DISCRETE MODELING OF STRAIN ACCUMULATION IN GRANULAR SOILS UNDER CYCLIC LOADING

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1 11th Worl Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th European Conference on Computational Flui Dynamics (ECFD VI) E. Oñate, J. Oliver an A. Huerta (Es) DISCRETE MODELIG OF STRAI ACCUMULATIO I GRAULAR SOILS UDER CYCLIC LOADIG goc-son guyen, Stijn François an Geert Degrane KU Leuven, Department of Civil Engineering, Structural Mechanics Kasteelpark Arenberg 40, 3001 Leuven, Belgium Key wors: Granular soils, cyclic loaing, strain accumulation, iscrete element metho Abstract. This paper presents a stuy of strain accumulation in granular soils uner vibration by using the iscrete element metho. A loose an a meium ense sample compose of a relatively large number of spheres are simulate. A series of stress controlle cyclic triaxial tests with ifferent excitation amplitues an frequencies is performe on these samples at ifferent static stress states. The stuy focuses on the influence of ifferent factors on strain accumulation such as the sample ensity, the cyclic excitation amplitue an frequency an the static stress state. In aition, the evolution of the internal structure of the granular samples is also investigate. 1 ITRODUCTIO Roa an railway traffic inuce vibrations cause the stress state in soils to vary cyclically with low amplitue compare to the stress state an relatively high frequency. The excitation frequency can be typically aroun Hz for roa traffic inuce vibrations [1] an aroun Hz for railway traffic inuce vibrations [2]. Uner the action of vibrations, granular soils uner the founation of builings accumulate strain, which might cause ifferential settlement, hence amage to builings. A profoun unerstaning of strain accumulation in granular soils is very important to avance the ability to preict this phenomenon. Strain accumulation in granular soils has been stuie in the laboratory by using cyclic tests performe at low frequency [3, 4]. At high frequency (typically higher than 10 Hz), the accuracy of cyclic tests is significantly reuce [5]. The numerical simulation with the iscrete element metho (DEM) pioneere by Cunall an Strack [6] can complement laboratory experiments. The DEM is suitable to simulate cyclic tests at high frequency. Moreover, this metho allows an investigation of the microscopic behavior of granular samples uring cyclic excitation as it is possible to access to information at the particle level. 1

2 This paper presents a stuy of strain accumulation in granular soils inuce by vibrations by using the DEM. umerical simulations are performe with the software PFC3D [7]. 3D granular samples compose of a relatively large number of spheres (about 10000) are consiere. This stuy aims at analyzing the influence of ifferent factors such as the sample ensity, the amplitue an frequency of the cyclic excitation an the static stress state on strain accumulation in granular soils. In aition, the local behavior of granular samples uring low amplitue cyclic excitation is investigate. This paper is organize as follows. Section 2 presents two numerical samples consiere in the current stuy. The behavior of these samples uring triaxial compression tests is briefly iscusse in section 3. Strain accumulation in these samples uring low amplitue cyclic triaxial tests is presente in section 4. 2 UMERICAL SAMPLES Two samples A an B with ifferent ensities are create, each of which is compose of spheres with mass ensity ρ = 2650 kg/m 3. The linear contact moel [7] with normal stiffness k n = /m, tangential stiffness k s = k n an friction coefficient µ = 0.6 is aopte in the current stuy. o viscous amping is ae at the contact points; therefore, only friction issipates energy in the samples Figure 1: Sample containe by a parallelepipe compose of 6 rigi walls. The particles of each sample are ranomly generate in a parallelepipe compose of 6 rigi walls (figure 1). The samples are then compacte by compression in the three irections until reaching a given target stress state. To obtain ifferent ensities, the friction coefficient µ for samples A an B is set to 0.6 an 0.3 uring the compaction phase, respectively. When about 90% of the target stress state is reache, µ is reset to its original value. After compaction, the porosity n is 0.43 for sample A an 0.41 for sample 2

3 B at a confinement stress σ 11 = σ 22 = σ 33 = σ o = 50 kpa an both samples are isotropic. The stress tensor σ an strain tensor ε of each sample are efine from the contact forces applie by the walls on the sample an the isplacement of the walls [8]. The sign convention use in this paper is that tensile stresses an strains are positive. For triaxial loaing, the mean stress p an the eviatoric stress q are efine as p = (σ σ 33 )/3 an q = σ 11 σ 33. The volumetric strain ε v an the eviatoric strain ε are efine as ε v = ε 11 +2ε 33 an ε = 2 ε 11 ε 33 /3. 3 TRIAXIAL COMPRESSIO TESTS Quasi-static triaxial compression tests are performe on samples A an B by approaching slowly the top an bottom walls an keeping the lateral stresses σ 22 an σ 33 equal to the confinement stress σ o. Figure 2 shows the stress ratio η = q/ p an the volumetric strain ε v versus the axial strain ε 11 for two tests performe on samples A an B at σ o = 50 kpa. The figure shows that sample A presents the behavior of a loose granular sample, while sample B presents the behavior of a meium ense one. These samples first contract an then ilate. The state at which this transition occurs is calle the characteristic state. For sample A the characteristic state occurs at large eformation (ε ), while for sample B this state occurs at small eformation (ε ). Lines L A an L B rawn in figure 2 epict the characteristic state for samples A an B, respectively. L B L A q/ p B2 B3 A2 A3 B1 A1 εv ε 11 Figure 2: Stress ratio η = q/ p an volumetric strain ε v versus axial strain ε 11 for two triaxial compression tests applie on samples A (ashe line) an B (soli line) at a confinement stress σ o = 50 kpa. 3

4 Points A1, A2 an A3 rawn on the curve for sample A an points B1, B2 an B3 rawn on the curve for sample B represent the stress states at which cyclic excitations are applie (these loaings are presente in section 4). The arrows inicate the volumetric behavior of each sample at the corresponing stress states. The volumetric behavior of a sample at a given stress state is quantifie by the strain increment ratio ε v / ε compute with a strain increment tensor ε starting from this stress state. Table 1 shows the stress ratio η an the strain increment ratio ε v / ε at these points. Sample A tens to contract less from point A1 to point A3; the same tenency is observe for sample B from point B1 to point B3. In particular, at point B3, sample B slightly ilates. Sample Point η ε v / ε A A A A B B B B Table 1: Stress ratio η an strain increment ratio ε v / ε at points A1, A2, A3, B1, B2 an B3 in figure 2. 4 LOW AMPLITUDE CYCLIC TRIAXIAL TESTS To perform a cyclic triaxial test on a sample, the sample is first consoliate by triaxial compression until the stress state reaches the target average stress state (σ 11 = σ 11 an σ 22 = σ 33 = σ o with σ 11 the target average axial stress). The axial stress σ 11 is then cycle between the lower an upper values σ 11 ±σ cyc 11 with σ cyc 11 the cyclic stress amplitue an the lateral stresses σ 22 an σ 33 are kept equal to the confinement stress σ o. The cyclic stress σ 11 is applie by moving cyclically the top an bottom walls inwar an outwar until σ 11 reaches the upper an lower values, respectively. By oing so, the amplitue σ cyc 11 can be controlle exactly; however, the frequency f can only be approximately controle within a given range by trial an error. A cyclic triaxial test has four parameters: the average stress ratio η = q/ p, the confinement stress σ o, the cyclic stress amplitue σ cyc 11 an the excitation frequency f. A low amplitue cyclic excitation correspons to a small value of the ratio ζ cyc = σ cyc 11 / p. Table 2 recapitulates the cyclic triaxial tests performe on samples A an B at ifferent static stress states an with ifferent excitation frequencies an amplitues. Figure 3 illustrates the time history of the cyclic stress σ 11 applie on sample B in test 4

5 Sample Test η σ o (kpa) σ cyc 11 (kpa) f (Hz) TA TA TA3 24 A TA TA TA6 30 TA TB TB2 25 B TB TB4 100 TB Table 2: Cyclic triaxial tests performe in the current stuy. TB2. Uner this cyclic excitation, sample B accumulates strain as shown clearly in figure 4. For clarity, this plot is split into three subplots with equal intervals of the axial strain ε 11. The strain accumulation is large for the first two cycles an slows own as the cyclic loaing continues. σ11 (kpa) t (s) Figure 3: Time history of the cyclic stress σ 11 applie in test TB2. In the following, the influence of the sample ensity, the cyclic excitation amplitue an frequency an the average stress ratio on strain accumulation is analyze. 5

6 σ11 (kpa) σ11 (kpa) σ11 (kpa) ε 11 Figure 4: Axial stress σ 11 versus axial strain ε 11 in test TB Influence of the sample ensity The influence of the sample ensity on the intensity an the irection of strain accumulation is analyze by consiering test TA3 performe on sample A an test TB2 performe on sample B. The frequencies of these tests are about 24 Hz. Figure 5 shows the accumulate volumetric strain v an eviatoric strain for samples A an B versus cycle number. Strain accumulates much more strongly in sample A than in sample B. Both samples accumulate more eviatoric than volumetric strain. In particular, sample A accumulates much more volumetric strain than sample B. This means that loose granular soils are more likely to be compacte uring low amplitue cyclic loaing than ense soils. As observe experimentally by many authors [3, 9, 10], the volumetric behavior of a granular sample uring cyclic excitation epens on its volumetric behavior at the average stress state. If the sample tens to contract at the average stress state then it contracts uring the cyclic excitation. The opposite is observe if the sample tens to ilate at the average stress state. In particular, if the cyclic excitation is applie at the characteristic state then the strain accumulates in the sample with no volume change. These experimental observations can be confirme by numerical simulation. Inee, at point A2 in figure 2, sample A contracts strongly; as a result it contracts strongly uring test TA3. On the other han, at point B2 which is near to line L B, sample B contracts weakly uring test TB2. 6

7 v (a) (b) Figure 5: Accumulate (a) volumetric strain v for samples A (soli line) an B (ashe line). an (b) eviatoric strain versus cycle number 4.2 Effect of the cyclic stress amplitue σ cyc 11 Tests TA2, TA3 an TA6 are performe on sample A at point A2 with ifferent cyclic stress amplitues σ cyc 11 = 3, 6 an 12 kpa. Figure 6 shows that σcyc 11 influences greatly the strain accumulation in sample A. Both accumulate volumetric an eviatoric strains v 11 strain accumulates an increase as σ cyc 11 increases. Inaition, at ahigher valueof σ cyc more rapily, particularly for about the first 100 cycles. The increase in v an with σ cyc 11 can be explaine by the fact that a higher excitation amplitue causes more sliing motion between particles to issipate energy. v (a) (b) Figure 6: Accumulate (a) volumetric strain v an (b) eviatoric strain of sample A versus cycle number for a cyclic stress amplitue σ cyc 11 = 3 kpa (thin soli line), 6 kpa (ashe line) an 12 kpa (bol soli line). The ensity of a granular sample at the microscopic scale is escribe by the coorination number efine as the average number of contacts per particle = 2 c p, (1) with c the number of contacts an p the number of particles. Kuhn [11] introuce the effective coorination number eff by removing all the floating particles (particles that 7

8 have no more than 3 contacts with their neighbors) from the sample when calculating the coorination number. Figure 7 shows the evolution of the effective coorination number eff of sample A uring cyclic loaing for ifferent values of the cyclic stress amplitute σ cyc 11. A marke increase of eff is observe for σ cyc 11 = 12 kpa, while eff remains almost constant for σ cyc 11 = 3 kpa. The epenence of eff on σ cyc 11 results from the fact that the sample ensifies more strongly at a higher value of σ cyc 11 (figure 6). eff Figure 7: Effective coorination number eff of sample A versus cycle number for a cyclic stress amplitue σ cyc 11 = 3 kpa (thin soli line), 6 kpa (ashe line) an 12 kpa (bol soli line). 4.3 Influence of the cyclic excitation frequency f Tests TA3, TA4 an TA5 are performe on sample A at ifferent excitation frequencies f = 24, 67 an 93 Hz an tests TB2, TB3 an TB4 are performe on sample B at f = 25, 68 an 100 Hz. The magnitue of the accumulate strain is efine as Frobenius normoftheaccumulate straintensor. Fortriaxialloaing, isiagonal; therefore, = F = ( 11 ) 2 +( 22 ) 2 +( 33 ) 2. (a) (b) Figure 8: Accumulate strain versus cycle number (a) for sample A at a cyclic excitation frequency 24 Hz (bol soli line), 67 Hz (ashe line) an 93 Hz (thin soli line) an (b) for sample B at 25 Hz (bol soli line), 68 Hz (ashe line) an 100 Hz (thin soli line). The magnitue of the accumulate strain is plotte versus the cycle number for samples A an B in figure 8. The cyclic excitation frequency f has little effect on strain 8

9 accumulation in these samples. For sample A, is about at 24 Hz after 1000 cycles, compare to a value of at 93 Hz. This result might be explaine by the fact that, for the consiere values of the excitation frequency, samples A an B are still in the quasi-static regime. 4.4 Effect of the average stress ratio η Tests TA1, TA4 an TA7 on sample A at points A1, A2 an A3 an tests TB1, TB3 an TB5 on sample B at points B1, B2 an B3 are consiere to analyze the effect of the average stress ratio η on strain accumulation. ote that η = 0.2 at points A1 an B1, η = 0.4 at points A2 an B2 an η = 0.6 at points A3 an B3 (figure 2). These tests are performe at a high frequency of approximately 65 Hz. The number of cycles is about 8000 for the tests performe on sample A an 2000 for the tests performe on sample B. v (a) (b) Figure 9: Accumulate (a) volumetric strain v an (b) eviatoric strain versus cycle number for sample A at an average stress ratio η = 0.2 (thin soli line), 0.4 (ashe line) an 0.6 (bol soli line). The accumulate strain in sample A epens strongly on the average stress ratio η, particularly for the accumulate eviatoric strain, as inicate in figure 9. Strain accumulates much more strongly in sample A for η = 0.6 than for η = 0.2 an 0.4, in particular for the first 10 cycles. After 6000 cycles, reaches a large value of 0.06 for η = 0.6, compare to a value of for η = 0.4 an 0.01 for η = 0.2. In aition, for η = 0.6 strain still accumulates substantially in sample A after 7000 cycles, while for η = 0.2 an 0.4, strain accumulation almost ceases after 1000 cycles. A similar effect of the average stress ratio η on strain accumulation is observe on sample B (figure 10). The accumulate volumetric strain v is negligible compare to the eviatoric strain. The accumulate strain is almost zero uring the cyclic excitation applie at η = 0.2 as the behavior of sample B is highly elastic at this average stress state (figure 2). The fabric of a granular sample is escribe by the following tensor: H ij = 1 c n k in k j, (2) c 9 k=1

10 v (a) (b) Figure 10: Accumulate (a) volumetric strain v an (b) eviatoric strain versus cycle number for sample B at an average stress ratio η = 0.2 (thin soli line), 0.4 (ashe line) an 0.6 (bol soli line). where n k i is the i-th component of the unitary normal vector at contact k [12]. For triaxial loaing, H 11, H 22 an H 33 are the three principal values an H 22 H 33. In this case, the anisotropy of a sample can be measure by H = H 11 H 33. The effect of the average stress ratio η on the evolution of the anisotropy of sample A is shown in figure 11. The average stress ratio η affects greatly the anisotropy inuce by the consoliation phase, but not the anisotropy inuce by low amplitue cyclic excitation. The anisotropy of the sample remains almost constant uring the applie cylic excitations whatever the value of η is. H Figure 11: Anisotropy measure H versus cycle number for sample A at an average stress ratio η = 0.2 (thin soli line), 0.4 (ashe line) an 0.6 (bol soli line). 5 COCLUSIOS A series of simulations with the DEM was carrie out to stuy strain accumulation in granular materials subjecte to low amplitue cyclic loaing. A loose an a meium ense sample compose of about spheres were consiere in these simulations. The stuy has shown that the loose sample accumulates much more strain than the ense one uring cyclic excitation. The strain accumulation increases with the cyclic stress amplitue an the average stress ratio; however, it is not affecte by the cyclic excitation frequency 10

11 up to 100 Hz. At the microscopic scale, the internal structure of the samples evolves slightly uring cyclic loaing. An increase in the coorination number observe for the loose sample is ue to its ensification uring cyclic excitation. However, the anisotropy of these samples inuce by low amplitue cyclic excitation is negligible compare to the anisotropy inuce by the consoliation phase. The DEM is able to reprouce, at least in a qualitative sense, the strain accumulation phenomenon observe in laboratory experiments. To avance the use of the DEM to stuy this topic, more realistic particle shapes shoul be accounte for in simulations. References [1] G. Lombaert an G. Degrane. The experimental valiation of a numerical moel for the preiction of the vibrations in the free fiel prouce by roa traffic. Journal of Soun an Vibration, 262: , [2] G. Lombaert, G. Degrane, J. Kogut, an S. François. The experimental valiation of a numerical moel for the preiction of railway inuce vibrations. Journal of Soun an Vibration, 297: , [3] T. Wichtmann, A. iemunis, an Th. Triantafylliis. Strain accumulation in san ue to cyclic loaing: raine triaxial tests. Soil Dynamics an Earthquake Engineering, 25: , [4] T. Wichtmann. Explicit accumulation moel for non-cohesive soils uner cyclic loaing. PhD thesis, Des Institutes Für Grunbau Un Boenmechanik Der Ruhr- Universität-Bochum, Bochum, Germany, [5] C. Karg. Moelling of strain accumulation ue to low level vibrations in granular soils. PhD thesis, Ghent University, Ghent, Belgium, [6] P.A. Cunall an O.D.L. Strack. A iscrete numerical moel for granular assemblies. Géotechnique, 29(1):47 65, [7] Itasca Consulting Group Inc. PFC3D: Theory an Backgroun, [8] C. O Sullivan. Particulate Discrete Element Moelling: A Geomechanics Perspective. Applie Geotechnics Volume 4. Spon Press, [9] M.P. Luong. Mechanical aspects an thermal effects of cohesionless soils uner cyclic an transient loaing. In IUTAM Conference on Deformation an Failure of Granular Materials, pages , Delft, [10] C.S. Chang an R.V. Whitman. Draine permanent eformation of san ue to cyclic loaing. Journal of Geotechnical Engineering, 114: ,

12 [11] M.R. Kuhn. Structure eformation in granular materials. Mechanics of Materials, 31: , [12] M. Satake. Fabric tensor in granular materials. In P. Vermeer an H. Luger, eitors, IUTAM Symposium on Deformation an Failure of Granular Materials, pages A.A. Balkema,