Elementary elasticity theory

Excellent treatises and textbooks on elasticity are abundant, for example, Landau and Lifshitz (1986), Timoshenko and Goodier (1970), and Timoshenko and Young (1962). It takes a lot of time to read and find useful information. This appendix contains an elementary treatment of elasticity regarding STM and AFM. 

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El theory In all materials (plastics, metals, wood, etc.) elementary mechanical theory demonstrates that some shapes resist deformation from external loads. This phenomenon stems from the basic physical fact that deformation in beam or sheet sections depends upon the mathematical product of the modulus of elasticity (E) and the moment of inertia (I), commonly expressed as EL This theory has been applied to many different constructions including sandwich panels. 

The cycle rank completely defines the connectivity of a network and is the only parameter that contributes to the elasticity of a network, as will be discussed further in the following section on elementary molecular theories. In several other studies, contributions from entanglements that are trapped during cross-linking are considered in addition to the chemical cross-links [23,24]. The trapped entanglement model is also discussed below. 

The basic postulate of elementary molecular theories of rubber elasticity states that the elastic free energy of a network is equal to the sum of the elastic free energies of the individual chains. In this section, the elasticity of the single chain is discussed first, followed by the elementary theory of elasticity of a network. Corrections to the theory coming from intermolecular correlations, which are not accounted for in the elementary theory, are discussed separately. 

The starting approach will be the elasticity theory (3). In the elementary theory of beams, the only component of the stress tensor differing from zero is Gxx = Ey/R, which, according to the theory developed for the elastic case, can be written as... 

The elementary surface excited states of electrons in crystals are called surface excitons. Their existence is due solely to the presence of crystal boundaries. Surface excitons, in this sense, are quite analogous to Rayleigh surface waves in elasticity theory and to Tamm states of electrons in a bounded crystal. Increasing interest in surface excitons is provided by the new methods for the experimental investigation of excited states of the surfaces of metals, semiconductors and dielectrics, of thin films on substrates and other laminated media, and by the extensive potentialities of surface physics in scientific instrument making and technology. 

The basic postulate of elementary molecular theories of rubber elasticity states that the elastic free energy of a network is equal to the sum of the elastic free... 

Historically, the pressurized bulge configuration was analyzed on the basis of the assumption that the applied pressure is resisted entirely by bending stiffness of the film. If any membrane stress is present, it is not coupled to the bending stress within the range of small deflection behavior. On this basis, elementary elastic plate theory leads to the expression... 

Plastic hinges are modeled by nonlinear torsional springs mounted over revolute joints. The nonlinear response of the torsional springs are described in terms of a quasi-static moment-angle relationship based on the curves for elastic-plastic behavior, where stiffness of the elastic part is obtained from the elementary beam theory [19] as... 

The normalized bending stresses at the Gauss points and centers of eight elements nearest to the fixed end are plotted as functions of the normalized z-coordinate in Fig.(3-7). The solid line represents the solution of ihe elementary beam theory, which serves as a reference for comparison. It is seen in Fig.3 that the linear elastic solution, which is obtained by multiplying the solution at F=0.01 by one hundred, matches the... 

Most materials scientists at an early stage in their university courses learn some elementary aspects of what is still miscalled strength of materials . This field incorporates elementary treatments of problems such as the elastic response of beams to continuous or localised loading, the distribution of torque across a shaft under torsion, or the elastic stresses in the components of a simple girder. Materials come into it only insofar as the specific elastic properties of a particular metal or timber determine the numerical values for some of the symbols in the algebraic treatment. This kind of simple theory is an example of continuum mechanics, and its derivation does not require any knowledge of the crystal structure or crystal properties of simple materials or of the microstructure of more complex materials. The specific aim is to design simple structures that will not exceed their elastic limit under load. 

The expressions given in this section, which are explained in more detail in Erman and Mark [34], are general expressions. In the next section, we introduce two network models that have been used in the elementary theories of elasticity to relate the microscopic deformation to the macroscopic deformation the affine and the phantom network models. 

The elastic free energy given by the elementary and the more advanced theories are symmetric functions of the three extension ratios Xx, Xy, and Xz. One may also express the dependence of the elastic free energy on strain in terms of three other variables, which are in turn functions of Xx, Xy, and Xz. In phenomenological theories of continuum mechanics, where only the observed behavior of the material is of concern rather than the associated molecular deformation mechanisms, these three functions are chosen as... 

It is easy to see that these models are all based on the same (microstructural) principle, viz. that there is an elementary structural unit that can be described and then used for calculation. Remember that the corresponding unit cell for foamed polymers is the gas-structure element8 10). Microstructural models are a first approximation to a general theory describing the deformation and failure of gas-filled materials. However, this approximation cannot be extended to allow for all macroscopic properties of a syntactic foam to be calculated 166). In fact, the approximation works well only for the elastic moduli, it is satisfactory for strength properties, but deformation... 

The Clusius-Dickel column is shown schematically in Figure 2. A wire is mounted at the axis of a cylinder. The wire is heated electrically and the outer wall is cooled. This sets up a radial thermal gradient which leads to a thermal diffusion separation in the x direction. As a result of the radial temperature gradient, a convection current is established in the gas, which causes the gas adjacent to the hot wire to move up the tube with respect to the gas near the cold wall. The countercurrent flow leads to a multiplication of the elementary separation factor. For gas consisting of elastic spheres, the light molecules will then concentrate at the top of the column, while the heavy molecules concentrate at the bottom. The transport theory of the column has been developed in detail (3, iS, 18) and will not be presented here. In a later section we shall discuss the general aspects of the multiplication of elementary separation processes by countercurrent flow. 

DAL influences practically all stages of the elementary flotation act. Buoyancy velocity of bubbles of definite size with retarded and non-retarded surfaces can differ from each another by a factor of about 2 (see Fig. 8.2). According to the theory of quasi-elastic (Section 10.1) and inelastic (Section 10.2) collisions, a smaller film thickness h corresponding to the beginning... 

The elastic free energy given by the elementary and the more advanced theories is symmetric functions of the three extension ratios and A.. One may also... 

This being so the engineer who has a problem to solve may have to resort to a model merely to obtain a solution from the deductive calculus of the theory. The use of photoelasticity is an example. Consider a photoelastic specimen used to model the theory of simple beam behaviour. This is for illustration purposes only, of course, since the deductive calculus of a simple elastic beam is easily solved. Both model and theory assume at least the following Newtonian mechanics elastic behaviour of materials symmetrical bending and no resultant forces on the system. The theoretical derivation of the elementary equations of... 

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