International Conference on Geotechnical Engineering. 2013 Analysis of Dynamic Loading and Penetration of Soils Applications to Site Investigation and Ground Improvement J. P. Carter, M. Nazem Australian Research Council Special Research Centre for Geotechnical Science and Engineering, The University of Newcastle, Callaghan NSW, Australia and D.W. Airey Centre for Geotechnical Research, The University of Sydney, Sydney, NSW, Australia ABSTRACT: A robust numerical method is suggested for dealing with the complex and difficult problem of analysing dynamic soil penetration and compaction. The approach is based on the Arbitrary Lagrangian-Eulerian (ALE) method of analysis, whose main features and challenges are briefly described. The method is then employed to perform a parametric study of a variety of penetrometers free-falling into layers of inhomogeneous clay deforming under undrained conditions. The effect of the mechanical properties of the clay soil on the penetration characteristics is discussed and some of the predictions are compared with the results of laboratory experiments. How these results might be used in the interpretation of in situ penetration tests is then suggested. The numerical method is also adopted to examine the fundamental compaction behaviour of soils and the trends in predicted results are broadly compared with the results of physical experiments. 1 INTRODUCTION Successfully analysing the dynamic compaction and penetration of soil deposits, by either free falling or propelled objects, is one of the most challenging problems in geomechanics. To date most methods used to predict the effects of an object impacting and possibly penetrating the ground surface have been empirical, based on observations in field trials and laboratory experiments. However, there is now a real need for rigorous theories to describe the fundamental response of the soil in this situation, theories that are soundly based on the principles of impact mechanics and theoretical soil mechanics. One aim of this paper is to describe such a theoretical approach. The use of a falling weight (or “pounder”) is the most common means of applying a dynamic impact to the soil surface in order to cause compaction of the underlying material, although in recent times the use of impact rollers has increased in this area of ground treatment and improvement. The analysis of a free-falling mass will be described in later sections of this paper. Obviously the theory describing this type of behaviour has much in common with that required to analyse the impact and subsequent penetration of a soft soil. Penetrometers are widely used to investigate the mechanical properties of soils and to embed objects in soil deposits, especially anchors and other items in the sea floor. Free falling penetrometers have been employed to provide information on the strength of seabed sediments. Such tests can be envisaged as the dynamic equivalent of a static cone penetration test (CPT). They can provide useful data such as the total depth and time of penetration and the deceleration characteristics of the falling penetrometer. Potentially, these data can then be used to deduce strength parameters for the soil in situ. Other types of projectiles are also used to embed anchors and other objects deeply in the sea floor. Large scale anchors of this type have been employed for ship moorings and the mooring of floating (offshore) petroleum production facilities. To date, dynamic penetrometers have been used for offshore oil and gas industry applications, such as determining soil strengths for pipeline feasibility studies and anchoring systems, in military applications for naval mine countermeasures and terminal ballistic studies, in extra-terrestrial exploration, and they have also been proposed and investigated for deep sea nuclear waste disposal (Chow 2011). The specifications of a dynamic penetrometer, including its shape, geometry, mass, and initial velocity, normally depend on its application. For instance, torpedoes used to anchor flexible risers are typically 12-15 m long, weigh 240-950 kN in air, and are 762-1077 mm in diameter (Medeiros 2002), while the STING penetrometer, developed by Canada’s Defence Research Establishment Atlantic, to investigate the upper sea bed consists of a 1 m long 19 mm diameter steel rod with an enlarged tip at its base with diameters of 25-70 mm (Mulhearn 2002). Numerical and experimental studies have shown that the penetration characteristics of these penetrometers depend on their geometry and impact energy as well as the mechanical properties of the soil layer, particularly its undrained properties. Scott (1970), Dayal and Allen (1973), True (1975), and Beard (1977) are among the pioneers who presented the earliest applications of free falling penetrometers in providing information on the upper few metres of seabed sediments. At the time of these studies powerful numerical methods were not available to solve the corresponding theoretical problem and researchers had to rely on empirical correction factors to interpret empirical test data. Scott (1970) also demonstrated that technological limitations prevented reliable data from being obtained. Over the last decade, improvements in sensor technology have led to renewed interest in the dynamic penetration resistance of soil with several new penetrometer systems being developed to investigate soil strength and bearing capacity. One of the factors preventing the widespread use of this technology is the uncertainty in the method of interpretation. For example, experimental studies (Mulhearn, 2002, Abelev et al, 2009) with penetrometers of different diameters have indicated limitations of the widely used empirical methods of interpretation developed by True (1975), and have demonstrated the need for a better understanding of the factors affecting the dynamic soil resistance. For a variety of reasons, the numerical simulation of these types of impact compaction and penetration problems is probably one of the most difficult problems in theoretical geomechanics. In the case of poorly draining material the analyst must consider the relative incompressibility of the soil during the short testing times required for full penetration and the likely inhomogeneity and rate dependence of the soil deposited on the seabed. For free-draining materials the volume changes resulting from the passage of stress waves through the soil must also be considered. These aspects of the soil response require a robust stress-integration algorithm to accurately evaluate the stresses for a given strain field. Inertia effects may also be important in cases where the penetration occurs rapidly. Moreover, the penetration of objects into layers of soil usually involves severe mesh distortion caused by the large deformations. More importantly, the boundary conditions of the problem do not remain constant during the analysis since the contact surface between the penetrometer and the soil changes continuously during penetration. Finally, an advanced timeintegration scheme must be employed to accurately predict the highly nonlinear response of the soil. This paper describes a robust numerical method for dealing with such complex and difficult problems. The favoured approach is based on the Arbitrary Lagrangian-Eulerian (ALE) method of analysis, whose main features and challenges are described briefly in the paper. Application of this method to the simulation of the dynamic compaction of soils and the dynamic impact compaction and penetration of instruments into undrained layers of non-uniform soil is discussed, and comparisons with experimental observations are made. 2 2.1 NUMERICAL APPROACH Background The Arbitrary Lagrangian-Eulerian (ALE) method, being well established in fluid mechanics and solid mechanics, has been shown to be robust and efficient in solving a wide range of static as well as dynamic geotechnical problems involving large deformations. Nazem et al. (2006) presented a robust mesh optimisation procedure used with the operator-split technique for solving large deformation problems in geomechanics. This technique refines a mesh by relocating the nodal points on all material boundaries followed by a static analysis. Later, Nazem et al. (2008) presented an ALE formulation for solving elastoplastic consolidation problems involving large deformations. These authors compared the performance of the ALE method with the Updated Lagrangian (UL) method and showed the efficiency and robustness of the ALE method by solving some classical problems such as the consolidation of footings and cavity expansion. Sheng et al. (2009) addressed a successful application of the ALE method, incorporated with an automatic load stepping scheme and a smooth contact discretisation technique, in solving geotechnical problems involving changing boundary conditions, such as penetration problems including the installation of piles. The application of the ALE method in the dynamic analysis of geotechnical problems was also addressed by Nazem et al. (2009a). 2.2 Operator-split technique The Arbitrary Lagrangian-Eulerian (ALE) method has been developed to eliminate the mesh distortion that usually occurs in the Updated Lagrangian (UL) method by separating the material and mesh displacements. In a UL formulation, all variables are calculated at the end of the last equilibrium configuration. This assumption necessitates updating the spatial coordinates of all material points according to the incremental displacements at the end of each time step. While capable of solving some problems with relatively large deformations, the UL method often fails to succeed due to the occurrence of excessive mesh distortion and the entanglement of elements. Mesh distortion usually starts with elements twisting or otherwise distorting out of their favourable shapes. For instance, in a domain consisting of triangular elements it is usually preferred that all triangles remain roughly equilateral during the analysis. However, material displacements, especially in regions with higher deformation gradients, will gradually cause distortion of elements, decreasing the accuracy of the analysis or even ultimately resulting in a negative Jacobian. One way to overcome this shortcoming of the UL method is to separate the material and mesh displacements, leading to the development of the ALE method. The equilibrium equation in the ALE method can be written in two different forms. It is possible to write the governing global system of equations in terms of two sets of unknown mesh and material displacements, leading to the so-called coupled ALE method. A supplementary set of equations in terms of the material and mesh displacements needs to be established through a mesh motion scheme and the two sets of unknown displacements are then solved simultaneously. Alternatively, in the decoupled ALE method or the operator-split technique, the analysis can be performed in two separate steps: a UL step followed by an Eulerian step. In the UL step, the governing equations are only solved for the material displacements in order to fulfil the requirements for equilibrium. This step usually results in a distorted mesh. In the subsequent Eulerian step, the main goal is to minimise the mesh distortion by refining the mesh. Mesh refinement can be achieved by generating a new mesh for the entire domain or by moving current nodal points into new positions. Regardless of which strategy is adopted for mesh refinement, the topology of the domain, i.e., the number of nodes, number of elements, and the connectivity of elements should not be changed. The Eulerian step is particularly important if significant mesh distortion occurs during the UL step. After mesh refinement, all state variables at nodal points as well as at Gauss points must be mapped from the distorted mesh to the new mesh. This remapping is usually performed using a first order expansion of Taylor’s series as ∂f fr = f + vi − vir ⋅ ∂xi ( ) (1) and f denote the time derivatives of an arbitrary function f with respect to the mesh where f r and material coordinates respectively, vi is the material velocity, and vir represents the mesh velocity. Other important aspects of the ALE method are discussed as follows. 2.3 Updated Lagrangian formulation During each increment of the operator split technique, the analysis starts with a UL procedure at time t. The main goal of the UL step is to find the incremental displacements, velocities and accelerations that satisfy equilibrium at time t+Δt. The matrix form of equilibrium is usually derived from the principle of virtual work. The weak form of this principle states that for any virtual displacement δu, equilibrium is achieved provided " − ∫ σ δε dV − ∫ δ u ρu dV − ij ij k i i k V V ∑$$$+ k δ u b dV + δk u f dS k # ∫Vk i i k ∫ Sk i i k +∫ Sc (t N ∫ Vk δ ui cui dVk % ' '' & δ g N + tT δ gT )dSc = 0 (2) where k is the total number of bodies in contact, σ denotes the Cauchy stress tensor, δε is the variation of strain due to virtual displacement, u, u and u represent material displacements, velocities and accelerations, respectively, ρ and c are the material density and damping, b is the body force, q is the surface load acting on area S of volume V, δgN and δgT are the virtual normal and tangential gap displacements, tN and tT denote the normal and tangential forces at the contact surface Sc. The solution of Equation (2) requires the discretisation of the domain as well as the contact surfaces. In this study we adopt the so-called node-to-segment contact discretisation as shown in Figure 1, in which a node on the slave surface may come into contact with an arbitrary segment of the master contact surface. With this assumption the last two terms in Equation (2), representing the virtual work due to normal and tangential contact forces, can be written in the following form: Master surface Slave node Active master segment Figure 1. Node-to-segment contact discretisation ∫ Sc ∫ nc t N δ g N dSc ≈ ∑ δ uiT FNci i =1 t δ g N dSc ≈ ∑ δ uiT FNci Sc N (3) nc i =1 (4) in which nc is the total number of slave nodes, and FNc and FTc represent the normal and tangential nodal forces of the contact element, respectively. The equation of motion for a solid can be obtained by linearisation of the principle of virtual displacement and is written in the following matrix form (e.g., Wriggers 2006) t+Δt t+Δt + Cu t+Δt + R t+Δt = Fext Mu (5) where M and C represent respectively the mass and damping matrices, R is the stress divergence term, u denotes the displacement vector, and Fext is the time-dependent external force vector. Note that the right-hand super-script denotes the time when the quantities are measured and a superimposed dot represents the time derivative of a variable. The solution of the momentum equation requires a step-by-step integration scheme in the time domain. Explicit and implicit methods are available for this purpose. In explicit methods the solution at time t+Δt depends only upon known variables at time t. These methods, such as the central difference method, do not require a factorization of the effective stiffness matrix and are easy to implement. However, explicit methods are conditionally stable, i.e., the size of the time steps must be smaller than some critical time step. Implicit methods, on the other hand, require the solution of a nonlinear equation at each time step and have to be combined with another procedure such as the Newton-Raphson method. The main advantage of implicit methods is that they can be formulated to be unconditionally stable, allowing the analyst to use a bigger time step than used in the explicit methods. In this approach we adopt the implicit generalised-α method presented by Chung and Hulbert (1993) to integrate the momentum equation over the time domain. This method has been shown to be very efficient in solving dynamic problems in geomechanics (e.g., Kontoe et al. 2008 and Nazem et al. 2009a). In the generalised-α method, two integration parameters α f and α m , are introduced into the momentum equation to compute the inertia forces at time t + (1 − α m ) Δt and the internal and damping forces at time t + (1 − α f ) Δt , respectively. The accelerations and velocities are approximated by Newmark equations according to t+Δt = u 1 1 t 1− 2 β t ut+Δt − ut − u − u 2 βΔt 2β βΔt u t+Δt = ⎛ α⎞ ⎛ α α ⎞ t ut+Δt − ut + ⎜ 1− ⎟ u t + ⎜ 1− ⎟ u βΔt ⎝ β⎠ ⎝ 2β ⎠ ( ( ) ) (6) where α and β are the Newmark integration parameters and Δt represents the time step. By introducing a tangential stiffness matrix ⎛ ∂R ⎞ K T ( uit +Δt ) = ⎜ t +Δt ⎟ ⎝ ∂u ⎠uti +Δt (7) we can use the Newton-Raphson method to solve the momentum equation in the following form for the ith iteration $ ' α 1− α f & 1− α m ⋅ M + ⋅ C + 1− α f ⋅ K T (i−1) ) ⋅ Δu(i ) = &% β ⋅ Δt 2 )( β ⋅ Δt ( t+Δt ext (1− α ) ⋅ F f ) ( ) t t+Δt + α f ⋅ Fext − 1− α f ⋅ Fint (i−1) ( ) $ 1− α 1− α m t ' t+Δt t m ⋅ u − u − ⋅ u ) & 2 (i−1) β ⋅ Δt & β ⋅ Δt ) t −α f ⋅ K T (i−1) ⋅ u − M & ) * 1− α - t m &− , ) −1/⋅ u )( . %& + 2β ( ) $ α 1− α * α - t' f t & ⋅ u(t+Δt − u + 1− 1− α ⋅ u ) , f / i−1) & β ⋅ Δt + β . ) −C& ) t &−Δt *, α −1-/ 1− α ⋅ u ) f &% )( + 2β . ( ) ( ) ( ( ) u(t+Δt = u(t+Δt + Δu(i ) i) i−1) where ) , u(t+Δt = ut 0) (8) and where Fint , the internal force vector, is obtained according to the Cauchy stress tensor, σ, and the nodal forces at contact surfaces from t +Δt Fint =∫ V nc ( BT ⋅ σ t +Δt dV t +Δt − ∑ FNc i + FTci t +Δt i =1 ) (9) Note that the tangential stiffness matrix in Equation (7) includes the contribution of the material stiffness, Kep, the stiffness due to geometrical nonlinearity, Knl, and the stiffness due to normal and tangential contact, KNs and KTs, i.e., KT = K ep + K nl + K Ns + KTs (10) In the problems studied here the penetrometer is idealised as a rigid body, i.e., its size and shape remain unchanged during penetration. To achieve this condition all nodal points on the penetrometer are prescribed to undergo equal vertical displacements while horizontal displacement is prohibited. 2.4 Definition of contact To describe the contact at the interface between two bodies, constitutive equations must be provided for both the tangential and normal directions. Among several strategies available in contact mechanics we use the penalty method to formulate these constitutive relations. The normal contact in the penalty method is described by tN = ε N g N (11) where εN is a penalty parameter for the normal contact. In general, the response in the tangential direction is captured by the so-called stick and slip actions. In the former, no tangential relative movement occurs between the bodies whilst the latter represents a relative displacement of bodies in the tangential direction. This assumption facilitates splitting the relative tangential velocity between the bodies, g T , into a stick part gTst and a slip gTsl part, as in the following rate form gT = gTst + gTsl The stick part can be used to obtain the tangential component of traction by (12) tT = ε T gT (13) where εT is a penalty parameter for tangential contact. Note that the assumption in (12) is analogous to the theory of plasticity in which the incremental strains are divided into an elastic and a plastic part. Following this analogy, we must provide a slip criterion function. This can be achieved by writing the classical Coulomb friction criterion in the following form f s ( t N , tT ) = t T − µt N ≤ 0 (14) where µ is the friction coefficient. For the numerical examples considered later in this paper, the interface between the penetrometer and the soil is assumed, for simplicity, to be perfectly smooth. This situation corresponds to the special case where µ = 0. 2.5 Material behaviour Undrained behaviour of the soil to be penetrated is represented by an elastoplastic Tresca material model with an associated flow rule or by the Modified Cam Clay (MCC) stress-strain model, which is formulated in terms of effective stresses. For cases involving the fully drained response of the soil the MCC model has been adopted. In a large deformation analysis, the stress-strain relations must be frame independent. This requirement, known as the principle of objectivity, can be satisfied by introducing an objective stress rate, such as the Jaumann stress rate σ ∇J , into the stress-strain relations as ep dσ ij∇J = Cijkl ⋅ d ε kl (15) ep where C represents the elastoplastic constitutive matrix and ε is the strain vector. Among several available options to integrate equation (15), Nazem et al. (2009b) showed that it is slightly more efficient to apply rigid body corrections during integration of the constitutive equations. This strategy is adopted in this study. It is noted that the undrained shear strength of cohesive soils often increases with the rate of straining. This phenomenon, termed the strain rate effect, usually introduces uncertainties in predicting the shear strength of soils in dynamic penetration tests. Graham et al. (1983) proposed a simple equation which prescribes approximately 5-20% increase in shear strength for each order of magnitude increase in the rate of shear strain. This effect can be expressed by the following widely used equation ⎡ ⎛ γ su = su,ref ⎢1+ λ log ⎜ ⎢⎣ ⎝ γ ref ⎞⎤ ⎟⎥ ⎠ ⎥⎦ (16) where su is the undrained shear strength of the clay, su,ref is a reference undrained shear strength measured at a reference strain rate of γ ref , γ denotes the shear strain rate and λ is the rate of increase of strength per log cycle of time. Typically γ ref = 0.01 per hour for clays (Einav and Randolph 2006), and this value was adopted in the current study. For simplicity, the shear strength of a material can be assumed to be constant while integrating the constitutive equations during an individual time-step of a nonlinear finite element analysis. However, the shear strength parameters must be updated according to the known shear strain rates at the end of each increment. Note that the parameter γ appearing in equation (16) should be evaluated using a frame-independent quantity such as the maximum plastic shear strain. In cone penetration tests it is found that the penetration resistance in clays increases as the rate of penetration (v) or the diameter of the penetrometer decreases. In static tests this is mainly due to partial consolidation occurring in advance of the cone tip, which in turn increases the effective stresses around the cone followed by an increase of the sleeve friction. Usually a normalised velocity, V, is used to assess this degree of consolidation according to V= vd cv (17) where cv represents the coefficient of consolidation of the soil (e.g., Lehane et al. 2009). Values of the normalised velocity between 10 (O'Loughlin et al. 2004) and 100 (Brown and Hyde, 2008) have been recommended to ensure that the soil behaviour is undrained. In the penetration problems studied here the value of normalised velocity is generally above 10,000 due to the fast penetration of the object (see Section 4), and the relatively short penetration time (0.1-0.9 s). Therefore, we assume undrained conditions throughout the penetration, and thus neglect the shear strength increase due to partial consolidation. Centre line FFP Soil (a) Boundary between soil and object before deformation. (b) Distorted boundary (c) Boundary after nodal relocation Figure 2. Nodal relocation on the boundary 2.6 Mesh refinement In the ALE operator split technique the material displacements are obtained at the end of the UL step and will normally result in a distorted mesh. Refining the mesh at the beginning of each Euler step is very important since a distorted mesh can lead to inaccurate results. Most mesh refinement techniques are based on special mesh-generation algorithms, which must consider various parameters such as the dimensions of the problem, the type of elements to be generated and the regularity of the domain. Developing such algorithms for any arbitrary domain is usually difficult and costly. Moreover, these algorithms do not necessarily preserve the number of nodes and the number of elements in a mesh and they may cause significant changes in the topology. A robust method for mesh refinement, based on the use of a simple elastic analysis, was presented by Nazem et al. (2006). The method has been implemented for two-dimensional plane strain problems as well as axi-symmetric problems. To obtain the mesh displacements, we first re-discretise all the boundaries of the problem, which include the boundaries of each discrete body, the material interfaces and the loading boundaries, resulting in prescribed values of the mesh displacements for the nodes on these boundaries. Each boundary node is then relocated along the boundary as necessary. With the known total displacements of these nodes on the boundaries, an elastic analysis is then performed using the prescribed boundary displacements to obtain the mesh displacements for all the internal nodes and hence the optimal new mesh. An important advantage of this mesh optimisation method is its independence of element topology and problem dimensions. The method does not require any mesh generation algorithm, does not change the topology of the problem, and hence can be easily implemented in existing finite element codes. This method is potentially a good candidate for three-dimensional mesh optimisation. The relocation of nodes along the boundaries plays a key role in the mesh refinement scheme proposed by Nazem et al. (2006). This relocation demands an efficient mathematical representation of the boundaries, which strongly depends on the physical problem being simulated. Nazem et al. (2008) proposed an efficient method for nodal relocation based on a quadratic spline technique. This method was successfully employed for solving many geotechnical problems such as consolidation and bearing capacity of soils under a footing. However, it is not suitable for simulating boundaries in a FFP problem due to the linear shape of the penetrating object, which consequently introduces abrupt changes in the slope of the straight lines defining the boundary. Instead, we introduce a simpler, but more efficient, algorithm for nodal relocation along the boundary of the soil in contact with the penetrometer, which is based on the assumption of linear boundary segments. This procedure is depicted schematically in Figure 2. The boundary between the soil and the penetrating object at the beginning of a time step (Figure 2a) may become distorted after the UL step, as shown in Figure 2b. Such cumulative distortion, if not prevented, can potentially cause significant distortion of elements in that region ultimately indicated by a negative Jacobian. Thus, nodal relocation on the boundary can be performed as shown in Figure 2c. Note that the nodes on a curved boundary are only permitted to move in the tangential direction of the boundary, i.e., the normal component of the convective velocity on a boundary is zero, but not necessarily the tangential component. However, on a multi-linear boundary the nodes may be allowed to move along a line defining a linear segment. The nodal relocation on linear segments is shown schematically in Figure 3, where node i must be moved into a new position, i′ , along line 1 or line 2 such that l = li . Note that li represents a normalised length of the segment and can be calculated at the beginning of a time step before the mesh is distorted. The Cartesian coordinates of the new location can be calculated explicitly. For brevity, the full mathematical details are not provided here but the procedure for relocating all nodes along a boundary is described in Nazem et al. (2012). 2.7 Energy absorbing boundaries One of the well-known issues in computational dynamic analysis of soil-structure-interaction (SSI) problems is how to simulate an infinite medium. Employing the finite element method, for instance, with boundaries that are not infinitely distant, one must guarantee that the outgoing waves from the source (usually a structure) do not reflect back from the finite boundaries toward the source, since in reality these waves should propagate to infinity and dissipate at a far distance from the source. If it is allowed to occur, such reflection will most probably affect the accuracy of the numerical results. To assure no waves are reflected back from truncated computational boundaries, it is common to use artificial boundaries that absorb the energy of incoming waves. A simple, but efficient, boundary was developed by Lysmer and Kuhlemeyer (1969), which is known as the “standard viscous boundary” in the literature. The standard viscous boundary is probably the most popular artificial boundary since it possesses an acceptable dissipation characteristic at a low computational cost. Kellezi (1998) suggests that absorbing boundaries must not be located closer than (1.2 - 1.5)λs (where λs is the length of the shear waves) from the excitation source. A recent study by Kontoe (2006) showed that the standard viscous boundary is capable of absorbing dilatational waves (P-waves) as well as shear waves (S-waves) in the analysis of plane strain and axisymmetric problems. When required, this type of boundary was used in the problems solved in this study. y l=li i’ i i+1 i-1 x Figure 3. Linear nodal relocation 3 PENETRATION INTO NON-UNIFORM SOIL In previous studies (e.g., Nazem et al. 2012), we considered a smooth free-falling object penetrating into a uniform soil layer, in which the material properties do not change in any direction. However, in many natural settings the shear strength of soils actually increases with depth. This variation can be expressed by a linear equation as su ,ref ( y ) = su ,ref ( 0 ) + ks y (18) where su,ref(y) represents the reference undrained shear strength of soil at depth y, ks indicates the increase in shear strength per unit depth, and su,ref(0) is the reference shear strength of soil at the ground surface. Table 1. Parameter values adopted in Section 3 Parameter Unit Value d M vo su,ref(0) G/su,ref L ksd/su,ref(0) m kg m/sec kPa 0.04, 0.06,0.08, 0.1 0.5, 1.0 5, 10 1.0 67 0.2 0, 0.25, 0.5, 1.0 - The effect of the shear strength increase parameter ks, on soil penetration was studied by analyzing more than 50 different problems with parameters summarized in Table 1. Note that in all cases the ratio between the shear modulus of the soil, G, and its undrained shear strength, su,ref, is assumed to be 67 at all integration points, and the rate parameter, λ, is 0.2. Depending on the site, the rate of strength increase with depth ks, usually varies between 1-3 kPa/m. Adopting these typical values for a free-falling penetrometer (FFP), the effect of ks on the penetration characteristics will not be significant, mainly due to the small diameter (size) of the penetrating object. Alternatively, a normalized parameter, ksd/su,ref(0), is adopted in this study, which quantifies the significance of the rate of shear strength increase on the penetration. The finite element mesh and boundary conditions of this axi-symmetric problem are depicted in Figure 4. The finite element mesh includes 10,252 nodal points and 4,988 6-node triangular elements each containing 6 integration points. For simplicity, the material damping and friction between the soil and penetrometer are assumed to be zero. Note that the right-hand and bottom boundaries are able to dissipate the energy of oncoming waves. m, v0 Smooth energy absorbing boundary 10d 16d Smooth energy absorbing boundary 7.5d Figure 4. Finite element mesh of free falling penetrometer In a previous study, Nazem et al. (2012) showed that an appropriate strategy for studying the dynamic penetration problem is to plot the initial impact velocity of the penetrometer, 0.5mv2, normalised by 0.25πd3su,ref(0), versus the total depth of penetration, normalized by d. Adopting this strategy, the normalised impact energy of the penetrometer is plotted versus the normalised depth of penetration in Figure 5. According to Figure 5, the depth of penetration of objects with an identical impact energy decreases as the normalised parameter ksd/su,ref(0) increases. Figure 5 also shows that there is a linear relation between the normalised depth of penetration and the normalised Normalised impact energy impact energy, but the slope of such lines depends on the value of ksd/su,ref(0). This dependency, for the specimens analysed in this example, can be eliminated by plotting the initial impact energy, normalised by 0.25πd3su,ref(d), versus the normalised depth of penetration, as represented in Figure 6. Note that su,ref(d) indicates the reference shear strength of the soil at depth d. Similar independence for other material properties and penetrometer parameters is yet to be investigated. ksd/su,ref = 0.0 ksd/su,ref = 0.25 ksd/su,ref = 0.5 ksd/su,ref = 1.0 Penetration in diameters Normalised impact energy Figure 5. Initial impact velocity of the penetrometer, normalised by 0.25πd3su,ref(0), versus the total depth of penetration, normalised by d Penetration in diameters Figure 6. Initial impact velocity of the penetrometer, normalised by 0.25πd3su,ref(d), versus the total depth of penetration, normalised by d To study the deceleration characteristics of the penetrometer, its variation of velocity versus penetration is plotted in Figure 7 for specimens where m = 0.5 kg, d = 0.04 m, v0 = 10 m/sec, and ksd/su,ref(0) = 0.0, 0.25, 0.5, and 1.0. The plots in Figure 7 indicate that the velocity of the object reduces approximately quadratically as it decelerates and penetrates further into the non-uniform layer of soil. Note that the quadratic reduction of velocity with penetration has also been reported in experimental studies as well as numerical analysis results of penetration into a uniform clay layer (Nazem et al. 2012). It is also notable that for all cases studied here the total time of penetration is less than 0.3 sec. Velocity (m/sec) Penetration in diameters ksd/su,ref = 0.0 ksd/su,ref = 0.25 ksd/su,ref = 0.5 ksd/su,ref = 1.0 Figure 7. The velocity of the penetrometer versus the penetration in diameters 4 TORPEDO ANCHORS An interesting application of dynamic penetration problems is the numerical analysis of the installation of torpedo anchors, which have proved to be efficient for deep-water anchoring systems due to their relatively easy and economic installation. Torpedo anchors are usually 10-15 m long, 0.75-1.2 m in diameter, and may weigh between 250-1,000 kN (Randolph & Gourvenec 2011). Assuming that the installation phase is relatively fast, we consider only undrained soil behaviour here. For simplicity, material damping and the friction between the soil and anchor are ignored. The impact velocity of the anchor is assumed to be less than its terminal velocity in a fluid, i.e., the torpedo may accelerate after impacting the soil due to the gravitational force. To investigate the soil behaviour during torpedo penetration, 27 specimens were analysed with the properties summarised in Table 2. The finite element mesh and the boundary conditions are the same as those used in the previous example, and are shown in Figure 4. Table 2. Values of parameters used in torpedo anchor analysis Parameter Unit Value d m vo su,ref (0) G/su,ref λ ksd/su,ref (0) m kg m/sec kPa - 1.0 1,000 5, 10 1.0 67 0.25, 0.5 0, 1.0, 2.5 The numerical results for all specimens, in terms of the total depth of penetration, p, and the total time of penetration, tp, are summarised in Table 3. According to Table 3 the total penetration time of a torpedo anchor is comparatively higher than the penetration time of a miniature freefalling penetrometer (FFP). In addition, the strength rate and shear strength increase with depth parameters, λ and ks, significantly influence the total depth and time of penetration. Table 3. Summary of numerical results for torpedo anchors No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 ks (kPa/m) 0 0 1 1 1 1 1 1 2.5 2.5 2.5 2.5 2.5 2.5 0 0 1 1 1 1 1 2.5 2.5 2.5 2.5 2.5 2.5 λ 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 m (kg) 2,000 2,000 2,000 2,000 5,000 5,000 10,000 10,000 2,000 2,000 5,000 5,000 10,000 10,000 2,000 2,000 2,000 5,000 5,000 10,000 10,000 2,000 2,000 5,000 5,000 10,000 10,000 v0 (m/sec) 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 10 5 10 5 10 5 10 5 10 5 10 p (m) 5.34 14.34 1.96 3.19 4.15 6.07 7.67 10.24 1.49 2.31 2.69 3.87 4.24 5.87 2.21 6.93 2.46 3.01 4.71 5.56 7.93 1.22 1.85 2.06 3.11 3.23 4.66 tp (sec) 1.85 2.89 0.53 0.50 0.93 0.83 1.43 1.26 0.38 0.35 0.63 0.54 0.88 0.77 0.71 1.41 0.39 0.74 0.66 1.16 1.03 0.31 0.28 0.50 0.44 0.72 0.62 To study the penetration properties of a torpedo anchor in more detail, the velocity variations of Specimens 17 and 23 are plotted versus time as well as the normalised penetration in Figures 8 and 9, respectively. For these two specimens all input parameters, except ks, are identical. According to Table 3, the value of ks in Specimens 17 and 23 are 1.0 and 2.5 kPa/m, respectively. Figures 8 and 9 indicate that the torpedo accelerates at the early stages of penetration. This behaviour was observed in all specimens, indicating that, unlike miniature FFPs, the acceleration of the torpedo changes approximately linearly with time. To study the soil behaviour due to torpedo penetration we consider two different cases where ks = 0 (uniform soil) and ks > 0 (non-uniform). For a uniform soil, Nazem et al. (2012) showed that the dynamic soil resistance due to penetration of a FFP can be estimated according to: qdyn = Ndp su ,ref ( 0) (19) where ⎛ G N dp = 2.321 − 50.488λ + (1.690 + 34.546λ ) ln ⎜ ⎜s ⎝ u ,ref ⎛ p⎞ ⎜ ⎟ ⎝d⎠ −1 ( −2.698 − 148.36λ + 410.27λ ⎛ ( 2.399 + 46.52λ − 103.91λ ) ln ⎜⎜ s G 2 2 ⎝ u , ref ⎞ ⎟⎟ − ⎠ + ⎞⎞ ⎟⎟ ⎟ ⎟ ⎠⎠ (20) Velocity (m/sec) Time (sec) Figure 8. The velocity of a torpedo anchor versus time Velocity (m/sec) Penetration in diameters Figure 9. The velocity of torpedo anchor versus the penetration normalised by diameter Using the values in Table 3, the dynamic penetration factors, Ndp, for specimens 2 and 16 are estimated as 31.75 and 53.01, respectively. On the other hand, the dynamic soil resistance versus the normalised penetration predicted by numerical analysis is plotted in Figure 10 for the same specimens. According to Figure 10, the predicted normalised dynamic soil resistance is in good agreement with the value predicted by Equation (19) for Specimen 2, and is slightly lower than 53.01 for Specimen 16. This suggests that the validation of Equations (19) and (20) for torpedo anchors penetration into a uniform soil layer requires further investigation. Unlike uniform soils, the dynamic resistance of a non-uniform layer of soil is not likely to converge to a constant value as the normalised penetration, p/d, approaches infinity. For Dynamic soil resistance / su,ref (0) Specimens 8 and 14, where ks is respectively 1.0 and 2.5 kPa/m, the normalised dynamic soil resistance is plotted versus the normalised penetration of the torpedo in Figure 11. It is evident that the soil resistance due to penetration increases approximately linearly with penetration. This observation also indicates that the force acting on the anchor due to soil resistance changes with time as the object penetrates into soil, and hence the deceleration of torpedo is not constant during penetration. Penetration in diameters Dynamic soil resistance / su,ref (0) Figure 10. Normalised dynamic soil resistance versus normalised penetration of torpedo into a nonuniform soil layer Penetration in diameters Figure 11. Normalised dynamic soil resistance versus normalised penetration of torpedo into a uniform soil layer 5 VALIDATION OF PENETRATION PREDICTIONS Previously, Carter et al. (2010) reported the validation of the finite element results predicted by the method of penetration analysis by comparing the numerical predictions with experimental test results undertaken at the University of Sydney, Australia (Chow 2012). These tests involved model penetrometers being dropped into pots of normally consolidated clay with constant undrained shear strength. An illustration of the experimental apparatus used in these tests is presented in Figure 12. Some of the results of the testing program conducted to validate the numerical approach are presented in Table 4. This table also provides the results obtained from the ALE finite element analysis, which were all conducted assuming λ = 0.2. Table 4 shows that there is generally good agreement between the experimental results and the numerical predictions. Adjustable release mechanism Wall 400 2450 Penetrometer 315 Kaolin pot Note : All dimensions in mm Drawing not to scale (a) (b) Figure 12. (a) Side elevation of laboratory free falling test setup; (b) A completed free falling penetrometer drop into the kaolin pot 6 USE OF PENETRATION PREDICTIONS TO INTERPRET IN SITU TESTS Some examples of the predictions of the final penetration of an object free falling into a uniform clay deposit obtained using the dynamic ALE method were presented in the previous sections. For relatively uniform cohesive soil deposits deforming under undrained conditions, four material parameters are required to define completely the response of the soil to penetration. These are G, su,ref, γ ̇ref, and λ. Table 4. Comparison between experimental results and ALE predictions Test su,ref (kPa) d (m) m (kg) vo (m/s) p/d (Measured) p/d (ALE) Difference ALE and tests (%) 1 5.15 0.02 0.26 4.77 4.33 3.99 7.7 2 5.15 0.02 0.35 4.77 4.80 5.22 8.7 3 5.15 0.02 0.45 4.75 5.20 6.50 25.0 4 5.15 0.02 0.54 4.74 7.05 7.80 10.6 5 5.15 0.02 0.63 4.76 7.94 9.36 17.9 6 7.46 0.04 0.71 4.77 1.14 1.39 22.2 7 6.91 0.04 0.71 4.75 1.39 1.44 3.7 8 6.91 0.03 0.74 4.75 2.75 2.71 1.4 9 4.45 0.02 0.26 4.75 4.45 4.38 1.6 For undrained deformations Poisson’s ratio of the soil takes the value 0.5 because the soil deforms at constant volume. It was found from this study that the final penetration depth was relatively insensitive to G, but was very dependent on the uniform strength su,ref, somewhat dependent on the rate parameter λ and less dependent on the rigidity index, G/su,ref. Provided reasonable assumptions can be made for the values of the parameters G/su,ref, γ ̇ref and λ, the final penetration depth can be used to estimate the uniform value of the rate-independent undrained strength, su,ref. Hence the normalisation of the numerical predictions shown, for example, in Figure 6 proves a useful means of interpreting that value of undrained shear strength from the measured final penetration of the object into the soil deposit. This process is illustrated schematically in Figure 13. Figure 13. Schematic illustraion of the interpretation method for the falling penetrometer test 7 ANALYSIS OF IMPACT COMPACTION In this example numerical analysis of the dynamic compaction of a soil layer under a free-falling weight is considered. The problem domain, axi-symmetric finite element mesh, boundary conditions, and material properties are shown in Figure 14. The water table is located at the ground surface. A fully coupled consolidation analysis under dynamic loading conditions has been carried out. The Modified Cam-Clay (MCC) material model, which facilitates plastic volumetric changes, was used to model soil stress-strain behaviour. The material parameters shown in Figure 14 are defined as follows: λ = the slope of the normal compression line (NCL) in the space of the logarithmic mean stress lnp′ versus the void ratio e, κ = the slope of the unloading-reloading line (URL) in the lnp' - e space, e0 = the initial void ratio, OCR = the over-consolidation ratio of the soil, φ = the friction angle of soil, γ = the unit weight of the soil, k=the hydraulic conductivity of the soil, and ν = Poisson’s ratio of the soil. Smooth / impermeable 8R Smooth / permeable Smooth / impermeable R=1 m γ=17 kN/m3 ν=0.3 φ=250 λ=0.25 κ=0.05 e0=1.8 k=10-9 m/s OCR=1 Smooth / impermeable 8R Figure 14. Dynamic compaction problem The finite element mesh includes 1447 nodal points and 690 6-node triangular elements. The analysis includes three stages. In the first stage, a body force loading due to the self-weight of the soil was used to establish an initial stress field over a long period of time, allowing all excess pore pressure to dissipate. After generating the nonzero stress field, the location of the yield surface at each integration point was adjusted according to the initial effective stresses and the designated value of the overconsolidation ratio (OCR). The second stage of analysis included applying an overburden pressure of p0 = 5 kPa. Finally, the weight was dropped onto the soil layer at an initial velocity, and removed as soon as it comes to rest. The process of releasing the falling weight and allowing it to impact the soil surface was repeated in quick succession, a further four times, thus applying 5 dynamic impact blows to the surface of the soil layer. Four different values of the impact velocity, viz., 1, 2, 3 and 4 m/s, were used in the numerical simulations. To avoid further complexities, the object was assumed to be rough so that tangential displacement of the soil at the falling weight-soil interface was not permitted. It should be noted that the value selected for the hydraulic conductivity of the soil (10-9 m/sec) is typical of a fine-grained soil such as clay. Each successive blow from the falling weight occurs almost immediately after the weight comes to rest during the previous blow. Furthermore, the period of the each impact is only a fraction of a second, so that the overall period to effect 5 blows is very short, on the order of 1 sec. Thus, in this particular example the behaviour of the soil is effectively undrained, i.e., it will occur under the constraint of constant volume deformation. Some results of these dynamic consolidation analyses are presented in Figures 15 to 18. Figure Time (sec) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Velocity (m/sec) 0 -0.5 Impact velocity = 1 m/sec -1 -1.5 Impact velocity = 2 m/sec -2 Figure 15. Velocity versus time Settlement / R 0 0.1 0.2 0.3 0.4 0.5 Velocity (m/sec) 0 -0.5 Impact velocity = 1 m/sec -1 -1.5 Impact velocity = 2 m/sec -2 Figure 16. Velocity versus normalized settlement 15 indicates the velocity of the falling weight from the time it first impacts the soil surface until it comes to rest in the soil. After each blow it is raised virtually instantaneously and allowed to impact the soil again with the same velocity as the previous blow. Results are included in Figure 15 for initial impact velocities of 1 and 2 m/sec. The same results are presented in Figure 16 in the form of the velocity of the falling weight versus the settlement it causes in the soil immediately in Dynamic soil resistance (kPa) 70 Impact velocity = 1 m/sec 60 Impact velocity = 2 m/sec 50 40 30 20 10 0 0 0.1 0.2 0.3 0.4 Settlement / R Figure 17. Dynamic soil resistance versus normalised settlement contact with the falling weight. Both Figures 15 and 16 indicate the cumulative effects of each successive impact. The final settlement after 5 blows is dependent on the initial impact velocity, and for the two cases considered here is either almost 20% or 40% of the radius of the impact weight. The variation of the soil resistance with penetration of the falling weight into the soil, for each impact blow, is illustrated in Figure 17. The resistance is plotted as the average contact pressure between the soil and the falling weight. It is of interest to observe that the peak in this resistance increases with each blow. It would also be of interest to predict whether this increase continues with blows beyond the mere 5 studied here. However, such calculations have not yet been conducted and are the subject of an on-going research work. Because the impact loading occurs relatively rapidly on this saturated clay-like soil, it is also of interest to predict the excess pore water pressures generated in the soil and how they vary as a result of repeated impact blows. Figure 18 indicates contours of total pore water pressure (static plus excess) initially, before any impact loading, and immediately after each impact blow from the falling weight, for the case of an initial impact velocity of 2 m/sec. The steady accumulation of excess pore water pressure with each blow is indicated by the regular and increasing distortion of the pattern of initial static pore water pressure. The increasing penetration of the falling weight with successive blows is also evident from the deformed shapes plotted in Figure 18. It would also be of interest to predict the further deformation of the soil as these excess pore water pore water pressures generated by the repeated impact dissipate and the soil consolidates. Such predictions are also the subject of on-going work. 8 EXPERIMENTAL RESULTS FOR IMPACT COMPACTION To enable comparison with the numerical results model scale dynamic compaction tests have been conducted. These have been performed in a 2-D configuration in a tank with a Perspex face, shown schematically in Figure 19, allowing observation of the mechanism of dynamic compaction. These tests have been limited to dry and partially saturated sands and sand-silt mixtures and thus are not directly comparable to the numerical analyses of the impacts on clay. The tests have made use of high speed photography and digital image correlation (DIC) techniques to investigate the deformation patterns, calculate soil strains and observe strain localisation. Additional instrumentation has included accelerometers on the plunger and buried in the soil, and load cells. The testing apparatus, shown in Figure 19, consists of a fabricated container fixed to a steel frame open at the top with a clear acrylic (Perspex) front. The metal a.Before impact c. After 2nd drop e. After 4th drop b. After 1st drop. d. After 3rd drop f. After 5th drop Figure 18. Contours of pore water pressure (kPa) – Impact velocity = 2 m/sec frame extends to a height of 2.8 m above the floor with a fixed pulley system resting on an axle positioned at the top of the frame. The internal dimensions of the soil container are 350 mm x 150 mm x 750 mm. A 25 mm thick Perspex sheet is bolted to the front of the frame. A high speed camera was positioned 1.2 m from the testing apparatus. The field of view of the 1024 x 1024 pixel image is 403 mm x 403 mm giving an object space pixel size of 0.39 mm. Photographs have been taken at rates between 1000 frames per second (fps) and 3000 fps. Further details of the photographic technique are provided in Nazhat et al (2011). Figure 19. Schematic of test configuration The guiding trolley and attached plunger are able to free fall up to 1 m, although in the tests discussed here drop heights ranged between 150 mm and 300 mm (impact velocities of 1.7 to 2.4 m/s). The weight of the free falling assembly ranged from 50 N to 180 N, depending on the width of the tamper which varied between 35 mm and 110 mm. The tamper length (150 mm) was equal to that of the container to ensure a 2-D configuration. A typical test has involved 5 to 8 drops of the tamper, with high speed photography taken over a time interval of 3.2 seconds for each drop. The test programme has included tests of dry Sydney sand at different relative densities, tests to investigate the effect of adding non-plastic fines to the sand, tests to investigate the effect of degree of saturation, and tests to investigate the effect of tamper geometry. The soils used have been Sydney sand, which has the properties d50 = 0.3 mm, coefficient of uniformity cu = 3, maximum and minimum voids ratios emax = 0.8, emin = 0.58, and non-plastic feldspar fines which has been mixed with the sand to form a 2:1 sand:silt mixture with properties d50 = 0.2 mm, cu = 22, emax = 0.89, emin = 0.51. The results of high speed photography combined with DIC have been used to investigate the pattern of deformation in the soil. Figure 20(a) shows a typical displacement pattern after an impact on a dry loose sand specimen. It is noticeable that the impact mobilises a general bearing capacity mechanism with much of the sand displaced sideways and upwards as the plunger penetrates the soil. It can also be observed that there is a region beneath the plunger where the soil is displaced downwards, and this is the region of most interest for dynamic compaction in practice. If the region beneath the plunger is looked at in more detail the strain pattern shown in Figure 20(b) is obtained. This figure shows the cumulative shear strains 0.01 sec after the impact and indicates a sequence of shear (compaction) bands that have been shown (Nazhat and Airey, 2012) to propagate down from the plunger. Volumetric strains in this region are only significant within these shear strain bands. After a compaction band has passed any location the shear and volumetric strains return almost to zero. However, at the end of the process the compaction bands are locked in the soil. In a series of dynamic impacts, each impact causes additional shear (compaction) strain bands to propagate and accumulate in the soil so that after several impacts the cumulative strain contours appear as in Figure 21a. The pattern associated with these strain bands depends on the soil type and tamper geometry (Nazhat and Airey, 2012). As an example, the effect of 6 impacts on a compressible sand-silt mixture is shown in Figure 21(b). In this case the general bearing capacity mechanism is absent and the energy from the impact is used more effectively to compact the soil beneath the plunger. Figure 20. (a) Displacement vectors produced by an impact on loose sand, and (b) Cumulative shear strain contours 0.01 s after impact beneath the plunger Figure 21. Cumulative shear strains after 6 impacts on (a) loose sand, (b) loose sand-silt mixture In addition to the photography accelerometers and load cells have been placed on the plunger and buried in the soil. For comparison with the numerical results for clay a typical result showing the average stresses mobilised by the plunger during an impact and during slow static loading is shown in Figure 22. This figure shows that very similar responses are observed, at least during the early stage of soil penetration. It is expected that these particular results will be more alike than would be the case for clay, because rate effects are less significant in sand. It has also been observed that the mechanism for sands is similar in both static and dynamic loading (Nazhat, 2013), the primary difference is the presence of the compaction bands at depth, but these represent only a small part of the total energy dissipation. 120 100 Average applied stress (kPa) Dynamic Test Static Test 80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 100 Displacement (mm) Figure 22. Soil resistance versus depth of penetration for static and dynamic loading The existence of shear/compaction bands observed in these experiments presents a significant challenge for numerical modelling, and is unlikely to be predicted by conventional dynamic analyses adopting the more common, continuum-based constitutive models used regularly to represent the static behaviour of soils. Further research is under way to understand these shear bands and to develop numerical analyses capable of reproducing this behaviour. 9 CONCLUSIONS Finite element solutions of problems involving impact compaction and dynamic penetration of soil deposits, by either free falling or propelled objects, have been studied. A robust numerical method was described for dealing with such complex and difficult numerical problems. The method was employed in a parametric study of a variety of penetrometers free-falling into layers of inhomogeneous clay deforming under undrained conditions. The effect of the mechanical properties of the clay soil on the penetration characteristics was discussed and some of the predictions were compared with the results of laboratory experiments. It was found that these smaller penetrometers decelerated at approximately uniform rate soon after initial impact with the soil. A penetrometer of much larger physical dimensions, viz., a torpedo anchor typical of those used in offshore moorings, was also investigated numerically. The key difference in response for these more massive penetrometers was the prediction of a non-uniform rate of deceleration during penetration of the soil. As for the smaller penetrometers the depth of final penetration was a function of the kinetic energy with which the penetrometer first impacts the soil. How the numerical results for essentially cylindrical penetrometers with a conical tip might be used in the interpretation of in situ penetration tests was then suggested. It was found that the known initial impact energy and final penetration depth could be used to estimate the undrained shear strength of the soil. Finally, the numerical method was also adopted to examine the fundamental compaction behaviour of soils and the trends in predicted results were compared, at least broadly, with observations made in physical experiments. It was noted that further research is required to understand the deformation mechanisms observed in the compaction experiments and to develop numerical analyses capable of reproducing such behaviour. It appears that current methods based on continuum models of soil behaviour are unable to reproduce the behaviour observed in the physical experiments. 10 REFERENCES Abelev, A., Simeonov, J. & Valent, P. 2009, Numerical investigation of dynamic free-fall penetrometers in soft cohesive marine sediments using a finite element approach, Oceans 2009, IEEE/MTS conference, Biloxi. Beard, R.M. 1977, Expendable doppler penetrometer, Technical Report, Naval Construction Battalion Center, Port Hueneme, California. 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True D.G. 1975, Penetration of Projectiles into Sea Bed Floor Soils, Civil Engineering Laboratory, Technical Report R-822, Port Hueneme, CA. Wriggers, P. 2006, Computational Contact Mechanics, Springer-Verlag, Austria. 11 ACKNOWLEDGEMENTS The authors would like to acknowledge the support of the Australian Research Council in funding some of the work described in this paper.
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