New Constitutive Material Models in RS 2

Transcription

1 New Constitutive Material Models in RS The Softening-Hardening Material Model Experimental evidence indicates that plastic deformation in soils starts from the early stages of loading. The typical elasto-perfect plastic models are not adequate for capturing such behavior in a constitutive model. To simulate such behavior, constitutive models that utilize a hardening law after initial yielding are required. The Softening-Hardening Model in RS 2 has been developed to meet the afore-mentioned need. The model can utilize up to three yield surfaces that include deviatoric (shear), volumetric (cap), and tension cut off. The yield surfaces and hardening characteristics of this model are illustrated in Figure 1 below. The formulations of these three mechanisms, definitions of yield surfaces, and their corresponding plastic potential and hardening laws are presented in the following sections. q 1 q 1 M f M f Yield Surface Yield Surface Hardening Hardening (a) p c p (b) p c p Figure 1. The yield surfaces of the Softening-Hardening model; a) deviatoric yield surface (red) and the vertical cap (green); b) the deviatoric yield surface (red) and elliptical cap (blue)

2 1.1 The Deviatoric Mechanism The deviatoric yield surface is very similar to the yield surface of the Mohr-Coulomb model. The equation for the Mohr-Coulomb yield surface (Pietruszczak, 2010) using the,, invariants is given by equation 1 (the convention is compression positive). 0 ; Eq. 1 The hardening for this yield surface is considered for the mobilized friction angel (and cohesion) and it is attributed to plastic distortion. In the above equation is the ultimate/failure friction angle and and are the mobilized cohesion and friction angle, respectively. There are two types of hardening laws considered for this model. The first one uses a relationship between tan and the deviatoric plastic strain, presented in equation 2. tan tan Eq. 2 where: is the deviatoric plastic strain, and is the hardening parameter (positive and constant). The second hardening law uses custom tabular piecewise linear values for the mobilized friction angel and cohesion with deviatoric plastic strain,, &,. The advanced option for plastic potential is to define a compaction-dilation angle in such a way that when the mobilized friction angle () is less than this angle () the volumetric plastic strain is positive (compaction) and when the mobilized friction angle is greater than the volumetric plastic strain is negative (dilation). 1.2 The Volumetric Mechanism There are two options for the yield surface of the volumetric mechanism, or the cap: vertical and elliptical. The yield surfaces are defined as follows. (Vertical Cap), 0 (Elliptical Cap) Eq. 3 Where is the location of the intersection of this yield surface with the axis.

3 The elliptical cap is very similar to the yield surface of the modified Cam-Clay model with an offset to consider the cohesion. The hardening for these yield surfaces is considered for and it is attributed to volumetric plastic strain. The build in function for the hardening follows the same hardening law as in the Modified Cam-Clay model. Tabular hardening law which uses custom tabular pricewise linear values for versus volumetric plastic strain, is also available for this model. The flow rule is associated for this yield surface. 1.3 The Tension Cut-Off Mechanism This mechanism is to incorporate the tensile strength of the material to this model. In this mechanism the minor principal stress is limited to the tensile strength of the material. The flow rule is associated and the mechanism has no hardening. 2.0 Numerical Examples The applicability of the Softening-Hardening model is demonstrated in simulations of tunnels and retaining walls in the following sections. 2.1 Tunnel with Nearby Building Load The 2D model ground consists of a mix of 1.6 and 3.0 mm aluminum rods in the ratio of 3:2 by weight. The ground is prepared by piling up the stack of aluminum rods from the bottom (Nakai, 2013). Figure 2. Material used in building 2D scaled model The biaxial test results of this material are presented in the following figure. The simulated result using the Hardening Softening model is in very good agreement with the observed behavior. Since the material is

4 inherently frictional, the deviatoric mechanism would be dominant for this material and thus the cap yield surface is not activated for modelling the behavior of this material. Figure 3. Biaxial Compression Test results (σ2=19.6 kpa) The scaled model for tunnel with nearby building is illustrated in Figure 4. Figure 4. Apparatus for 2D foundation and tunnel model tests

5 The bearing capacity of the foundation without excavation of the tunnel is presented in Figure 5. The RS 2 results are in very good agreement with the observed behavior. The numerical results published in Nakai (2013) overestimate the bearing capacity. Figure 5. Load-displace and bearing capacity of the rigid strip shallow foundation The surface settlement profiles for the tunnel excavation for the case of greenfield and nearby surface load are presented in Figure 6. The distribution of deviatoric strain in the ground, which is an indication of slip surfaces, is demonstrated in Figure 7. Figure 6. Surface settlement profile; Tunnel with nearby building load

6 Figure 7. Distributions of the deviatoric strains in the ground, Fixed Invert The failure pattern and settlement profile is in good agreement with the observations. 2.2 Tieback Retaining Wall The benchmark exercise described by Schweiger (2002), consisting of a deep excavation problem in Berlin sand, is the basis for this comparison. Figure 8 shows the geometry and the excavation steps of the problem. Figure 8. Problem geometry and excavation stages (Schweiger, 2002)

7 To identify the material properties the material properties reported for a Hardening Soil model in the PLAXIS Material Models Manual were used to first simulate some triaxial test data. Then the simulated triaxial tests were used to identify the material properties of the Softening-Hardening model. The results of the triaxial test on layer 1 is presented in Figure 9. Since the material is Sand the deviatoric mechanism is dominant and so the cap is not used in the constitutive modelling of the materials in this problem. Figure 9. Drained Triaxial Compression Test results (σ 3 =100 kpa) The results of the excavation problem is presented in the form of the deflection of the retaining wall. Figure 10 shows the observed deformation versus the simulation result produced by RS 2, FLAC and PLAXIS. Wall deflection (m) Measured Plaxis Flac RS Depth below the surface (m) Figure 10. Lateral wall deflection at the final stage of excavation

8 3.0 The User Defined Material Models The latest version of RS 2 has the option of accepting user defined material models by dll. The dll file that is included in the install folder includes the Double-Yield, ChSoil, and CySoil of FLAC as well as the Hardening Soil and HSS Small soil of PLAXIS. The excavation problem in Berlin sand was simulated by the CySoil model in RS 2 and the results are presented in Figure 11 below. Wall deflection (m) Measured Flac RS2 CySoil Depth below the surface (m) Figure 11. Lateral wall deflection at the final stage of excavation 4.0 References Nakai, T. Constitutive Modeling of Geomaterials: Principles and Applications, Taylor & F. Boca Raton: CRC Press, 2013 Pietruszczak, S. Fundamental of Plasticity in Geomechanics, Taylor & Francis Group, Leiden/London/New York, Schweiger, H.F. Benchmarking in Geotechnics, Institute for Soil Mechanics and Foundation Engineering, Graz University of Technology, Austria 2002.