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

Soil Dynamics & Earthuake Engineering (Vol. 7, No., pp. -8, 7) Strain accumulation in sand due to cyclic loading: drained cyclic tests with triaxial extension T. Wichtmann i), A. Niemunis, Th. Triantafyllidis Institute of Soil Mechanics and Foundation Engineering, Ruhr-Uniersity Bochum, Uniersitätsstraße, 8 Bochum, Germany Abstract This paper presents results of numerous drained cyclic tests with triaxial extension. The influence of the strain amplitude, the aerage stress and the number of load cycles on the accumulation rate was studied. A simple cyclic flow rule was obsered. Most findings confirm a preious study with cyclic triaxial compression tests. The test results sere as the basis of an explicit accumulation model. Key words: Cyclic triaxial tests; triaxial extension; strain accumulation; residual strains; sand Introduction In order to predict the residual settlements in noncohesie soils under high-cyclic loading (= number of cycles N > ) an explicit accumulation model was deeloped []. The model is based on numerous cyclic triaxial tests [] and multiaxial Direct Simple Shear (DSS) tests []. The cyclic triaxial tests presented in [] coered uniaxial in-phase [] stress cycles that superposed triaxial compression. At a constant lateral stress σ the axial stress σ was oscillating with an amplitude σ ampl = ampl about the aerage alue σ a. The aerage deiatoric stress was a = σ a σ. The experiments were performed with a uniform medium coarse uartz sand. Tests with a simultaneous oscillation of σ and σ ( off-phase cycles) were published in []. As already shown in [] for a the cyclic flow rule, i.e. the ratio of the accumulated olumetric (ε acc = ε acc + ε acc ) and the accumulated deiatoric (ε acc = /(ε acc ε acc )) strain is a simple function of the aerage stress ratio η a = a /p a only with p a = (σ a + σ )/ being the aerage mean pressure. The strain amplitude, the aerage stress and the oid ratio do not influence the direction of the strain accumulation rate, while some slight changes with the number of cycles occur. In [] it is demonstrated that the i) Corresponding author. Tel.: + 9--8; fax: +9- -; e-mail address: torsten.wichtmann@rub.de direction of the strain accumulation rate can be well approximated by the flow rules of constitutie models for monotonous loading (e.g. modified Cam clay model). The intensity of the strain accumulation rate ε acc = ε acc / N with ε = ε = (ε ) + (ε ) was found proportional to the suare of the strain amplitude, i.e. ε acc ( ). The accumulation rate decreases exponentially with the aerage mean pressure p a and increases exponentially with the aerage stress ratio η a = a /p a. Under cyclic loading a loose soil is compacted faster than a dense one. The relationship ε acc ln(n) holds for N < while the increase of the accumulated (residual) strain with N is faster than logarithmic for higher numbers of cycles. The loading freuency does not influence the accumulation. Tests on four different grain size distributions demonstrated an increase of ε acc with an increasing uniformity index U = d /d and a decreasing mean grain diameter d []. To the authors best knowledge drained cyclic triaxial extension tests, i.e. tests with an aerage stress a lying below the p-axis in the p--plane (Fig. ), hae not been presented yet in the literature. This paper presents such tests. Different stress amplitudes ampl (Section.), aerage mean pressures p a (Section.) and aerage stress ratios η a = a /p a (Section.) were tested. The influence of the number of cycles is discussed in Section.. The test results are compared to the triaxial compression tests presented in [].

Wichtmann et al. Soil Dyn. Earthu. Eng. (Vol. 7, No., pp. -8, 7) a M c ( p ) M c ( c ) aerage stress a p a p ampl ampl = M e ( c ) M e ( p ) a = a / p a Fig. : State of stress in a test with triaxial extension (M e(ϕ c), M c(ϕ c): inclinations of critical state line, M e(ϕ p), M c(ϕ p): inclinations of failure line from monotonous tests) Test procedures and results. Test procedure In all tests the lateral stress σ was kept constant and the axial component σ was cyclically aried, i.e. only uniaxial in-phase stress cycles were tested. The same medium coarse uniform sand (d =. mm, U = d /d =.8, maximum and minimum oid ratios e max =.87, e min =.77) as in the tests with triaxial compression was used. The grain size distribution is gien in []. The specimens were prepared by pluiating dry sand out of a funnel through air into halfcylinder moulds and after that saturated by de-aired water. The test deice and the measurement techniue are described in detail in []. In all tests, cycles were applied with a freuency of f B =. Hz. This lower freuency compared to the tests in [] (f B = Hz) was chosen due to technical reasons (adapter for tension loading).. Influence of the stress/strain amplitude Seen tests with an identical aerage stress (p a = kpa, η a = -.) and a similar initial density index (.9 I D = (e max e)/(e max e min ).) were performed applying different stress amplitudes kpa ampl kpa. Figure shows, that the strain amplitude decreased during the first cycles (this so-called conditioning phase was significant especially in the case of the higher stress amplitudes) and after that remained nearly constant (een a slight reincrease of was measured for N > ). A similar behaiour was obsered for cyclic compression []. In Fig. it is shown, that the strain amplitudes,, and γ ampl = (ε ε ) ampl are linear proportional to the stress amplitude ampl for ampl kpa. For higher stress amplitudes the strain amplitudes increase oer-linearly with ampl. Strain amplitude [ - ] 8 7 p a = kpa, η a = -., I D =.9 -. σ ampl [kpa] =. Number of cycles N [-] Fig. : Strain amplitude as a function of the number of cycles N for different stress amplitudes ampl Strain amplitudes [ - ] 7 γ ampl p a = kpa η a = -., I D =.9 -. Stress amplitude ampl [kpa] Fig. : Strain amplitudes (mean alues oer cycles) in dependence of stress amplitude ampl Fig. presents the total accumulation ε acc (N). We distinguish between the first, irregular cycle and the subseuent regular ones since the deformation during the first cycle may significantly differ from that during the following cycles. The explicit accumulation model [] describes the strain due to the regular cycles only. An implicit model (a conentional σ-ε model) is responsible for the calculation of the irregular cycle. Fig. presents solely the accumulation during the regular cycles. From Fig. it is obious, that larger stress amplitudes produce higher accumulation rates. The residual strain grows almost proportionally to the logarithm of N, except for the stress amplitudes ampl

Wichtmann et al. Soil Dyn. Earthu. Eng. (Vol. 7, No., pp. -8, 7) kpa, where for N, the accumulation is faster than ln(n). The function f N describing the shape of the cures ε acc (N) is further discussed in Section.. Accumulated strain ε acc [%]....8... σ ampl [kpa] =. p a = kpa, η a = -., I D =.9 -. Number of cycles N [-] Fig. : Accumulation cures ε acc (N) In Fig. the residual strain after N cycles is plotted ersus the suare of the strain amplitude, which was calulated as the mean alue = ( N εampl (N) dn)/n. In order to remoe the effect of the slightly arying initial densities and the different compaction rates, ε acc has been normalized, separating the influence f e of the oid ratio as proposed in []: acc / fe [%] N =, N =, N =, N = N = p a = kpa η a = -., I D =.9 -. ( ampl ) [ -7 ] Fig. : Residual strain ε acc normalized by f e as a function of ( ) acc [-] acc / Strain ratio ω = -. -.8 -. -. p a = kpa, η a = -., I D =.9 -. N = N =, N = N =, f e = (C e e) + e + e ref (C e e ref ) () -. Strain amplitude ampl [ - ] With the material constant C e =. and the reference oid ratio e ref = e max =.87 the function f e describes the influence of the oid ratio e on the intensity of accumulation. In Fig. the bar oer f e indicates that f e has been calculated for a mean alue of the eoling e(n) dn)/n. Independent of N, the proportionality ε acc ( ) turns out to be alid also for triaxial extension. Thus, the function [] oid ratio, namely ē = ( N f ampl = ( / ref ) () with the reference amplitude ref = is assumed alid for all stress states. Fig. presents the ratio ω = ε acc /εacc for different numbers of cycles. No significant influence of the strain amplitude on the direction of strain accumulation was found. This conclusion coincides with []. Similarly as in the tests with triaxial compression, a slight increase of the olumetric component of the direction of strain accumulation with N was obsered (Fig. ). Fig. : Strain ratio ω as a function of the strain amplitude (mean alue oer cycles). Influence of the aerage mean pressure Six tests with η a = -., an amplitude ratio of ζ = ampl /p a =. and initial densities. I D. were performed at different aerage mean pressures kpa p a kpa. The cyclic stress paths are shown in Fig. 7a and the strain amplitudes (mean alues oer cycles) are presented in Fig. 8. The non-linear dependence of stiffness on pressure leads to an increase of the strain amplitudes with p a for ζ = ampl /p a = constant. From the shear strain amplitudes, Fig. 8, the stress-dependence of the hysteretic shear modulus was found to be G hyst = τ ampl /γ ampl (p a ) n with n =. (solid cure in Fig. 8). Figure 9 presents the effect of p a on the accumulated strain ε acc. The residual strains were normalized by f ampl and f e. Analogously to [] the intensity of accumulation decreases with p a. The exponential func-

Wichtmann et al. Soil Dyn. Earthu. Eng. (Vol. 7, No., pp. -8, 7) a) [kpa] p [kpa] η = M e ( p ) η = M e ( c ) η = -. b) [kpa] η = M e ( η = M e ( p ) p [kpa] Fig. 7: Stress paths in the tests on the influence of the aerage stress: a) ariation of p a, b) ariation of η a tion f p = exp [ C p (p a /p ref )] () with the reference pressure p ref = p atm = kpa proposed in [] remains alid for extension for kpa p a kpa, Fig. 9. The material constant. C p. was found to increase with N. In [] it was demonstrated that this N-dependence can be neglected without noticeable aggraation of the accuracy of the acumulation model. For η a =.7 a alue C p (N = ) =. was reported in [], which is slightly higher than C p (N = ) =. for η a =.. An application of E. () with C p =. seems reasonable also for a <. Howeer, the accumulation rate at p a = kpa was larger than it could be expected from the test results at higher pressures (Fig. 9, the test at p a = kpa was repeated and similar rates were obtained). A more detailed study of the cumulatie behaiour at small pressures p a < kpa (for η a < as well as for η a ) seems necessary. Similarly as for triaxial compression the direction of strain accumulation ω (cyclic flow rule) has been found to be independent of p a (Fig. ). Strain amplitudes [ - ] γ ampl η a = -., ζ =., I D =. -. Aerage mean pressure p a [kpa] Fig. 8: Strain amplitudes (mean alues oer cycles) in the tests with different aerage mean pressures p a c ) acc / (fampl fe) [%].... C p =..9.8.. η a = -., =., I D =. -. N =, N =, N =, N = N = Aerage mean pressure p a [kpa] Fig. 9: Accumulated strain ε acc normalized by f ampl and f e ersus aerage mean pressure p a acc [-] acc / Strain ratio ω = -. -.8 -. -. N = N =, N = N =, η a = -., ζ =., I D =. -. -. Aerage mean pressure p a [kpa] Fig. : Strain ratio ω as a function of the aerage mean pressure p a. Influence of the aerage stress ratio tests with p a = kpa, initial densities.9 I D. and different aerage stress ratios.88 η a. were performed. In most tests an amplitude ratio of ζ = ampl /p a =. was chosen. For η a = -.7 and η a = -.88 the amplitude ratio was reduced towards ζ =. and ζ =., respectiely in order to keep the minimum deiatoric stress min = a ampl aboe the failure line determined from monotonous tests. The stress paths of the tests with η a are shown in Fig. 7b. In Fig. the strain amplitudes in the tests with ζ =. are plotted ersus η a. For triaxial compression the strain amplitudes decrease with η a while for triaxial extension they increase with η a. While for η a the deiatoric strain amplitudes are significantly larger than the olumetric ones ( ), this difference decreases with decreasing η a and for η a <. both

Wichtmann et al. Soil Dyn. Earthu. Eng. (Vol. 7, No., pp. -8, 7) amplitudes are almost identical. This reeals that for the amplitudes tested, the hysteretic stiffness significantly depends on the stress ratio η a. This stress ratio dependence is usually less pronounced for smaller strain amplitudes (e.g. for γ ampl []) Strain amplitudes [ - ] p a = kpa, ζ =., I D =.9 -. γ ampl acc / (fampl fe) [%]....8... = M e ( c) N =, N =, N = N = E. ()..8...8. Stress ratio Y a [-] = M c ( c) p a = kpa, ζ =. -., I D =.9 -. Fig. : Accumulated strain normalized by f ampl and f e ersus aerage stress ratio Ȳ a -.8 -...8. Aerage stress ratio a = a /p a [-] Fig. : Strain amplitudes as a function of the aerage stress ratio η a The dependence of the accumulated strain ε acc on η a is plotted in Fig.. The accumulation rate increases when the stress ratio η a increases, both for triaxial compression and extension. Howeer, ε acc ( η a ) ε acc (η a ), see Fig.. acc / (fampl fe) [%]....8... = M e ( c) N =, N =, N = N = -.8 -...8. Stress ratio p a = kpa, ζ =. -., I D =.9 -. a = a /p a [-] = M c ( c) Fig. : Accumulated strain normalized by f ampl and f e ersus aerage stress ratio η a In Fig. ε acc /(f ampl f e ) is plotted ersus Ȳ a, which is used as an alternatie measure of the stress ratio []. At η a = the stress ratio is Ȳ a = and on the critical state line Ȳ a = holds. For the tests with η a the function f Y = exp ( C Y Ȳ a) () could be fitted to the data resulting in. C Y. (solid lines in Fig. at η a ) which coincides well with. C Y. determined in []. Howeer, the application of E. () with C Y =. (proposed in []) to the tests with η a < oerestimates the accumulation rates (in Fig. this is shown for N = ). In order to describe the tests with triaxial extension more precisely, a modification of E. () could be used: f Y = exp [ C Y (Ȳ a ) CY ] () A cure-fitting of E. () to the tests with η a (solid lines in Fig. at η a ) resulted in. C Y. and. C Y.7. The material constants are therefore proposed as {. for η C Y = a. for η a < {. for η C Y = a. for η a < Howeer, E. () with the proposed constants still oerestimates the accumulation rates in the tests with. η a (Fig. ). The following hypothesis could explain the difference between () and (). It is based on the assumption that a static preloading (similarly as a cyclic preloading) of a sample reduces its rate of accumulation. Although, as yet, we hae not collected sufficient experimental data for a uantitatie description of this phenomenon, the ualitatie obserations seem plausible. The accumulation rate is slower for an isotropically preloaded sample (static preloading) compared to the accumulation rate of a freshly pluiated sample with the same stress and density (at the beginning of cyclic loading). It is not straightforward to determine the surface of static preloading for sands experimentally. Monitoring acoustic emissions (the intensity of grain cracking

Wichtmann et al. Soil Dyn. Earthu. Eng. (Vol. 7, No., pp. -8, 7) or frictional contact changes) a surface similar to the one in Fig. can be constructed []. For technical reasons the cyclic loading in the compression regime was preceded by the monotonic loading and in the extension regime by. As we see, for small stress ratios in triaxial extension (dashed line) the cyclic loading is performed in a oerconsolidated state and therefore the obsered accumulation is slower than for an analogous stress ratio in compression. This could be responsible for the unsymmetry in the diagrams in Figs. and. For larger stress ratios in extension, the aerage stress is normally consolidated and therefore the rates of accumulation are similar to the ones of triaxial compression. CSL CSL yield surface Fig. : Surface of static preloading and stress paths in the cyclic tests The obsered difference in the cumulatie behaiour for triaxial compression and extension may be also attributed to an influence of the direction of deposition in respect to the directions of the major and the minor principal stresses. In the triaxial compression tests the major principal stress is parallel to the direction of deposition while it is perpendicular in the case of triaxial extension. The influence of the direction of deposition and the polarization of cyclic loading will be studied more detailed in future. p In Fig. a for the tests with η a the strain ratio ω = ε acc /εacc is plotted ersus the aerage stress ratio η a. For small alues of η a the presentation of the reciprocal alue /ω = ε acc /ε acc in Fig. b is more conenient. As in the tests reported in [] the direction of strain accumulation becomes more deiatoric (i.e. ω decreases) with an increasing obliuity of the aerage stress. While at an aerage stress lying on the p-axis (η a = ) the accumulation is pure olumetric ( ε acc =, ω ), a pure deiatoric accumulation ( ε acc =, ω = ) is obtained on the critical state line (η a = M e (ϕ c ) =.88). For η a > M e (ϕ c ) cyclic loading leads to a densification of the material, at an aerage stress beyond the CSL (η a < M e (ϕ c )) a dilatie material behaiour is expected. As for η a [] the direction of strain accumulation can be well approximated by the flow rules of constitutie models for monotonous loading (modified Cam clay model, hypoplastic model, Fig. ). a) acc acc Strain ratio ω = ε / ε [-] b) acc acc Strain ratio /ω = ε / ε [-] - - - - a = M e ( c) N =, N =, N = N = hypoplasticity mod. Cam Clay dilatancy contractancy - -. -.8 -. -. -. - - - - Aerage stress ratio a = a /p a [-] dilatancy contractancy a = M e ( c) Aerage stress ratio a = a /p a [-] p a = kpa, =. -., I D =. -. N =, N =, N = N = hypoplasticity mod. Cam Clay - -. -.8 -. -. -. Fig. : a) Strain ratio ω = ε acc /ε acc and b) reciprocal alue /ω = ε acc /ε acc as a function of the aerage stress ratio η a, comparison with the flow rules of the modified Cam clay model and the hypoplastic model Fig. in [] has been extended by now coering the tests with η a, Fig.. The direction of strain accumulation is shown as a unit ector in the p--plane starting at (p a / a ) with the inclination /ω towards the horizontal. From Fig. the strong dependence of the cyclic flow rule on the aerage stress ratio (and also the slight N-dependence) is obious.. Influence of the number of cycles Figure 7 contains the accumulation cures ε acc (N) measured in the tests on the influence of, p a and η a normalized with f ampl, f e, f p and f Y (E. ()). The tests with. η a were excluded from Fig. 7

acc / (fampl fp fy fe) [%] Wichtmann et al. Soil Dyn. Earthu. Eng. (Vol. 7, No., pp. -8, 7) Accumulated strain up to cycle: N =, N =, N =, N = N = I D =.7-.9 Mc( η =. p ) η =.7 η =. = Mc( η =. η =.7 c)...... Tests for f ampl Tests for f p Tests for f Y Test with N =, bandwidth of tests in [] E. () η =. Number of cycles N [-] η =. Fig. 7: Accumulation cures ε acc (N) normalized with f ampl, f p, f Y and f e [kpa] - - - acc η = M e ( acc p) p [kpa] η = -. η = -.88 = M e ( η = -. η = -. η = -.7 η = -. η = -.7 c) Fig. : Direction of accumulation in the p--plane: tests with different aerage stresses σ a (see the remarks in Sec..) and a test with N =, cycles at p a = kpa, η a =. and I D =. was added. Mostly all normalized accumulation cures fall into the bandwidth of the tests presented in Fig. 8 of []. Therefore, the approximation f N = C N [ln( + C N N) + C N N] () with the material constants C N =., C N =. and C N =. (solid cure in Fig. 7 and in Fig. 8 of []) is justified also for triaxial extension. Summary and conclusions Numerous drained cyclic tests with triaxial extension ( a < ) hae been performed and compared to cyclic tests at a published in []. The main findings are: The direction of accumulation does not depend on the strain amplitude and on the aerage mean pressure p a. The cyclic flow rule is a simple function of the aerage stress ratio η a = a /p a only and can be well approximated by the flow rules of constitutie models for monotonous loading []. The intensity of accumulation ε acc = ε acc / N increases proportionally to the suare of the strain amplitude and decreases exponentially with p a. The alidity of the functions f ampl, f p and f N found in the tests with triaxial compression [] and implemented into the accumulation model [] could be extended to a <. The accumulation rate ε acc increases if the aerage stress ratio η a increases, but ε acc ( η a ) ε acc (η a ). A modification of the function f Y for η a < was proposed in this paper, but two alternatie explanations of this effect are still to be examined. Acknowledgements This study is a part of the project A8 Influence of the fabric change in soil on the lifetime of structures of the collaboratie research center 98 Lifetime oriented design concepts of DFG (German Research Council). The authors are grateful to DFG for the financial support. References [] Niemunis A, Wichtmann T, Triantafyllidis T. A highcycle accumulation model for sand. Computers and Geotechnics, ;():-. 7

Wichtmann et al. Soil Dyn. Earthu. Eng. (Vol. 7, No., pp. -8, 7) [] Wichtmann T, Niemunis A, Triantafyllidis T. Strain accumulation in sand due to cyclic loading: drained triaxial tests. Soil Dynamics and Earthuake Engineering (in print). [] 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 CBS, Bochum, March/April, Balkema, :7-. [] 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, ;():789-8. [] Tanaka Y. New findings in recent experimental studies, In: Oda M, Iwashita K. Mechanics of Granular Materials, Balkema, Rotterdam, 999;7:7. 8