Collisions engineering: theory and applications (Q342618)

The introduction to this highly interesting book perfectly describes its main ideas as well as its structure. ``The investigation of collisions occurring in the motion of solids has captured the attention of scientists from the beginning of the establishment of the mechanical sciences. Today the topic is still active. That just shows how many aspects it has. It is even related to the very bases of mechanics. In this book, we consider collisions which may be assumed instantaneous. This subjective choice leaves numerous openings for science and engineering. As we are going to see, it is not very restrictive. We develop the idea that a system of solids is deformable. Indeed, moving solids form a mechanical system. And the form of this system changes because the solids move one's with respect to the others. The first element of the theory is to identify and quantify the deformation of the system. When this is achieved, we define internal generalized forces with either a power or a work as in mechanical parlance or with a duality product in mathematical parlance. This point of view gives the equations of motion derived from the principle of virtual work. To complete the theory, we need constitutive laws for the internal generalized forces. Once the expression of the work of the internal generalized forces is chosen, the constitutive laws are obtained with experiments which are guided by the laws of thermodynamics. These relationships exhibit the quantities which have to be related and measured. As the constitutive laws are infinitely numerous, we facilitate their derivation by using pseudo-potentials of dissipation introduced by Jean Jacques Moreau. Many phenomena are described by pseudo-potentials of dissipation. These behaviours have a physical feature: the effect, for instance the percussion in two solids colliding is roughly proportional to the cause, here the deformation velocity of the system made of the two solids. A phenomenon which has such a property may fall in the formalism of pseudo-potentials of dissipation. In this case, a linear constitutive law complemented with the reactions to the internal constraints, mainly the impenetrability conditions, is sufficient to capture the core of the physical properties. This is only in the enhancement of the modelling that non linear constitutive laws may by introduced. From our point of view, the essence of physics has to appear in the combination of the equations of motion and of linear constitutive laws with the mandatory nonlinear reactions to the internal constraints. Of course, all the phenomena do not fall within the scope of pseudo-potential of dissipation. There are situations where the effect is not clearly proportional to the cause. This is the case of the Coulomb's friction law which is in someway related to collisions as we will see in Chap. 2. The friction phenomena are not progressive. A small action does not produce a small sliding velocity and a large actions does not produce a large sliding velocity. There is a threshold depending on the pressure which is applied. In this situation, pseudo-potentials of dissipation are unsuitable. To keep the versatility of pseudo-potentials, we may use dissipation functions which are no longer convex functions. These functions keep a part of the good properties of pseudo-potentials, for instance the second law is still satisfied, but the equations are no longer monotone, unicity of possible solutions is problematic. But this not always a drawback. The mathematical aspect of the predictive theories is not addressed. Each time it is possible, we mention experimental and numerical results to illustrate the presentation. After the bases of the theory are presented in the first Chapter after this introduction, Chap. 2, developments and applications are described in the following Chaps. 3--10. In Chap. 2, we present the basic ideas predicting the motion of a point above an immobile plane. The concepts, the velocity of deformation, and the internal percussion are introduced. In Chap. 3 we use the adaptability of the basic ideas to develop the theory for the motion of two points moving on an axis. The thermal effect of collisions is investigated. Because we assume the collision is instantaneous, we are induced to assume the temperatures are discontinuous with respect to time. The collisions being dissipative, it results the temperatures tend to increase. And diffusion tends to equalize the temperatures. Those two effects intervene in the theory. Chapters 4 and 5 are devoted to enhancements of the results to the motions of disks in a plane and the motion of balls above a plane and in a box. The numerical results show the variety of the possible motions in a box. We see in these Chapters that the attractive notion of coefficient of restitution well fitted for collision of two balls is ill-suited to different systems. Chapter 6 is devoted to the motions of crowds: pedestrians are assumed to be either points or disks with or without interactions at a distance to modelize children holding hands of their parents. The pedestrians may collide or avoid to collide. The examples, people getting out of a theater, getting off a train, pedestrians walking on a bridge... illustrate the possibilities and the versatility of this theory. Chapter 7 investigate the collisions of deformable solids with obstacle, with and without thermal phenomena. The theory predicts the micro-rebounds of a steel bar which vibrates after colliding an obstacle. Chapter 8 describes collisions of solids and fluids. the theory predicts the large painful percussion pressure a diver experiments in a belly flop. It predicts also that the behaviour of a flat stone colliding the surface of a lake depends on the relative importance of the horizontal velocity and of the falling velocity. When the horizontal velocity is larger than the falling velocity, the stone ricochets, when it is not, it does bounce. This is what we experiment when skipping stone on a lake. Chapter 9 investigate the collision of debris flows with structures. It is shown that a smooth damping protection may be used to protect buildings. In Chap. 10, we investigate collision of a solid made of shape memory alloy with an obstacle. The thermal effects produce both and increase of temperature and possibly an austenite-martensite phase change. In the following motion, the solid may recover its initial position.'' Because of its diversity with respect to applications this monograph should find a large number of readers.