Prediction of torsion shear tests based on results from triaxial compression tests

Transcription

1 Prediction of torsion shear tests based on results from triaxial compression tests P.L. Smith 1 and N. Jones *2 1 Catholic University of America, Washington, USA 2 Geo, Lyngby, Denmark * Corresponding Author ABSTRACT The Single Hardening Model (SHM) has previously been shown to predict the stress-strain response of frictional materials loaded by principal stresses and depicted in the principal stress space. The SHM is based on concepts from elasticity and work-hardening plasticity theories, and requires a total of 12 parameters derived from a minimum of three triaxial compression tests and one isotropic compression test. Presented here, is the prediction of the stress-strain behavior from four torsion shear tests performed on loose Santa Monica Beach Sand. During the torsion shear tests, shear stresses are applied to a hollow cylinder specimen, resulting in rotation of the principal stresses. Three loading conditions have been examined. 1) The shear stress is increased while the deviator stress is held constant, 2) the shear stress is increased while the deviator stress is also increased, and 3) the shear stress is increased while the deviator stress is decreased. The third loading condition gives rise to either loading or unloading depending on the loading direction relative to the yield surface. The predictions include the stress-strain response in the vertical direction, the volumetric strain and the shear It is shown that the SHM is capable of predicting the sand behavior during rotation of the principal stresses with overall good accuracy. 1 INTRODUCTION Understanding the stress-strain response during rotation of the principal stresses in connection with direct simple shear tests and triaxial tests, often performed in connection with offshore wind farms, is important. The effect of rotation of the principal stresses becomes further evident in an anisotropic soil. A method to understanding the behavior is numerical modeling the stress-strain response during rotation of the principal stresses. 2 THE SINGLE HARDENING MODEL The Single Hardening Model (SHM) introduced by Lade and Kim (1988 a, 1988 b ) and Kim and Lade (1988) has predicted the behavior of sand, clay and concrete under a variety of loading conditions, e.g. the three-dimensional behavior of remolded normally consolidated clay (Lade 1990), the drained behavior of sand and concrete (Lade & Kim 1995), and instability and liquefaction of silty sands (Yamamuro & Lade 1999). In the SHM the total strain increments under loading are calculated from the summation of the elastic and plastic strain increments. The elastic strain increments are calculated from Hooke s law using a nonlinear variation of Young s modulus developed by Lade and Nelson (1987). The failure criterion is expressed in terms of the first and third stress invariants. The crosssection of the failure surface on an octahedral plane is shaped like a triangle with smoothly curved edges. A non-associated flow rule is used to calculate the plastic strain increments. Plastic work is used as the hardening parameter to define the location and shape of the yield surface. The plastic potential and yield functions are expressed in terms of the three stress

2 invariants. The plastic potential surface is shaped as an asymmetric cigar with a smoothly rounded triangular cross section similar but not identical to those for the failure surface. The yield surface is shaped as an asymmetric teardrop in the triaxial plane with a smoothly rounded triangular cross section. For normally consolidated, cohesionless soil eleven parameters are required to fully describe the behavior. If the soil has cohesion, a twelfth parameter, a is used for translation of the stress space. All parameters needed to calibrate the model can be extracted from standard experiments consisting of three triaxial tests at different confining pressures and one isotropic compression test. Details of parameter determination can be found in e.g. Lade (2005). 3 TESTS Two series of tests have been used for this study. Triaxial tests performed by Boonyachut (1977) and hollow cylinder torsion shear tests performed by Geiger (1979). Both test series are performed on loose Santa Monica Beach Sand with the purpose of studying the behavior of cohesionless soil during large stress reversals. Santa Monica Beach Sand and is composed of subangular to subrounded grains consisting mainly of quartz and feldspar with the classification parameters shown in Table 1. The triaxial tests used for the parameter determination consist of three tests with confining pressures of 120 kpa, 240 kpa, and 480 kpa. The stress-strain curves are shown in Figure 3. Furthermore, the isotropic compression phase prior to shearing is used for the parameter determination instead of an independent isotropic compression test. The parameters determined for loose Santa Monica Beach sand are shown in Table 2. The hollow cylinder torsion shear tests are performed with a constant confining pressure of 200 kpa. Figure 1 show the stress path (normalized with respect to atmospheric pressure) followed in the four hollow cylinder torsion shear tests. Furthermore, the failure surface predicted from the triaxial tests is indicated. It is seen that none of the tests were continued to failure before the stresses were reversed. In all the tests, the point of stress reversal is marked by an A. The point A is shown in the corresponding stress-strain curves in Figure 4 to Figure 7. In the first test (L12) the vertical stress is increased corresponding to triaxial loading, followed by a phase where the shear stress is increased while the vertical stress is held constant. At the end of the test, the vertical stress drifted (decreased). The offset of this is marked by the point B, also found on the stress-strain curves. In the second and third test (L15 and L10), the specimen is loaded with both vertical stress and shear stress in a ratio of approximately 0.75 and 2.25, respectively. After stress reversal the second test (L15) has the shear stresses reduced while the vertical stress is kept constant. The third test (L10) has both shear stresses and vertical stresses reduced at a constant stress ratio of Table 1. Classification parameters for Santa Monica Beach Sand. Parameter Value Min. void ratio, e min 0.58 Max. void ratio, e max 0.87 Specific gravity, G s 2.66 Mean diameter, D 50 [mm] 0.27 Coefficient of uniformity, C u 1.58 Void ratio, e Boonyachut (1977) Geiger (1979) Figure 1. Stress path followed in the four hollow cylinder torsion shear tests, together with the predicted failure surface. The point of stress reversal for each test is marked with A. The last test (L14) consists of triaxial loading where the vertical stress is increased, followed of a phase where a shear stress is increased while the vertical stress is decreased. The predicted yield surface at stress reversal (point A) is shown in Figure 2. It is seen that the loading in the last phase consists of

3 loading both inside and outside this yield surface, the result being elastic unloading and plastic loading, respectively. 4 MODEL CALIBRATION The SHM is executed in the Scilab software and the results presented here are predicted using one finite element representing the whole test specimen. The stresses are used as input and the corresponding strain response is predicted. The parameters determined the from triaxial compression tests can be seen in Table 2. Table 2. Parameters for the Single Hardening Model determined for loose Santa Monica Beach Sand. Parameter Value Elastic behavior ν 0.26 λ 0.27 M 600 Failure criterion m η a 0 Plastic potential ψ μ 2.26 Hardening function C p 1.55 Yield criterion h conditions, triaxial loading performed at the beginning of some of the hollow cylinder torsion shear tests were compared with the triaxial tests. This is done in Figure 3, where the stress-strain relations from the tests are compared. The tests performed in the torsion shear apparatus with a confining pressure of 200 kpa are initially stiffer than the tests performed in the triaxial apparatus at 240 kpa. The volume change from the torsion shear tests follows the same patters as the triaxial tests. The difference in stiffness is found to have no significant effect on most parameters determined for the SHM, except the yield surface parameter α. To be able to predict the torsion shear tests, the parameter α is determined using the stress-strain relations from triaxial loading phases in the torsion shear tests. 5 RESULTS The predicted strains in the first test, L12 are compared to the measured values in Figure 4. The predicted behavior shows good agreement with the measured data. At point B, the decrease in vertical stress results in an elastic unloading in vertical direction, despite the specimen being sheared at the same time. Prediction of this behavior is studied further in the last test (L14). Figure 2. Stress path in test L14 compared with the predicted yield surface at stress reversal (point A). To be able to predict the results using only one finite element, the boundary conditions in the triaxial test and the torsion shear tests should be similar. The triaxial tests were performed with lubricated ends and the torsion shear tests were performed with rough ends. This produces higher apparent stiffness in the torsion shear tests (Rowe & Barden 1964). As seen in Table 1, the void ratio is a little lower in the torsion shear tests than in the triaxial tests. To overcome this difference in void ratio and boundary Figure 3. Comparison of stress-strain relation from the triaxial tests and triaxial loading in the torsion shear tests.

4 The predicted and measured strains for test L15 are compared in Figure 5. The loading with a constant ratio between the vertical stress and the shear stress of 0.75 results in larger shear strains than vertical strains. This is captured with the predicted strains. However, the predicted strains are a little smaller than the measured strains. After stress reversal at point A, the measured behavior shows a substantial amount of both normal and shear strain despite unloading of the specimen. This is believed to be caused by time effects due to the close proximity to failure. Time effects are not included in the current SHM and the observed behavior can therefore not be modeled accurately right after stress reversal. The volumetric strain right after stress reversal shows a little dilation, then a substantial amount of contraction, also in discrepancy with the expected and modeled behavior. In test L10, the specimen is loaded with both vertical stress and shear stress in a ratio of approximately 2.25, then unloaded in a ratio of approximately The time effects observed after stress reversal in test L15 is also present in test L10. Figure 4. Comparison of predicted and observed behavior for test L12. a) Stress-strain relation. b) Vertical strain vs. volumetric c) Shear stress-shear d) Shear strain vs. volumetric There is good agreement between the observed and predicted behavior during loading, c.f. Figure 6. Right after stress reversal the time effects are not captured. After the time effects are diminished, the behavior of the vertical strain and the shear strains is captured by the SHM after stress reversal.

5 Figure 5. Comparison of predicted and observed behavior for test L15. a) Stress-strain relation. b) Vertical strain vs. volumetric c) Shear stress-shear d) Shear strain vs. volumetric A comparison of the predicted and measured strain for the last test, L14 is shown in Figure 7. There is good consistency between the predicted and measured strains. Figure 6. Comparison of predicted and observed behavior for test L10. a) Stress-strain relation. b) Vertical strain vs. volumetric c) Shear stress-shear d) Shear strain vs. volumetric After the triaxial compression phase, the vertical stress is reduced and the shear stress is increased. This results in elastic behavior immediately after point A. This can be seen in Figure 7 a), where the decrease in vertical stress, results in elastic unloading and in Figure 7 c), where the increase in shear stress results in elastic loading. When the stresses start expanding the yield surface again, the observed increases in both shear and vertical strains are captured by the predicted plastic behavior.

6 yielded good agreement between the predictions and the measured strains. ACKNOWLEDGEMENT This research was funded in part by the American Chemical Society Petroleum Research Fund under grand PRF # AC9. REFERENCES Figure 7. Comparison of predicted and observed behavior for test L14. a) Stress-strain relation. b) Vertical strain vs. volumetric c) Shear stress-shear d) Shear strain vs. volumetric Boonyachut, S. 1977, Experimental study of the behavior of cohesionless soil during stress reversals, Ph.D. dissertation, University of California, Los Angeles Geiger, E. 1979, Experimental study of the behavior of cohesionless soil during large stress reversals and reorientation of principal stresses, M.S. dissertation, University of California, Los Angeles Kim, M.K. & Lade, P.V. 1988, Single hardening constitutive model for frictional materials I. Plastic Potential Function, Computers and Geotechnics, 5, Lade, P.V. 1990, Single-Hardening model with application to NC clay, Journal of Geotechnical Engineering, 116, Lade, P.V. 2005, Single hardening model for soils: Parameter determination and typical values, Soil constitutive models, (Eds: Yamamuro, J.A. and Kaliakin, V.N.), , Lade, P.V. & Kim, M.K a, Single hardening constitutive model for frictional materials II. Yield criterion and plastic work contours, Computers and Geotechnics, 6, Lade, P.V. & Kim, M.K b, Single hardening constitutive model for frictional materials III. Comparisons with experimental data, Computers and Geotechnics, 6, Lade, P.V. & Kim, M.K. 1995, Single hardening constitutive model for soil, rock and concrete, International Journal of Solids and Structures, 32, Lade, P.V. & Nelson, R.B Modeling the elastic behavior of granular materials, International Journal for Numerical and Analytical Methods in Geomechanics, 11, Rowe, P.W. & Barden, L. 1964, Importance of free ends in triaxial testing, Journal of the Soil Mechanics and Foundations Division, ASCE, 90, 1-27 Yamamuro, J.A. & Lade, P.V. 1999, Experiments and modelling of silty sands susceptible to static liquefaction, Mechanics of Cohesive-frictional Materials, 4, CONCLUSIONS There is an overall good agreement between the predictions and the experiments. Stress paths with different combinations of vertical and shear stresses