Theory of elasticity

In the conventional theory of elastic image fomiation, it is now assumed that the elastic atomic amplitude scattering factor is proportional to the elastic atomic phase scattering factor, i.e. 

Landau L D and Lifshitz E M 1970 Theory of Elasticity (Course of Theoretical Physics vol 7) (Oxford Pergamon)... 

We have already observed that the entropy theory of elasticity predicts a modulus of the right magnitude and possessing the proper temperature coefficient. Now let us examine the suitability of Eq. (3.39) to describe experimental results in detail. 

Parton V.Z., Perlin P.I. (1981) Methods of the mathematical theory of elasticity. Nauka, Moscow (in Russian). 

Timoshenko S.P., Goodier J.N. (1951) Theory of elasticity. McGraw-Hill, New York. 

I. N. Sneddon and M. Lowengmb, Crack Problems in the Classical Theory of Elasticity,]ohn Wiley Sons, Inc., New York, 1969. 

S. P. Timoshenko and ]. N. Goodier, Theory of Elasticity, McGraw Hill, 1970, Chap. 1. 

Timoshenko, S. (1934) Introduction to the Theory of Elasticity for Engineers and Physicists (Oxford University Press, London). 

The Hertz theory of contact mechanics has been extended, as in the JKR theory, to describe the equilibrium contact of adhering elastic solids. The JKR formalism has been generalized and extended by Maugis and coworkers to describe certain dynamic elastic contacts. These theoretical developments in contact mechanics are reviewed and summarized in Section 3. Section 3.1 deals with the equilibrium theories of elastic contacts (e.g. Hertz theory, JKR theory, layered bodies, and so on), and the related developments. In Section 3.2, we review some of the work of Maugis and coworkers. 

S. G. Lekhnitskil, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, 1963. 

Obviously, the assumptions involved in the foregoing derivation are not entirely consistent. A transverse strain mismatch exists at the boundary between the fiber and the matrix by virtue of Equation (3.8). Moreover, the transverse stresses in the fiber and in the matrix are not likely to be the same because v, is not equal to Instead, a complete match of displacements across the boundary between the fiber and the matrix would constitute a rigorous solution for the apparent transverse Young s modulus. Such a solution can be found only by use of the theory of elasticity. The seriousness of such inconsistencies can be determined only by comparison with experimental results. 

N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, The Netherlands, 1953. 

S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Government Publishing House for Technical-Theoretical Works, Moscow and Leningrad, 1950. Also P. Fern (Translator), Holden-Day, San Francisco, 1963. 

Fedorov A. F. I., Theory of elastic waves in cristals, Plenum... 

Love A. E. H., A Treatise on the mathematical theory of elasticity, Dover, 1944. [PyccKHii nepeaoA npeaufl. Hsfl. JI H B A., MaTeMaTHqecKBfl leopna ynpyrocTH, FTTH, M., 1935.]... 

The theories of elastic and viscoelastic materials can be obtained as particular cases of the theory of materials with memory. This theory enables the description of many important mechanical phenomena, such as elastic instability and phenomena accompanying wave propagation. The applicability of the methods of the third approach is, on the other hand, limited to linear problems. It does not seem likely that further generalization to nonlinear problems is possible within the framework of the assumptions of this approach. The results obtained concern problems of linear viscoelasticity. 

Stresses may also exist in the interior of solid bodies, and are considered in the theory of elasticity. 

Thimoshenko, S., Goodier, J. N. Theory of Elasticity, N, York McGraw-Hill (1970)... 

Landau, L., Lifehitz, E., Theory of Elasticity, Mir, Moscow, Russia (1967). 

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Pergamon Press, New York, 1986. 

Southwell, R. V. (1913) Phil. Trans. 213A, 187. On the general theory of elastic stability. 

Timoshenko, S. (1936) Theory of Elastic Stability (McGraw-Hill). 

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 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. 

This is possible because the result must conform to the traditional theory of elasticity when crystallinity is zero (w = 0). This happens if r is interpreted as an average chain vector (made up of N links), the components x, y, z of which deform affinely, so that in simple elongation along z... 

The deformation of a specimen during indentation consists of two parts, elastic strain and plastic deformation, the former being temporary and the latter permanent. The elastic part is approximately the same as the strain produced by pressing a solid sphere against the surface of the specimen. This is described in detail by the Hertz theory of elastic contact (Timoshenko and Goodier, 1970). 

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