Our aim in this chapter is to treat several classical problems of elastostatics. Strictly speaking, elasticity problems such as those described below cannot be solved exactly in general: they can be solved exactly in very particular cases (e.g., special geometry); otherwise, approximate numerical solutions are obtained using computers. However, in the examples treated below, we are going to find approximate solutions giving an idea of the exact solution under some reasonable conditions that will be made precise in each case (by using, in particular, the Saint-Venant principle described in Section 14.7).

For each of the mechanical problems that we will consider, we will find (by guessing) an exact solution for a modified problem related to the one under consideration. By the uniqueness theorem for elastostatics, there is no other solution to the modified problem. Then, the relation between the solutions of the initial and modified problems is made precise using the Saint-Venant principle. We will also interpret the mathematical results from the mechanical point of view, which leads in general, but not always, to conclusions that are consistent with practical intuition.

Longitudinal traction–compression of a cylindrical bar

We consider an elongated cylindrical bar in traction (or in compression). We assume that the axis of the cylinder is parallel to Ox1 (see Figure 14.1).

For the study of the problem, we formulate the following simplifying assumptions that are realistic when the bar is long enough and when we remain far enough from its ends:

• The volume forces are negligible;

• The external forces on the lateral surface vanish;

• […]

In this last chapter of Part 1, we return to the fundamental concepts of continuum mechanics and develop two new independent subjects.

On the one hand, we introduce some thermodynamical concepts, namely, internal energy, heat, and temperature to express the energy conservation principle, which leads to a new equation.

On the other hand, we study shock waves: contrary to the regularity assumptions consistently made until now, we consider here the case in which some physical and mechanical quantities are piecewise regular, that is, everywhere regular except at the crossing of some surfaces. It is this framework that is used, for instance, in perfect fluid mechanics, to model the shock waves produced by planes flying at transsonic or supersonic speeds.

Heat and energy

We consider a material system S that fills the domain Ωt at time t.

Definition 6.1.For every material system S and at each time t, there exists a measure carried by Ωtof the form e(x, t) dx, where e is nonnegative. By definition

is the internal energy of S at time t, e(x, t) is the mass density of specific internal energy of S at time t, and ρe is the volume density of internal energy.

Definition 6.2.The energy of the system S at time t is the sum of its kinetic energy and of its internal energy:

The energy ε is sometimes called the total energy of the system and is thus defined by its volume density

Remark 6.1: For fluids, thermodynamics yields relations between ρ, p, and e (p is the pressure). In particular, it postulates the existence of a relation, called the equation of state, of the form e = g(p, ρ).

The purpose of this chapter and of the next one is to present complex problems of fluid mechanics involving other physical phenomena such as electromagnetism (for this chapter) and chemistry (for the next one).

In this chapter, we consider flows of fluids conducting electricity in the presence of electric currents and of electromagnetic fields; the purpose of magnetohydrodynamics (MHD) is to study such flows. The MHD equations consist of the Maxwell equations, which govern the electromagnetic quantities, and of the equations of fluid mechanics in which we include the electro-magnetic forces.

In this chapter, we will limit ourselves to a single fluid flow: in certain circumstances (e.g., very high temperatures, intense electromagnetic fields), the conductive fluid is ionized and becomes a plasma; it then becomes desirable to study the flows of the positive and negative particles separately and, in such a case, one has to deal with a multifluid flow, which is a situation similar to the flow of the mixture of two miscible fluids. That case does not present any particular difficulty; multifluid flows are considered in the next chapter.

The modeling of the electromagnetic phenomena leads to the Maxwell equations, and it can be conducted essentially like the modeling of the mechanical phenomena in Chapters 1 to 5: describing the physical quantities and writing the physical conservation laws, partial differential equations, and boundary conditions. It would be too long, and not particularly instructive from the modeling point of view, to conduct this study here; we will be content with describing the Maxwell equations and will refer the reader to the specialized literature for their physical justification.

Introduction

Regular gravitational potential has a stabilizing effect in that it brings the liquid volume toward the bottom of its container. When the body force diminishes, the liquid volume assumes any location inside its container in an unpredictable manner. The problems of liquid sloshing dynamics under microgravity are different from those encountered under a regular gravitational field. These problems include liquid reorientation and the difficulty of moving and handling, since the body force is almost negligible. Under microgravity, surface tension forces become predominant. The Bond number, given by the ratio of the gravitational to capillary forces, plays a major role in the free-liquid-surface characteristics. For very small values of the Bond number, capillary forces predominate and the free surface will not be flat, but will rise around the vertical walls of a container.

The problems of liquid behavior at a low and zero gravity include the mechanics and thermodynamics of capillary systems, heat transfer in cryogenic tanks and mechanisms of energy transport, capillary hydrostatics, low-gravity sloshing, and fluid handling. The early work on free-liquid-surface behavior under a low gravity field considered different problems of the surface–vapor interface (Petrash and Otto, 1962, 1964, Shuleikin, 1962, Petrash, et al. 1962, 1963, Petrash and Nelson, 1963, Neu and Good, 1963, Paynter, 1964a, b, Zenkevich, 1965, Abramson, 1966a, Seebold et al., 1967 and Hastings and Rutherford, 1968).

Introduction

An impulsive acceleration to a liquid can result in impact hydrodynamic pressure of the free surface on the tank walls. It can also occur during maneuvering or docking of spacecraft in an essentially low gravity field. Methods for estimating liquid impact and its hydrodynamic pressure are not well developed and are only identified by experimental studies (Pinson, 1963, Dalzell and Garza, 1964, and Stephens, 1965). It was found that when hydraulic jumps or traveling waves are present, extremely high impact pressures can occur on the tank walls. A hydraulic jump may occur in a liquid container undergoing oscillatory motion if the liquid height is relatively shallow and the excitation frequency is close to the natural frequency of the free surface.

Typical pressure–time history records under sloshing impact condition were reported by Cox, et al. (1980). Feng and Robertson (1971, 1972) studied the liquid propellant dynamics during docking. Rumyantsev (1969a) analyzed the collision of a body containing a viscous liquid with a rigid barrier. Hamlin, et al. (1986) indicated that the hydraulic jump could create localized high impact pressures on the container walls, which has a direct effect on the container dynamics and may result in structural damage. Verhagen and WijnGaarden (1965) reported that close to resonance a hydraulic jump was observed, which travels periodically back and forth between the container walls. This hydraulic jump is a nonlinear phenomenon, analogous to the shock wave appearing in one-dimensional gas flow under similar resonance conditions (Chester, 1964, 1968).

The objective of this chapter is the study of mixtures of reactive gaseous or liquid fluids, that is to say fluids undergoing chemical reactions (fuel, oil and oxygen, various chemical species, etc.). The equations and models that we present are commonly used nowadays for the study of phenomena as different as the propagation of chemical pollution in the atmosphere or in water (seas, rivers, estuaries) or the combustion of mixtures of fuel, oil, and oxygen in cars, planes, or rocket engines. The number N of chemical species taken into account can vary from a few units to several hundreds.

We start, in Section 11.1, by describing the framework for mixtures of nonnecessarily reactive fluids. Section 11.2 is devoted to the equations of chemical kinetics. We then formulate in Section 11.3 the general equations of combustion and describe a typical problem of combustion. In Section 11.4, we introduce the Stefan–Maxwell equations corresponding to an even more complex problem of chemical kinetics and combustion. Finally, we finish, in Section 11.5, with an overview of the case in which the mixture only contains two chemical species; although much simpler, this case already leads to very complex mathematical problems that we will be only able to describe briefly.

Equations for mixtures of fluids

We consider in this section a mixture composed of N different miscible species.

Solitons are a type of nonlinear wave whose discovery and study are relatively recent. Solitons appear in numerous wave propagation phenomena, and they should eventually play a major role in telecommunications by optical fibers. Two well-known equations possess soliton solutions: the Korteweg–de Vries equation and the nonlinear Schrödinger equation; we will present these equations in Chapters 19 and 20, respectively. These equations may be obtained by passing to the limit in various equations (e.g., fluid mechanics, electromagnetism). Let us mention here two remarkable properties of solitons. On the one hand, these are waves that propagate by keeping a constant shape; on the other hand, when two solitons that propagate at different velocities meet, they interact for some time and then recover their initial shapes and continue their propagation with their own initial velocity.

Solitons appear in particular in the small-amplitude motions of the surface of a shallow fluid; they are then governed by the Korteweg–de Vries equation. In this chapter, we write different water-wave equations and show how the Korteweg–de Vries equation may be deduced from the Euler equation of incompressible fluids; we finally present the soliton solutions of the Korteweg–de Vries equation (KdV).

Solitons were discovered by the Englishman John Scott Russell. He noticed the appearance of waves with constant shape propagating over large distances in shallow canals, and he used to follow them on horseback along the banks of the canal (J.S. Russell, 1884). The theoretical (analytical) study of these waves was then undertaken by Korteweg and de Vries, who introduced the equation bearing their names (Korteweg and de Vries (1895)).

In this chapter, we present some principles of energy minimization that characterize equilibria in elastostatics; we also introduce two related concepts, namely the concept of duality (duality for the variational principles and duality between displacements and stresses), and the concept of variational formulations. Apart from their relevance to mechanics, these concepts also play an important role in the numerical approximation of the problems we consider (but we do not address this here of course), and, in other related forms, in many other areas of science.

Elastic energy of a material

In an isotropic elastic material, we can define a quadratic functional w = w(ε), called the energy function, such that:

for every tensor field ε, and thus Eq. (15.1) is then identical to the stress–strain law of the material. This function is thus defined by

hence,

or else

Consequently, w is a positive definite quadratic form in ε (we will always assume that μ and κ = λ + 2μ/3 are strictly positive).

If we now consider a tensor field ε defined on, we can associate with it the functional

We have the following result, which provides a physical interpretation of W and justifies its introduction:

Proposition 15.1.We consider an elastic system S that is in its natural state (ε = 0) and fills the volume Ω at time t = 0 and that is at time t = t1in the deformation state corresponding to the deformation tensor ε. Then, –W(ε) is, to first approximation, the work of the internal forces between times 0 and t1. Thus, W(ε) is the amount of energy stored in stresses between 0 and t1.

Weather forecasting in meteorology is based on two complementary approaches:

1. Accumulating a very large amount of data and interpreting these statistically (wind velocity, humidity, and temperature measured over very large intervals of time and over large regions of the earth). Here, the mathematical techniques that are necessary for the assimilation and the exploitation of these data are those of statistics and of stochastic processes.

2. Modeling of atmospheric phenomena by ordinary and partial differential equations and the numerical simulation of these equations. From the computational point of view, one obtains, by discretization of such partial differential equations, systems of equations with millions, even billions, of unknowns whose numerical resolution could saturate the most powerful computers currently available. The memory size and computational speed capacities needed for such calculations are very high even in the context of the teraflop (1012 operations per second), which is the next step in the foreseeable future.

In this chapter, we are interested in the second approach, and we intend to give a very modest description of the most fundamental equations widely accepted in the field. The atmosphere is a fluid whose state is described by the velocity vector, the temperature, the density, and the pressure at every point in the Eulerian description. The equations are essentially variants of the Navier–Stokes and of the temperature equations that take into account the particular aspects of the problem.

Section 12.1 is devoted to various preliminaries.

Before studying, in Chapters 19 and 20, some nonlinear wave equations occuring in mechanics, we consider in this chapter several linear wave equations arising in mechanics and study some fundamental aspects of the corresponding vibration phenomena.

In Section 18.1, we start by recalling the fundamental wave equations that appeared in the previous chapters: the equations of linear acoustics that appeared in Chapter 8 in the context of fluid mechanics and the Navier equation that appeared in Chapter 13 in the context of linear elasticity. We also specialize these equations to specific phenomena such as sound pipes and vibrating cords and membranes.

In Section 18.2, we show how to solve the one-dimensional wave equation considered in the whole space ℝ. Then, in Section 18.3, we are interested in bounded intervals, leading us to introduce the normal (self-vibration) modes, which also depend on the boundary conditions; some typical examples of boundary conditions are considered and the corresponding eigenmodes made explicit. In Section 18.4, we show how to solve the wave equation in a general bounded domain of ℝ3 by using the corresponding eigenmodes again; however, in this case, the solution is not complete because we cannot compute the eigenmodes, in general. Finally, in the last section of this chapter, Section 18.5, we give some indications on other important vibration phenomena such as superposition of waves, beats, and wave packets.

Returning to the equations of linear acoustics and of linear elasticity

Returning to the wave equations of linear acoustics

The equations of linear acoustics have been introduced in Chapter 8, Section 8.4.