NON-NEWTONIAN FLUID PARAMETERS CALIBRATION FOR. NUMERICAL MODELLING OF LANDSLIDES. CALIBRATION DES PARAMETRES D'UN FLUIDE ...
Fifth International Young Geotechnical Engineering Conference - 5iYGEC’13
NON-NEWTONIAN FLUID PARAMETERS CALIBRATION FOR NUMERICAL MODELLING OF LANDSLIDES CALIBRATION DES PARAMETRES D’UN FLUIDE NON-NEWTONIEN POUR LA MODELISATION NUMERIQUE DES GLISSEMENTS DE TERRAIN 1
Adrian Andronic 1 Technical University of Civil Engineering, Bucharest, Romania
ABSTRACT - Common geotechnical engineering practice deals with landslide modelling during the triggering and in some cases immediate post-failure phases. This is generally carried out using either Limit Equilibrium Method or Lagrangian formulation for Finite Element Method. The paper describes a method for assessing both the aforementioned phases and propagation of landslides, modelling the sliding mass by the methods of Eulerian formulation specific to Computational Fluid Dynamics. The equation of state parameters governing the fluid-soil equivalence, as well as the ones describing the velocity-shear strain behaviour are found both by numerical matching (the former) and a newly developed laboratory testing method (the latter). Similarities and differences with respect to the classical approach are pointed out as conclusions.
1. Introduction The study of landslides and of the consequences they produce has become a multi-disciplinary subject, involving, besides the engineering expertise, the geographical, pedological and urban planning aspects. The engineering approach is currently focusing on the triggering phase, in the attempt to avoid the phenomena from happening. For this purpose, the methods employed contain Limit Equilibrium Method or the modelling of solid masses using the Lagrange formulation implemented in the Finite Element Method. The analysis of the propagation phase challenges the limits of the classical methods because of the large deformations suffered by the sliding soil mass, generating Finite Element convergence problems. The method proposed to overcome these model difficulties treats the sliding soil mass as a highviscosity fluid using the Eulerian formulation classically used in Computational Fluid Dynamics Method and the civil engineering bedrock as a solid modelled as a Lagrange media. The method employed for analysing the EulerLagrange Coupling (solid-liquid coupling) uses a serial solver that separates the media into two stages. The Lagrange step is solved first, allowing the mesh to deform according to the Lagrange parts. The solution for this step is superimposed on the Euler mesh and the new step is computed according to the parameters of the liquid phase.
2. Euler-Lagrange Coupling method parameters used in the analysis The Euler-Lagrage Coupling method consists in modelling the solid part of the analysis as a Lagrangian solid, using the classical formulation of
the Finite Element Method, and the liquid part (the sliding soil mass) as an Eulerian material. The boundary conditions imposed for the two parts assure the coupling of Lagrange and Euler parts. The parameters used in the case of the solid part are common for Finite Element Method for modelling the behaviour of soils, namely the density, which provides the mass of the volume, the elastic behaviour parameters (Young modulus and Poisson’s ratio) and the plastic behaviour parameters corresponding to the Drucker-Prager hardening criterion derived from the Mohr-Coulomb parameters. The determination of these parameters represents common geotechnical practice and the methods don’t constitute the purpose of this article. For the case of the liquid part, the parameters are the density, the elastic behaviour parameters and the plastic behaviour parameters corresponding to the Drucker-Prager hardening criterion, similar to the model for the Lagrage part, but besides these, the viscosity of the liquid and the Hugoniot linear form (Us-Up) of the Mie-Grüneisen equation of state are introduced to model the Euler part.
3. The procedure for viscosity determination The viscosity is a parameter characterizing the shearing stress response to an applied shearing strain rate. In common geotechnical practice, the shearing behaviour is studied in terms of shearing variation with respect to an applied set of normal stresses (as in the case of the triaxial test) or to an applied displacement (as in the case of direct shear test). The direct shear test can be considered to be a viscometer as long as the shearing stresses with the applied displacements at various rates are measured. Moreover, the equipment is provided
Fifth International Young Geotechnical Engineering Conference - 5iYGEC’13 with a loading ram for applying the normal stresses, so the tests have been conducted also varying this parameter. In common fluids with incompressible isotropic behaviour, the spherical stress has no influence upon the viscosity, the classical models being given in Figure 1. Yet, in the case of the soils, it was discovered the drastic influence of this parameter. Since for the time being there is no constitutive model to consider the variation of viscosity with spherical stresses, it was decided to use the viscosity as point-wise values and to correct it by means of Mie-Grüneisen law varying the material density on vertical direction. As a research direction, the request of a coherent constitutive model for these situations is trying to be met.
The values obtained for the shear stresses were normalized with the normal stresses and plotted with respect to displacement velocities. The graphical representation of the Casson equation was added and the values were compared. The results were very similar, proving that the method for determining the viscosity using the direct shear apparatus is similar with the classical methods for viscosity determination (Figure 3).
Figure 3. Maximum shear stresses normalized with the normal stresses plotted with respect to the displacement velocities
Figure 1. Different types of viscous behaviour (J. Locat, D Demers, 1988) The tests were performed at various displacement velocities (0.01 mm/min, 0.02 mm/min, 0.05 mm/min, 0.1 mm/min, 0.2 mm/min, 0.5 mm/min, 1 mm/min, 2 mm/min, 5 mm/min) at constant normal stresses for three loading cases (50 kPa, 100 kPa, 150 kPa). The results were graphically represented in terms of maximum shear stress with respect to normal stresses. For each linear representation, for the values of one test it was obtained a linear equation plotted also on the graph (Figure 2).
Figure 2. Maximum shear stress plotted with respect to normal stresses for different displacement velocities
4. The determination of the parameters of the Hugoniot linear form of the Mie- Grüneisen equation of state In the absence of laboratory tests which provide the parameters for the equation of state, the method employed is numerical matching. For this purpose, several calibration models have been analysed, using both classical Lagrage numerical analysis and Euler-Lagrange coupling. The Mie-Grüneisen equation of state parameters were considered according to Chapman et al.(2006) and the results obtained through the analysis of the calibration models were compared and the parameters were altered so that the results were convergent. The simplest model implied the Dynamic Explicit analysis of two parallelepipeds defined as soils placed between two solid concrete beams. The first model was analysed with the classic Lagrage formulation and the latter using the Euler-Lagrage Coupling. The deformations of the two models, taking into account that both suffered small deformations, were convergent. The general displacement of the Euler-Lagrange Coupling model could not be computed because the nodes of the Euler part don’t have an actual displacement, but this fact can be corrected using a control Lagrange part. The development of plastic zones is more obvious in the case of the Euler model because the influence of the efforts applied on the instance simulates more accurately the real behaviour.
Fifth International Young Geotechnical Engineering Conference - 5iYGEC’13
Figure 4. The shear stress for Euler-Lagrage Coupling model (left) and Lagrange model (right) The second model used for calibration consists in studying the stability of a soil mass subjected to gravity. The results were convergent in both types of analyses, but it is considered that the EulerLagrange model behaves more accurate compared to the real case. Figure 5 presents the difference between the two models in terms of von Mises stresses (top representing the Euler – Lagrange coupling and the bottom the classic Lagrange model).
Figure 6. The von Mises stresses in the soil for Euler-Lagrage Coupling model (top) and Lagrange model (bottom) If the stress state of the soil is considered (Figure 6), the results are significantly close: Smax = 116.4kPa reached at the base of the soil, mostly due to its own weight, for the Lagrange model, while Smax = 142.9kPa for the coupled model. Some areas of the soil column present integration problems, but the overall response of the model is fairly similar.
Figure 5. The von Mises stresses for Euler-Lagrage Coupling model (top) and Lagrange model (bottom) After calibrating the parameters using the simple models presented above, the final verification was made for the behaviour of a foundation raft, analysing the soil stress state and the contact pressures on the raft.
Figure 7. The contact pressures on the base of the raft for the Euler – Lagrage method
Fifth International Young Geotechnical Engineering Conference - 5iYGEC’13 Us-Up linear equation and calibrating them in the case of an actual landslide through in-situ monitoring techniques.
6. References
Figure 8. The contact pressures on the base of the raft for the Lagrage method Considering the contact pressures between the foundation system and the soil, as it can be observed in Figure 7 and Figure 8, some differences appeared. Although from the mechanical point of view, the outer part of the raft surface is subjected to higher efforts, the EulerLagrange method proves several discontinuities of integration, resulting in aberrant values. If we consider the maximum values, in the case of the Lagrange modelling, an effort of 210.7kPa has been obtained on the most compressed area, while using the Euler-Lagrange coupling, the maximum effort reached 610.4kPa. If we disregard these errors, we obtain a similar maximum value of 254kPa. The difference between the two models lies in the constitutive laws of the material and phenomena type. Therefore, while in the Lagrange model an instantaneous calculation is conducted using an elasto-plastic model, in the EulerLagrange coupling, a transient phenomenon took place and the calculation achieved a total time of 10s. This time value does not permit any creep, hardening or softening effects to appear. Overall, both models achieved a similar response in terms of variation of the deflection of the raft.
5. Conclusions Comparing the results obtained from the direct shear test for the viscosity determination and the usual viscous behaviour models, the conclusion is that this method can be used, but taking into account the limitations due to variable normal stresses applied. A constitutive model taking into account the spherical stress could provide better results. The equation of state parameters have been determined using the numerical matching and the results obtained from the classical Lagrange model and from the Euler-Lagrange couple are convergent. These parameters can be used for modelling landslides behaviour and in general all stability problems, using the models chosen for calibration to determine them for each case in particular. The following research directions will be a laboratory method for determining the Hugoniot
Alonso, E. et al (2010). Geomechanics of Failure. Advanced Topics. Springer; 1st Edition Arrigada, M.C.S. et al. (2002). Numerical Modeling of Coupled Phenomena in Science and Engineering: Practical Use and Examples (Multiphysics Modeling). Taylor & Francis; 1st edition. Chapman, D.J. et al. (2006). The behaviour of dry sand under shock-loading. American institute of Physics. Locat J, Demers D. (2006). Viscosity, yield stress, remolded strength, and liquidity index relationships for sensitive clays. Can Geotech J 25(4):799–806. Pastor, M. et al. (2003). Modelling of Landslides: (I) Failure Mechanisms. Centro de Estudios y Experimentacion de Obras Publicas, Madrid, Spain. Pastor, M. et al. (2003). Modelling of Landslides: (I) Propagation. Centro de Estudios y Experimentacion de Obras Publicas, Madrid, Spain.
