Mechanics concepts

Eracture mechanics concepts can also be appHed to fatigue crack growth under a constant static load, but in this case the material behavior is nonlinear and time-dependent (29,30). Slow, stable crack growth data can be presented in terms of the crack growth rate per unit of time against the appHed R or J, if the nonlinearity is not too great. Eor extensive nonlinearity a viscoelastic analysis can become very complex (11) and a number of schemes based on the time rate of change of/have been proposed (31,32). 

The fluid mechanics origins of shock-compression science are reflected in the early literature, which builds upon fluid mechanics concepts and is more concerned with basic issues of wave propagation than solid state materials properties. Indeed, mechanical wave measurements, upon which much of shock-compression science is built, give no direct information on defects. This fluids bias has led to a situation in which there appears to be no published terse description of shock-compressed solids comparable to Kormer s for the perfect lattice. Davison and Graham described the situation as an elastic fluid approximation. A description of shock-compressed solids in terms of the benign shock paradigm might perhaps be stated as ... 

The discussion of fracture mechanics will be divided in two parts. First, basic principles of fracture mechanics will be described. Second, the application of fracture mechanics concepts to composite materials will be discussed. In both parts, the basic approach is that of Wu [6-12],... 

The mechanical concepts of stress are outlined in Fig. 1, with the axes reversed from that employed by mechanical engineers. The three salient features of a stress-strain response curve are shown in Fig. la. Initial increases in stress cause small strains but beyond a threshold, the yield stress, increasing stress causes ever increasing strains until the ultimate stress, at which point fracture occurs. The concept of the yield stress is more clearly realised when material is subjected to a stress and then relaxed to zero stress (Fig. Ih). In this case a strain is developed but is reversed perfectly - elastically - to zero strain at zero stress. In contrast, when the applied stress exceeds the yield stress (Fig. Ic) and the stress relaxes to zero, the strain does not return to zero. The material has irreversibly -plastically - extended. The extent of this plastic strain defines the residual strain. 

We can consider the spreading of a sessile drop on a soft, lossy substrate rather like the advance of a negative crack and thus use fracture mechanics concepts, as was the case in the derivation of Eq. (15) for the separation of an elastomer from a rigid solid. The term negative is used since the spreading of a drop leads to the creation of solid/liquid interface rather than separation. 

As noted in the previous section, spin is a purely quantum-mechanical concept there is no classical-mechanical analog. 

Chapter 4 discusses the well-known VSEPR model. Although this model can be regarded as an empirical model that does not directly use quantum mechanical ideas, its physical basis is to be found in the Pauli principle. This dependence on a quantum mechanical concept has not always been clearly understood, so we emphasize this aspect of the model. We have tried to give a rather complete and detailed review of the model, which has been somewhat modified over the years since it was first proposed in 1957. 

For the description of systems with conjugated double bonds force field calculations of the kind described here are not very useful since, in principle, they only allow the description of relatively localised valence effects and of pairwise nonbonded interactions. Effects of delocalisation as occurring in conjugated vr-systems represent a new element for whose description quantum-mechanical concepts are appropriate. 

Finally Baeyer s theory is based on a mechanical concept of valency and served its purpose in stimulating research in the field of cyclic compounds. Now we have the electronic theory and the reactivity of olefines has been attributed to the n electrons and not on the basis of strains. 

The problem of a priory assessment of stable structure formation is one of the main problems of chemical physics and material science. Its solution, in turn, is directly linked with the regularities of isomorphism, solubility and phase-formation in general. Surely, such problems can be cardinally solved only based on fundamental principles defining the system of physical and chemical criteria of a substance and quantum-mechanical concepts of physics and chemistry of a solid suit it. 

These were bold and simple statements. To put them in a modern context, the discovery of triphenylmethyl combined the novelty of something like bucky balls with the controversial nature of something like polywater or cold fusion. Thus Gomberg was soon to find that the triphenylmethyl problem was attractive and complex enough to occupy him and many others for a long time. A first period lasted until about 1911 when the phenomena observed had been clarified to the satisfaction of a majority of the research community. Theoretically, little understanding was possible before the advent of the electron pair bond and, in particular, theory based on quantum mechanical concepts. This meant that the theory available... 

The models of optical properties in this book are strictly classical. However, modern theoretical work aimed at understanding in detail the microphysics of optical properties is mostly quantum mechanical. Therefore, in this section we briefly discuss a few relevant quantum-mechanical concepts and also show that there is an analogy between the classical and quantum-mechanical descriptions of optical properties. 

Note Lemerys reference to mechanical concepts which he does not systematically develop in a separate section of the book. 

The experimental trends in bonding and structure which we have discussed in the previous chapter cannot be understood within a classical framework. None of the elements and only very few of the thousand or more binary AB compounds are ionic in the sense that the electrostatic Madelung energy controls their bonding. And even for ionic systems, it is a quantum mechanical concept that stops the lattice from collapsing under the resultant attractive electrostatic forces the strong repulsion that arises as the ion cores start to overlap is direct evidence that Pauli s exclusion principle is alive and well and hard at work ... 

The concepts which we need for understanding the structural trends within covalently bonded solids are most easily introduced by first considering the much simpler system of diatomic molecules. They are well described within the molecular orbital (MO) framework that is based on the overlapping of atomic wave functions. This picture, therefore, makes direct contact with the properties of the individual free atoms which we discussed in the previous chapter, in particular the atomic energy levels and angular character of the valence orbitals. We will see that ubiquitous quantum mechanical concepts such as the covalent bond, overlap repulsion, hybrid orbitals, and the relative degree of covalency versus ionicity all arise naturally from solutions of the one-electron Schrodinger equation for diatomic molecules such as H2, N2, and LiH. 

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