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Francesca Ceccato

Università degli Studi di Padova - DICEA

Lars Beuth

Deltares – Delft (The Netherlands)

Pieter Vermeer

Deltares – Delft (The Netherlands)

Paolo Simonini

Università degli Studi di Padova - DICEA


Large deformation problems are quite common in geotechnical engineering, e.g. pile driving, landslides, underground excavations. Because of mesh distorsions, Finite Element Method (FEM) which takes into account large deformation effects, such as Updated Lagrangian FEM, is not suitable to analyze these problems. Advanced techniques have recently been developed to overcome mesh distortion drawbacks. The Material Point Method is one of them. In the MPM the continuum is discretized by a cloud of material points (MP) which moves through a background mesh, thereby reproducing the large deformations of the solid. The MP carry all the properties of the continuum. This note shows the application of the MPM to the penetration of a piezocone in undrained clay.


Problemi di grandi deformazioni sono piuttosto comuni in ingegneria geotecnica e i più diffusi metodi numerici come i FEM soffrono d’inaccuratezza a seguito di grandi distorsioni della mesh. Per superare questo limite sono state sviluppate altre tecniche (Arbitrary Lagrangian Eulerian, metodi particellari e meshless, ecc); tra queste risulta particolarmente promettente il Material Point Method, che può essere considerato un’evoluzione del classico FEM Lagrangiano. Il corpo è discretizzato con un insieme di punti materiali, aventi tutte le proprietà del continuo, che possono muoversi attraverso una mesh computazionale che può essere arbitrariamente modificata. In questa nota viene presentata un’applicazione del metodo alla penetrazione del piezocono in condizioni non drenate.

1 Introduction

In the Lagrangian formulation the nodes of the mesh move together with the material; on the contrary in the Eulerian formulation the material flows through a fixed mesh. In the first case the element mesh


can suffer from heavy distortions through which the accuracy of calculations deteriorates significantly. As a remedy remeshing techniques can be employed. In this case state variables must be mapped from the distorted to the new mesh, but the procedure is not straightforward and additional errors might be introduced. On the other hand, the Eulerian formulation is not suitable for problems with history dependent materials, as state variables are not traced for material points, but fixed point in space (Sulsky et al., 1994). The need to overcome these limitations encouraged research on advanced numerical methods such as combined Lagrangian-Eulerian methods e.g. Arbitrary Lagrangian Eulerain method (ALE), meshfree and meshless methods (Li & Liu, 2002).

The Material Point Method (MPM) can be regarded as an extension of the classical Updated Lagrangian Finite Element Method. It has been successfully applied to many problems of solid mechanics in the last 20 years. It was first applied to granular material by Więckowski et al. (1999, 2004) and has since then been introduced to geotechnical engineering, see e.g. Coetzee et al. (2005), Beuth et al. (2007), Alonso & Zabala (2011), Al-Kafaji (2013). This paper presents an application of the Material Point Method to the simulation of cone penetration in undrained conditions.

Cone penetration testing is a widely-used in-situ test, since it is cost-effective and can be applied to a wide range of soils; moreover it allows to recognize soil type and estimate mechanical properties. A deeper understanding of the penetration process leads to a more accurate interpretation of measurements and a more aware use of the soil parameters in practice.

In section 2 a brief introduction of the MPM is given; a detailed presentation of the formulation exceeds the purpose of this note. The model for undrained conditions is explained in section 3. The following section is dedicated to CPT modeling. Here the geometry of the model is described as well as the features of the contact algorithm, which has been improved in the frame of this research to model cone penetration in clay. Preliminary results are presented in section 4.4. Conclusions and future developments close the note.

2 The Material Point Method

The MPM is a sophisticated numerical technique specifically developed to model large deformations (Coetzee et al., 2005). The continuum is discretized by a cloud of Lagrangian points, called material points. Large deformations are modeled by MP moving through an Eulerian fixed mesh. The MP carry all the physical properties of the continuum such as mass, momentum, material parameters, strains, stresses as well as external loads, whereas the Eulerian mesh carries no permanent information.

Through this approach, MPM combines the advantages of both, Lagrangian and Eulerian formulations, while avoiding shortcomings of them. Indeed, mesh distortion is prevented and numerical diffusion associated with the convective terms in the Eulerian approach does not appear in MPM solution (Al-Kafaji, 2013).

For an applied load increment, the underlying finite element grid is used to solve the system of equilibrium equations. Once strains are computed at the locations of the material points, the mesh is usually reset into its original state. However it might be changed arbitrarily. The assignment of MP to finite elements is updated after mesh adjustment (Fig. 1). The finite element grid used with the MPM must cover not only the solid in its initial configuration, but the entire region of space into which the solid is expected to move. However, only those elements which contain material points (active elements) contribute to the equation of motion.

It should be emphasized that in contrast to meshless Lagrangian methods such as the Discrete Element Method, the material points represent subregions of a solid body and not individual particles such as sand grains.

The MPM code used in this study is being developed to solve dynamic large deformation problems in the field of geotechnical and hydromechanical engineering (Vermeer, 2013). However it is also suited for analyses of quasi-static problems, i.e. problems where inertia of a deforming body plays no significant role, as assumed for CPT. The code features a contact formulation to model soil-structure interaction, as presented in section 4.2, and it supports different soil models (Tresca, Mohr-Coulomb, Hypoplasticity and Strain Softening). It has recently been complemented by absorbing boundaries to prevent reflection of waves at mesh boundaries. Furthermore it has been extended by a two-phase


formulation which allows consolidation analysis of large deformation problems.

Figure 1 Movement of material points through the computational grid for a time step; (left) initial configuration; (centre) incrementally deformed mesh; (right) reset mesh (Beuth, 2012)

3 Undrained conditions

In this study numerical analyses of cone penetration testing are performed for fully saturated clay. When the rate at which a load is applied on soil is much faster than the rate at which pore pressure is able to dissipate out of the soil, volume change is prohibited and pore water pressures are generated that balance, together with effective stresses, the applied load.

With the performed analysis, the total mean stress rate of the undrained soil, ̇, is split into effective mean stress, ̇, and the excess pore pressure, ̇ (effective stress principle). Consideration of strain compatibility gives:

̇ ⁄ ̇ and ̇ ̇ ,

where Kwater is the bulk modulus of water, n is the porosity and K’ is the bulk modulus of the soil skeleton. The term Kwater/n can also be written as

( ) ( )( )

where u and ’ are the undrained and the effective Poisson ratio respectively. Incompressibility of the soil implies u close to 0.5.

In this study it is assumed Kw = 28987 kPa, n = 0.3, ’ = 0.25, u = 0.49, and K’ = 3355 kPa. The strength of undrained clay is modeled by the Tresca model, assuming cu = 20 kPa.

4 Modeling Cone Penetration Testing

The cone penetration test is performed by pushing a measuring device, the cone penetrometer, that is attached to the tip of a steel cylindrical rod, into the ground with a constant rate of penetration of 2 cm/s. The standardized cone penetrometer has a conical tip with an apex angle of 60° and a base area of 10 cm2 (corresponding to D = 3.57 cm).

Generally, the tip resistance is related to an undrained shear strength cu by:

where Nc is the cone factor and 0 the overburden stress. Empirical cone factors are available on the basis of field investigations. They are supported by theoretical correlations based on analytical or numerical models of the cone penetration process (see e.g. Robertson, 2009). Analyzing the correlations between soil properties and tip resistance enables more reliable predictions of soil properties. This study aims at contributing to this goal.

4.1 Geometry

A 20° slice of the axisymmetric cone penetration problem is considered. The cone is slightly rounded in order to circumvent numerical problems induced by a discontinuous edge at the base of the cone. Apart from this modification, the dimensions of the penetrometer correspond to those of a standard penetrometer: the apex angle is 60º and the horizontal base area is 10 cm2.


Figure 2 Computational mesh (a) and initial particle discretization (b)

The radius of the discretized domain is about 4D. It extends 5D below the tip (Fig. 2a). Roller boundaries are prescribed at lateral surfaces, the bottom is fixed.

The cone is pushed into the ground, through prescribed velocities, down to 2D. Here the penetrometer itself experiences no deformations. The steady state solution is assumed to be reached after a penetration of about 6D (Beuth, 2012). Thus this shallow penetration is suitable for testing the algorithm, but not to obtain the cone factor.

In MPM the computational mesh does not align with the boundary of the solid and this makes the application of boundary conditions not straightforward. The difficulties arise in particular when dealing with nonzero boundary conditions (tractions and prescribed velocities). This can be avoided by the concept of the moving mesh which is a procedure that ensures that the computational mesh aligns with the surface where traction is applied (Beuth, 2012). The moving mesh zone is attached to the penetrometer and moves with the same displacement as the cone. The elements of this zone keep the same shape throughout the computation, while the elements in the compressed zone reduce their vertical length. It should be emphasized that the elements of the compressed zone must have a reasonable aspect ratio in the initial configuration to avoid mesh distortion.

A coarse mesh is used for the performed initial test computations, consisting of 5324 4-noded tetrahedral elements which have been extended by a strain smoothing algorithm to prevent locking. 40612 material points are placed inside the initially activated elements (Fig. 2b).

4.2 The contact algorithm

The contact between soil and cone is modeled by Bardenhagen’s algorithm (Bardenhagen et al., 2001). It might be viewed as a predictor-corrector scheme, in which the velocity is predicted from the solution of each body separately and then corrected using the velocity of the coupled bodies.

The equations of motion of the two bodies as well as that of the combined bodies are solved separately. If at a certain node the velocity of the body (vnB) is equal to the velocity of the combined


system (vnS) there is no need for

correction, else it must be checked if the bodies are approaching or separating. If they are separating there is no need for correction, else sliding condition must be checked. In our case the bodies are sliding if the contact force is greater than the specified adhesion, a. To this end, the existing frictional contact formulation has been extended in the frame of this study to take into consideration adhesion. If the bodies stick to each other, then the velocity is equal to the one of the combined system. Otherwise the normal component is corrected in order to avoid interpenetration and the tangential component is corrected in such a way that the tangential force is equal to the maximum contact force. A flowchart of the contact algorithm is presented in Figure 3.

4.3 Reference computations

A MPM code has been developed by Vermeer and Beuth specifically for the

analyses of quasi-static problems (Beuth, 2008). The quasi-static Material Point Method has been successfully used to simulate cone penetration in undrained clay by Beuth (2012). Contact is modelled by special interface elements (Vermeer et al., 2009). Calculations were performed with Tresca, Mohr-Coulomb and the AUC material model. The latter has the capability of considering the load-type dependency of the undrained shear strength and the strength anisotropy of natural clay. The relationship between cone tip resistance and the strength of undrained clay has been investigated and compared to results obtained with the ALE method (Van den Berg, 1994).

In the presented study explicit instead of implicit time integration and an alternative contact formulation are applied to simulate the cone penetration. Results are compared to previous MPM analyses.

4.4 Preliminary results

Preliminary results prove the capability of the used MPM code to simulate CPT in undrained clay. As can be seen from Figure 4, the tip stress increases with the penetration depth and the adhesion factor. Results agree with those presented by Beuth (2012), although the steady state solution is not reached with the performed preliminary analysis.

Principal stress directions are shown in Figure 5. For smooth contact (a = 0 kPa) the principal stress 1 is perpendicular to the surface, while for rough contact (a=20 kPa) it is rotated 45° with respect to the cone surface.


Figure 4 Tip stress for different values of adhesion Figure 6 Principal effective stress near the tip for rough (left) and smooth (right) contact.

5 Conclusions and future developments

The method seems to be very promising for simulating large deformation problems because of its capability of avoiding the drawbacks of both Lagrangian and Eulerian FEM. The computational costs are higher than with the FEM, but considerably less than with meshless methods.

Preliminary results of the performed CPT analyses are promising, but further investigations are necessary. Mesh refinement will be investigated to reach an optimal compromise between computational cost and accuracy.

It is very important to study the effect of the boundary conditions, in particular the introduction of absorbing boundaries and spring boundaries, capable of simulating the stiffness of the real unbounded domain. Obviously a deeper cone penetration will be considered to obtain the appropriate cone factor. Furthermore the available consolidation formulation is presently made ready for application to the large deformation analysis of CPT. It will be particularly interesting to simulate CPT penetration in partially drained conditions which are very common in soils like silts, but are not fully understood yet. Many problems in geotechnical engineering are multiphase phenomena and the formulation of a multiphase MPM, capable to describe large deformations problems, is of great interest.


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Figure 2 Computational mesh (a) and initial particle discretization (b)

Figure 2

Computational mesh (a) and initial particle discretization (b) p.4
Figure 3 Flow chart explaining the contact algorithm.

Figure 3

Flow chart explaining the contact algorithm. p.5
Figure 4 Tip stress for different values of adhesion           Figure 6 Principal effective stress near the tip for  rough (left) and smooth (right) contact

Figure 4

Tip stress for different values of adhesion Figure 6 Principal effective stress near the tip for rough (left) and smooth (right) contact p.6