The following section presents topics related to nonlinear continuum mechanics, and are es-sential for the understanding of the nonlinear finite element method, a tool that is widely used for this thesis.
Continuum mechanics is concerned with the deformation and motion of materials modeled as a continuous mass (continuum), and ignores inhomogeneities like molecular, grain and crystal structures. The objective is to establish models for the macroscopic behaviour of both fluids, solids and structures [16].
To develop a model for material behaviour, it is necessary to describe the state of the body, in other words, its configuration. The domain of the initial configuration is denoted Ω0, and for the current configuration, Ω (see Figure 2.8). Equations will be referred to a reference configuration and can be the same as both the initial and the current state of the body. The initial configuration will often be the undeformed state.
Figure 2.8: Initial and current configuration of a body.
CHAPTER 2. THEORY
The position of a material point in the initial configuration is given by X and the current is given by x. u is the deformation and φ is the mapping between the initial and the current configurations, given by x=φ(X,t).
2.2.1 Eulerian and Lagrangian description
When analysing the deformation and motion of a continuum, two different descriptions are used: a Lagrangian (material) and an Eulerian (spatial). The Lagrangian description uses the initial configuration as a reference, while the Eulerian uses the current configuration.
Consequently, the Eulerian description focuses on a fixed point in space describing particles passing through the point, while the Lagrangian focuses on the movement of individual particles [16]. The behaviour of solids is usually dependent on the deformation history and the undeformed configuration must therefore be known. A Lagrangian description is therefore dominant in solid mechanics. An Eulerian description is typically used for fluid mechanics, since it is often impossible to describe a fluid on the basis of the initial configuration. In addition, fluids are generally independent of deformation history, making it unnecessary to use a Lagrangian description.
Figure 2.9: Deforming block with Lagrangian (L) and Euler (E) elements.
CHAPTER 2. THEORY Figure 2.9 illustrates the difference between an Eulerian and a Lagrangian mesh. For the Eulerian mesh, the coordinates of the nodes are fixed, whereas for the Lagrangian mesh the, node coordinates are independent of time. The white dots in Figure 2.9 illustrates the element material points. In a Lagrangian mesh, the nodes and the material points will stay coincident with the element integration points. Because of this, the boundary nodes will remain on the boundary when the body deforms. This makes it easier to apply boundary conditions in Lagrangian meshes. Adding boundary conditions is much more complicated for an Eulerian mesh, since the boundary nodes do not remain coincident with the boundary.
On the other hand, since the elements deforms with the material, elements in a Lagrangian mesh can become severely distorted for large deformations. In an Eulerian mesh, the element length remains constant and large deformations will not be a problem.
2.2.2 Arbitrary Lagrangian-Eulerian Formulation (ALE)
The Arbitrary Lagrangian-Eulerian (ALE) formulation is a hybrid technique combining the advantages of Eulerian and Lagrangian formulations while seeking to minimize the disadvan-tages of the different methods. ALE will not be discussed in detail here, since the method has not been applied in any numerical studies for this thesis. It is however important to know about the method, since ALE is an alternative approach for solving problems that will be presented later.
In an ALE method, the nodes can either move with the material in a Lagrangian sense, be kept in place as in an Eulerian description or the nodes can move arbitrarily to accommo-date rezoning needs and avoid mesh entanglement (see Figure 2.10)[17]. In a fluid-structure analysis involving ALE, the fluid mesh follows the deformation of the (Lagrangian) structure.
This is in contrast to an embedded analysis, which will be discussed further in Section 2.3.
CHAPTER 2. THEORY
Figure 2.10: 1D example of ALE mesh and particle motion [17].
2.2.3 Conservation laws
The conservation laws (balance laws) must be satisfied in all physical systems, and are conse-quently an essential tool in continuum mechanics. In short, they state that certain properties of an isolated physical system do not change as the system evolves. For a thermomechanical system, the following conservation laws are relevant [16]:
• Conservation of mass
• Conservation of linear momentum
• Conservation of energy
• Conservation of angular momentum
The conservation laws can be used to derive the conservation equations, which may be of either Eulerian or Lagrangian description. Derivation of the equations can be found in the literature [16].
CHAPTER 2. THEORY
CHAPTER 2. THEORY
The following describes the different variables used for the conservation equations.
ρ, ρ0: Current and original density v: Velocity field
J: Determinant of Jacobian between spatial and material coordinates, J=det[δxi/δXj] σ: Cauchy stress
b: Body force P: Nominal stress q: Heat flux
w: Hyperelastic potential on reference configuration s: Specific heat source term
F: Deformation gradient, Fij =δxi/δXj
Conservation equations in an ALE description is similar to those of an Eulerian description, with the only difference being a material time derivative DfDt. Please refer to the literature for more details [16].
2.2.4 Constitutive equations
To mathematically describe the behaviour of a material, a constitutive equation must be used.
A response of the material is then approximated by relating physical quantities specific to that material. For example, in solid mechanics, the constitutive equation is often referred to the stress-strain relation of a material. These relations are often purely phenomenological.
There exist numerous constitutive equations, some more comprehensive than others. Only a short description of the constitutive relation adopted for a fluid (air) will be presented here.
The constitutive relations for the solid (concrete) will be discussed in Section 3.3, where selected material models are presented.
CHAPTER 2. THEORY
The ideal gas law
The pressure P of an ideal gas is given by the equation:
P =ρ(γ−1)eint (2.16)
where ρ is the density,γ = CCp
v is the heat capacity ratio and eint the specific internal energy.
The gas may also be described by the equation:
P =ρRspecT (2.17)
where Rspec is the specific gas constant and T is the temperature.
The conservation equation listed in Section 2.2.3 makes up an indeterminate set of equa-tions. In order to close the system, one extra equation is required, namely a constitutive equation such as the ideal gas law. The ideal gas law, together with the conservation laws, are able to describe a compressible flow, e.g. a flow that experiences significant changes in fluid density [18].