effects, because in thiscase the grains cease to be interpretable as inert particles.


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TIME DEPENDENT MECHANICAL BEHAVIOUR OF LOOSE SANDS 47On the contrary, in this paper, the previous aspects will be completely disregarded and the delayedplasticity of the m echanical behaviour of sands will be experimentally and theoretically investigated atcommon confining pressures. A similar experimental programme, concerning Toyoura sandmechanical response with time, has already performed by Murayam a et The experimental resultswere interpreted by me ans of a classical reological creep model.

3. EXPERIMENTAL RESULTS3 .1 . Experimental programmeTo investigate the material experimental response with reference to time, a series of triaxial loadcontrolled tests was camed out: i.e. a series of classical creep tests were performed. In Figure 1 asimplified scheme of the employed triaxial apparatus is show n; the axial stress is imposed by means ofa dead load applied to a jack.All the tests have been performed on specimens (140 mm high, 70 mm wide) of loose Houston RFsand (DrW 20%).* It is a silica sand characterised by a relative percentage of 99.The specimens have been created by using th e moist tamping techn ique; this initial very low relativedensity has been obtained by adding a very low percentage of water, approximately 2%by mass.Thanks to this, the internal microstructure completely changes and becomes flocculate, by inc reasingthe void ratio.The experimental results, shown in the following, concern both saturated and dry sand specim ens. Inthe first case, to reach at least the 95 % of saturation, the COz method has been em ployed: the B valuemeasurements indicated 0.95 or better for most specimens. In the seco nd one, the water is not made toseep through the specimen.In this case, the material will be defined 'dry' even if the authors are aw are that the suction of thecapillary water may quantitatively influence the material's mechanical response. Nevertheless, thiseffect will be neglected, as with Lanier et al.9 and Skopek et a1." because the experimental resultsrelative to the dry spec imen s will be only qualitatively taken into considera tion. Moreover, in section 6only the exp erimental results relative to saturated sand specim ens will be numerically simulated.Two different types of triaxial tests have been carried out, schem atically illustrated in Figure 2. Thepath n.1 of Figure 2 is a drained standard triaxial compre ssion test, obtained by increasing the axialload and by keeping constant the radial stress; the second one is characterized by a first standarddrained triaxial compression phase and a following q constant effective stress path. This second triaxial

Figure I . Load control system, applied on a classical triaxial apparatus


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48 C. DI PRISCO AND S. IMPOSIMATO

0 10 20 30 40 50 60 70 80 90 100 110 12 0 130 140 150 160 170 180 190P IkPal

Figure 2. Drained compression triaxial tests: standard effective stress-paths and effective stress-pathsof q constant tests

phase is obtained by keeping the axial load constant and, on saturated specimens, by increasing thepore pressure, on dry specimens, by directly decreasing the cell pressure.Some exp erimental results, concerning these types of tests, have been already shown in Matiotti etal . The collapse points and the particular unstable phenomena, which suddently take place during theloading paths will not be taken here into account, and only the time-dependency of the materialmechanical behaviour will be analysed.The stress paths are obtained by imposing finite incremental steps of stresses (Figure 3); the timeperiods between two following stress increments and the amplitude of these stress increments havebeen chosen to be variable: every test has a particular time history.Finite load increments have been chosen to record the following material delayed strain responseand, in particular, to clarify the material mechanical behaviour with reference to the previous historyand to the current stress increment. In fact, if the time dependency of the mechanical behaviour isconsidered, the stress-strain relationship is the final result of a complex superimposition, orconvolution, of the previous load history. If a continuous stress increment is imposed, the timedepende ncy of the matcrial mechanical behaviour is implicit, because, to evaluate the current straineffects caused by the previous load increments, the appropriate convolution rule should be defined. Onthe contrary, if !kite load increments are imposed, and if the time pe riods between two load steps aresufficiently long, it is possible to assum e the current strain effects associated to previous load steps tobe quantitatively negligible in comparison with the strain effect associated to the current loadincrement.3.2. Standard drained triaxial compression testsWe can start by anaiysing a standard triaxial compression test (as = 100@a, T100a) on a loosesaturated sand specimen, up to the stress level corresponding to a mobilised fictio n angle (Om)of 16(Figure 4a); in Figure 4b the relative axial load time history is illustrated: the time pe riods between twofollowing load steps are con stant ( 5 min), but the load increment values are variable. In Figure 4c, dthe global stress-strain curve and the volumetric behaviour are shown respectively. The points depictedon these figures correspond to the value recorded just before the new stress incrcment. By collecting


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TIME DEPENDENT MECHANICAL BEHAVIOUR OF LOOSE SANDS

I!-

10 -

EM of perloa time 1120 mn

10 -5 -0 1 ,----

0 20 40 60 80 100 120 140 160 180 20 0 220 240 260 28 0 300CLm Imin.1

49

Figure 4. Drained standard triaxial test [a: = 100 H a (TIOOa)]. (a) Effective stress path, (b) axial load-time history, (c) stress-strain relationship, (d) volUmetric behaviour


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50 C. DI PRISCO AND . IMPOSIMATO(4 MobRi6.d lricfia angla I I

0 1 2 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 60

0 10 20.3

g 0 4f 0 5

0.6

-c-

0. 70.8 4I0.9 j, i

73 i

, .- ,0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6

Mobliredtrktlon mcJe I O 1Figure 4. (continued)

the axial-strain-time curves correspo nding to each load creep testing step, Figure 5 is obtained (forclarity, in Figure 5 only the load steps charackrised by a dq = 5 kPa are shown).Figure 5 clearly shows that the material strain response to the instantaneous stress increment isdelayed in time. The axial strain one minute after the load increment is largely less than 50% of theaxial strain after five minutes. Moreover, from the same figure it is possible to note that the final strainvalue and the initial slope associated to each curve regularly increase with the stress level. It isimportant to underline that, during the time between two load increments, the pore-pressure, except forthe first seconds, remains constant and equal to the back-pressure imposed at the end of theconsolidation phase,Consequently, this particular time-dependent behaviour is not due to the water flow within thespecimen, i.e. to the con solidation phenom enon, but uniquely to the m aterial mechanical properties. To


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TIME DEPENDENT MECHANICAL BEHAVIOUR OF LOOSE SANDS 51time 1min.l

0 1 2 3 4 5

00 2

0.04 --0.06 .ID

0.12 10.14

- 1 12.6O--O- 41 1 3 9-- 5114.4O--1 15.P- - - 71 16O I

Figure 5. Axial strain-versus time curves [c$ 100 @a] (TlOOa): axial stress incremements d q = 5 kP a

measure the pore pressure inside the specimen and to allow the water drainage during the test, the valvelocated on the top is open, but the on e at the bottom of the specimen is connected to a m anometer.Test TlOOa, previously described $igure 4). is characterized, at a stress level corresponding to amobilised friction angle of 16", by a load increment followed by a time period of 17 h (Figure 4b). InFigure 6 the axial strain versus time curve recorded du ring t h i s long time period is illustrated; to showthat, under a constant load, the axial strain slowly but continuously increases, the same curve isreproduced in a semilogarithmic scale (Figure 6b). The axial strain after one hour is the 80% of thefinal recorded one.The presented experimental results have been confirmed by a large number of tests performed atdifferent cell pressures and at different stress rates.As anticipated in the introduction, the experimental programme and the constitutive model havebeen developed at the same time; for this reason, only after the constitutive model and the theoreticalinterpretation framework presentation, the rem aining experimental results w ill be shown. Moreover,many experimental results have been used as confirmations of theoretical assum ptions and, hanks tothe constitutive model, many sand's mechanical behaviour aspects have been highlighted.

4. THE ELASTOVISCOPLASTIC CONSTITUTIVE MODEL4. I . Theoretical outlinesTo reproduce, by means of a constitutive relationship, the previously illustrated time dependency, anelastoviscoplastic constitutive law has been introduced.The material mechanical response is assumed to be characterised by the superimposition of aninstantaneous elastic strain increment and a delayed plastic strain. Consequently, the followingrelationship may be w ritten

dEij = d&t + ds?, (1)where d&t s the elastic strain increment tensor and dE7 is the viscoplastic one.The elastic strain response is defined as follows:

d~! = C& dt&,


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52 C. DI PRISCO AN D S. MPOSIMATO(4 tinmlminl

0 120 240 360 480 600 720 84 0 960 1080.o r-----

0.25II0.3 '

(b)1

0

0.05

0.12-i 0.15E5

0.2

0.25

0.3

10lim min.1

100 1o m

Figure 6 . Mobilised friction angle 16";axial stress increments 5 Wa, lOOa test; (a) axial strains versus time; (b) axial strainsversus rime: semilogarithmic scale

where Cg! is the elastic compliance tensor, independent of time. Moreover, the viscoplastic strainincrement% may be defined in the following way:

wheref s the yield function, g the plastic potential, Oy) he viscous nucleus and y is a constitutiveparameter.The plastic potential defines the direction of the viscoplas tic strain increment tensor and the yieldfunction influences its modulus by means of the viscous nucleus 0 .By using the present definition of the viscoplastic strain increment tensor, it is not necessary tointroduce a consistency rule, because the viscoplastic strain rate modulus depends directly on thefunction describing the visco us nucleus and not on the plastic multiplier.


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53IM E DEPENDENT MECHANICAL BEHAVIOUR OF LOOSE SANDSBy not assuming the consistency rule, the yield fun ction may be positive or negative, without anyconstraint, i.e. the stress state may be external or internal to the yield locus.From (3), it derives that the viscoplastic strains develop with tim e, also by keepin g the effective stateof stress constant. Moreover, if a strain hardening rule is assumed, also the state variables will chan gewith time and, consequently, the yield locus will evolve, even if the effective state of stress is keptconstant.From a numerical point of view, this approach, already introduced with reference to differentm ate ria ls by P e r ~ y n a , ' ~ * '~dachi et aZ.,'4 does not allow us to define a loading-unloading criterion.This makes the numerical implementation of this kind of constitutive models into FEM codes moresimple; on the contrary, the num erical convergence to the solution is deeply linked to the chosen timestep. The su itable time step depends on many factors: the stress rate, the cu rrent stress level, the stressincrement intensity, i.e. the current value of the viscous nucleus @.v). he time step should becontrolled by the value: whenIf the viscoplastic strain is assumed to increase only when the state of stress is outside the yieldfunction, i.e. whenf> 0, it is possible to prove (Appendix A) that

is small, Az may be increased, and vice versa.

where St$ is the corresponding plastic strain increment tensor. In other words, in this case theviscoplasticity may be interpreted as a natural extension of incremental plasticity.The parameter y of (3) influences the strain rate and consequently the rapidity with which theasymptotic strain value is reached. In order to clarify this dependency, in Figure 7b some strain-timecurves, associated with different y values, but to the sam e instantaneous stress increments (Figure 7a),are schematically drawn. By increasing the y value, the initial curve slopes become steeper: wheny 3 00 the limit value 6$ is instantaneously reached.

4 .2 .1 . The viscous nucleus. We have observe d that the choice of the viscous nucleus is important indescribing the material's mechanical response in the time; consequently, many trial viscous nuclei havebeen taken into account. We have chosen00 ear. ( 5 )

because it appeared to be simple and capable to reproduce the experimental evidence well.splasticity and plasticity continue to coincide, if the viscous nucleus(Dossumes thatBy using ( 5 ) in Appendix B, equation (4) is discussed. It is analytically demonstrated that vico-

To clarify physically the mechanical consequences deriving from the condition expressed by (6b) itis interesting to analyse the experimental curve of Figure 6. It is evident that, in the considered timeperiod, the axial strain does not reach an asymptotic value. If the experimental result is theoreticallyinterpreted by means of the proposed viscoplastic approach , two different hypotheses may be assumed:

(a) in the considered time elapse, the yield locus (f= 0) has not yet been reached by the effective(b) the constraint expressed in (6b) is not physically correct.state of stress;


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54 C.DI PHSCO A ND S . IMPOSIMATO

-_l = t 0 t

Figure 7. (a) Considered stress increments versus time. (b) Theoretical materials strain response, by introducingdifferent value s of y

If (6b) is not assumed to be valid, plastic strains may take place within the yield locus , too; in fact theMoreover, if the constraint expressed in (6b) is removed, the following analytical consequenceschosen viscous nucleus ( 5 ) is no t zero whenf < 0.derives:

i.e. the plastic strain does not reach a finite value, when the time period becomes unlimited, asdemonstrated in (B5) Appendix B).Naturally, this may be physically meaningless, but, in the considered analytical approach, other


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I50


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TIME DEPENDENT MECHANICAL BEHAVIOUR 01: LOOSE SANDS 57

5 0.1

O

I .-0.1 t ------- ---- I0 10 20 30 40 50 60 70 80 90

t'm rntn.l

0 5 10 IS 20 25 30 35 40 45 50 55 80 65 70 75 80 85tim Imin.1

Figure 9. (continued)

be w ithin the yield locus and the followings, by showing a relevant plastic strain response, appear to beoutside.Figure 9c shows the material volumetric behaviour and Figure 9d the corresponding axial strainversus time curve: wh en the effective stress state remains within the y ield locus, the global volumetricor axial strain is small; on the contrary, when the effective stress state reaches the yield locus (point Cof Figure 9c, d), it increases. But both inside and outside, a time dependent volumetric behaviour isobservable. Consequently, hypothesis (b) of Section 4.2.1 seems to be experimentally confirmed.Thanks to the previously described experimental observations, it seemed to be reasonable to neglecttheconstraint expressed by (6b), i.e. delayed plastic strains will arise also within the yield locus and the


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0.3

0.25

0.2

0.1

0.05

01

59IME DEPENDENT MECHANICAL BEHAVIOUR OF LOOSE SANDS

10 1w 1000 toowtime Imh.1

Figure 10. Material strain-time response: cxperimental results and numerical simulations:semi-logarithmic plane

5 . FURTHE R EXPERIM ENTAL OBSERVATIONSFrom the experimental evidence the constitutive model has been derived, but only by mean s of thetheoretical approach, it ha s been possible to put in evidence some par ticular aspects of the materialmechanical behaviour; consequently, the following experimental results have been obtained just tocorroborate the theoretical assumptions.As previously illustrated, the theoretical model assumes a continues change, with time, of theviscous nucleus value; when the state of stress is outside the yield locus, (Figure 1la), 0 0 ssumeshigh values, otherwise it rapidly decreases. When a long time period between two following load stepselapses, thanks o the experimental expression of the chosen viscous nucleus, the effective state ofstress, at the end of the time period results in the interior of the yield locus. If a new load step isapplied, the effective state of s tress may remain within the yield locus (Figure 11b); if this happens , thecorresponding strain increment is sm all, because the 0 0 alue is small.To verify this theore tical observation, after the long time period o f the Test TI OOa, already shown inFigure 6a, two new load steps have been analysed. In Figue 12 three axial strain versus timeexperimental curves are collected; curve number 1 is relative to the first thirty minutes of the curveshown in Figure 6a, i.e. it is relative to the load increment followed by a very long time period. Curvesnumber 2 and 3 are relative to the two following load increments, as shown in Figure 13a when thecomplete load time history of test TlOOa is drawn. The stress increment of the three load steps isconstant, but the global strain increments are very different.Curves 2 and 3 of Figure 12, show a steeper slope in comparison with line 1 , because the long timeperiod relative to the step n.1 makes the mjcro structure more stable. This mechanical stiffening ismore im portant on step n.2, than on the following one (step n.3).The sam e effect is observable in Figure 13b, by considering the pointed out area. In this figure theaxial strain versus stress level curves of Test TlOOa is drawn.Point (a) is relative to the axial strain recorded 5 min after the load increment application, point (b)is relative to the axial strain recorded just before the folJowing load inc rement (17 h later ), poin t (c), inthe sam e manner, is relative to the final axial strain of load step n.2 and po int (d) of load step n.3. Theconsidered strain behaviour clearIy shows the effect of time on the materials mechanical response; the


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60 C.DI PRISCO AN D S. IMPOSIMATO

P'

Figure 1 1 . Yield b c t i o n and effective stress state: (a) high stress rate; (b) low stress rate. At is the time period elapsing betweentw o different stress increments

0

006

zf 0.1f20.15

0.2

Figure 12. Long time elapsing effect on the two following load steps: TIOOa, dq= 5 kPa


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00.2

0.406

1.4

1-

MDbdicedtrictlon nmle I a I

i

IIi

1 10

(b)0 2 4 6 8 141 20

ii1TIIIILL

61

1.6 4i

l : IFigure 13. (a) Load increments versus time, relative to the two previous described triaxial tests. (b) Comparison between twodifferent drained triaxial compression tests: [a: = 100 kPa], corresponding to two different stress rates: stress-strain behaviour

distance between points (a) and (b) quantitatively underlines this effect. Moreover, the materialsbehaviour from point (b) to (d) appears to be elastic, even if it is not about a reloading.Figure 13b shows the comparison between Test TlOOa and another drained standard triaxialcompression test (TlOOd); the difference consists of the elapsing time between the load increments. InFigure 13a the load increments versus time, with reference to the two different tests, are shown: up to aq value of 76 W a, the test TlOOa follows the same load versus time history of the test TlOOd, but in thefollowing Test T 1OOd has been m ore quickly performed.It is notic eable that poin t (d) of Test TlOOa reaches the curve of Test TlOOd, because the effect of theprevious long time period (step n.1 of Figure 13a) is erased; consequently the time effect on thematerial mechanical behaviour seems to influence a limited region in the effective stress space. Theglobal response appears to be not influenced by the stress rate.


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62 C.DI PRISCO AND S.IMPOSIMATOTo confirm this result, on the same material some strain controlled drained triaxial compression testshave been carried out. The tests are characterised by d ifferent strain rates; among them, in Figure 14only three experimental curves are drawn. In Figure 14a the stress-strain curv es and in Figure 14b therelative volumetric behaviour are shown. It appears e vident that the global mechanical response is notdeeply inff uenced by the strain rate. For this reason, the sand mechanical behaviou r is usually assumedto be independent on the strain rate and consequen tly on the time factor.These ex perimental observations do not have a general validity; in fact, by analysing the m echanicalresponse obtained by means of load controlled tests, it is po ssible to confirm that the global stress-strain response s may be different, if the im posed stress rates are also very different.In Figure 15b, where the stress-strain curves relative to o ther two test (T100b and T100c ) are drawn,the relative load time histories are illustrated in Figure 15a. The Tests TlOOb and TlOOc are twostandard triaxial compression tests on dry loose sand specimens.

i25 0 'I200

100

50

0

0

0.51

zE'2 2.5l 3.54

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 IS 16 17 18 19 20axial strain 1%1

(b) axie4 swam 1%10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20,--uR--L--' ' '

4.5 1j5 -

IaJ swam raw. 1 mmlminfbl strain rate: 0.333 nnvrnin, -fcl swain rate: 0.0078 mmlminI-

Figure 14. Strain controlled drained triaxial compression tests on loose saturated sand specimen at different strain rates:experimental results. (a) Stress-strainrelationship, (b) volumetric behaviour


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0.1

0.2zI 0.380.4

0.5

TIME DEPENDENT MECHANICAL BEHAVIOUR OF LOOSE SANDS

2

10

MobiinrdMhnab I I

i t( ItlI ,l iL L

4 6 8 12 1410 16

63

i i i

loo00

18

IIIIt1

Figure 15. Tests TlOOb and TlOOc (standard triaxial compression tes ts on dry oose sand specimens). (a) Load time histories. (b)Comparison between the two global stress-strain responses. (c) Comparison between the two tests results relatively to the loadincrement N.1 ( d q = S Wa, 'M= 11 . 1" ) . (d) Comparison between the two tests results relatively to the load increment N.2(dq=S kPa,'l&15.5")In Figure 15b the m ore slow ly performed experimental test is characterised by a less rigid curve; at arnobilised friction angle of 12", the axial strain difference is about 25%, but by increasing the stresslevel the percentage strongly decreases. The percentage has the expression belt, where E is therecorded strain, relative to on e of the two curves, and A& is the difference between two curves at thesame stress level. Because the axial strain E increases more than he, this ratio decreases; consequently,

when e is large, i.e. at high stress levels, this stress rate effect may becom e negligible.The experimental results, being the tests on dry sand specimens, confirm that all the previouslymechanical observations are independent on the water presence within the specimen . Consequently, theobserved time effect cannot be attributed at all to a c onsolidation process. To analyse the stress rate


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64 C . DI PRJSCO AN D S. MPOSIMATO

II0 1 '

(4 hm c Imin.10 90 120 15 0 180 2100 60

--I 1100~ o -5lo - 73 Pal kPa. swess level 1 5 5 ' 1008 JI

0. 1 iFigure IS. conn'nued)

effect, the comp arison between th e two previously illustrated tests, but concerning a single load step,may be interesting.In Figure 15c the m echanical response relative to the two tests is compared with reference to a loadstep corres ponding to a mob ilised friction angle of 1 l o (n. of Figure 15a). The steeper curve is relativeto the more rapid stress rate test; when the specimen is quickly loaded, it tries to more quick ly react. InFigure 15d the load step following the long time period elapsing at q = 73 kPa of the TlOOc test isshown . With reference to the step n.2 of Figure 1Sa, the comparison between the m echanical responsecorrespon ding to the slow and rapid tests is drawn.


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65IM E DEPENDENT MECHANICAL BEHAVIOUR OF LOOSE SANDSThe experim ental results of Figure 15c, d may be better understoo d if they are interpreted by meansof the mathematical model. When the specimen is rapidly loaded, the effective state of stresscontinuously remains outside the yield locus, consequently the @malue is high and the strainincrement is quantitatively relevant. Otherwise, if the stress rate is reduced, the yield locuscontinuously reaches the current effective stress state, the @(f ) value remains small and the initialslope of the axial strain versus time curve is less steep.

6. NUMERICAL SIMULATIONSTo simulate numerically the previously shown experimental results, the elastoviscoplastic model,presented in Section 4, has been used. The loose sand specimen has been interpreted as a macroelement, by having disregarded any disuniformity of the strain and stress state within it and anyconsolidation process following each load step.To show the good capability of the model to simulate, not only qualitatively but also quantitatively,the experimental results, in F igure 16a, the exp erimental data, already shown in Figure 6a, of TI OOaTest (Figures 4a and 13) and the corresponding numerical simulations, are drawn .The results are relative to the step number 1 of Figure 13a; the axial load increment is about 5 kPa,the stres s level is characterised by a mobilised friction angle of 16. At the same time, it is possible toshow the numerical simulations of two among the three curves previously illustrated in Figure 12(Figure 16b).

Cwve (a) is relative to a load step followed by a long time period (17 h), curve (b), instead, isrelative to the following step. The stiffening, induced by the first long time period is well reproduced,because the stress state is located within the yield locus and the load increment is not large enough tobring it out.If the unloadings of Figure 8 are numerically simulated, Figure 17 is obtained. The numericalsimulations are charac terised by an ins tantaneous axial elastic strain decrem ent, followed by acontinuous axial strain increasing.To clarify the transition from the elastic instantaneous response and the delayed plastic one, inFigure 18a the axial strain versus time numerical curve, relative to 477 Test (Figure 9), is shown.Thanks to the chosen viscous nucleus, the point, at which the yield locus is reached, is not adiscontinuity point of the material mechanical response. At any rate, from the num erical simulations, arapid change of the strain response clearly appears; in fact, when f > 0 (6a) the viscous nucleusassumes large positive values.On the contrary, if an elastoplastic model had been assumed to interpret the m aterial mech anicalbehaviour, and if a yield function, as schem atically drawn in Figure 18b, is introduced at the beginning ,the state of stress would be inside the yield function. Only by decreasing the effective mean pr essurepwould the boundary be reached consequently a material mechanical response discontinuity wouldresult.In practice, the assump tion that the viscoplastic strains take place even within the yield locus im pliesthat the transition between the stiff response of the material, typical of unloading-reloadings and thesofter response, typical of virgin loadings, is smooth, at variance with the classica l elastoplastic theoryfor which such a trans ition, ruled by the loading-unloading criterion, is sharp.By com paring Figure 18 a with Figure 9d, it is evident that the numerical simulations are not capableof quantitatively reproducing the experim ental strain-time behaviour. If the experim ental trend of thecurve shown in Figure 9d, with reference to the load steps following point C, s taken into account, thisdiscrepancy may be explained. In fact, after point C, the single load increments axial strain responsesare character ized by a very pa rticular trend: at the beginning, the strain rate increases, but subsequentlydecreases. This strain process is associated with a large strain development, which the model is not


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66 C. D1 PRISCO AN D S. MPOSIMATOtime Imin.1

0 120 24 0 360 480 600 720 840 98 0 I080(4---.-i--.-

0.05 t

(b) Ikm Imb.10 5 10 15 20 25 30

0.2 J

Figure 16. Test TlOOa (drained standard triaxial, load controlled, test on loose saturated sand specimen (see Figure 4)). (a)Comparison between experimental results and numerical simulations relatively to the load increment N.l of Figure 13a(dq= kPa, C& = 16") @) Long time elapsing effect on the following load steps (see Figure 15): experimental results andnumerical simulations relative to the load increment N.2 of Figure 13aable to simulate at all, because it has been conceived to reproduce the stable strain increasing,characterising the experimental mechanical sand response associated with low stress levels.According to us, the mechanical response is stable when the strain rate, during the time periodfollowing the single load increm ent, continuously decreases, and unstable when the acceleration mayassume positive values too.In Figure 19 the effect of the stress rate on the strain response of the material is numerically

reproduced. The numerical results concern the standard experimental triaxial tests (T 1OOa and TiOOd)shown in Figures 4a and 13; the axial strain versus the mobilised friction angle curve is drawn.Moreover in Figure 20, the comparison between experimental data and numerical simulations isillustrated. Figure 20a show s the mechanical behaviour relative to the low stress rate te st (TIOOa), in


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TIME DEPEND ENT MECHANICAL BEHAVIOUR OF LOOSE SANDS 67

.0.01

0

g 0.01IB 0.021-

0.03

0.04

linw Imh.10 15 30 45 60 75 90 105 120 135 15 0 165 1 80 195 210I

Figure 17. Comparisonbetween experimental esultsand numerical simulations, corresponding o the unloading steps of Figure 8

1

1.02

1 M

(4 t h e Inh.10 10 20 30 40 50 60 70 80 90: -

iI1.1 -

PFigure 18. (a) Test 477, numerical simulations: axial-skaain-time curve (q constant phase). (b)Theoretical interpretationof a q constant test


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68 C. DI PRlSCO AND S. IMPOSIMATOMobased frictionangle I I

0 2 b 6 8 10 I2 1 4 16 18 20 2 2

0.5

2

I2. 5 -Figure 19. Stress rate effect: numerical simulations

Figure 20b, c, the results concern the more rapid stress rate test (T100d). Generally, both theexperimental results and the numerical simulations show quantitatively a good agreement betweeneach other.Finally, in Figure 21 the com parison between nu merical results and experimental data, presented inFigure 14 and concerning the strain controlled drained triaxial tests, is illustrated. In Figure 21, forclarity, only one experim ental curve (Test a, Figure 14) and two numerical sim ulations (relative to T estsa and c of Figure 14) are shown. Th e two numerical sim ulations are not distinguishable, because thetwo lines are sup erimpos ed. This result confirms the theoretical observation introduced in Section 5.7. CONCLUSIONS

The time dependency of the loose san ds mechanical behaviour has been analysed. A series of triaxialload controlled tests has been performed; finite load incremental steps have been imposed.The experimental results clearly show that the strain response is delayed in time. This dependencycould be a priori associated both to a consolidation phenomenon or to a delayed microstructuralevolution. But the consolidation process and the m icrostructural rearrangement have been shown to beinterpretable as two uncoupled phenomena, because the timescales characterising the two differentphysical processes are very different.Moreover, by testing dry loose sand specimens, it has been demonstrated that the materialmechanical behaviour time dependency is not linked to the water presence w ithin the specimen: thetime dependency is exclusively caused by a m icrostructural evolution.The time dependency, characterising the exp erimental data, has been highlighted by applying finiteload incremental steps.The time delayed response is characterised by a convolutive evolution: if the material is slowlyloaded, the strain increments are slow and vice versa.

By having assumed a separation between the elastic instantaneous material response and the delayedplastic one, co rroborated by experimental results, an elastoviscoplastic model h as been conceived. Inthis model, by following the Perzyna approach, the plastic strains are not instantaneo us, but, to takeplace, they need time to elapse. Consequently, the state variables physically related to the micro-


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TIME DEPENDENT MECHANICAL BEHAVIOUR OF LOOSE SANDS 69

120

100

80-580

40

20

0

140

120

100

805D 60

40

20

01 2 3.5 3.5 4 4.50.5 1.5

ax id rtrah 1%1

+ --------I --Y----c------t-----+----+0 5 1 5 2 5 3 3 5 4 4 50 1 2

a n a l wmn l%I

Figure 20. Comparison between experimental results and numerical simulations. (a) T100a: stress-strain curves. @) T100d:stress-strain curves. (c) T100d: olumetric behaviour


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70 C. DI PRlSCO AN D S. IMPOSIMATO300 1

25 0

0 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20Bxial swain

5 1

1 - 1

----

numerical simulation smn ram 1

' numema1 lmdaucn' sv m rateI 0 078 mm/mm//0 Y W , . - T ww 1O S y0 1 2 3 4 5 6 7 6 9 10 1 1 12 13 1 4 15 >6 17 18 19 20

aaial stram 1%1Figure 21. Strain controlled drained triaxial compression tests on loose saturated sand specimen at different strain rates:comparison between experimental data and numerical simulations. (a ) Stress-strain relationship,(b) volumetric behaviour

structural fabric rearrangement, also evolve with time. The presented theoretical approach may beapplied to any elastoplastic constitutive model and allows to avoid the consistency rule.The explicit expression of the delayed plastic strain increment numerically simplifies its usage. Butthis simplification is coupled with a non-negligible dependenc e of the numerical solution on the timestep.Moreover, it is important to underline that the time dep endency of the sand mechan ical behaviour,analysed in this paper, is not due to particular chemical or physical micro phenomena, but concerns thetransient condition associated to the mechanical disturbances . When a load increment is applied, beforereaching the updated cond ition, a com plex phenomenon takes place.The elastoplastic theory photographs only the starting steady condition and the final one, bydisregarding the transient condition. On the contrary, the delayed plasticity theory is able to describealso the phenomenon during its time evolution.


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TIME DEPENDENT MECHANICAL BEHAVIOUR OF LOOSE SANDS 71According to us, this transient phase comprehension may be important in order to highlight thematerial unstable or dynamical mechanical behaviour.Finally, to obtain num erical simulations of the experimental data,an elastoplastic constitutive model,recently presented by di Prisco, has been modified, by adding only two new constitutive parameters, toreproduce the observed time dependency. The obtained numerical results show a good agreem ent bothquantitatively and quantitatively with experimental reality.The model capability to reproduce the experimental results confirms the validity of the theoreticalapproach; consequently the viscoplastic constitutive models, commonly implemented in the finiteelement codes to regularise the numerical solutions gain a clear physical meaning, not only withreference to cohesive materials but also to granular ones.

ACKNOWLEDGEMENTSThis research was conducted within the framework of Project 2, Localisation phenomena inGeomechanics, of the A .L.E.R.T. Geomaterials Programm e, funded by the E.U. (Human Capital andMobility). Financial support from Italian C.N.R. and M.U.R.S.T. is also gratefully acknowledged.We acknowledge Professor R. Nova and Ing. R. M atiotti for their helpful support and Mr E. Iscandrifor his support in perform ing laboratory tests.

APPENDIX ATo demonstrate (4), the viscous nucleus @,Cnmay be assumed, for simplicity, to be (f , where (.)denotes the M acCauley brackets which are understood as

( f ) =f when f > 0,(f) 0 when f < 0.

Moreover the yield functionf s assumed, as usual, to be a functrion of a state variables set ( au )andof the effect stress state (oh).If we assum e the stress increment to be instantaneous, after the time O+, only the state variables ( a i i )change and the effective stress state does not; consequently:

and, being the effective stress constant, H (the plastic hardening modulus) may be assumed to beconstant, too.By integrating (A2):f ( t ) =f(to+ )e-"Yz (A31

d& T(r, t) =f(to+)Y e-HY'ag/aab dt . (A41then, by means of (3), ( A l ) and (A3),

1If the plastic potential gradient (ag/ao;) may be assumed not to be dependent on the time then


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72 C . DI PRISCO A N D S. MPOSIMATOBut, since dcr, = da,(deyf) and dez corresponding to to' = 0 t follows that

I f f ( to-) is negligible the result is

f ( to - ) is negligible, for instance, when the last load increment is followed by a sufficiently long timeperiod, in this case the yield surface has reached the point corresponding to the current stress state.APPENDIX B

analogously to (A2):

and when 0 0 e"

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Time dependent mechancial behvior of loose sand di prisco imposimato 1996