Beams are slender structural elements, usually defined by their axis and cross-sections. This permits two basically different ways of modeling beams, one based on the translation and rotation of the points on the beam axis and the connected cross-sections (Fig. 4.1), the other treating the beam as a special example of a fully three-dimensional continuum. This chapter and the next develop fully non-linear beam models of the first type. In this chapter the fully non-linear theory of a beam represented as a curve with elastic properties – a so-called elastica – is developed, and its finite element implementation discussed. This theory is in principle unique, and can therefore be called the theory of the elastica. It describes the deformation of an elastic space curve in terms of total translation and rotation, and relies heavily on the properties of finite rotations, presented in Chapter 3. An alternative formulation of non-linear beams is presented in Chapter 5. In that description the motion is decomposed into a local deformation described in a frame of reference following the beam, and a motion of the local frame of reference. The local frame of reference moves with each beam element, and this type of formulation is therefore often called co-rotating. Simple beam theories can be used within a co-rotating formulation, where finite rotation contributions are added. The application-oriented reader may want to go directly to the co-rotational formulation in Chapter 5.

Many problems of practical interest involve non-linear behavior of solids and structures. In the present context a solid means a body with a firm shape, as opposed to a fluid, while a structure refers to a solid composed of slender elements such as beams, plates and shells. Typical problems are the motion of robots, collapse scenarios of structures, metal forming processes in industrial production, and material deformation and failure in geotechnical engineering. These problems typically involve a considerable change of shape, often accompanied by non-linear material behavior.

The finite element method is an important tool for the analysis of nonlinear problems, such as geometrical and material non-linear behavior of solids and structures. The solution of non-linear problems by the finite element method involves modeling, leading to the formulation of an appropriate set of non-linear equations describing the problem, followed by an appropriate strategy for the numerical solution of these equations. In contrast to linear problems, where the solution strategy reduces to the solution of a system of linear equations, the solution phase in a non-linear problem typically involves an iterative procedure.

Non-linear modeling and analysis is a very active research area with many engineering applications. The many different aspects involved are not covered in any single text. However, some central references to general texts should be given here. A brief introduction to some of the basic problems of non-linear finite element analysis of solids and structures is included in the book by Cook et al. (1989).

The finite element method has been the method of choice for modeling and analysis of structures and solids for several decades. The basic idea is that the structure (or solid) is considered as an assembly of elements, and that each element is modeled in a standard format that permits repetitive use of the individual element formats. Bar elements only contain a single internal degree of freedom – the elongation – and they are therefore a convenient means for introducing and illustrating the basic features of geometrically non-linear finite element analysis.

In a geometrically non-linear problem the first question to arise is the definition of a non-linear measure of deformation, the strain. This is addressed in Section 2.1. When a structure is assembled from the individual elements, use is generally made of the principle of virtual work. The principle of virtual work is a restatement of the equilibrium equations in scalar form. It turns out that once a non-linear strain definition has been adopted, the corresponding definition of stress follows from the formulation of the principle of virtual work. This is the subject of Section 2.2. The tangent stiffness matrix of a geometrically non-linear bar element is derived in Section 2.3 in global coordinates.

The derivation of the equilibrium and stiffness relations of the bar element is quite simple because the strain is constant within the element. In order to illustrate the relation to more complex problems involving other types of elements, the tangent stiffness matrix is re-derived by use of shape functions in global coordinates.

Beams and shells can be considered as three-dimensional solids, in which two or one of the dimensions are small compared to the remaining dimensions. This has given rise to beam and shell models in which the representation of the transverse displacement components is simplified. These models are formulated in terms of the translation and rotation of a reference curve or surface in the case of beams and shells, respectively. As it will appear, there are fundamental differences between the representation of translations and rotations. It is important to account for the special characteristics of rotations in the formulation and analysis of kinematically non-linear theories for beams and shells, and this chapter provides a concise presentation of the special properties of rotations needed for the development of general non-linear beam and shell theories. A detailed discussion of rotations and their various representations has been given by Argyris (1982) and by Géradin and Cardona (2001).

The translation of a point is described by a vector u, and the result of a sequence of translations Δu1, Δu2, … is simply the sum of the individual translation vectors. The result is independent of the order of the individual terms. Finite rotations cannot be described in this simple manner, as may be seen by considering two rotations of 90° about orthogonal axes, and comparing the result with that of the same rotations taken in the opposite order (Goldstein, 1980).

Abstract

We consider the problem of the existence and finite dimensionality of attractors for some classes of two-dimensional turbulent boundarydriven flows that naturally appear in lubrication theory. The flows admit mixed, non-standard boundary conditions and time-dependent driving forces. We are interested in the dependence of the dimension of the attractors on the geometry of the flow domain and on the boundary conditions.

Introduction

This work gives a survey of the results obtained in a series of papers by Boukrouche & Łukaszewicz (2004, 2005a,b, 2007) and Boukrouche, Łukaszewicz, & Real (2006) in which we consider the problem of the existence and finite dimensionality of attractors for some classes of twodimensional turbulent boundary-driven flows (Problems I–IV below). The flows admit mixed, non-standard boundary conditions and also time-dependent driving forces (Problems III and IV). We are interested in the dependence of the dimension of the attractors on the geometry of the flow domain and on the boundary conditions. This research is motivated by problems from lubrication theory. Our results generalize some earlier ones devoted to the existence of attractors and estimates of their dimensions for a variety of Navier–Stokes flows. We would like to mention a few results that are particularly relevant to the problems we consider.

Most earlier results on shear flows treated the autonomous Navier–Stokes equations. In Doering & Wang (1998), the domain of the flow is an elongated rectangle ω = (0, L) × (0, h), L ≫ h.

Abstract

Since the 1970s the use of statistical solutions of the Navier–Stokes equations has led to a number of rigorous results for turbulent flows. This paper reviews the concept of a statistical solution, its role in the mathematical foundation of the theory of turbulence, some of its successes, and the theoretical and applied challenges that still remain. The theory is illustrated in detail for the particular case of a two-dimensional flow driven by a uniform pressure gradient.

Introduction

It is believed that turbulent fluid motions are well modelled by the Navier–Stokes equations. However, due to the complicated nature of these equations, most of our understanding of turbulence relies to a great extent on laboratory experiments and on heuristic and phenomenological arguments. Nevertheless, a number of rigorous mathematical results have been obtained directly from the Navier–Stokes equations, particularly in the last two decades.

Of great interest in turbulence theory are mean quantities, which are in general well behaved, in contrast to the corresponding instantaneous values, which tend to vary quite dramatically in time. The treatment of mean values, however, is a delicate problem, as remarked by Monin & Yaglom (1975). In practice time and space averages are the most generally used, while in theory averages with respect to a large ensemble of flows avoid some analytical difficulties and have a more universal character.