Modelling Progressive Failure with MPM

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1 Modelling Progressive Failure with MPM A. Yerro, E. Alonso & N. Pinyol Department of Geotechnical Engineering and Geosciences, UPC, Barcelona, Spain ABSTRACT: In this work, the progressive failure phenomenon is studied using the Material Point Method (MPM). MPM is categorized as a method in between the classical Finite Element Method (FEM) and the truly particle methods. It is capable of modeling large deformations, displacements and contact between different bodies. Two different geotechnical applications are presented: the pull-out of a pile; and the failure and runout of a slope due to a rise of the phreatic level. In both cases a strain-softening Mohr-Coulomb constitutive model is used to simulate the brittleness of the materials. 1 INTRODUCTION The progressive failure phenomenon is a common problem in the geotechnical engineering field and it has been studied by different authors (Potts et al., 1990; Dounias et al., 1996; Potts & Zdravković, 2001; Jardine et al., 2004). Zabala & Alonso (2010) applied the MPM to analyze the brittle failure of Aznalcóllar dam. Progressive failure mechanisms are directly associated with brittle materials. These are characterized by means of a strain softening constitutive law. Strain softening causes a reduction of the strength from peak to residual conditions. The strength loss does not occur simultaneously along a potential rupture surface because the rate of softening is linked with the magnitude of shear strain. Therefore the progression of failure is controlled by the stress strain behavior of both, the localized shearing surface(s) and the continuum medium itself. The Material Point Method (MPM) was initially developed by Sulsky et al. (1994) and it is categorized as a method between the classical Finite Element Method (FEM) and the particle methods. MPM discretizes the media in two different ways: a set of material points (Lagrangian mesh) which move through a fixed finite element grid (Eulerian mesh). This dual description provides MPM a few specific features: it is capable of modeling large displacements and large deformations without mesh distortion problems; the contact between different bodies is automatically solved and the frictional contact can be easily implemented. These properties provide advantages to analyze a number of geotechnical problems such as landslides and the process of installation of structures (pile driving, caisson sinking, etc). Two different geotechnical applications are presented in this paper. On the one hand, the interaction between a pile and the adjacent soil, under pull-out conditions is modelled. On the other hand, the failure and run-out of a slope due to a rise of the phreatic level is simulated. In both cases a strainsoftening Mohr-Coulomb constitutive model is used to simulate the brittleness of the materials. 2 OUTLINE OF MPM FORMULATION The MPM (Sulsky et al., 1994) represents the material as a collection of unconnected points so-called material points (Fig.1). The mass of a region is considered to be concentrated in a material point. Then, the density of the mixture can be expressed as x, x - x (1) in which m p and x p are the mass and the position of the p th material point, δ(x) is the Dirac delta function, and N p is the total number of material points. An important assumption is that the mass assigned to each material point remains fixed during all the calculation, thus assuring mass conservation. Other quantities such as velocities, strains and stresses, are also carried by the material points. Otherwise the governing equations are solved at the support numerical mesh, which covers the full domain of the problem. The standard linear shape functions pro-

2 vide the relationship between the material points and the nodes of any point of the domain. The MPM formulation for a mechanical problem was presented in Sulsky & Schreyer (1996), where basically the equation of dynamic momentum balance is discretized. MPM has also been extended to solve coupled hydro-mechanical problems based on the well-known equations described in Zienkiewicz & Shiomi (1984) and Verruijt (2010). In this case, the solid velocity-liquid velocity formulation developed by Jassim et al. (2012) for MPM has been considered. Figure 1. Discretization in material points and a finite element mesh used in MPM. 3 STRAIN SOFTENING CONSTITUTIVE MODEL This paper extends the basic non-associated Mohr- Coulomb law by introducing strain softening plasticity with the aim of modeling the strength loss that occurs after peak strength conditions. Moreover, in order to reduce the singularities of the Mohr- Coulomb yield surface, the modifications proposed by Abbo & Sloan (1995) have been implemented. The Mohr-Coulomb yield surface can be written in the following way, (1) where 2 ; 2 (2) and being the maximum and minimum effective principal stresses. The softening behavior is accounted for by allowing the strength parameters (friction angle φ, and cohesion c ) to decrease with the accumulated equivalent plastic strains ɛ p eq according to the softening rules, (3) (4) The model requires the specification of peak (c p,φ p ) and residual (c r,φ r ) strength parameters, and an additional parameter η. 4 LOCALIZATION AND MESH DEPENDENCE It is well known that the inclusion of strain-softening features in standard continuum numerical methods (such as FEM and MPM) leads to the strainlocalization phenomenon (Oliver & Huespe, 2004), which is the concentration of strains in narrow bands. This might be understood as a way to model displacement discontinuities. However, mesh dependent regularizations are needed to avoid meshdependence problems. In this work, the smeared crack approach (Rots et al., 1985) has been used as a regularization technique. In this procedure the plastic work dissipated by a mesh element is made equal to the fracture energy dissipated at the crack. Therefore the plastic softening modulus is dependent on the mesh size. The strength of this smeared surface is assumed to be a function of the relative displacement at the discontinuity. In the exponential term of the softening rules (Eq.3 and Eq.4), the parameter is calibrated to obtain a specific relation between shear stress and relative displacement for the fixed size of the mesh element. 5 PULL-OUT OF A PILE A pull-out of a pile has been modelled with the purpose of studying the interaction between the structure and the surrounding terrain. Plane strain conditions are considered and only half space has been modelled in order to optimize the computational time. The pile is 10 m deep and it has a total width of 0.25 m (see geometry in Fig.3). The installation effects have not been taken into account, which means that all materials involved in the problem have an initial stress due to the gravity when the pile starts to be pulled-out. The horizontal displacements along the vertical boundaries and the vertical displacements at the bottom contour have been prevented. The pull-out has been modelled considering that the top of the pile moves upward with a constant velocity of m/s. Neither contact elements nor contact algorithms have been implemented in this calculation. The pile is considered as an elastic material and the surrounding terrain has been modelled as a brittle rock with the strain softening Mohr-Coulomb constitutive model previously described. The following material parameters have been used:

Table 1. Material parameters. Pile Dry unit weight (kn/m3) 20 Young modulus (MPa) 30000 Poisson s coefficient 0.2 Terrain Dry unit weight (kn/m3) 25 Young modulus (MPa) 1000 Poisson s coefficient 0.

3 Table 1. Material parameters. Pile Dry unit weight (kn/m3) 20 Young modulus (MPa) Poisson s coefficient 0.2 Terrain Dry unit weight (kn/m3) 25 Young modulus (MPa) 1000 Poisson s coefficient 0.33 Cohesion (peak/residual) (kpa) 65 / 0.5 Friction angle (peak/residual) (º) 25 / 20 Calibration parameter, 1000 the residual conditions. In Figure 2 the distribution of the total shear strain is represented at two times (1 s and 2 s). It shows how the strain localizes at the pile-soil interface and the progression of the shear band towards the pile tip. A more detailed analysis of the development of the failure has been done using the mobilized shear strength concept, which is a measure of the intensity of shear (q) in a particular point. Moreover, in order to compare stress-evolutions of different material points at different depths, the mobilized shear strength has been normalized with respect to the volumetric stress p*: tan (5) sin (6) In this way, the normalized strength can be understood in terms of a mobilized friction angle (Eq.6). Under peak and residual conditions the mobilized friction angles coincides with the peak and residual friction angles respectively. The mobilized shear strength of 10 material points distributed along the shear band (at depths of 0.5 m, 1.5 m, 2.5 m, 3.5 m, 4.5 m, 5.5 m, 6.5 m, 7.5 m, 8.5 m, 9.5 m) have been determined for different times. The resulting distributions are shown in Figure 3. These plots provide a clear view of the progressive failure phenomenon. The strength loss does not occur simultaneously, but according to the stress state and the accumulated plastic shear strain. The stress path of the material point located at the pile-soil interface at a depth of 7.5 m is represented in Figure 4. The peak and residual Mohr-Coulomb yield surfaces have also been included. From an initial elastic state of stresses due to the gravity (which is a coincidence that matches the residual yield surface), the confinement of the material point reduces. Then, the shear stress rises until the peak yield surface is reached. At this point the plastic shear strain increases with the pulling of the pile, initiating the softening process and drifting the strength down to Figure 2. Geometry of the problem and distribution of the total shear strain at different times: (a) t = 1 s; (b) t = 2 s. The results are plotted on the mesh. Figure 3. Distribution of the mobilized shear strength (normalized) along the failure for different times.

As a result, the gravitational stresses are sufficient to induce a progressive failure that initiates at the foot of the slope and propagates upwards. Subsequently, the instability occurs.

4 Figure 4. Stress path of a material point located 7.5 m deep in the shear band. slope. This fact leads to some points to reach the peak conditions. After that, the strain softening effect, implicit in the constitutive model, decreases the strength parameters of the plastic zones down to the residual yield surface. As a result, the gravitational stresses are sufficient to induce a progressive failure that initiates at the foot of the slope and propagates upwards. Subsequently, the instability occurs. The mobilized shear strength defined in Eq.6 is plotted in Figure 5 along the shear surface developed. The progressive mechanism starts in the lower part of the slope (point 0 ) and propagates upwards. The development of the failure and run-out of the slope is illustrated in Figure 6, where the contours of the shear strain provide the evolution of the shear band. 6 SLOPE INSTABILITY This example consists on simulating the instability of a slope, 6 m high and 37º steep, due to a rise of the phreatic level. To do that, an increase of the pore pressure of 40 kpa has been imposed at the lower boundary. This is a plane strain simulation, where the horizontal displacements along the vertical contours have been prevented and the lower boundary has been completely fixed. The mesh has been refined in the domain where the failure is expected in order to get more accurate results and optimise the computational cost. The strain softening Mohr-Coulomb constitutive model has been used to simulate the brittle behaviour of the soil. The properties of the material are presented in Table 2. Figure 5. Distribution of the mobilized shear strength (normalized) along the initial shear band for different times. Table 2. Soil parameters of the slope. Soil parameters Porosity Intrinsic permeability (m3/s2) Dry unit weight (kn/m3) Young modulus (kpa) Poisson s coefficient Cohesion (peak/residual) (kpa) Friction angle (peak/residual) (º) Calibration parameter / / Initially, a hydrostatic pressure distribution is assumed and saturated conditions are considered during all the calculation. The calculation starts with the increase of pore pressure along the lower boundary, which is linearly increased during 10 seconds, up to 40 kpa. After this time, the water pressure on the boundary is maintained constant during all the calculation. The excess pore pressure induced by the boundary condition reduces the effective stresses in the Figure 6. Distribution of the total shear strain at the material points in different times. The range of shear strain increases with time: (a) from 0 to 0.2; (b) from 0 to 3; (c) from 0 to 8.

5 Figure 7. Evolution of the run-out (total displacement) of 3 different material points (P1, P2, and P3) originally located in the initial shear band. Figure 7 shows the run-out of three material points located along the initial shear band. This plot helps to identify the different phases of an instability process. During the initial phase, up 20 seconds, the slope is stable. Although there is no significant motion, the progressive failure process is taking place (see Figure 5), leading to the failure mechanism. In the course of the second phase, the mobilized mass accelerates and moves forward. This implies the development of large displacements and deformations until it the slope reaches a new stable profile. In this calculation, the maximum displacement after stabilization is 10.9 m. Conference. (R.J. Jardine, D.M. Potts, K.G. Higgins, eds.). Thomas Telford, London (Vol. 1, pp ). Jassim, I., Stolle, D., & Vermeer, P. (2012). Two- phase dynamic analysis by material point method. International Journal for Numerical and Analytical Methods in Geomechanics. doi: /nag Oliver, J., & Huespe, A. E. (2004). Continuum approach to material failure in strong discontinuity settings. Computer Methods in Applied Mechanics and Engineering, 193(30-32), doi: /j.cma Potts, D. M., & Zdravković, L. (2001). Finite Element Analysis in Geotechnical Engineering: Theory & Application (p. 427). Thomas Telford. R Rots, J. G., Nauta, P., Kuster, G. M. A., & Blaauwendraad, J. (1985). Smeared Crack Approach and Fracture Localization in Concrete. Sulsky, D., Chen, Z., & Schreyer, H. L. (1994). A particle method for history-dependent materials. Computer Methods in Applied Mechanics and Engineering, 118(1 2), doi: / (94) Sulsky, D., & Schreyer, H. L. (1996). Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems. Computer Methods in Applied Mechanics and Engineering, 139(1 4), doi: /s (96) Verruijt, A. (2010). Theory and applications of Transport in Porous Media. In Springer Verlag (Ed.), An Introduction to Soil Dynamics. Berlin, Heidelberg. Zabala, F., & Alonso, E. (2010). Modelación de problemas geotécnicos hidromecánicos utilizando el método del punto material. Universitat Politècnica de Catalunya. Zienkiewicz, O. C., & Shiomi, T. (1984). Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution. International Journal for Numerical and Analytical Methods in Geomechanics, 8, CONCLUSIONS The progressive failure phenomenon has been studied with the MPM. A strain softening Mohr- Coulomb constitutive model has been used to simulate the brittleness of the material, and two different numerical examples have been solved: a pull-out of a pile and the instability of a slope. The paper shows the capability of the MPM to simulate the whole instability process in a unified calculation, which encompasses small deformations during the initial loss of instability and large deformations and displacements during run-out. 8 REFERENCES Abbo, A., & Sloan, S. (1995). A smooth hyperbolic approximation to the Mohr-Coulomb yield criterion. Computers & structures, 54(3), Dounias, G. T., Potts, D. M., & Vaughan, P. R. (1996). Analysis of progressive failure and cracking in old British damsno Title. Géotechnique, 46(4), Jardine, R. J., Gens, A., Hight, D. W., & Coop, W. R. (2004). Developments in understanding soil behaviour. Advances in geotechnical engineering. The Skempton

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