Fabien Béchet

Singularities of Hyperbolic Inhibited Shells

This chapter is devoted to hyperbolic shells, whose principal curvatures are of opposite sign. Consequently, at each point of the middle surface, there are two asymptotic directions. Concerning the singularities emerging when \(\varepsilon \searrow\) 0, some aspects are very similar to the case of parabolic shells. For instance, singularities along characteristics are more singular than the loading f3 (at least for the normal displacement u3) and propagate.

Generalities on Boundary Conditions for Equations and Systems: Introduction to Sensitive Problems

This chapter constitutes a general heuristic study of sensitive problems, and in particular of sensitive elliptic shell problems, i.e. elliptic shells clamped (or fixed) by a part Γ0 of the boundary and free by the rest Γ1. Note that such sensitive problems or “ill-posed problems”, have already been considered in general in [70], and in some very particular case of shells in [10][23][83].

Singularities of Elliptic Well-Inhibited Shells

The theoretical analysis developed in Chapters 2 and 5 revealed the singular displacements which can appear in the internal and boundary layers when the loading is singular. These singularities and internal layers are linked to the loss of regularity of the bending displacement \(u^{\varepsilon}_3\), solution of the Koiter model, when the thickness e tends to zero.

Singularities and Boundary Layers in Thin Elastic Shell Theory

The aim of this chapter is to study the asymptotic behavior of the Koiter shell model when the relative thickness e tends to zero, and the singularities of the displacements, resulting from a singular loading, which appear in the internal and boundary layers.

Geometric Formalism of Shell Theory

Shells are classically described geometrically by a middle surface (a two dimensional surface embedded in ℝ3) and a thickness which is generally constant. In this book, we will focus on thin shells whose thickness is small with respect to the characteristic dimension of the middle surface. Moreover, we will restrain our analysis to the linear elastic framework. Thus in this book, the thin elastic shells studied will be described by Koiter shell model, or at the limit by the membrane model, which are two-dimensional models involving only the variables of the middle surface.

Examples of Non-inhibited Shell Problems (Non-geometrically Rigid Problems)

In this chapter, we shall see how the numerical anisotropic adaptive procedure of remeshing reacts when the shell is not inhibited. In that case, the middle surface naturally deforms with inextensional displacements, and the Koiter model tends to a pure bending problem when e tends to zero (see section 2.3.5 of chapter 2). Indeed, in order to minimize elastic energy, the natural trend of the shell (for small e) is to deform by “pure bendings”: it avoids the (large) membrane energy, and only uses the (small) bending energy. In these very particular deformations, involving inextensional displacements, the asymptotic lines of the surface play a peculiar role, leading to an anisotropic behavior.

Numerical Simulation with Anisotropic Adaptive Mesh

Chapters 1 and 2 revealed all the diversity of the limit problems and of the associated singularities existing in shell theory when the thickness e tends to be zero. Obviously, this induces serious difficulties for numerical computations of Koiter shell model (or of any other shell model).

Numerical Simulations for Sensitive Shells

Chapter 7 was devoted to the study of singularities for elliptic well-inhibited shells, or equivalently elliptic shells clamped all along their lateral boundary. In chapter 8, we considered elliptic shells having a part of their boundary which is free. In that case, they are ill inhibited (and even “sensitive”, i.e. V m is not a space of distribution), and the problem is more complex. A pathological behavior emerges progressively when e tends toward zero. A complexification phenomenon (large oscillations corresponding to a new kind of instability) appears on the free boundary.

Anisotropic Error Estimates in the Layers

As we saw in the last chapter, the normal displacement u3, as a solution of the limit membrane model, is at best in L2(Ω), whereas the solution \(u^\varepsilon_3\) of the Koiter model is in H2(Ω) (for e> 0). Consequently, boundary and internal layers, leading to singularities of the displacements inside the layers, may appear during the asymptotic process when e decreases toward zero.