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.

Turbulence is the state of vortex fluid motion where the velocity, pressure and other properties of the flow field vary in time and space sharply and irregularly and, it can be assumed, randomly.

Turbulent flows surround us, in the atmosphere, in the oceans, in engineering systems and sometimes in biological objects. A contrasting class of fluid motions, when the fluid moves in distinguishable layers (laminae in Latin) and the flow-field properties vary smoothly in time and space, is known as laminar flow. In Figure 11.1 an example of the time dependence of the velocity in a turbulent flow is presented. For laminar flow the time dependence would be a smooth line.

Leonardo da Vinci already knew about and clearly distinguished these two types of flow. Leonardo even used the term “turbulenza”. However, his observations and thoughts were buried in his notebooks, solemnly and carefully preserved in the Royal and Papal archives. They were not published until recently and therefore most regrettably did not influence future studies.

The systematic scientific study of turbulence began only in the nineteenth century, and here two names should be mentioned in particular, those of the French applied mathematician Joseph Boussinesq and the British physicist Osborne Reynolds. Boussinesq was a student of A. Barré de Saint Venant, who, in his turn, was a devoted disciple of C. L. M. H. Navier, the originator of the mathematical models for both Newtonian viscous fluid flows and the deformation of perfectly elastic bodies.

The mathematical theory of elasticity based on the idealization presented in this chapter is a remarkable classical branch of the mechanics of continua, which has advanced far in the more than 200 years it has been studied. In this chapter a concise presentation of the fundamentals of this theory will be given, bearing in mind readers who have not met it before, and it will also serve as a preparation for the next chapter, where we discuss the mathematical modeling of fracture phenomena, which nowadays is the principal area of attention.

The fundamental idealization

A crucially important property of a deformable solid continuum is that it is possible for it to possess non-trivial stress distributions even when the body is at rest, i.e. when the velocity is everywhere equal to zero.

The theory of elasticity as a science is older than fluid mechanics. Its basic law, which was developed to a fundamental model, was formulated by Robert Hooke more than 300 years ago, in the article Hooke (1678).

Readers already know the formulation of Hooke's law for an elastic rod. A rod is an elastic body whose length l is substantially larger than its cross-sectional size s and which has a constant cross-section area S (see Figure 5.1, taken from the book of Galileo Galilei (1638)). Let us take the longitudinal direction of the rod as the x1 axis of a system of orthonormal Cartesian coordinates.

In his preface to this book, Professor G. I. Barenblatt recounts the saga of the course of mechanics of continua on which the book is based. This saga originated at the Moscow State University under the aegis of the renowned Rector I. G. Petrovsky and moved with the author first to the Moscow Institute for Physics and Technology, then to Cambridge University in England, then to Stanford University, until it reached its final home as a much loved and appreciated course at the mathematics department of the University of California, Berkeley. Those not fortunate enough to have been able to attend the course now have the opportunity to see what has made it so special.

The present book is a masterful exposition of fluid and solid mechanics, informed by the ideas of scaling and intermediate asymptotics, a methodology and point of view of which Professor Barenblatt is one of the originators. Most physical theories are intermediate, in the sense that they describe the behavior of physical systems on spatial and temporal scales intermediate between much smaller scales and much larger scales; for example, the Navier–Stokes equations describe fluid motion on spatial scales larger than molecular scales but not so large that relativity must be taken into account and on time scales larger than the time scale of molecular collisions but not so large that the vessel that contains the fluid collapses through aging.

This chapter covers several additional methods used to form micro- and nanofibers. Some of them have already achieved maturity, such as the island-in-the-sea method discussed in Section 6.1, melt fracture in meltblowing processes (Section 6.2) and the flash spinning process (Section 6.3). Some others are still relatively exotic or under development, like the two methods based on Couette shear flow described in Section 6.4, or the centrifugal spinning method in Section 6.5. Nontraditional materials used for nanofiber formation, discussed in Section 6.6, include liquid crystals, conducting polymers, biopolymers and denatured proteins. Finally, Sections 6.7 and 6.8 discuss the specifics of drawing glass optical fibers, and in particular, polarization-maintaining optical fibers with multilobal cladding (Section 6.8).

Island-in-the-sea multicomponent fibers and nanofibers

Microscopic bi- and multicomponent fibers can be formed using melt spinning (Section 1.2 in Chapter 1), meltblowing (Section 4.1 in Chapter 4), or integrated processes such as spunbonding (Section 1.5 in Chapter 1). The additional polymer components are supplied to the main polymer through separate inner spinnerets inserted into the main outer spinneret similarly to formation of core–shell bicomponent fibers in co-electrospinning (Section 5.8 in Chapter 5) and solution blowing (Section 4.8 in Chapter 4). Cross-sections of such bi- and multicomponent fibers are reminiscent of islands in the sea, which explains the name of this technology (Nakajima 2000). In some cases the islands can merge and form winged structures, as seen in Figure 6.1a.

This chapter deals with the mechanisms and electrohydrodynamic modeling of the physical processes resulting in electrospinning of nanofibers with cross-sectional diameters approximately in the range 100 nm to 1 µm. These involve the physical nature of fluids used in electrospinning, leaky dielectrics, discussed in Section 5.2, and the formation of the precursor of electrospun jets, the Taylor cone, described in Section 5.3. Polymer jets in electrospinning possess an initial straight section, which is discussed in Section 5.4. Experimental observations of the key element of the electrospinning process, the electrically driven bending instability, which is similar to the aerodynamically driven jet bending of Chapters 3 and 4, are covered in Section 5.5. Section 5.6 describes the theory of the bending instability in electrospinning. Multiple jet interaction in electrospinning and needleless electrospinning are discussed in Section 5.7. Co-electrospinning and emulsion electrospinning of core–shell fibers (Section 5.8) are based on similar physical principles to electrospinning of monolithic nanofibers. The electrostatic field-assisted assembly techniques developed with the aim of positioning and aligning individual nanofibers in arrays and ropes are discussed in Section 5.9. Melt electrospinning of polymer fibers is briefly outlined in Section 5.10.

Electrospinning of polymer solutions

Electrospinning of polymer solutions, liquid crystals, suspensions of solid particles and emulsions employs an electric field of the strength about 1 kV cm−1. The first US patent on electrospinning was issued to Formhals (1934), but interest in this process was dormant until electrified jets of polymer solutions and melts were investigated as routes to the manufacture of polymer nanofibers (Baumgarten 1971, Larrondo and Manley 1981a–c, Doshi and Reneker 1995, Reneker and Chun 1996). In electrospinning, the electric force results in an electrically charged jet flowing out from a pendant or sessile droplet (see Figure 5.1). After the jet flows away from the droplet in a nearly straight line, it bends into a complex path and other changes in shape occur, during which electrical forces stretch and thin it by very large ratios, quite similar to the effects of the aerodynamic forces in melt- and solution blowing discussed in Chapter 4. After the solvent evaporates, solidified nanofibers are left.