SIMULATION OF PIEZOCONE PENETRATION IN SATURATED POROUS MEDIUM USING THE FE REMESHING TECHNIQUE

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1 F O U N D A T I O N S O F C I V I L A N D E N V I R O N M E N T A L E N G I N E E R I N G No Darius MARKAUSKAS *, Rimantas KAIANAUSKAS *, Rolf KATZENBACH ** * Vilnius Gediminas Technical University ** Darmstadt University of Technology SIMULATION OF PIEZOCONE PENETRATION IN SATURATED POROUS MEDIUM USING THE FE REMESHING TECHNIQUE The simulation of piezocone penetration in saturated soil modelled as porous medium has been presented herein. The non-linear transient coupled soil-pore fluid model and the finite element method (FEM) have been applied. To handle very large distortion of medium surrounding the cone tip, a finite element remeshing technique for deformed domain involving combined transfer of the state variables has been developed. The moving least square method was utilized for the transfer of Gauss points variables while the simple interpolation method was used for the transfer of nodal points variables. This technique was implemented into the post-processor type software compatible with a standard FEM code. The proposed remeshing technique enables the simulation of the penetration process with penetration depth of several diameters until steady state penetration is reached. The validity of the remeshing is justified by comparing the simulation results in porous and homogeneous media. Key words: porous medium, remeshing technique, finite element method, penetration test 1. INTRODUCTION The cone penetration test is an important tool to evaluate in-situ properties of soil. One of the primary applications of cone penetrations tests (CPT) was to measure soil resistance at the tip of the penetrometer and the friction along the sleeve of the tool. The correlations for the assessment of the cone testing results Publishing House of Poznan University of Technology, Pozna 2005 ISSN

2 104 Darius Markauskas, Rimantas Kaianauskas are obtained by combining the results of different experimental, analytical and numerical methods. Most of the existing correlations were determined for fine grained or coarse grained soils in which cone penetration is performed in fully drained or undrained conditions. To account for the generated pore pressure, the cone penetration test with pore pressure measurement or piezocone test (CPTU) is widely used. The piezocone penetrometer differs from a conventional static cone penetration device by the presence of a porous element that allows pore pressure measurements at multiple locations. Nowadays, however, most of the efforts are focused on the correlation and correction of measurement results or classification of soils, rather than physical interpretation of factors influencing the pressure and effective stress fields around the cone. A comprehensive review on the CPT and CPTU may be found in the book of Lunne et al. [12]. The improvement of CPT is still continued while a miniature cone penetration test system is presented by Tumay and Kurup [20]. An efficient way of the interpreting of the soil-cone interaction is numerical modelling. The finite element method widely used in engineering besides existing semi-empirical approaches, becomes the most attractive alternative to handle the penetration problem and the influence of various factors on cone resistance. Simulation of cone penetration taking into account the natural material non-linearity is, in fact, a complicated geometrically non-linear problem, since the cone must be pushed numerically into the soil with a vertical displacement of several times the diameter of the cone. For most of the early simulations conventional small strain displacement finite elements have been explored for the cone penetration analysis [4]. Kousis et al. [8] presented a large strain formulation and its application to the analysis of the cone penetration test in clay soil. The large cone penetration analysis is also presented by Sheng et al. [17]. However, in the cases when partial drainage of the pore water occurs during penetration the soil should be considered a porous media, where coupling of the skeleton model and pore water flow has to be taken into account. The phenomenological approach used by Biot [3] in developing a theory for elastic porous medium forms the basis for various extended theories. The extension of Biots s theory into the non-linear range was probably first proposed by Prevost [16]. The coupled soil-pore fluid FE formulation of fully saturated soil behaviour including inertia forces was considered by Zienkiewicz et al. [25] and [26]. Many mechanical and numerical aspects of flow in porous media are covered in a comprehensive study of Lewy and Schrefler [10]. Voyiadjis and Abu-Farsakh [23] adopted and implemented the updated Lagrangian FE formulation originally introduced by Bathe [2] to simulate cone penetration into the non-linear porous medium. This approach was later explored by Voyiadjis et al. in a series of works for simulation of different cone

3 Simulation of piezocone penetration in saturated porous medium 105 penetration problems in porous media, for example, to investigate the permeability of soils using the piezocone [18] or to take into account viscoplastic properties of skeleton [24]. However, large distortion of the finite element geometry during penetration leading to ill-conditioned equations and failure of iterative process restricted the application of the FEM to CPT modelling. Difficulties associated with this kind of problems are discussed by van den Berg [21], being also pointed out by Song et al. [18], Markauskas et al. [14] and Voyiadjis and Kim [24]. Some details regarding jumping of soil-cone interface nodes of deformed homogeneous medium due to sliding are illustrated by Markauskas et al. [13]. However, the validity of the conventional porous medium non-linear analysis presented by Voyiadjis and Kim [24] was limited to the penetration depth of 3.5 diameter of the cone. To avoid large mesh distortion that occurs in the deep cone penetration analysis, van den Berg [21] and van den Berg et al. [22] used an arbitrary Lagrangian-Eulerian formulation to uncouple the nodal point displacements and velocities from material displacements and velocities. Van den Berg [21] presented some results of the cone penetration simulation in partially drained soil by using a simplified uncoupled FEM analysis. Another alternative to avoid the large mesh distortion is a remeshing technique involving transfer operation of state variables from the deformed mesh to the new mesh started to be applied in non-linear analysis. Theoretical aspects of this technique were probably first generalized by Peric et al. [15]. Most of the applications of remeshing are focused on metal forming and fracture problems, however, the application of remeshing to solving geotechnical problems is still limited [6, 11]. Susila and Hryciw [19] have applied remeshing technique to study cone penetration in drained sand using FE explicit solution algorithm. The FE remeshing based on stress patch recovering technique [27] have been implemented and applied to cone penetration in fully undrained soil using the FE implicit solution algorithm by the authors in [13]. All above mentioned applications of remeshing in geotechnical problems were limited to modelling of soil as homogeneous medium. This paper presents a FE remeshing technique for the deformed domain involving a combined transfer of the state variables for a coupled soil-pore fluid problem. A major goal of the presented work is to illustrate the capability of the developed remeshing technique to capture steady state penetration of piezocone in saturated porous medium with penetration depth of several diameters of the cone and to investigate the influence of soil permeability on cone resistance and pressure fields. The FE formulation for the coupled soil-pore fluid problem with remeshing is presented in Section 2. The FE remeshing technique involving a combined transfer of the state variables is briefly described in Section 3. Finally,

4 106 Darius Markauskas, Rimantas Kaianauskas the illustration of the numerical simulation performed and discussion on the results are reported in Section THE FE COUPLED SOIL-PORE WATER MODEL WITH REMESHING The non-linear transient coupled soil-pore fluid model is applied to present saturated porous medium. In the framework of the current investigation, porous medium is considered as the two-phase medium consisting of the solid grains (skeleton) and the pore water[10]. Each phase is regarded as individual continuum following its own behavior governed by different equations. The skeleton is assumed to be elastic-plastic solid undergoing large strains and large displacements, while the flow of pore-water through the voids is assumed to follow Darcy s law. The behavior of porous solid is predicted by the influence of pressures of pore fluid filling the voids and the concept of effective stress is employed to describe the effect of strong coupling. The coupled soil-pore fluid model of porous medium presents the equilibrium equations of the skeleton-fluid mixture which can be written using the principle of virtual work, as it is usually done in mechanics of non-linear solids, and the balance of mass, which simply means a transient seepage equation. By omitting the details, it may be sufficient to consider a general matrix form of coupled transient static finite element equations, expressed in terms of the nodal displacements u, nodal pressures p and their velocities u and p [26]: 0 Q T 0u Sp t t K 0 Qu Hp t t F - q t t. (2.1) Here, K presents the structural stiffness matrix, the fluid matrices S and H present compressibility and permeability, while Q couples the field of pressures in the equilibrium equations. In saturated soil the compressibility is neglected, therefore matrix S is assumed to be a zero. The right-hand vector stands for external forces F and q including boundary terms. The original updated Lagrangian FE formulation of coupled equation (2.1) may be presented as follows [23]: t t t t K Q u F -. t T t t t (2.2) Q Ht p q

5 Simulation of piezocone penetration in saturated porous medium 107 Actually, the FE equation (2.2) is an incremental equation, where stands for the increment, while a left-side superscript t indicates the configuration related to time t in which the quantity occurs. The original formulation of eq. (2.2) assumes the constant finite element mesh to be fixed during the entire operation interval. As the mesh is regenerated with respect to an appropriate criterion, the incremental solution procedure, in general, cannot be recomputed from the initial state t = 0, but has to be continued from the previously computed state t. The outline of current presentation is focused on the issues of remeshing applied during the simulation process. Let us consider matrix t K as a representative characteristic matrix (or vector) of equation (2.2). When the values of the scalars, vectors and matrices at time t related to the mesh h are denoted by the right-side subscript h, matrix t K means the matrix at time t related to mesh h. For the sake of simplicity, it may be expressed as an integral over the volume t V of the matrix product t k. After remeshing, the new matrix t K h1 has to be calculated in the new mesh h+1 and may be expressed as follows: t t t t K d t h1 k h1 h 1, h1 V, (2.3) t Vh 1 where matrix t k h1 to be integrated depends on the Cauchy stress t h1 and if necessary on the internal variables t h1 defined at time t for the new mesh h+1. For elastic plastic solid the hardening parameters play the role of internal variables. Actually, the Cauchy stress and internal variables are referred to old mesh h, therefore, a transfer operation from the old mesh h to the new mesh h+1 is required. The remainder matrices in equation (2.2), have to be obtained in the same manner, with the pore pressure additionally involved as an interdependent internal variable. The integration methodology of expression (2.3) over deformed domain is, generally, mesh independent, therefore, the overall remeshing remains a postprocessing procedure to be developed. h 3. COMBINED REMESHING TECHNIQUE The basics of the remeshing technique put forward to tackle the coupled model (2.2) for cone penetration problem are outlined in the following manner. Generally, it may be considered as regeneration of the mesh together with the transfer of the state variables to a new mesh and comprising a loop of the

6 108 Darius Markauskas, Rimantas Kaianauskas following operations performed until the required overall penetration has been reached: 1. Solution of non-linear analysis problem with a given mesh and prescribed values of the internal variables. 2. Regeneration of the new mesh using the deformed domain boundary. 3. Transfer of the internal variables from the deformed initial mesh into the new mesh. However, there are several different remeshing criteria found in the literature [9, 5] which are based on geometrical or local error observations. Because penetration actually presents the motion of the cone driven by displacement increments having constant speed with respect to a regular structured mesh, the regular remeshing with the frequency predefined about 1/2 size of the smallest element was reasonable. Since the regeneration of mesh is a standard pre-processing procedure, the main attention was focused on the transfer of variables. The coupled model contains two types of internal variables which have to be transferred to a new mesh. The pressure field is described by primary variables defined at nodes, while the stress field described by Cauchy stress and internal variables presenting hardening parameter are defined at Gaussian points, therefore, a combined transfer technique with two different procedures has to be used. The moving least square method using superconvergent patch recovery (SPR) technique [27] has been applied to transfer Gauss points variables, i.e. stresses and the hardening parameter, while pore pressure is transferred from the nodes of the old mesh to the nodes of the new mesh using the polynomial function (interpolation method). The transfer operation using the moving least square method consists of the following steps: constructing the patches from the Gauss points of the old elements mesh, finding in which patch the Gauss point of the new element is, transfer of the variables from the Gauss points of the old mesh to the Gauss points of the new mesh using the polynomial function and the least-squares method. The moving least square and interpolation methods are illustrated in Fig. 1.

7 Simulation of piezocone penetration in saturated porous medium 109 a) b) Old deformed mesh New unformed mesh Gauss point of old mesh Gauss point of new mesh Fig. 1. Illustration of the transfer of state variables: a) moving least square method, b) interpolation method The proposed combined remeshing technique presents post-processor software compatible with conventional FE codes, while current investigation is implemented into ABAQUS [1] environment. 4. SIMULATION OF CONE PENETRATION The proposed FE remeshing technique is applied to simulate the piezocone penetration in saturated soils as porous medium with a wide range of water permeability. A standard cone penetrometer with the diameter d = 2r 0 of 35.7 mm, cone tip angle 60 and penetration speed equal to 2 cm/s is used in this study. The penetrometer is modelled as a rigid surface. The friction between a rigid surface and soil was not taken into account, which is a case of smooth penetrometer. For normally consolidated soil the modified Drucker-Prager/Cap model (DPC) [1] with the following major parameters is used to describe the properties of the skeleton: Young s modulus E = 5 MPa, Poisson s ratio = 0.2, friction angle = 30, initial hydrostatic compression yield stress p b,0 = 80 kpa. The cap hardening law is defined by the piecewise linear function relating the hydrostatic compression yield stress, p b, and the corresponding volumetric plastic strain,, as follows: 10 kpa 0.0, 300 kpa , 500kPa , 1000 kpa pl vol 0.004, 1500 kpa Since cone is penetrating by vertical loading, the 3D problem may be reduced to axi-symmetric problem with 2D domain (Fig. 2). To avoid boundary effects at the start of the analysis, the cone is placed into a pre-bored hole having

8 110 Darius Markauskas, Rimantas Kaianauskas the initial depth h = 0.55 m, with the surrounding soil still in its in situ stress state. r0 p 0 Rigid cone F h h Contact surface Infinite domain H Soil D Fig. 2. 2D solution domain The soil region divided by finite elements should be considerably large to avoid the influence on the result of the replacement of the infinite half-space by a limited size zone. After verification by numerical experiments with a larger domain, the size of the FE domain was taken equal to D = m, h+h = 1.8 m. Since deep penetration is considered and only a limited area around the cone is modelled by finite elements, therefore the gradient of the vertical stress from soil weight and pore pressure is of secondary importance and a homogeneous initial state of stress and pore pressure can be introduced into the model at the start of analysis v,0 = h,0 = 80 kpa, p 0 = 80 kpa. The soil domain was discretised using structured mesh into 1378 bilinear four nodded elements, while soil-penetrometer interaction is described by contact surface. This mesh provides high quality of solution of coupled problem leading to minimal pore pressure oscillations in the worst case with very small water permeability value. The numerically obtained loading curve relating resultant load F acting on the tip of cone and cone displacement u as well as distortion of the initial finite element mesh are the most important characteristics and, at the same time, quality indicators of numerical analysis. The penetration process is initiated by applying displacement u to the rigid surface of the penetrometer, until a steady state is reached. In our investigation the simulation was stopped at the displacement value u = 260 mm or relative value u = 7d with respect to cone diameter d.

9 Simulation of piezocone penetration in saturated porous medium 111 The validity of the remeshing is justified by comparing simulation results in porous media described by the coupled model (2.2) and homogeneous media. For the sake of comparison, the cone penetration analysis in fully undrained and fully drained soil is made. The porous media is characterized by small permeability value k =10-11 m/s for undrained soil and by high permeability value k = 10-5 m/s for fully drained soil. The cone penetration analysis in homogeneous media is made using uncompressible Mises (MIS) type and DPC type models, respectively. The data of homogeneous media DPC model are the same as they were in porous media, while the shear strength c u in MIS model was taken c u = 48 kpa which corresponds to the undrained shear strength of the DPC model used. The obtained smoothed resulting loading curves together with the results of cone penetration in homogeneous media results are presented in Fig. 3. The character of curves proves that the developed remeshing technique provides high quality modelling of deep penetration, since the steady-state behavior is reached and subsequently followed with practically unlimited penetration of the cone. The modelling results also show that, in limited cases, the differences between the porous media and the homogeneous media are small, which confirms the validity of the coupled model. It can be seen that cone resistance obtained from the numerical simulations increases rapidly until it reaches the steady state conditions at a depth of about 2d when soil permeability is m/s, while the steady state is reached at the depth of about 4d when permeability is k = 10-5 m/s. When the soil permeability is m/s the obtained cone resistance is about two times lower than that found for soil permeability of 10-5 m/s. F, MPa DPC, k=1e DPC, k=1e DPC, no pore pressure MIS, no pore pressure u, mm Fig. 3. Loading curves In addition, the validity of numerical results for homogeneous undrained soil is confirmed by the investigation of the cone factor N obtained from the loading curve. The obtained value is compatible with other theoretical solutions and provides good agreement with experimental results. The details of this comparison made by the authors may be found in [13].

10 112 Darius Markauskas, Rimantas Kaianauskas Plotting of the results of numerical simulation versus soil permeability is used here to interpret the influence of soil permeability to cone resistance q t. The resulting curve showing the dependence of the measured cone resistance is given in Fig. 4. It clearly indicates the presence of two threshold values 10-9 m/s and 10-6 m/s for fully drained and undrained normally consolidated soils, respectively, and in-between range of the partially drained soil. q t, MPa E-11 1E-10 1E-9 1E-8 1E-7 1E-6 1E-5 k, m/s Fig. 4. Variation of the cone resistance vs soil permeability Excess pore pressure, kpa p1 p2 1E-11 1E-10 1E-9 1E-8 1E-7 1E-6 1E-5 k, m/s Fig. 5. Variation of the excess pore pressure vs soil permeability The curves exposing the generated pore pressure on the cone, p 1, and behind the neck of the cone, p 2, dependent on soil permeability are given in Fig. 5, while profiles of the pore pressure corresponding to permeability values 10-7 m/s and 10-8 m/s for penetration depth of 216 mm are depicted in Fig. 6. Using results presented in Fig. 5 adjustment factor K = p 2 -p 0 /p 1 -p 0 is equal to when k = m/s. This value is in a typical range of values for normally consolidated clays [7]. It may be concluded that no excess pore pressure is generated and cone penetration at standard speed is performed in drained conditions when the coefficient of permeability is higher than 10-5 m/s. The result obtained is in good agreement with the conclusion made in [21]. But in the considered case the generated excess pore pressure does not make any strong influence on cone resistance, when soil permeability is greater than 10-6 m/s. Fully undrained

11 Simulation of piezocone penetration in saturated porous medium 113 condition is achieved when permeability is smaller then m/s but the increase of permeability does not affect cone resistance considerably when permeability is lower than 10-9 m/s. Cone penetration in intermediate soils is performed in the partially drained conditions. The character of the curve and permeability thresholds agree well with the numerical results obtained and discussed by Song et al. [18]. a) r / r r / r b) r / r 0 Fig. 6. Pore pressure, u = 216 mm, p 0 = 80 kpa: a) k = 10-7 m/s, b) k = 10-8 m/s, The results relating to the generated pore pressure are relevant only for normally consolidated or lightly overconsolidated soil. The degree of soil overconsolidation can largely affect the generated pore pressure because of soil dilatation. r / r DISCUSSION AND CONCLUSIONS The finite element remeshing of deformed geometry involving the transfer operation combining both the moving least square method based on SPR technique and the interpolation method for transfer of state variables is developed and implemented into post-processor type software compatible with standard FEM code. On the basis of the numerical investigation of the CPTU tests in saturated porous medium the following conclusions have been drawn: 1. The developed remeshing technique shows good performance in modelling deep penetration of piezocone into elasto-plastic saturated porous media reaching steady-state behavior, with a practically unlimited value of cone displacement. In our investigation, the simulation was stopped at the displacement value u = 260 mm or relative value u = 7d with respect to cone diameter d.

12 114 Darius Markauskas, Rimantas Kaianauskas 2. The validity of the remeshing is confirmed by comparing the simulation results in porous and homogeneous media. The coupled solutions obtained for undrained soil are compatible with other theoretical solutions and experimental results. 3. Investigation of measured cone parameters versus permeability clearly indicates two threshold values 10-9 m/s and 10-6 m/s for fully undrained and drained, normally consolidated soils, respectively. Between these threshold values there exists a partially drained state, where measured cone parameters strongly depend on the permeability of soil. An exact profile of the above relations is dependent, however, on the properties of soil, therefore comprehensive investigation is still required. 4. The numerical simulation of piezocone penetration in saturated porous media provides information for the evaluation of semi-empirical relations. In particular, in further analysis, it can help to establish the correlations between the measured cone resistance and the effective soil parameters in soils where during cone penetration the partially drained conditions occur. BIBLIOGRAPHY 1. ABAQUS / Standard: User s manuals. Version 5.8, Habbitt, Karlsson & Sorensen, Inc Bathe K.: Finite element procedures, Englewoods Cliffs, Prentice-Hill Biot A.M.: Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys, 26 (1955) de Borst R., Vermeer P.A.: Finite element analysis of static penetration tests, in: Proc. 2nd Eur. Symp. on Penetration Testing, Vol 2, 1982, Erhard T., Taenzer L., Diekmann R., Wall W.A.: Adaptive remeshing issues for fast transient, highly nonlinear processes, in: Proc. of ECCM-2001, Cracow, volume on CD-ROM, 2001, Hu, Y., Randolph, M. F.: Deep penetration of shallow foundation on nonhomogeneous soil, Soils and Foundations, 38, 1 (1998) Karakouzian M., Avar B., Hudyma N., Moss J.: Field measurements of shear strength of an underconsolidated marine clay, Engineering Geology, 67 (2003) Kiousis P. D., Voyiadjis G. Z., Tumay M. T.: A large strain theory and its application in the analysis of the cone penetration mechanism, Int. J. Numerical and Analytical Methods in Geomechanics, 12 (1988) Lackner R., Mang H.A.: Calculation strategies in adaptive nonlinear FE analysis, in: Proc. of ECCOMAS 2000, Barcelona, volume on CD-ROM, 2000, 19.

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