As scientific meetings got bigger some attendees were not able to give a talk because there was not enough time. The scientific poster was invented to allow these disadvantaged persons to display their work.

Poster sessions have been common for only a few decades, so procedures and poster design techniques are still evolving. The poster is the least formal of the three main communication methods, and corralling viewers for it is competitive. It has the same relationship to the formal article as a painted miniature does to a Vermeer portrait. The advantage of presenting a poster (at anything other than a small meeting for specialists) is that you have an opportunity to interest people outside your special field. It will be clear then that visual presentation is the key to designing a poster that gets noticed. Of the three forms of presentation the poster, as the name implies, is the one that has the largest element of deliberate advertising in it. You need to attract attention. Of course the substance of what you present must be interesting too.

You will probably get an opportunity to talk about your work before you have to write about it. You use some of the same evidence in a talk as in an article, though prepared differently, mostly in visual form with tables and figures as ‘slides’. But speaking is a performance art in real time and needs different skills from writing. When writing you have time to reconsider and revise; when giving a talk you have only one chance to get it right. Detailed preparation and at least some practice are essential. Giving a successful talk that interests the audience can be a satisfying, if nerve jangling, experience. It is a rapid way to recognition amongst peers in your subject .

When writing or showing a poster you are competing for attention, but a talk is different. Your audience is captive. This advantage is also a responsibility: the members of your audience have come hoping to learn something and, perhaps, be entertained too. The stakes are high.

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.

The problem of structural integrity

The science of studying the strength of structures (i.e. structural integrity), traditionally known as the strength of materials (this term, as we will see, is not quite adequate) appeared like Athena from Zeus' head. It was created by Galileo Galilei, and was presented in his book Dialogues Concerning Two New Sciences (Galilei, 1638), which appeared, by the way, not in his home country, Italy, but in Leiden, The Netherlands. He was able to overcome the Catholic Church's prohibition to publish anything by sending parts of his manuscript to Leiden via his friends. Galilei gave the following definition of the subject of this new science: “New science, treating the resistance which solid bodies offer to fracture.” It is remarkable that now, nearly 400 years since its publication, this definition sounds quite modern. Roughly speaking, the problem is to determine the limiting load which a structure is able to carry. The goal of the tensile test, shown in Figure 5.1 and taken from his book (Galilei, 1638), was to determine the limiting load.

Robert Hooke's fundamental paper published 40 years later, in 1678, which as we saw in the previous chapter laid the foundation of the theory of linear elasticity, marked, strictly speaking, a deviation from the path formulated by Galilei. In fact, this paper founded the science treating the deformation of elastic solid bodies due to the action of loads applied to them. This is a remarkable, very important, but different problem.

In the first chapter we introduced the important concept of an observer and formulated the invariance principle, which states the equivalence (equal rights in mathematical modeling!) of observers. In this chapter we consider what follows from the equivalence of observers whose units of measurement are of the same physical nature but of different magnitudes.

This looks simple but in this case the consequences of the invariance principle are far from trivial. Indeed, we will show that it follows from the principle of equivalence of observers that the functions describing physical laws have a fundamental property which is called generalized homogeneity. This property allows a reduction in the numbers of arguments of these functions and simplifies their determination in a numerical computation or in an experiment. The corresponding procedure is called dimensional analysis. Dimensional analysis is closely related to the rules of the modeling of physical phenomena, which make up the essence of the theory of physical similitude. Dimensional analysis and the theory of similitude will be presented in this chapter in sufficient detail for their use throughout the whole book. More detailed presentation of the subject of this chapter can be found in the author's book (Barenblatt, 2003).

Examples

Example 1. In the autumn of 1940, when the development of atomic weapons was beginning, a fundamental question arose concerning the mechanical action of the energy released during an atomic explosion. An outstanding American expert in explosives, G. B. Kistyakovsky, reported that even if such a weapon were created all its energy would go to radiation and would have essentially no mechanical effect.

The idealized model of a continuous medium

In the mechanics of continua the most important invention (whose fundamental value, however, is not always appreciated because it seems so natural) is the very concept of a continuous modeling of real materials. More precisely, the truly fundamental discovery was recognizing that to a knowable degree of accuracy the motion, deformation, fracture and/or equilibrium of real bodies can be based on an idealization (a model), that of a continuous medium.

In fact, we intend to study, i.e. to make models of, the motions, deformations, flows, fracture etc. of real bodies. These bodies consist of specific materials: honey, milk, petroleum, metals, polymers, ceramics, rocks, composites etc. If we look at these materials with the naked eye they very often seem continuous and homogeneous. But, when viewed through a microscope or telescope these materials (see Figures 1.1–1.7) display a developed microstructure at various scales – from atomic to essentially macroscopic ones – having a huge diversity of shape. How can we account for this diversity of shapes and properties of the elements of microstructures? Let us forget for the moment that we do not know the equations governing the equilibrium or motion. We do know, however, that taking into account the shape of the elements of a microstructure should mean accepting certain conditions at the boundaries of these odd formations. Let us imagine that by some miracle we know all these odd shapes. It is easy to show that it is impossible to write down the conditions at the boundaries of the microstructural elements even for the simplest problems.

Physical meaning of the velocity potential. The Lavrentiev problem of a directed explosion

We now must clarify the direct physical meaning of the velocity potential: without understanding this it is impossible to formulate the Dirichlet boundary value problem: we have to prescribe the velocity potential at the boundary, but we do not know yet what the potential is.

Consider a body in a continuous medium which at t = t0 is at rest. Assume that at t = t0 each particle experiences a pressure pulse such that the pressure varies according to the law

Here θ(x) is a function of the position of the particle, and δ(z) is the generalized Dirac function. According to the simplest definition of this function, which is all we need for now,

for arbitrarily small positive ∊.

The motion begins from a state of rest before the pressure pulse starts. Therefore uf tge system of mass forces acting on the medium is a potential one, the Lagrange–Cauchy integral holds in the ideal incompressible fluid approximation:

We put (4.1) into (4.3) and integrate from t = t0 − ε to t = t0 + ε.