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Figure from:A SANISAND model with anisotropic elasticity

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Fig. 4. Comparison of simulations obtained from various approaches with experimental results of an undrained loading/reverse loading test on a dense (e=0.735) sample of Toyoura sand (data taken from Verdugo and Ishihara [3]). assume that sand behavior at the very beginning of loadings or reverse loadings is mainly elastic. The tangent to the stress path (namely q/p) at point A, and the slope of unloading-reloading curve at points B, and C are practically vertical (to fortify reasoning, the tangent to stress path at point C is shown in Fig. 1 by dashed line). Also, deviation of unloading-reloading curve at points D and E from vertical axis is not relatively significant. Phase transformation took place at point F. Noting that phase transformation point is defined as where contraction turns into dilation or vice versa [28]. At point G, located in dilative zone of behavior, the last unloading was conducted on the sample. As illustrated by the dashed line constructed at point G, deviation of the tangent line to the unloading path from the vertical direction is considerable which cannot be explained by the theory of isotropic elasticity (i.e., Eq. (7)). The similar pattern of behavior has been reported in many other experimental studies (e.g., [3,19,20,22-24]). Graham and Houlsby [17] paid attention to this particular behavior and attributed it to the development of anisotropic elasticity in soil mass. At this time, this conclusion has been adopted by many researchers in this field of study (e.g., [7, 20,21]). The pattern shown indicates that anisotropy evolves with shear loading and evolution of anisotropy intensifies when stress state approaches phase transformation. Immediately after the stress state passes the phase transfor- mation point and steps into the dilative regime of behavior, the slope of the very beginning point of the loading/unloading curve reaches an ultimate limit and remains unchanged upon the

Figure 4 Comparison of simulations obtained from various approaches with experimental results of an undrained loading/reverse loading test on a dense (e=0.735) sample of Toyoura sand (data taken from Verdugo and Ishihara [3]). assume that sand behavior at the very beginning of loadings or reverse loadings is mainly elastic. The tangent to the stress path (namely q/p) at point A, and the slope of unloading-reloading curve at points B, and C are practically vertical (to fortify reasoning, the tangent to stress path at point C is shown in Fig. 1 by dashed line). Also, deviation of unloading-reloading curve at points D and E from vertical axis is not relatively significant. Phase transformation took place at point F. Noting that phase transformation point is defined as where contraction turns into dilation or vice versa [28]. At point G, located in dilative zone of behavior, the last unloading was conducted on the sample. As illustrated by the dashed line constructed at point G, deviation of the tangent line to the unloading path from the vertical direction is considerable which cannot be explained by the theory of isotropic elasticity (i.e., Eq. (7)). The similar pattern of behavior has been reported in many other experimental studies (e.g., [3,19,20,22-24]). Graham and Houlsby [17] paid attention to this particular behavior and attributed it to the development of anisotropic elasticity in soil mass. At this time, this conclusion has been adopted by many researchers in this field of study (e.g., [7, 20,21]). The pattern shown indicates that anisotropy evolves with shear loading and evolution of anisotropy intensifies when stress state approaches phase transformation. Immediately after the stress state passes the phase transfor- mation point and steps into the dilative regime of behavior, the slope of the very beginning point of the loading/unloading curve reaches an ultimate limit and remains unchanged upon the

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Constitutive equations of SANISAND model in triaxial space [15,25]. Table 1
Constitutive equations of SANISAND model in triaxial space [15,25]. Table 1
Fig. 1. Stress path of a typical undrained triaxial test including loading reversals. After isotropic consolidation, the shear stress was applied to the sample. As a result, the stress state moves along AB path. Then it was unloaded and reloaded along B>C-B path. Similarly, other unloading and reloading cycles were applied on sample at points D and G. test was ended at point H (Data taken from Ishihara et al. [28]).
Fig. 1. Stress path of a typical undrained triaxial test including loading reversals. After isotropic consolidation, the shear stress was applied to the sample. As a result, the stress state moves along AB path. Then it was unloaded and reloaded along B>C-B path. Similarly, other unloading and reloading cycles were applied on sample at points D and G. test was ended at point H (Data taken from Ishihara et al. [28]).
Fig. 2. Stress path of a sample of Toyoura sand subjected to undrained cyclic loading (data taken from Towhata and Ishihara [37]) 64 is the Kronecker’s delta that equals 1 when i=j, and 0 otherwise. Triaxial mode of shearing is considered here. In soil testing, usually direction of o; is defined along the vertical direction (i.e., gravity), and that of o3 is along horizontal plane. The same assumption is usually applied for the strain field. In this situation, Let us assume that as a constraint, volume change is not allowed when shear stress is applied (i.e., undrained shear). As a consequence, Eq. (7) implies that p =0, and q/p =o. Table 2
Fig. 2. Stress path of a sample of Toyoura sand subjected to undrained cyclic loading (data taken from Towhata and Ishihara [37]) 64 is the Kronecker’s delta that equals 1 when i=j, and 0 otherwise. Triaxial mode of shearing is considered here. In soil testing, usually direction of o; is defined along the vertical direction (i.e., gravity), and that of o3 is along horizontal plane. The same assumption is usually applied for the strain field. In this situation, Let us assume that as a constraint, volume change is not allowed when shear stress is applied (i.e., undrained shear). As a consequence, Eq. (7) implies that p =0, and q/p =o. Table 2
Amount of parameters used in simulations. Table 3 Physical properties of Toyoura and Fuji river sands.
Amount of parameters used in simulations. Table 3 Physical properties of Toyoura and Fuji river sands.
Fig. 3. Model surfaces in q-p triaxial stress space [15,25]. Fig. 1 illustrates the stress path of typical sand subjected to some cycles of shear reversals in triaxial apparatus. Description of the testing method and stress path is outlined in Fig. 1. It is logical to
Fig. 3. Model surfaces in q-p triaxial stress space [15,25]. Fig. 1 illustrates the stress path of typical sand subjected to some cycles of shear reversals in triaxial apparatus. Description of the testing method and stress path is outlined in Fig. 1. It is logical to
Fig. 5. Comparison of simulations obtained from various approaches with experimental results of a drained loading/reverse loading test on a loose (e=0.810) sample of Toyoura sand (data taken from Verdugo and Ishihara [3]).
Fig. 5. Comparison of simulations obtained from various approaches with experimental results of a drained loading/reverse loading test on a loose (e=0.810) sample of Toyoura sand (data taken from Verdugo and Ishihara [3]).
Fig. 6. Simulations versus experiments in undrained loading/reverse loading test on medium-loose (e=0.833) samples of Toyoura sand (data taken from Verdugo and Ishihara [3]).
Fig. 6. Simulations versus experiments in undrained loading/reverse loading test on medium-loose (e=0.833) samples of Toyoura sand (data taken from Verdugo and Ishihara [3]).
Fig. 7. Simulations versus experiments in undrained loading/reverse loading test on dense (e=0.735) samples of Toyoura sand (data taken from Verdugo and Ishihara [3]) It can be observed that @, only appears with z in constitutive relations (see Eq. (17)). Recalling that @, is a scalar constant In the establishment of constitutive equations using the representation theorem of isotropic functions, physical considera- tion based on experimental observation is always essential and recommended (e.g. [31,32]). Experimental studies have revealed that the pattern of stress path in the reverse loading is generally similar to that of loading. Let us consider a constant volume cyclic test with identical amplitudes of shear stress in compression and extension sides. An example of such experiment is demonstrated in Fig. 2. As a result of the condition of equality of shear stress amplitudes, microstructural evolutions are relatively the same upon the compression and extension sides in the butterfly loop. In Fig. 2, the sample starts to demonstrate butterfly loop from the extension side in the third cycle of loading (i.e., from point B). Usually, the microstructural fabric rapidly reaches the saturate state in a butterfly zone. Hence, it is logical to assume that the amounts of z at points A and B are Zmax, and —Zmax, respectively. At the very beginning of an unloading process, the response is nearly elastic. Ignoring the small contribution of plastic strains, if exist,
Fig. 7. Simulations versus experiments in undrained loading/reverse loading test on dense (e=0.735) samples of Toyoura sand (data taken from Verdugo and Ishihara [3]) It can be observed that @, only appears with z in constitutive relations (see Eq. (17)). Recalling that @, is a scalar constant In the establishment of constitutive equations using the representation theorem of isotropic functions, physical considera- tion based on experimental observation is always essential and recommended (e.g. [31,32]). Experimental studies have revealed that the pattern of stress path in the reverse loading is generally similar to that of loading. Let us consider a constant volume cyclic test with identical amplitudes of shear stress in compression and extension sides. An example of such experiment is demonstrated in Fig. 2. As a result of the condition of equality of shear stress amplitudes, microstructural evolutions are relatively the same upon the compression and extension sides in the butterfly loop. In Fig. 2, the sample starts to demonstrate butterfly loop from the extension side in the third cycle of loading (i.e., from point B). Usually, the microstructural fabric rapidly reaches the saturate state in a butterfly zone. Hence, it is logical to assume that the amounts of z at points A and B are Zmax, and —Zmax, respectively. At the very beginning of an unloading process, the response is nearly elastic. Ignoring the small contribution of plastic strains, if exist,
Fig. 8. Simulations versus experiments in undrained loading/reverse loading test on three dense samples of Toyoura sand starting at pi,=1000 kPa (data taken from Verdugo and Ishihara [3]).
Fig. 8. Simulations versus experiments in undrained loading/reverse loading test on three dense samples of Toyoura sand starting at pi,=1000 kPa (data taken from Verdugo and Ishihara [3]).
As an advantage of the approach of this study, Eq. (21) provides a simple straightforward method for calculation of Zmax. This may be important considering that direct and simple
As an advantage of the approach of this study, Eq. (21) provides a simple straightforward method for calculation of Zmax. This may be important considering that direct and simple
Fig. 10. Simulations versus experiments of two medium-loose samples of Fuji river sand (data taken from Ishihara et al. [28]) All approaches predict the same response on the loading path. Unlike loading, the predicted stress paths are not the same upon reverse loading. Among simulations, the less favorable result is obtained from the conventional SANISAND model which ignores the effect of fabric evolution. As a consequence, the decrease in mean principal effective stress is underestimated. Moreover, the predicted stress path obtained from Dafalias and Manzari [15] initially lags behind experiments as a result of the assumption of isotropic elasticity. Afterwards, in moderate shear stress levels, the predicted stress path surpasses the experiment results. The predicted stress path by the present study well fits the experi- mental results immediately after reverse loading. The mentioned agreement persists till moderate levels of shear stress (say 1000 kPa). This deficiency may be improved by relating dilatancy In total, the model has 15 parameters. A systematic calibration procedure for these parameters has been described in [13,15,16,25,38,39]. Besides, a simple calibration method for Zmax was given in Section 3. Using the results of 25 tests reported by Verdugo and Ishihara [3], Ishihara and Okada [40], Ishihara et al. [28], Papadimitriou et al. [38], and Yoshimine and Hosono [41] on Toyoura and Fuji river sands, the model is evaluated in the following subsections. The model parameters used in simulations and the physical properties of Toyoura and Fuji river sands are given in Tables 2 and 3, respectively.
Fig. 10. Simulations versus experiments of two medium-loose samples of Fuji river sand (data taken from Ishihara et al. [28]) All approaches predict the same response on the loading path. Unlike loading, the predicted stress paths are not the same upon reverse loading. Among simulations, the less favorable result is obtained from the conventional SANISAND model which ignores the effect of fabric evolution. As a consequence, the decrease in mean principal effective stress is underestimated. Moreover, the predicted stress path obtained from Dafalias and Manzari [15] initially lags behind experiments as a result of the assumption of isotropic elasticity. Afterwards, in moderate shear stress levels, the predicted stress path surpasses the experiment results. The predicted stress path by the present study well fits the experi- mental results immediately after reverse loading. The mentioned agreement persists till moderate levels of shear stress (say 1000 kPa). This deficiency may be improved by relating dilatancy In total, the model has 15 parameters. A systematic calibration procedure for these parameters has been described in [13,15,16,25,38,39]. Besides, a simple calibration method for Zmax was given in Section 3. Using the results of 25 tests reported by Verdugo and Ishihara [3], Ishihara and Okada [40], Ishihara et al. [28], Papadimitriou et al. [38], and Yoshimine and Hosono [41] on Toyoura and Fuji river sands, the model is evaluated in the following subsections. The model parameters used in simulations and the physical properties of Toyoura and Fuji river sands are given in Tables 2 and 3, respectively.
Fig. 11. Comparison of simulated behavior with the measured behavior of an undrained cyclic test on a dense sample of Fuji river sand (data after Ishihara et al. [28]). Comparing simulations with experimental results in the reverse path, it is observed that the conventional model initially predicts dilation that is not in agreement with the observed behavior. Besides, also in this case, the model of Dafalias and Manzari [15] overestimates the contraction occurring in reverse loading. Finally, it can be observed that the simulation obtained from the approach of this study is nearly parallel to that of experiments. It is worth mentioning that, in both Figs. 4 and 5, Verdugo and Ishihara [3] performed a complete set of triaxial tests on Toyoura sand covering wide ranges of density and mean principal effective stress. Samples were prepared by wet tamping method. Simulations by the modified model with anisotropic elasticity for eleven medium-loose and dense samples of Toyoura sand are compared with the measu respectively. In each series of tests s red data in Figs. 6-8, hown in Figs. 7 and 8, samples were isotropically consolidated to p=100, 1000, 2000, and 3000 kPa, first. Then, shear stress was ap loading was applied when the amount of plied on samples. Reverse axial strain reached 25%. Finally, the predicted behaviors of three dense samples with different amounts of initial void ratios, but the same initial mean effective stress (p=1000 kPa) are compared with experiments in Fig. 8. Again, the present approach shows its capability after
Fig. 11. Comparison of simulated behavior with the measured behavior of an undrained cyclic test on a dense sample of Fuji river sand (data after Ishihara et al. [28]). Comparing simulations with experimental results in the reverse path, it is observed that the conventional model initially predicts dilation that is not in agreement with the observed behavior. Besides, also in this case, the model of Dafalias and Manzari [15] overestimates the contraction occurring in reverse loading. Finally, it can be observed that the simulation obtained from the approach of this study is nearly parallel to that of experiments. It is worth mentioning that, in both Figs. 4 and 5, Verdugo and Ishihara [3] performed a complete set of triaxial tests on Toyoura sand covering wide ranges of density and mean principal effective stress. Samples were prepared by wet tamping method. Simulations by the modified model with anisotropic elasticity for eleven medium-loose and dense samples of Toyoura sand are compared with the measu respectively. In each series of tests s red data in Figs. 6-8, hown in Figs. 7 and 8, samples were isotropically consolidated to p=100, 1000, 2000, and 3000 kPa, first. Then, shear stress was ap loading was applied when the amount of plied on samples. Reverse axial strain reached 25%. Finally, the predicted behaviors of three dense samples with different amounts of initial void ratios, but the same initial mean effective stress (p=1000 kPa) are compared with experiments in Fig. 8. Again, the present approach shows its capability after
Fig. 12. Simulations versus experiments of four undrained triaxial tests on dense to very loose dry deposited samples of Toyoura sand: (a) and (b) stress paths; (c) and (d shear stress versus shear strain curves (data taken from Papadimitriou et al. [39]).
Fig. 12. Simulations versus experiments of four undrained triaxial tests on dense to very loose dry deposited samples of Toyoura sand: (a) and (b) stress paths; (c) and (d shear stress versus shear strain curves (data taken from Papadimitriou et al. [39]).
Fig. 13. Simulations versus experiment results for an undrained triaxial test on a dense (e=0.741) sample of Toyoura sand: (a) and (b) experiment results; simulations obtained from: (c) and (d) Manzari and Dafalias [25] conventional model; (e) and (f) Dafalias and Manzari [15] approach with anisotropic dilatancy; (g) and (h) the modified model with anisotropic elasticity; (i) and (j) the modified model with both anisotropic elasticity and dilatancy (data taken from Yoshimine and Hosono [41]).
Fig. 13. Simulations versus experiment results for an undrained triaxial test on a dense (e=0.741) sample of Toyoura sand: (a) and (b) experiment results; simulations obtained from: (c) and (d) Manzari and Dafalias [25] conventional model; (e) and (f) Dafalias and Manzari [15] approach with anisotropic dilatancy; (g) and (h) the modified model with anisotropic elasticity; (i) and (j) the modified model with both anisotropic elasticity and dilatancy (data taken from Yoshimine and Hosono [41]).
Without changing parameters, the stress paths of each test is simulated and demonstrated together with corresponding experi- ment in Fig. 10. Shear stress versus shear strain behavior of tests
Without changing parameters, the stress paths of each test is simulated and demonstrated together with corresponding experi- ment in Fig. 10. Shear stress versus shear strain behavior of tests