Strain accumulation in sand due to cyclic loading: drained triaxial tests

Transcription

1 Soil Dynamics & Earthuake Engineering (25, in print) Strain accumulation in sand due to cyclic loading: drained triaxial tests T. Wichtmann i), A. Niemunis, Th. Triantafyllidis Institute of Soil Mechanics and Foundation Engineering, Ruhr-Uniersity Bochum, Uniersitätsstrasse 5, 448 Bochum, Germany Abstract Our aim is the prediction of the accumulation of strain and/or stress under cyclic loading with many (thousands to millions) cycles and relatiely small amplitudes. A high-cycle constitutie model is used for this purpose. Its formulas are based on numerous cyclic tests. This paper describes drained tests with triaxial compression and uniaxial stress cycles. The influence of the strain amplitude, the aerage stress, the density, the cyclic preloading history and the grain size distribution on the direction and the intensity of strain accumulation is discussed. Key words: Cyclic tests; Triaxial compression; Uniaxial cyclic stress paths; Accumulation; Residual strain; Explicit model; Sand Introduction Cyclic loading leads to an accumulation of strain and/or excess pore water pressures in soils because the generated stress or strain loops are not perfectly closed. Such cumulatie phenomena are of importance in many practical cases. In the seismic areas a limited number of load cycles with large amplitudes (i.e. ε ampl > 3 ) during an earthuake may lead to an accumulation of high pore water pressures and conseuently to a dramatic loss of the bearing capacity of structures (liuefaction). Also offshore foundations may loose their bearing capacity in a similar way during a storm. Under drained conditions if the amplitude is small (i.e. ε ampl < 3 ) and the number of cycles is high the accumulation of settlements becomes the main concern (long-term sericeability, e.g. railways, watergates, tanks). Some structures are extremely sensitie to differential settlements, which must be kept within an extremely small range in order to ensure the operational reuirements. In this case an accurate prediction is reuired for seeral decades of intensie traffic. For the finite element (FE) prediction of residual settlements two different numerical strategies can be distinguished. The implicit (incremental or low-cycle) models need hundreds of load increments per cycle and the residual settlement appears as a by-product of clasi) Corresponding author. Tel.: ; fax: ; address: torsten.wichtmann@rub.de sically calculated stress - strain loops. The accumulation results from the fact that the cycles are not perfectly closed. The implicit approach is suitable for a relatiely low number of cycles (say N < 5) and general purpose constitutie models can be used (usually multisurface plasticity models, Mróz and Norris [], Dafalias [2]). The present research is deoted to another so-called explicit (high-cycle) model. In such a model only the first two cycles are calculated conentionally with strain increments (see Fig. ). The strain amplitude is determined from the second cycle. The first, so-called irregular cycle is not suitable for this purpose, because it may differ significantly from the following so-called regular ones. The accumulation due to the following cycles is determined from a direct (explicit) formula. The material model predicts the residual strain due to a package of e.g. N = 2 cycles by the piece without tracing the oscillating strain path during the indiidual cycles. The strain amplitude is assumed to be constant during this calculation. In order to update the strain amplitude after a possible densification or re-distribution of stress, the explicit calculation can be interrupted by a so-called control cycle (semi-explicit approach). For a high number (e.g. seeral thousands) of cycles explicit models turn out to be more useful and accurate because the accumulation of systematic errors in implicit models is of the order of the physical accumulation.

2 Soil Dynamics & Earthuake Engineering (25, in print) 2 implicit cycle (calculation of strain amplitude) explicit accumulation "cyclic creep" t, N for compression and extension corresponding both to Ȳ =. In all tests the aerage stress σ a was kept constant and the axial stress component σ was cyclically aried with an amplitude σ ampl (see Fig. 2). It is conenient to introduce the normalized size of the stress amplitude ζ = σ ampl /p a = ampl /p a. (7) irregular cycle regular cycles Fig. : Calculation procedure of explicit models The explicit constitutie model is based on experimental obserations. We hae performed numerous cyclic triaxial and cyclic multiaxial direct simple shear tests. The results of the tests with triaxial compression and uniaxial stress cycles are presented in this paper. The multiaxial tests are described in [3]. a aerage stress a Mc ( CSL M c ( p ) c ) ampl ampl = ampl ampl p = / 3 a = a / p a 2 Notation, preliminaries We use the effectie Cauchy stress σ (compression positie) and small strain components. The Roscoe ariables are p a M e ( c ) M e ( p ) p p = tr σ/3 = (σ + 2σ 3 )/3 () = 3/2 σ = σ σ 3 (2) and the work conjugated strain rates are ε = tr ε = ε + 2 ε 3 (3) ε = 2/3 ε = 2/3 ( ε ε 3 ) (4) The star designates the deiatoric part of. We denote the axial component with the index and the radial components with 2 and 3. Using the word rate we mean the deriatie with respect to the number of cycles N, that is = / N and not with respect to time. Stress obliuity is expressed with η = /p or using the yield function Y = I I 2 /I 3 of Matsuoka and Nakai [4] as Ȳ = Y 9 Y c 9 with Y = 27(3 + η) (3 + 2η)(3 η) (5) and with Y c = (9 sin 2 ϕ c )/( sin 2 ϕ c ). The critical friction angle is denoted by ϕ c and the one at peak as ϕ p. The inclinations of the critical state line (CSL, see Fig. 2) are M c = 6 sin ϕ c 3 sin ϕ c and M e = 6 sin ϕ c 3 + sin ϕ c (6) Fig. 2: State of stress in a cyclic triaxial test The strains under cyclic loading can be decomposed into a residual and a resilient portion denoted by the superscripts acc and ampl, respectiely. We abbreiate the norm of the former as ε acc = ε acc = (ε acc )2 + 2(ε acc 3 )2 (8) with the direction of strain accumulation (cyclic flow rule) ω = ε acc /ε acc. (9) ε acc and ε acc are the cumulatie olumetric and deiatoric strain components, respectiely. The strain amplitude generated by an uniaxial stress loop is described by ε ampl = ε ampl () or the olumetric (ε ampl ) and deiatoric (ε ampl ) components. Alternatiely also the shear strain amplitude is used: γ ampl = 3/2 (ε ) ampl = (ε ε 3 ) ampl () The tests were done at arious density indices I D = (e max e)/(e max e min ) = D r ϱ d,max /ϱ d (D r = relatie density, ϱ d = dry density).

3 Soil Dynamics & Earthuake Engineering (25, in print) 3 3 Need for a systematic study Most of the studies on the cyclic behaiour of sand concern a low number of cycles (e.g. N <, ) and it is unclear, if the obserations can be extended to the high-cycle behaiour. The few studies going beyond N =, are either not well documented (e.g. the strain amplitudes are not aailable) or their testing program is incomplete. Some of the published results are contradictory. There is a general consent that with increasing number of cycles N the residual strain ε acc increases without a clear limit while the rate of accumulation ε acc decreases. Different approximations of the experimental cures ε acc (N), howeer, were proposed. In cyclic triaxial tests Lentz and Baladi [5,6] obsered the increase of the residual ertical strain ε acc to be proportional to the logarithm of the number of cycles N. Similarly Suiker [7] found from cyclic triaxial tests on grael and sand that the accumulation rate ε acc decreases proportional to /N, howeer, with two proportionality coefficients c for N <, and c 2 for N >, with c > c 2. Helm et al. [8] obsered similar proportionality coefficients but with c 2 > c (cyclic triaxial tests on medium coarse and fine sand). An accumulation faster than the logarithm of the number of cycles was also obsered by Gotschol [9] in cyclic triaxial tests on grael. The effect of the strain amplitude was mainly studied in direct simple shear (DSS) tests. Youd [] reported a considerable increase of the rate of accumulation with increasing shear strain amplitude γ ampl. For amplitudes γ ampl < 4, howeer, he obsered no accumulation of residual strain. Similar experimental results were obtained by Siler and Seed []. Sawicki and Świdziński [2,3] performed DSS tests with different shear strain amplitudes. For a gien initial density they demonstrated a uniue cure ε acc = C ln(+ C 2 Ñ) (which they called common compaction cure ) wherein Ñ = N (γampl /2) 2. These test results may be affected by the disadantages of DSS tests, namely the inhomogeneous distribution of strain oer the specimen olume and the accumulation of lateral stresses, which are usually not measured (c.f. Budhu [4]). Furthermore, the applied number of cycles was rather small in all these tests. The common compaction cure was not confirmed by our triaxial tests with a large number of cycles, see Section 6. From cyclic triaxial tests performed by Marr and Christian [5] one could conclude a relationship ε acc (σ ampl ) b with the exponent b in the range.9 b Marr and Christian [5] proided data up to, cycles. In the aboe mentioned studies only deiatoric or pre-dominant deiatoric strain amplitudes were applied. Ko and Scott [6] performed complementary tests (on cubical specimens) using isotropic stress amplitudes. After a small initial compaction no further accumulation was obsered. Ko and Scott [6] suggested to neglect the effect of the olumetric amplitude on the accumulation rate. Experiments of Wichtmann et al. [3] hae demonstrated this false. The olumetric strain amplitude does affect the accumulation, although not as strongly as the deiatoric component. The influence of the space encompassed by the strain loop and its polarization (Pyke et al. [7], Ishihara and Yamazaki [8], Yamada and Ishihara [9], Wichtmann et al. [3]) has been reported elsewhere. Seeral experimental studies with cyclic simple shear tests (e.g. Siler and Seed [], Youd [], Sawicki and Świdziński [2]) ended with the conclusion that the aerage mean pressure p a does not influence the accumulation of residual strain. Our own tests, Section 7, show a significant influence of the aerage stress. Marr and Christian [5] obsered a slightly higher accumulation rate at a higher p a (keeping ζ = σ ampl /p a constant) and an increase of the accumulation rate with the aerage stress ratio η a. Howeer, the different strain amplitudes in the tests due to the stress-dependence of the stiffness were not taken into account (the strain amplitudes may differ significantly with p a or η a, see Figs. and 5). From numerous cyclic simple shear tests (e.g. Siler and Seed [], Youd []) and cyclic triaxial tests (e.g. Marr and Christian [5]) it is eident that the initial soil density is an important parameter for the prediction of accumulation. The influence of the freuency of cyclic loading on the accumulation rate was found to be negligible in some studies (Youd [] for.2 f.9 Hz, Shenton [2] for. f 3 Hz) and to be significant in others (Kempfert et al. [2]). The direction of accumulation (cyclic flow rule) was studied by Luong [22] in drained cyclic triaxial tests. Small packages with 2 load cycles were applied successiely to the same sand specimen. The aerage stress was aried from package to package and the accumulated strain was monitored. The so-called characteristic threshold (CT) line in the p - - plane was postulated such that σ a below the CT line leads to densification and σ a aboe the CT line to dilatie accumulation. Chang and Whitman [23] found that the direction of accumulation could be well expressed using the flow rule of the modified Cam Clay model ω = (M 2 c (η a ) 2 )/(2η a ). They proposed to identify Luong s CT line with the critical state line (CSL). The CT line was shown to be independent of p a, of the stress amplitude and the density. Howeer, no experimental data for the cyclic flow rule for N >, 5 was presented. We inestigated whether the slight increase of ω with N in the diagrams of Chang and Whitman

4 Soil Dynamics & Earthuake Engineering (25, in print) 4 [23] continues beyond N >, 5. 4 Testing deice, preparation of samples and tested materials back pressure Volume measuring unit cell pressure differential pressure transducer pneumatic cylinder load piston displ. transd. ball bearing inner rods outer rods plexiglas cylinder water in the cell soil specimen (d = cm, h = 2 cm) membrane ball bearing pressure transds. (pore + cell press.) drainage force transd. Fig. 3: Scheme of cyclic triaxial test deice In our study we used fie triaxial deices of the type presented in Fig. 3. The axial load was applied by a pneumatic loading system and measured at a load cell inside the pressure cell below the lower specimen end plate. The axial deformation of the specimen was monitored by a displacement transducer of a high accuracy which was fixed to the load piston. Due to the smearing of the end plates the bedding error was carefully subtracted. The inhomogeneity of the resilient and the cumulatie deformation in a triaxial sample was inestigated by Niemunis [24] using the PIV-techniue. These inhomogeneities remain een after a large number of cycles, both in the amplitude and in the accumulation rate. Local axial strain measurements with a short distance (< 5 cm) between the reference points can therefore lead to errors. The integral alue of the whole sample is thought to be more representatie (although there may be some disturbance at the top of the sample). For technical reasons it was easier to perform tests with saturated specimens than with dry ones. The results do not differ significantly. The olume changes of fully saturated specimens were measured by the sueezed out pore water using a differential pressure transducer. The lateral deformations of the dry specimens were determined with six local non-contact displacement transducers (LDT) distributed oer the specimen surface. Cell pressure and back pressure were monitored by means of pressure transducers. Specimens were prepared by pluiating dry sand out of a funnel through air into half-cylinder moulds. The funnel was being lifted so that the drop height was kept constant. Different initial densities were achieed by arying the outlet diameter of the funnel. If the specimen was supposed to be water-saturated it was first flushed with carbon dioxide and then saturated with de-aired water. Back pressures of 2 or 3 kpa were used. The uality of saturation was approed if B >.95 (B = parameter of Skempton). Starting from a small isotropic effectie stress, the cell pressure was isotropically increased to σ3 a and then the axial stress was raised to σ a. During the application of σ a the deformations were continuously measured. The cyclic loading with σ ampl was commenced after a one-hour consolidation period at σ a. In all tests (except those on the influence of the loading freuency, Section 9) the specimens were subject to 5 cycles at a freuency of Hz. The signals of all transducers were recorded oer the period of fie complete cycles after an idle (non-recording) phase of N cycles. The gaps N between the readings were inersely proportional to the accumulation rate (increase with N). Finer by weight [%] fine Sand medium coarse Grael fine d 5 =.52, U = 4.5 d 5 =.35, U =.9 d 5 =.45, U =.4 d 5 =.55, U = Diameter [mm] Fig. 4: Tested grain size distribution cures (d 5: mean grain diameter, U = d 6/d : uniformity index) This experimental study was performed with a uartz sand with subrounded grains. In all tests except those on the influence of the grain size distribution (see Section ) the siee cure No. in Fig. 4 with the minimum and maximum oid ratios e min =.577 and e max =.874 (German standard code DIN 826) and a critical friction angle ϕ c = 3.2 was used.

5 Soil Dynamics & Earthuake Engineering (25, in print) 5 5 The high-cycle accumulation model In this section the explicit accumulation model (see Niemunis et al. [25]) is briefly presented. The general stress-strain relation has the form σ = E : ( ε ε acc ) (2) wherein E denotes an elastic stiffness. The rate of strain accumulation ε acc is proposed to be ε acc = ε acc m (3) = f p f Y f e f π (f ampl f N A + f ampl }{{} f N B ) m }{{} ġ A ġ B with the direction expressed by the unit tensor m (for triaxial tests ω = 3/2 tr (m)/ m holds) and with the intensity ε acc = ε acc. While the cyclic flow rule m is a function of the aerage stress ratio η a = a /p a only, the intensity of strain accumulation ε acc depends on the amplitude, the shape and the polarization of the strain loop, the aerage stress σ a, the oid ratio e and the applied number of cycles N, i.e. the cyclic preloading history. In the proposed expression (3) the respectie functions describe the following contributions: f ampl : f p : f Y : f e : f N A, f N B: f π : amplitude, shape and polarization of the strain loop, summarized in the scalar ε ampl aerage mean stress p a aerage stress ratio Ȳ a aerage oid ratio e cyclic preloading = number of cycles N, if ε ampl = constant polarization changes These functions (except f π ) and the cyclic flow rule m were deried on the basis of the performed cyclic triaxial tests and are discussed in the following sections, see also Table. The number of cycles N is not a suitable state ariable for cyclic preloading since it contains no information about the amplitude of the preious cycles. Thus, a new state ariable g A = ġ A dn = f ampl f N A dn was introduced which considers both N and ε ampl in the past. In this paper only the special case of a onedimensional cyclic strain path is studied. In order to capture more complex strain loops the accumulation model uses a tensorial amplitude definition (see Niemunis [24] or Niemunis et al. [25]). For a one-dimensional strain loop, howeer, this new amplitude definition is identical to the classical one. The function f π was determined from cyclic multiaxial direct simple shear (CMDSS) tests and is presented elsewhere (Wichtmann et al. [3]). For the case that the polarization of the strain loop does not change during cyclic loading (e.g. in the performed cyclic triaxial tests) f π = holds. Function ( ) 2 f ampl = ε ampl /ε ampl ref ε ampl ref ( f N A = C g A ) NC N2 exp C N C N f ampl Mat. constants f B N = C NC N3 C N2.55 ( )] p a f p = exp [ C p p ref C N C p.43 p ref kpa f Y = exp ( C Y Ȳ a) C Y 2. f e = (C e e) 2 + e + e ref (C e e ref ) 2 C e.54 f π = if polarization = constant, else see Wichtmann et al. [3] e ref.874 Table : Summary of the partial functions f i and a list of the material constants C i for the tested sand 6 Influence of the strain amplitude The effect of the strain amplitude ε ampl on the accumulation rate was studied in a series of tests with stress amplitudes arying between.6 ζ.47 (2 kpa ampl 94 kpa). The aerage stress was p a = 2 kpa, η a =.75 in all tests and the initial relatie density lay within.55 I D.64, wherein initial means after the irregular cycle. Specimens were tested in the dry condition. Keeping ζ(n) constant a decrease of the strain amplitude ε ampl with N was noticed (Fig. 5), especially for larger amplitudes and during the first cycles (so-called conditioning phase). Fig. 6 presents the resulting accumulation cures ε acc (N) in a semi-logarithmical diagram. Only the accumulation during the regular cycles is of interest and the diagrams in this paper show only this portion. The accumulated strain was found to increase almost proportionally with the logarithm of the number of cycles up to N, and then oer-proportionally. The approximation of these cures by the functions f N A and f N B is discussed in Section. From Fig. 6 it is obious that higher amplitudes cause larger accumulation rates ε acc. In Fig. 7 the accumulated strain after different num-

6 Soil Dynamics & Earthuake Engineering (25, in print) 6 Strain amplitude ampl [ -4 ] p a = 2 kpa, η a =.75, I D = ampl [kpa] = exponents that hae been determined from the tests of Marr and Christian [5] in terms of stress amplitudes. acc / fe [%] N =, N = 5, N =, N =, N = N = 2 p a = 2 kpa, η a =.75, I D = Number of cycles N [-] Fig. 5: Strain amplitude ε ampl ersus the number of cycles N acc [%] Accumulated strain ampl [kpa] = p a = 2 kpa, η a =.75, I D = Number of cycles N [-] Fig. 6: Accumulation cures ε acc (N) in tests with different stress amplitudes ampl bers of cycles is plotted ersus the suare of the strain amplitude. Since ε ampl aries slightly with N (Fig. 5) the mean alue ε ampl = ( N εampl dn)/n is used in Fig. 7. The accumulated strain ε acc is normalized with the oid ratio function f e (Section 8) in order to remoe the effect of the slightly arying initial densities and the different compaction rates in the nine tests. The bar oer f e means that f e is calculated with a mean alue of the oid ratio ē = ( N e dn)/n. The cures in Fig. 7 reeal that the accumulation rate ε acc increases proportionally to (ε ampl ) 2, independently of N. The following amplitude function ( ) 2 ε ampl (4) f ampl = ε ampl ref is therefore proposed wherein the reference amplitude is ε ampl ref = 4. Function (4) is in agreement with the ( ampl ) 2 [ -7 ] Fig. 7: Accumulated strain ε acc / f e as a function of the suare of the strain amplitude ( ε ampl ) 2 acc [%] Accumulated strain ampl [kpa] = p a = 2 kpa, η a =.75, I D = ~ N = N ( ampl ) 2 Fig. 8: Accumulated strain ε acc in dependence on Ñ = N (γ ampl ) 2 as proposed by Sawicki and Świdziński [2,3] In Fig. 8 the accumulated strain is shown as a function of Ñ = N (γampl ) 2 and according to the common compaction cure proposed by Sawicki and Świdziński acc [2,3] the cures ε (Ñ) should coincide. Eidently, they do not. Fig. 9 presents the strain ratio ω as a function of the strain amplitude ε ampl. At higher alues of ε ampl the strain ratio is almost independent of ε ampl. At smaller strain amplitudes the accumulated olumetric and deiatoric strains were ery small and therefore the alues of ω could be inaccurate. Thus the direction of accumulation is thought to be independent of the strain amplitude. This is in accordance with the experimental results of Chang and Whitman [23] for lower cycle numbers. Plots of ω oer N demonstrate an increase of

7 Soil Dynamics & Earthuake Engineering (25, in print) 7 the olumetric component of the direction of accumulation with the number of cycles. acc [-] acc / Strain ratio ω = Accumulation up to: N = 2 N =, N = N =, N =, p a = 2 kpa, η a =.75, I D = Strain amplitude ampl [ -4 ] Fig. 9: Strain ratio ω as a function of the strain amplitude ε ampl 7 Influence of the aerage stress The aerage stress was aried between 5 kpa p a 3 kpa and.25 η a = a /p a.375 keeping ζ =.3 constant. Specimens were prepared with the initial densities.57 I D.69 and tested in the saturated condition. The obtained results contradict the assumption made by seeral researchers (Siler and Seed [], Youd [], Sawicki and Świdziński [2]), that the aerage stress does not influence the accumulation process. a) b) [kpa] Peak strength CSL p [kpa] 2 3 η =.75 [kpa] 2 p [kpa] Fig. : Stress paths in tests on the influence of the aerage stress, a) ariation of p a, b) ariation of η a The cyclic stress paths of a series of six tests with an identical aerage stress ratio (η a =.75) but different aerage mean pressures are presented in Fig. a. Figs. - 4 contain the results of these tests. The strain amplitudes slightly increased with increasing aerage mean pressure (Fig. ) due to the well known dependence of stiffness G p n (here: n =.75), i.e. slightly underproportional to p. Fig. 2 presents the accumulated strain as a function of the aerage mean pressure p a for different numbers Strain amplitudes [ -4 ] η a =.75, =.3, I D = ampl ampl 2 3 Aerage mean pressure p a [kpa] ampl ampl Fig. : Strain amplitudes in tests with different aerage mean pressures p a but ζ = const (mean alues during 5 cycles) acc / (fampl fe) [%] C p = η a =.75, =.3, I D = N =, N = 5, N =, N =, N = N = Aerage mean pressure p a [kpa] Fig. 2: Accumulated strain ε acc /( f ampl fe) in dependence on the aerage mean pressure p a of cycles. For comparison the residual strain has been purified from side effects, i.e. diided by the amplitude function f ampl and by the oid ratio function f e. It was interesting to obsere that the accumulated strain significantly increases with decreasing (!) aerage mean pressure. This pressure dependence is stronger for large N. The function f p is proposed in the simple form ( )] p a f p = exp [ C p. (5) p ref The atmospheric pressure p ref = p atm = kpa is used as a reference for which f p = holds. Euation (5) has been confirmed by the data in Fig. 2 for different numbers of cycles. The corresponding cures are shown as solid lines in Fig. 2 and the parameter C p is gien at each cure. C p turns out to increase with N. Howeer, in order to keep the number of material constants manageable, a constant mean alue of C p =.43

8 Soil Dynamics & Earthuake Engineering (25, in print) 8 is proposed. The diagram in Fig. 3 is similar to that in Fig. 2 with the only difference, that the dependencies on Ȳ a (see remarks later in this Section) and N (see Section ) hae been remoed. The cure resulting from euation (5) with C p =.43 in Fig. 3 demonstrates, that the loss of accuracy due to the assumption C p = const is acceptable. acc / (fampl fe fy fn) [-] η a =.75, =.3, I D = f p with C p =.43, E. (5) N =, N = 5, N =, N =, N = N = Aerage mean pressure p a [kpa] Fig. 3: Accumulated strain ε acc /( f ampl fe f Y f N ) in dependence on the aerage mean pressure p a The direction of accumulation was found to be relatiely insensitie to changes in the aerage mean pressure (Fig. 4). A slight increase of the strain ratio ω with a decreasing aerage mean pressure and with N could be obsered. The increase with N was obsered for dry (Fig. 9) and for fully saturated specimens (Fig. 4). Strain amplitudes [ -4 ] p a ampl ampl = 2 kpa, =.3, I D = ampl ampl Aerage stress ratio a = a /p a [-] Fig. 5: Strain amplitudes in tests with different aerage stress ratios η a (mean alues during 5 cycles) acc / (fampl fe) [%] N =, N = 5, N =, N =, N = N = 2 p a = 2 kpa, =.3, I D = Aerage stress ratio Y a [-] C Y = acc [-] acc / Strain ratio ω = η a =.75, =.3, I D = Accumulation up to: N = 2 N =, N = N =, N =, Fig. 6: Accumulated strain ε acc /( f ampl fe) in dependence on the aerage stress ratio Ȳ a The strain amplitudes slightly decrease with increasing η a as shown in Fig. 5. The accumulation increases with Ȳ a, Fig. 6. In the test with Ȳ a =.243 (η =.375) the stress cycles touched the failure line M c (ϕ p ) and the rate of accumulation exploded reaching nearly 2 % residual strain after 5 cycles. If the stress cycles do not surpass this limit the function f Y can be uite well approximated by Aerage mean pressure p a [kpa] Fig. 4: Strain ratio ω as a function of the aerage mean pressure p a The results of a series of eleen tests with.375 η a.375 (.88 Ȳ a.243) and p a = 2 kpa = const and ζ = const (see the scheme of the stress paths in Fig. b) are presented in Figs f Y = exp ( C Y Ȳ a) (6) The material constant C Y = 2. is approximately independent of the number of cycles (Fig. 6). For isotropic stresses (Ȳ a = ) f Y = holds. Fig. 7 shows the strain ratio ω plotted ersus η a. A decrease of ω with η a is eident. The critical stress ratio M c (ϕ c ) =.25 was determined from static tests. It coincides with a purely deiatoric accumulation. For η a < M c (ϕ c ) densification and aboe the critical state

9 Soil Dynamics & Earthuake Engineering (25, in print) 9 acc [-] acc / Strain ratio ω = densification dilatancy p a = 2 kpa, =.3, I D = Accumulation up to: N =, N =, N =, N = N = 2 hypoplasticity mod. Cam Clay a = M c ( c) Aerage stress ratio a = a /p a [-] Fig. 7: Strain ratio ω as a function of aerage stress ratio η a line a dilatie behaiour was obsered. This is in accordance with the experimental results of Luong [22] and Chang and Whitman [23]. The dependence ω(η a ) conforms with the flow rule of the modified Cam Clay model, as already obsered by Chang and Whitman [23], or with the hypoplastic theory, Niemunis [24], Fig. 7. Both flow rules (for monotonous loading) describe well the direction of accumulation obsered in the cyclic triaxial tests, especially for the higher numbers of cycles, e.g. N = 5. Next, Fig. 8, aerage mean pressures p a = kpa and p a = 3 kpa and different stress ratios η a > (ζ =.3 = const) were tested in order to study if a more complex description than f p f Y is needed. Fig. 8 confirms, that the accumulation increases with decreasing aerage mean pressure p a and increasing deiatoric stress ratio Ȳ a. For p a =, 2 and 3 kpa the same approximation (6) could be used with C Y = 2.. Analogously to Fig. 2, the parameter C p of E. (5) increases with N for all tested aerage stress ratios η a and took alues.25 C p (N = 5 ).66. Therefore, strictly speaking, C p is not a constant. It has local extremes for η a =.75 (maximum) and η a =.25 (minimum). For simplicity we disregard the ariability of C p with N and η a in the accumulation model. Fig. 9 presents cures of identical residual strains after N = 5 cycles with ε ampl = 3 4 and I D =.6. In a wide p--range the cures ε acc = const are almost parallel to the maximum shear strength inclination η = M c (ϕ p ). The direction of accumulation at different aerage stresses σ a is shown in Fig. 2. It is depicted as a unit ector starting at (p a, a ) and haing the inclination ε acc /ε acc towards the p-axis. acc / fampl [%] =.3, I D = , N =, 2 p a [kpa] Y a [-] Fig. 8: Accumulated strain ε acc after, load cycles for different aerage stresses σ a = σ - σ 3 [kpa] ε ampl = 3-4, I D =.6, N =, 5 % 2 % %.7 % p = (σ + 2σ 3 ) / 3 [kpa] M c ( p ) M c ( c ).5 % acc =.4 % Fig. 9: Cures of identical accumulation in the p--plane 8 Influence of the density In another series of tests the initial oid ratio was aried between.58 e.8 (.24 I D.99). The aerage stress (p a = 2 kpa, η a =.75) and the amplitude ratio (ζ =.3) were kept constant this time. The specimens were water-saturated. The condition ζ = ampl /p a = const implies slightly higher strain amplitudes for lower initial densities (Fig. 2), because the stiffness decreases with the oid ratio e. As it could be expected, the strain accumulation is larger in loose soils (Fig. 22). The following

10 Soil Dynamics & Earthuake Engineering (25, in print) ε acc / f ampl [%] N =, N = 5, N =, N =, N = N = 2 p a = 2 kpa, η a =.75, =.3 C e =.54 C e =.54 C e =.53 C e =.5 C e =.48 C e = oid ratio e [-] relatie density I D [-] Fig. 22: Accumulated strain ε acc / f ampl in dependence on the oid ratio e or the relatie density index I D, respectiely a [kpa] Accumulation up to: N =, N =, N =, N = N = 2 =.3, I D = M c ( η =.33 p ) η =.375 η =.25 = M c ( c ) η =. η =.75 η =.5 Strain amplitudes [ -4 ] ampl ampl ampl ampl p a = 2 kpa, η a =.75, = Initial oid ratio e [-] η =.25 Fig. 2: Strain amplitudes in tests with different initial oid ratios e (mean alues during 5 cycles) 2 p a [kpa] 3 4 Fig. 2: Direction of accumulation in the p--plane for different aerage stresses function was found to describe this dependence: f e = (C e e) 2 + e + e ref (C e e ref ) 2. (7) The material constant C e =.54 (= oid ratio for which ε acc = holds) is approximately independent of the number of cycles (Fig. 22). The reference oid ratio corresponding to f e = is chosen to be e ref = e max =.874. In Fig. 23 cures of identical accumulation ε acc = const are plotted in the e-ln(p)-diagram for N = 5, Void ratio e [-] ε acc = 2 %.5 % %.7 %.5 %.3 % Mean pressure p [kpa] ε ampl = 3-4, η a =.75, N =,.2 %. %.5 % Fig. 23: Cures of identical accumulation in the e-ln(p)- diagram

11 acc / (fampl fe) [%] Soil Dynamics & Earthuake Engineering (25, in print) ε ampl = 3 4 and I D =.6. From Fig. 23 it is obious that the rising inclination of the cures with ε acc = const is opposite to the inclination of the critical state line. Therefore, it does not seem possible to describe the accumulation rate by the distance e e c of the oid ratio from the CSL. The direction of accumulation was found to be independent of the oid ratio for the range of densities studied (Fig. 24). Howeer, for dense specimens (I D >.9) the scatter in the strain ratios ω was considerable (Fig. 24), probably because ω was calculated as a uotient of ery small alues (the accumulation was extremely slow). Strain ratio ω = acc / acc [-] p a = 2 kpa, η a =.75, =.3 Accumulation up to: N = 2 N =, N = N =, N =, Void ratio e [-] Fig. 24: Strain ratio ω as a function of the oid ratio e Strain amplitudes [ -4 ] ampl ampl ampl ampl p a = 2 kpa, η a =.75, I D = Loading freuency f [Hz] Fig. 25: Strain amplitudes as a function of the loading freuency f p a = 2 kpa, η a =.75, I D = N = 4 N = 3 N = N = 2 9 Influence of the loading freuency Loading freuency f [Hz] In order to study the effect of the loading freuency six tests with p a = 2 kpa, η a =.75, ζ =.3,.5 I D.6 and arying loading freuencies.5 Hz f 2 Hz were carried out. The loading freuency has almost no effect on the strain amplitudes, Fig. 25. The residual strains presented in Fig. 26 exhibit no dependence on the loading freuency. The direction of accumulation is also independent of f (Fig. 27). These obserations coincide with the work of Youd [] and Shenton [2] but contradict the results of Kempfert et al. [2]. Influence of the number of cycles The accumulation cures ε acc (N) normalized by f ampl, f p, f Y, f e and f π with f π = are shown in Fig. 28. The cures lie within a band, which can be approximated by the function f N : f N = C N [ln ( + C N2 N) + C N3 N] (8) Fig. 26: Residual strain ε acc /( f ampl fe) for different loading freuencies f acc [-] acc / Strain ratio ω = p a = 2 kpa, η a =.75, I D = Accumulation up to: N = 2 N =, N = N =, N =, Loading freuency f [Hz] Fig. 27: Strain ratio ω ersus loading freuency f

12 Soil Dynamics & Earthuake Engineering (25, in print) 2 f N = C NC N2 + C N2 N }{{} f N A + C N C N3 }{{} f B N (9) with the material constants C N = 3.4 4, C N2 =.55 and C N3 = The function f N consists of a logarithmic and a linear part. The logarithmic part is pre-dominant for N <, (see also Sawicki & Świdziński [2,3]), howeer, the linear part is necessary for the description of the accumulation at higher cycle numbers which is faster than logarithmic. The function f N could be also fitted to the results of tests up to N max = 2 6. ) [%] acc / (f ampl f p f Y f e f tests for f ampl tests for f p tests for f Y tests for f e f N, E. (8) logarithmic accumulation oer-logarithmic accumulation Number of cycles N [-] Fig. 28: Normalized accumulation cures As already mentioned the number of cycles N alone is not a suitable state ariable for the description of cyclic preloading. The amplitude of the preious cycles is also of importance. To simplify the matter the accumulation rate was splitted into a rate of structural accumulation ( f N A ) and a basic rate ( f N B). g A = ġ A dn = f ampl f N A dn is used as a state ariable for cyclic preloading depending on ε ampl and N. Replacing N in E. (9) by g A one obtains ( f N A g A ) = C N C N2 exp (2) C N f ampl The seere problem of the determination of cyclic preloading in situ is discussed by Triantafyllidis and Niemunis [26], Wichtmann and Triantafyllidis [27,28] and Triantafyllidis et al. [29]. Influence of the grain size distribution The presented tests on the uniformly graded sand No. (Fig. 4) are being supplemented by tests on the gradations 2 and 3 (Fig. 4), which hae similar uniformity indices.4 U = d 6 /d.9 but different mean grain diameters.35 mm d 5.45 mm. Also tests on the well-graded sand No. 4 (U = 4.5, Fig. 4) are being conducted, which has a similar d 5 compared to sand No.. Figure 29 presents the accumulation cures for an identical aerage stress, identical initial densities and similar strain amplitudes. The grain size distribution cure strongly affects the accumulation rate. An increase of ε acc with a decreasing mean diameter d 5 at U = const was found. At d 5 = const the intensity of accumulation significantly increases with U. Up till now the presented accumulation model can describe the behaiour of all sands by simply modifying the material constants. The tests on different gradations will be carried on in future. We expect that based on those tests we will be able to simplify the model (some constants might become redundant) or at least to correlate most constants to the granular properties (grain size distribution, grain shape, content of fines). acc [%] Accumulated strain Gradation: No. 4: d 5 =.52 mm, U = 4.5 No. 2: d 5 =.35 mm, U =.9 No. : d 5 =.55 mm, U =.8 No. 3: d 5 =.45 mm, U =.4 p a = 2 kpa, a =.75, I D = , ampl = Number of cycles N [-] Fig. 29: Accumulation cures ε acc (N) for different gradations 2 Summary, conclusions and outlook In order to deelop a high-cycle model for the accumulation of strain and/or stress in sand under cyclic loading numerous cyclic triaxial tests and cyclic multiaxial direct simple shear (CMDSS) tests were performed. This paper describes the model in brief and concentrates on the experimental results from the tests with triaxial compression and uniaxial stress cycles. The main conclusions from these tests are: The direction of accumulation (cyclic flow rule) does exclusiely depend on the aerage stress ratio η a = a /p a. A slight increase of the olumetric component of the direction of accumulation with the number of cycles N was obsered. The direction of accumulation is in agreement with the flow rule of material models for monotonous

13 Soil Dynamics & Earthuake Engineering (25, in print) 3 loading (modified Cam clay model, hypoplastic model). The accumulation rate is proportional to the suare of the strain amplitude, ε acc (ε ampl ) 2, at least within the range of the tested amplitudes.5 4 ε ampl 5 4. Our triaxial tests contradict the so-called common compaction cure (Sawicki and Świdziński [2,3]). The accumulation rate ε acc increases with decreasing aerage mean pressure p a and with increasing aerage stress ratio η a. ε acc grows with increasing oid ratio. ε acc does not depend on the loading freuency (.5 Hz f 2 Hz). For N > 4 the accumulated strain ε acc increases faster than the logarithm of the number of cycles N. The intensity of accumulation ε acc is strongly affected by cyclic preloading. The accumulation rate depends significantly on the grain size distribution cure. An increase of ε acc with a decreasing mean grain diameter d 5 was found. A well-graded soil is compacted much faster than a poorly-graded one. At present we are extending the model for triaxial extension. Furthermore, the influence of the polarization and the shape of the strain loop on the accumulation rate is being tested. Results of preliminary cyclic triaxial tests with arying stress loops in the p - - plane (uniaxial paths with different inclinations as well as elliptic ones) and some CMDSS test data can be found in [3]. Also the boundaries of the alidity of the explicit formulas hae to be checked and extended in further experiments (e.g. it was found that the relationship ε acc (ε ampl ) 2 oerestimates the accumulation for ε ampl > 3 ). A correlation of the material constants of the accumulation model with the grain characteristics (grain size distribution, grain shape, content of fines) will be tried out. Acknowledgements The authors are grateful to DFG (German Research Council) for the financial support. This study is a part of the project A8 Influence of the fabric change in soil on the lifetime of structures of SFB 398 Lifetime oriented design concepts. References [] Mróz Z, Norris VA. Elastoplastic and Viscoplastic Constitutie Models for Soil with Application to Cyclic Loading. Soil Mechanics - Transient and Cyclic Loads. Constitutie Relations and Numerical Treatment, Pande GN, Zienkiewicz OC (eds.), John Wiley and Sons, 982: [2] Dafalias, YF. Bounding surface elastoplasticityiscoplasticity for particulate cohesie media. Deformation and Failure of Granular Materials, Proceedings IUTAM Conf., Delft, 982:97-7. [3] Wichtmann T, Niemunis A, Triantafyllidis T. The effect of olumetric and out-of-phase cyclic loading on strain accumulation. Cyclic Behaiour of Soils and Liuefaction Phenomena, Proc. of CBS4, Bochum, March/April 24, Balkema, 24: [4] Matsuoka H, Nakai T. A new failure for soils in threedimensional stresses. Deformation and Failure of Granular Materials, Proc. IUTAM Symp. Delft, 982: [5] Lentz RW, Baladi GY. Simplified procedure to characterize permanent strain in sand subjected to cyclic loading. International Symposium on soils under cyclic and transient loading, 98: [6] Lentz RW, Baladi GY. Constitutie euation for permanent strain of sand subjected to cyclic loading. Transportation Research Record, 98;8:5-54. [7] Suiker ASJ. Static and cyclic loading experiments on noncohesie granular materials. Report No. -99-DUT-, TU Delft, 999. [8] Helm J, Laue J, Triantafyllidis T. Untersuchungen an der RUB zur Verformungsentwicklung on Böden unter zyklischen Belastungen. Beiträge zum Workshop: Boden unter fast zyklischer Belastung: Erfahrungen und Forschungsergebnisse, Institute of Soil Mechanics and Foundation Engineering, Ruhr- Uniersity Bochum, Issue No. 32, 2:2-22. [9] Gotschol A. Veränderlich elastisches und plastisches Verhalten nichtbindiger Böden und Schotter unter zyklischdynamischer Beanspruchung. PhD thesis, Uniersity Kassel, 2. [] Youd TL. Compaction of sands by repeated shear straining. Journal of the Soil Mechanics and Foundations Diision, ASCE, 972;98(SM7): [] Siler ML, Seed HB. Volume changes in sands during cyclic loading. Journal of the Soil Mechanics and Foundations Diision, ASCE, 97;97(SM9):7-82. [2] Sawicki A, Świdziński W. Compaction cure as one of basic characteristics of granular soils. Proc. of 4th Colloue Franco-Polonais de Mechaniue des Sols Appliuee, Grenoble, 987;:3-5. [3] Sawicki A, Świdziński W. Mechanics of a sandy subsoil subjected to cyclic loadings. International Journal For Numerical And Analytical Methods in Geomechanics, 989;3:5-29. [4] Budhu M. Nonuniformities imposed by simple shear apparatus. Canadian Geotechnical Journal, 984;2: [5] Marr WA, Christian JT. Permanent displacements due to cyclic wae loading. Journal of the Geotechnical Engineering Diision, ASCE, 98;7(GT8): [6] Ko HY, Scott, RF. Deformation of sand in hydrostatic compression. Journal of the Soil Mechanics and Foundations Diision, ASCE, 967;93(SM3):37-56.

14 Soil Dynamics & Earthuake Engineering (25, in print) 4 [7] Pyke R, Seed HB, Chan CK. Settlement of sands under multidirectional shaking. Journal of the Geotechnical Engineering Diision, 975;(GT4): [8] Ishihara K, Yamazaki F. Cyclic simple shear tests on saturated sand in multi-directional loading. Soils and Foundations, 98;2(): [9] Yamada Y, Ishihara K. Yielding of loose sand in three-dimensional stress conditions. Soils and Foundations, 982;22(3):5-3. [2] Shenton MJ. Deformation of railway ballast under repeated loading conditions. Railroad track mechanics and technology, Pergamon Press, 978: [2] Kempfert HG, Gotschol A, Stöcker T. Kombiniert zyklische und dynamische Elementersuche zur Beschreibung des Kurz- und Langzeiterhaltens on Schotter und granularen Böden. Beiträge zum Workshop: Boden unter fast zyklischer Belastung: Erfahrungen und Forschungsergebnisse, Institute of Soil Mechanics and Foundation Engineering, Ruhr-Uniersity Bochum, Issue No. 32, 2: [22] Luong MP. Mechanical aspects and thermal effects of cohesionless soils under cyclic and transient loading. Proc. IUTAM Conf. on Deformation and Failure of Granular Materials, Delft, 982: [23] Chang CS, Whitman RV. Drained permanent deformation of sand due to cyclic loading. Journal of Geotechnical Engineering, ASCE, 988;4():64-8. [24] Niemunis A. Extended hypoplastic models for soils. Habilitation. Institute of Soil Mechanics and Foundation Engineering, Ruhr-Uniersity Bochum, Issue No. 34, 23. [25] Niemunis A, Wichtmann T, Triantafyllidis T. Explicit accumulation model for cyclic loading. Cyclic Behaiour of Soils and Liuefaction Phenomena, Proc. of CBS4, Bochum, March/April 24, Balkema, 24: [26] Triantafyllidis T, Niemunis A. Offene Fragen zur Modellierung des zyklischen Verhaltens on nicht-bindigen Böden. Beiträge zum Workshop: Boden unter fast zyklischer Belastung: Erfahrungen und Forschungsergebnisse, Institute of Soil Mechanics and Foundation Engineering, Ruhr-Uniersity Bochum, Issue No. 32, 2:9-34. [27] Wichtmann T, Triantafyllidis T. Influence of a cyclic and dynamic loading history on dynamic properties of dry sand, part I: cyclic and dynamic torsional prestraining. Soil Dynamics and Earthuake Engineering, 24;24: [28] Wichtmann T, Triantafyllidis T. Influence of a cyclic and dynamic loading history on dynamic properties of dry sand, part II: cyclic axial preloading. Soil Dynamics and Earthuake Engineering, accepted for publication. [29] Triantafyllidis T, Wichtmann T, Niemunis A. On the determination of cyclic strain history. Cyclic Behaiour of Soils and Liuefaction Phenomena, Proc. of CBS4, Bochum, March/April 24, Balkema, 24:32-32.