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Hydro-mechanically Coupled Interface Finite Element for the Modeling of Soil-Structure Interactions: Application to Offshore Constructions

This chapter presents how the multiphysical soil-structure interactions can be modeled using the finite element method (FEM). This method takes into account different nonlinearities, due to geometry, material behavior, physical couplings and interfaces. The FEM allows the volume to be discretized into a finite number of elements and a finite number of degrees of freedom. The numerical techniques for solving a nonlinear problem in the framework of the FEM have been extensively studied and are now very robust.

The behavior of offshore foundations is highly dependent on interface properties. Upon large overturning moment or lateral load, the soil-structure interaction results in rotation, sliding or gap opening. Hydro-mechanical couplings have become particularly important in many applications, as the loading rate and soil permeability are such that the soil behavior is not completely drained or undrained.

A hydro-mechanically coupled interface element is proposed in this work to simulate the behavior of suction caissons embedded in sand. This interface element simulates the mechanical problem with a penalty method and reproduces the friction at the interface as a simple Coulomb model. The interface element introduces hydromechanical couplings based on the gap opening (storage, longitudinal permeability evolution) and the definition of an effective contact stress.

1.1. Introduction

The advent of numerical methods and, especially, finite element approaches has provided engineers with formidable tools to predict the behavior of constructions. Most of the time, the soil can be considered as a homogeneous medium behaving in a purely drained or undrained manner. However, in some cases, depending on the soil permeability and the loading rate, partially drained behaviors should be considered. In this case, hydro-mechanical couplings must be included in the finite element formulation for both volume and interface elements.

The development of renewable energy sources is the greatest challenge of the 21st Century. A large number of offshore wind farms have been developed throughout the last decade, especially in shallow waters in the North Sea where a sandy seabed is mostly encountered. The offshore wind industry is expected to grow exponentially, followed by the development of more recent wave and tidal energy devices. Whether they are bottom-fixed or floating, their foundations or anchors will have to be designed. The finite element simulation of anchors and foundations has become more popular among offshore wind farm designers. However, the modeling of the interface between the soil and the foundations is still very rudimentary.

From the mechanical point of view, shearing along the interfaces participates in the strength of the foundation against applied loads in any direction (horizontal, vertical or moment). When the maximum capacity is overcome, sliding occurs between the soil and the foundation. Each movement of the foundation results in a fluid flow, since it lies under the sea. This affects the interface behavior by modifying the maximum shear stress available or by inducing a suction effect, because of a gap opening (e.g. for suction caissons). The finite element modeling of these complex interactions at the interface is the topic of this chapter.

1.1.1. The finite element method (FEM)

The finite element method (FEM) is one of the most popular numerical methods for solving partial differential equations for boundary value problems [ZIE 00]. The main concept of the method is to decompose the entire continuum domain to be modeled into a collection of subdomains, called elements. The continuum field of any physical variable of interest (e.g. displacement, pore water pressure) is approximated over each element, based on a finite number of variables (physical unknowns) and pre-defined interpolation functions. The partial differential equations that must be solved over the entire continuum domain can then be approximated and discretized for each element. The set of discrete equations associated with each element can be combined into a single global system and solved algebraically. The solution of this set of equations is the one that minimizes the error of the approximation with respect to the actual solution. The FEM has many advantages, such as the ability to simulate complex geometries, incorporate different constitutive laws, involve multiphysical couplings and analyze stress and internal variables locally.

1.1.2. Review of existing contact formulations

Numerically solving the mechanical contact problem is not a recent topic, and several books have been dedicated to this issue [JOH 92, WRI 06]. Within the framework of finite element methods, two general approaches exist in order to manage contact between two solid bodies, namely the thin layer and the zero-thickness approach, as shown in Figure 1.1.

The first approach consists of explicitly modeling the contact zone with special finite elements, designed to encounter large shear or compression deformation [DES 84, SHA 93, WRI 13, WEI 15]. The second approach, adopted in the following, involves special boundary elements which have no thickness, namely zero-thickness elements [GOO 68, CHA 88, DAY 94, HAB 98, WRI 06]. They discretize a potential zone of contact. A gap between each side of the interface and a probable other solid is computed at each time step to detect contact. Three main ingredients are necessary to formulate this approach:

• - a criterion to rule the contact detection/loss and contact pressure evolution;
• - a constitutive law to describe the shear/normal behavior(s);
• - a technique to discretize the contact area between solids and to compute the gap function gN, namely the distance between two solids.

Figure 1.1. Comparison between the thin layer and zero-thickness approaches in the case of Hertzian contact (source: [CER 15])

The normal contact constraint ensures that two perfectly smooth solids in contact do not overlap each other. This mathematical criterion relates the gap function gN and the contact pressure pN. It states that when the gap is null, the solids are in contact and a contact pressure prevents their overlapping. When the pressure is equal to zero and the gap is greater than or equal to zero, there is no contact.

The relationship between gap and pressure evolution can be termed as high precision or low precision. In the first case, this relationship is physically based and the pressure increases with a gap that can be negative, leading to interpenetration (gN < 0). This is especially true in rock mechanics, where surfaces in contact are not perfectly smooth and the normal pressure increases with deformation of asperities [GEN 90].

A low-precision contact is ensured on a purely geometrical basis. Two surfaces of bodies in contact are considered to be perfectly smooth and interpenetration is theoretically not allowed (gN = 0) [WRI 04]. There are only two states: in contact or not in contact. The Lagrange multiplier method exactly ensures this condition [BEL 91, WRI 06]. Lagrange multipliers are introduced to enforce the constraint, and correspond to the contact pressures. On the other hand, the penalty method [HAB 98] regularizes the constraint by authorizing a limited interpenetration of the solids in contact. The contact pressure is linearly proportional to this interpenetration by a coefficient called the penalty coefficient. Both methods are compared in Figure 1.2.

Figure 1.2. Comparison between Lagrange multiplier and penalty methods on deformation and distribution of contact pressures (source: [CER 15])

The penalty method is adopted in the following for its simplicity, since it does not require us to introduce a variable number of unknowns (Lagrange multipliers) at each step. Both solutions are theoretically identical if the penalty coefficient tends to infinity. However, increasing this coefficient too much creates a badly conditioned stiffness matrix, which becomes difficult to solve.

The mobilization of shear stress along the interface is very important in many applications. The maximum shear stress that could be mobilized in a tangential plane is strongly dependent on the normal pressure. The most basic relationship between them is the classical Coulomb criterion. However, similar to the normal behavior, more complex constitutive laws may be defined: for example, to describe rock joints [ALO 13, ZAN 13] or soil-structure interfaces [LIU 06, STU 16], including critical state, we use dilative normal behavior and even cyclic degradation [LIU 08].

The normal contact constraint is defined as a continuous condition over the boundaries of the solids in contact. It must be discretized in the finite element framework. Computation of the gap (gN) and enforcement of the contact constraint are different depending on the method, as shown in Figure 1.3.

Figure 1.3. Comparison between the discretization methods of the contact area (source: [CER 15])

The simplest method, namely node-to-node discretization, imposes the constraint on a pair of nodes [KLA 88], limiting the problems to small relative tangential displacement. The...