numerical simulation of wave propagation in fully...

ABSTRACT: For many geotechnical problems involving fully saturated soil, the behavior of the soil has to be modeled as a two-phase material to reproduce the soil behavior realistically. Especially when considering dynamic loading conditions a two-phase approach is of importance in order to capture effects such as soil liquefaction or time-dependent soil compaction. The governing equations of a u-p formulation using Darcy’s flow law are derived. The u-p formulation is implemented within a dynamic total stress analysis using an explicit time integration rule. The approach is applied on the problem of one-dimensional wave propagation. The influence of different constitutive models for the solid phase as well as the influence of the relative density and the hydraulic conductivity on the wave propagation is studied. Effects such as the development of shock waves are investigated. Additionally, the problem of a vibrating foundation on a halfspace of fully saturated soil is investigated. Effects such as soil liquefaction and time-depended soil compaction can be modeled. In order to evaluate fully saturated soil under dynamic loading regarding soil liquefaction or soil compaction, the application of a coupled two phase analysis under partially drained conditions is of importance. Despite the limitations of the application of a u-p formulation in case of dynamic problems, a u-p formulation is an attractive way to model fully saturated soil under dynamic loading due to the much more simple formulation.

KEY WORDS: Soil dynamics; Fully saturated soil; u-p formulation; Wave propagation.

1 INTRODUCTION

Soil is a porous medium with voids often filled with a fluid, e.g. water. The behavior of such a two-phase material is important for many engineering problems. Especially considering dynamic loading, fully saturated soil can show a different behavior compared to dry soil or drained conditions with no development of excess pore pressure. This behavior can be important for the process of pile driving, analysis of liquefaction phenomena as well as earthquake loading. The theory of quasi-static and dynamic behavior of fully saturated porous media has been firstly presented by Biot [1]-[4]. Two dilatational waves are described [2]-[3]. The wave of the first kind, called the undrained wave, is a true wave due to compression of the fluid saturated porous medium. The solid phase and the fluid phase move practically with the same velocity [5]. The wave of the second kind arises due to a diffusion process of the fluid through the porous medium. This wave is highly attenuated due to the interaction between fluid and solid skeleton [2]. Thus, this wave can only be observed in the near-field of the applied load and can occur in stiff porous media such as rock [5]. If dissipation of fluid disappears due to fluid friction or a small permeability, only marginal relative motion between solid and fluid occurs and thus, the wave of the second kind disappears. Besides Biot's theory, different authors rederived the governing equations, differing from Biot’s theory only by some minor modifications, and developed discretised forms for use by the finite element method [6]. A simplified form of the governing equations is a so-called u-p formulation. By reason of assumptions and simplifications for derivation of this formulation, a u-p formulation is not able to capture both

dilatational waves due to negligence of the acceleration of the fluid phase relative to the solid skeleton [5]. Only the wave of the first kind can be modeled. Hence a u-p formulation is only suitable for modeling of the dynamic behavior of fully saturated soil if acceleration frequencies are low, e.g. earthquake loading, and if no high-frequency dynamic phenomena are considered [7]. One advantage of this formulation is the reduction in size of the equation set [8], which makes this formulation attractive for implementation in the finite element method.

In this paper the governing equations are derived and assumption and simplifications are outlined. The application of a u-p formulation implemented in a dynamic analysis procedure using an explicit time integration rule [9] on the problem of wave propagation in fully saturated soil is investigated. The influence of different parameters such as the relative density and the hydraulic conductivity as well as the influence of different constitutive models for the solid skeleton is studied.

2 MATHEMATICAL MODELS

There exist a variety of approaches to model porous media. All of them are leading to similar set of governing equations. These resulting equations can then be adapted to the present problem. As mentioned before the number of the governing equations and the variables can be changed by some assumptions. The basic idea of deriving the system of equations is to ensure the conservation of physical quantities, e.g. mass, momentum, energy for the constituents as well as for their mixture.

Numerical Simulation of Wave Propagation in Fully Saturated Soil Modeled as a Two-Phase Medium

J. Grabe1, T. Hamann1, A. Chmelnizkij1

1Institute of Geotechnical Engineering and Construction Management, Hamburg University of Technology, Harburger Schlossstraße. 20, 21079 Hamburg, Germany

email: [email protected], [email protected], [email protected]

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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A nice way to derive the u-p and u-U formulation from the u-w-p formulation has been shown in [11]. The u-p-w formulation consists of the following equations, which ensures mass and momentum conservation:

0

0

αmε 0

(1)

Where is a differential Operator which is used to compute the strain increments from the displacement increments. For the two dimensional case the Operator is defined as

0

0.

Furthermore the following notation has been used in (Equation 1),

displacement of the solid matrix

averaged velocity of the fluid

effective pressure

m total stress tensor with as the effective stress and as the Biot’s stress coefficient

porosity

1

density of the total composite with the sub-scripts , for the solid and fluid phase

ε strain of the solid skeleton

body force per unit mass viscous drag force

For the remaining notations the reader is referred to [11]. The three quantities (u,w,p) mentioned above are the main variables of this system. Under the assumption of constant densities and vanishingly small fluid-acceleration as well as the convective product

w it is possible to eliminate the variable . Furthermore for this purpose all thermal influences have to be neglected and the Darcy`s seepage law has to be assumed for R. Afterwards we obtain the u-p formulation consisting of the following equations:

0

αm 0 (2)

Where now the only variables are u and p. In Equation 2 the acceleration of the solid phase is renamed as and the

Darcy’s law is used.

The reason for different formulations is, to reduce computational effort by decreasing the number of variables and equations but at the same time to take into account the necessary terms. One problem occurs if using lumped mass matrices for the explicit time integration after the spatial discretization is done by the Finite Element Method. The consistent mass matrices in the u-p and u-p-w formulation are singular with zero rows. Therefore the integration of the system, which usually inverts the lumped mass matrix, can`t be done straightforward and special techniques have to be applied. One possibility is to perform the time integration for the mixture only and afterwards to split in the constituents in each time increment.

3 NUMERICAL MODELS

The u-p formulation derived above, is applied on the problem of one-dimensional wave propagation in a fully saturated sand column. Therefore the governing equations are implemented in the framework of the finite element code Abaqus/Explicit [10].

3.1 Analysis procedure Abaqus/Explicit

The dynamic analysis procedure Abaqus/Explicit solves the equation of motion at each node of the finite element model [10]. An explicit central-difference time integration rule is used to integrate the equation of motion explicitly through time. The solution at the end of a time increment can be advanced using known values of nodal acceleration, velocity and displacement at the beginning of a time increment. No iteration is necessary. The use of a lumped mass matrix makes the analysis procedure very inexpensive, since no simultaneous equations have to be solved. The explicit equation solver is conditionally stable, if the time increments are less than a critical time increment Δtcrit. The critical time increment ensures, that a dilatational wave cannot transit an element within a single time increment. The critical time increment depends on the characteristic element length Lchar and the dilatational wave speed cd and is defined as:

∆ (3)

3.2 u‐p formulation

A coupled pore fluid and stress analysis under dynamic loading is not supported by Abaqus. To extend the dynamic analysis procedure of Abaqus/Explicit for a u-p formulation, the approach proposed by Hamann and Grabe [9] is applied. A total stress analysis is carried out, solving the equation of motion for the mixture of solid and fluid. The behavior of the fully saturated soil, described by the u-p formulation, is implemented by use of a user-subroutine for constitutive models. Within the user-subroutine the effective stress state is calculated applying a constitutive model for the solid skeleton: linear elastic behavior or hypoplasticity. For calculation of the pore pressure pw a mass balance equation of the water phase is implemented (Equation 4). The fluid flow through the soil is modeled by Darcy's flow law. Calling the user-subroutine the current strain increments are passed into and the updated total stress state is returned.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

632

αm 0 (4)

with:

(5)

1 (6)

where kf is the hydraulic conductivity, ρw is the density of

water, b is a body force per unit mass (gravity), as is the current acceleration of the solid skeleton, is the Biot’s

constant, m is the second order unit tensor, s is the current

strain rate of the solid skeleton, Q is the bulk modulus of the mixture of soil and water, n is the porosity, Kw is the bulk modulus of water, Ks is the bulk modulus of the solid grains, E is the Young’s modulus of the solid skeleton and ν is the Poisson’s ratio.

3.3 Hypoplastic constitutive model

Linear elastic material behavior is only valid for a small strain range. In order to reproduce the inelastic and non-linear behavior a dependency of the stiffness on the stress state and void ratio is necessary. In the following a hypoplastic constitutive model according to Gudehus [12] and von Wolffersdorff [13] is used. The hypoplastic constitutive model is suitable to model the non-linear and inelastic behavior of granular materials realistically, e.g. sand. Specific properties like dilatancy, contractancy, different stiffness for loading and unloading as well as a dependency of the stiffness on the stress state and void ratio can be described. Taking account of accumulation effects and hysteretic material behavior in case of cyclic loading as well as small strain behavior of soils, an extension of intergranular strain was proposed for the hypoplastic model by Niemunis and Herle [14]. The constitutive model is rate independent in a rate-type formulation defined by the tensorial Equation (7):

, , : (7)

where T is the objective Jaumann stress rate, M is a fourth order tensor, T is the current Cauchy stress, e is the void ratio, is the intergranular strain and D is the strain rate.

4 PROBLEM OF ONE‐DIMENSIONAL WAVE PORPAGATION

4.1 Finite element model

The problem of one-dimensional wave propagation is investigated on a fully saturated sand column of a height of 100 m, as depicted in Figure 1. An axisymmetric analysis is carried out. For the behavior of the solid skeleton a linear elastic and a hypoplastic constitutive model is used. The material parameters are given in Table 1 and Table 2. The bottom and the sideways boundary are fixed in normal direction for the solid phase and are defined to be impermeable. The water level is specified at the upper permeable boundary of the model. Further, a surface load is applied on the upper surface of the model. The evolution of

the surface load over time follows half of a sine curve (load case 1) or has a stepwise progression (load case 2) as displayed in Figure 1.

Figure 1. Geometry of the soil column considered for one-dimensional wave propagation.

Table 1. Linear elastic parameters of the sand (ID=0.5).

Parameter Sand Description E 3.0e+4 Young’s modulus

[kN/m²] ν 0.3 Poisson’s ratio [-] Ks 3.7e+7 Bulk modulus solid

[kN/m²] Kw 2.1e+6 Bulk modulus water

[kN/m²] ρs 2.65 density of solid particles

[t/m²] ρw 1.0 density of water [t/m²] n 0.41 porosity [-] (ID=0.5) kf 1e-4 hydraulic conductivity

[m/s]

Table 2. Hypoplastic parameters of Karlsruher Sand.

Parameter Karlsruher Sand

Description

c 30 critical state friction angle [°] hs 5.8e+6 granular hardness [kN/m²] n 0.28 exponent ed0 0.53 minimum void ratio ei0 0.84 critical void ratio ec0 1.00 maximum void ratio 0.13 exponent 1.05 exponent R 0.0001 maximum value of inter-

granular strain mR 2.0 stiffness ration at a change of

load direction of 180° mT 5.0 stiffness ration at a change of

load direction of 90° R 0.5 exponent 6.0 exponent Ks 3.7e+7 Bulk modulus solid [kN/m²]


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Kw 2.1e+6 Bulk modulus water [kN/m²] ρs 2.65 density of solid particles [t/m²] ρw 1.0 density of water [t/m²] e 0.685 void ration [-] (ID=0.5) kf 1e-4 hydraulic conductivity [m/s]

4.2 Comparison of fully and partially drained conditions

The distribution of the excess pore pressure and the effective vertical stress at different times is depicted in Figure 2 (linear elastic material behavior) and in Figure 3 (hypoplastic material behavior). Considering linear elastic material behavior in case of fully drained conditions, no excess pore pressure arises and the external load is carried by the solid skeleton completely, as expected. The shape of the curve equals the applied external sinusoidal surface load.

Figure 2. Distribution of excess pore pressure (left) and

change of effective vertical stress (right) over the depth at different times, linear elastic material behavior, sinusoidal

loading.

In case of partially drained conditions, the pore water bears the main part of the external load due to the higher stiffness compared to the solid skeleton. Furthermore, the influence of dissipation can be observed, since the magnitude of the compression wave in the water phase decreases with ongoing propagation. The shape of the curve equals the applied external surface load, again. Since the relative acceleration of the water phase relative to the solid phase is neglected, the wave speed within the water and the solid phase is equal. Due to the higher stiffness of the fully saturated sand in case of partially drained conditions, a higher wave speed (v=1548 m/s) compared to fully drained conditions (v=174 m/s) can be found.

Describing the behavior of the sand by use of a hypoplastic constitutive model more realistically (Figure 3), the effect of the development of shock waves with a steep front and a flat back side can be observed in case of fully drained conditions [15],[16]. The compression wave consists of three parts: first a flat increase due to the small strain behavior (elastic strain range) followed by a shock front and a flat back side. Due to the dependency of the stiffness of the solid skeleton on the stress state, the stiffness of the sand and thus the wave speed increases temporarily while the compression wave is moving through the sand column. Due to the increase of the stiffness, the rear part of the wave is faster, catches up with the front part of the wave and is reflected at the front of the wave. Thus, the front of the wave becomes steeper. The magnitude of the wave decreases with ongoing propagation due to the permanent reflection at the shock front.

Figure 3. Distribution of excess pore pressure (left) and change of effective vertical stress (right) over the depth at different times, hypoplastic material behavior, sinusoidal

loading, relative density ID=0.5.

In case of partially drained conditions, the main part of the

external load is carried by the water phase again. Due to the stiffness dependency of the solid skeleton on the stress state, the stiffness of the sand increases over the depth due to its self-weight. Thus, the relation of the stiffness of the water and the solid phase changes over the depth and the load is transferred continuously from the water to the solid phase. Hence, the decrease of the magnitude of the compression wave in the water phase over the depth is more distinct compared to the linear elastic case. The development of a shock wave, as observed in case of fully drained conditions, can not be found. This is caused by the strong coupling of both phases, such that the wave moves with the same velocity


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and the same shape within both phases. The shape of the wave is dominated by the water phase due to its higher stiffness. The solid phase, which would form a shock wave, has only less influence on the shape of the wave. Similar to the case of linear elastic material behavior, the wave speed is higher in case of partially drained conditions (t=0.02 s: v=1638 m/s) compared to fully drained conditions (t=0.02 s: v=213 m/s). Due to stiffness dependency of the solid skeleton on the stress state, the wave speed increases over the depth. The wave speed increases from v=1556 m/s at a time of 0.01 s to v=1632 m/s (+4.8%) at a time of 0.04 s, see Table 3. The stiffness dependency of the solid skeleton on the stress state and thus, the load transfer from the water to the solid phase becomes more distinctly, when applying a stepwise surface load, see Figure 4. The increase of the magnitude of the compression wave in the solid phase over depth can be observed within the shape of the wave clearly.

Table 3. Wave speed at different times after beginning of load application, hypoplastic material behavior, sinusoidal loading,

relative density ID=0.5, hydraulic conductivity kf=10-4 m/s.

t

[s] Wave speed v [m/s]

Deviation from t=0.01 s [%]

0.01 1533 0.0 0.02 1554 2.5 0.03 1576 3.7 0.04 1604 4.8

Figure 4. Distribution of excess pore pressure (left) and change of effective vertical stress (right) over the depth at different times, hypoplastic material behavior, stepwise

loading, relative density ID=0.5, hydraulic conductivity kf=10-4 m/s.

4.3 Influence of the relative density on the wave propagation

Since the stiffness of the sand depends on the void ratio, the relative density of the sand influences the propagation of the wave additionally. In Figure 5 the relative density ID is varied and the wave propagation is displayed 0.02 s after beginning of load application. The relative density of the sand has a distinct influence on the wave speed and the distribution of

the loading on the water and solid phase. Increasing the relative density, the stiffness and thus the wave speed increase. The wave speed increases from v=1533 m/s in case of very loose sand (ID=0.0) to v=1679 m/s (+9.5 %) in case of very dense sand (ID=1.0), see Table 4. Furthermore, the stiffness relation between the water and the solid phase changes. Increasing the relative density, a higher part of the loading is transferred to the solid phase.

Figure 5. Distribution of excess pore pressure (left) and change of effective vertical stress (right) over the depth for different relative density ID, hypoplastic material behavior,

sinusoidal loading, hydraulic conductivity kf=10-4 m/s.

Table 4. Wave speed for different relative densities ID at a time of 0.02 s after beginning of load application, hypoplastic material behavior, sinusoidal loading, hydraulic conductivity

kf=10-4 m/s.

ID

[-] Wave speed v [m/s]

Deviation from ID=0.0 [%]

0.0 1533 0.0 0.2 1554 1.3 0.4 1576 2.8 0.6 1604 4.6 0.8 1638 6.9 1.0 1679 9.5

4.4 Influence of the hydraulic conductivity on the wave propagation

The influence of the hydraulic conductivity kf on the wave propagation is shown in Figure 6. Considering a hydraulic conductivity of 10-4 m/s ≤ kf ≤ 10-7 m/s, approx. no influence on the shape of the wave and on the wave speed as well as on the magnitude of the wave can be observed. Within this range of hydraulic conductivity, the soil shows a kind of undrained behavior, since dissipation is much slower compared to the wave speed. Considering a hydraulic conductivity of kf=10-3 m/s, the influence of dissipation becomes obvious. The shape of the wave becomes longer and the magnitude decreases.


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Figure 6. Distribution of excess pore pressure (left) and change of effective vertical stress (right) over the depth for

different hydraulic conductivity kf, hypoplastic material behavior, sinusoidal loading, relative density ID=0.5.

5 VIBRATING FOUNDATION ON A HALFSPACE

5.1 Finite element model

The problem of two-dimensional wave propagation is investigated on the example of a circular vibrating foundation on a halfspace, see Figure 7. The foundation has a diameter of 1 m. An axisymmetric analysis is carried. The behavior of the solid skeleton is described with the hypoplastic constitutive model. The hypoplastic parameters of the investigated Karlsruher sand (ID=0.5) are given in Table 2. The bottom and sideways boundary are assumed to be permeable and fixed in normal direction. The upper boundary is permeable and the pore pressure is set to zero. The vibrating foundation is simplified by applying a surface load with a sinusoidal evolution over time. Thus, a lift off of the foundation is not possible. A vibrating frequency of f=50 Hz and a dynamic surface load of 10 kN/m² are applied. To prevent the development of tension stress below the area of load application, the surface of the model is loaded with a static load of 15 kN/m² additionally.

Figure 7. Geometry of FE-modell of the vibrating foundation on a halfspace.

5.2 Results

The distribution of the effective vertical stress, the excess pore pressure and the void ratio after 4 s of vibration for the case of partially drained and fully drained conditions are depicted in Figure 8 and Figure 9. The distribution of the state variables on the axis of symmetry is shown in Figure 10. When starting the vibration, the sand is compacted due to cyclic loading as it can be observed in Figure 9 (fully drained conditions). In case fully drained conditions, the area of soil compaction reaches a depth of approx. 1 m, see Figure 10. The effective stress state changes such that areas of an increase as well as areas of a decrease of the initial stress state occur.

Considering partially drained conditions, the pore water has to flow out of the voids of the solid skeleton, before the sand can be compacted. Due to the flow resistance caused by the solid skeleton, excess pore pressure develop during vibration due to the constraint contractant behavior of the sand, see Figure 8 (top, right). As a result of the excess pore pressure the effective stress state becomes reduced (Figure 8, top, left). Down to a depth of 0.5 m below the foundation, soil liquefaction occurs with vanishing effective stresses and excess pore pressure up to 20 kN/m², see Figure 10. Only slight soil compaction right below the foundation can be observed (Figure 9 and 10), due to the resistance of the pore water. Considering partially drained conditions, soil compaction depends on the hydraulic conductivity of the soil. In order to evaluate fully saturated soil under dynamic loading regarding soil liquefaction or soil compaction, the application of a coupled two phase analysis under partially drained conditions is of importance.

Figure 8. Distribution of the effective vertical stress (left) and the excess pore pressure (right) after 4 s of vibration; top: partially drained conditions, bottom: drained conditions.


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Figure 9. Distribution of the void ratio after 4 s of vibration; left: partially drained conditions, right: drained conditions.

Figure 10. Distribution of the effective vertical stress (left), the excess pore pressure (center) and the void ratio (right) on

the axis of symmetry after 4 s of vibration.

6 CONCLUSION

The governing equations of a u-p formulation using Darcy’s flow law are derived. The u-p formulation is implemented within a dynamic total stress analysis using an explicit time integration rule. The approach is applied on the problem of one-dimensional wave propagation in a fully saturated sand column. The difference between fully drained and partially drained conditions is shown, such that the wave speed increases and the solid skeleton is less loaded by the wave in case of partially drained conditions. Effects such as the development of shock waves cannot be observed in case of partially drained conditions. Studying the influence of the hydraulic conductivity of the sand, an undrained soil behavior can be found in case of a hydraulic conductivity of kf≤10-4 m/s. When applying a hypoplastic constitutive model, a stiffness dependency of the solid skeleton on the stress state and the void ratio can be accounted for. The relative density of the sand has a distinct influence on the wave speed as well as on the distribution of the external load on the water and solid phase, since the stiffness of the solid phase increases with increasing relative density.

Investigating the problem of a vibrating foundation on a halfspace of fully saturated soil, effects such as soil liquefaction and time-depended soil compaction can be modeled. In order to evaluate fully saturated soil under dynamic loading regarding soil liquefaction or soil

compaction, the application of a coupled two phase analysis under partially drained conditions is of importance. Despite the limitations of the application of a u-p formulation in case of dynamic problems, a u-p formulation is an attractive way to model fully saturated soil under dynamic loading due to the much more simple formulation.

ACKNOWLEDGMENTS

The present research work on modelling of soil as a two-phase material is funded by the German Research Foundation (DFG) within the framework of the research training group GRK 1096 “Ports for Container Ships of Future Generations”. The funding is greatly acknowledged.

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