The purpose of the present book is to give an idea about fundamental concepts and methods, as well as instructive special results, of a unified intermediate asymptotic mathematical theory of the flow, deformation and fracture of real fluids and deformable solids. This theory is based on a quite definite and, we emphasize, idealized approach where the real materials are replaced by a continuous medium; therefore it is often called the mechanics of continua. It generalizes, and represents from a unified viewpoint, more focused disciplines: fluid dynamics, gas dynamics, the theory of elasticity, the theory of plasticity etc. For various reasons these disciplines underwent separate development for a long time. The splendid exceptions found in the work of A. L. Cauchy, C. L. M. H. Navier and A. Barré de Saint-Venant in the nineteenth century and L. Prandtl, Th. von Kármán, and G. I. Taylor in the twentieth century confirm rather than disprove the general rule. Therefore the teaching of the mechanics of continua and, more generally, the maintaining of interest in the mechanics of continua as a unified scientific discipline was, in this period of fragmentation, the job of physicists, who considered it to be a necessary part of a complete course of theoretical physics. So it was not by accident that among those who created courses in mechanics of continua were outstanding physicists: M. Planck, A. Sommerfeld, V. A. Fock, Ya. I. Frenkel, L. D. Landau and E. M. Lifshitz and, more recently, L. M. Brekhovskikh.

The physicists were more interested, however, in general ideas and methods rather than in the consistent presentation of special, even very important, results, which in fact give true shape to the subject.

(1) Mechanics, the science of the motion and equilibrium of real bodies, is the oldest natural science created by humankind. Unlike other scientific disciplines, mechanics had no predecessors. Prehistoric human beings started to take their first steps in mechanics – apparently even before starting to speak – when they invented and improved their first primitive tools.

Later, when humankind took its first steps in mathematics, mechanics was the first field of application. The value of mechanics for mathematics was clearly emphasized by Leonardo da Vinci: “Mechanics is the paradise for mathematical sciences because through it one comes to the fruits of mathematics.”

Nowadays mechanics is an organic part of applied mathematics – the art of designing mathematical models of phenomena in nature, society and engineering.

The development of mechanics, and the recognition of its value, has not gone smoothly over the centuries. An analogy of mechanics with the phoenix comes to mind. This legendary bird has appeared with practically identical magical features in the ancient legends of many cultures: Egyptian, Chinese, Hebrew, Greek, Roman, native North American, Russian and others. The names were different: the name “phoenix” was coined by the Assyrians; Russians called it “zhar-ptitsa” (fire-bird). According to the legend, unlike all other living beings the phoenix had no parents, and death could never touch it. However, from time to time, when it was weakened, the phoenix would carefully prepare a fire from aromatic herbs collected from throughout the world and burn itself. Everything superfluous is burnt in the fire and a new beautiful creative life opens to the phoenix.