Classical Theories of Failure

Classical theories of failure are based on concepts of maximum stress, strain, or strain energy and assume that the material is homogeneous and free from defects. Stresses, strains, and strain energies are typically obtained through elastic analyses. 

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In conclusion it seems proper to emphasize that Burrau s calculation of H2+ and the extension here to H2 constitute the first quantum-theoretic quantitative discussion of the binding of atoms into molecules by electrons— the valence forces of chemistry. The quantitative success of the new quantum mechanics in the face of the classical theory s failure must serve to lend strong support to the new methods. 

It is evident that the theory of failure of adhesion joints should be based on the general principles of solid destruction. However, the transfer of the classical concept of Griffith s theory to two-phase systems is very complex. The difficulties are related to determination of two main parameters in the equation for critical stress of fracture ... 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

The failure of classical mechanics in the analysis of physical phenomena, such as black-body radiation, is routinely discussed in elementary texts to emphasize the need of a quantum theory. The failure of classical mechanics to deal correctly with simple chemical systems, although rarely stated, is equally dramatic. 

It is noteworthy that Gibbs himself was acutely aware of the qualitative failures of 19th-century molecular theory (as revealed, for example, by erroneous classical predictions of heat capacities Sidebar 3.8). In the preface to his Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics (published in the last year of his life), Gibbs wrote ... 

Note that this extended semiclassical model leads to the correct ratio of magnetic moment to spin [Eq. (88)]. Failure of classical theory to account for the correct ratio is one of the main arguments in favor of a quantum model. However, it is noted that Eq. (88) results from a first approximation to a definitive model of a... 

Much has been written about the failure of classical theory in interpretating the specific heat of metals and the subsequent development of a quantum theory of the perfect solid state. It would be quite impossible to give an adequate account of this development here, and the... 

The peak observed in the frequency distribution of blackbody radiation is completely inconsistent with the predictions of classical electromagnetic theory. This failure of classical physics is called the ultraviolet catastrophe. 

If, contrary to the order of historical development, we have discussed the quantum theory of the atom before quantum statistics, we have our reasons. In the first place, the failure of- the classical theory displays itself in atomic mechanics—for instance, in the explanation of line spectra or the diffraction of electrons—even more immediately than in the attempts to fit the law of radiation into the frame of classical physics. In the second place, it is an advantage to understand the mechanism of the individual particles and the elementary processes before proceeding to set up a system of statistics based upon the quantum idea. 

We divided the six concepts into three pairs, the first dealing with space, the second with time, and the third with the classic discrete/continuous dichotomy, already evident in the distinction between arithmetic and geometry. Placing the members of each pair of concepts on opposing faces of the cube in Figure 1, each face is in contact with each of the others except for the face directly opposite. Adjacent faces of the cube represent phenomena which require both concepts, can be explained by either concept, or lead to a new conceptual development subsuming both as in the space-time continuum of relativity theory. The concept of velocity requires both direction in space (3D) and duration in time (t). As seen in the previous section, certain failures encountered in the structural theory of organic chemistry can be explained either in spatial terms... 

It is worthwhile to consider whether the classical theories (or criteria) of failure can still be applied if the stress (or strain) concentration effects of geometric discontinuities (eg., notches and cracks) are properly taken into account. In other words, one might define a (theoretical) stress concentration factor, for example, to account for the elevation of local stress by the geometric discontinuity in a material and still make use of the maximum principal stress criterion to predict its strength, or load-carrying capability. 

The classical description is based on the assumption that Gm is an analytic function of the mole fraction x and the temperature T for a fixed pressure p at and near the critical point of the mixture. In spite of rather good qualitative agreement with experimental results there is more and more evidence that the classical theory seems to be quantitatively inadequate at critical solution points, analogous with the failure of the classical theory at the critical point of a single component system. ... 

J. M. H. Levelt Sengers, Physica, 1974, 73, 73. This article includes an excellent history of the experimental failure of die classical theory, shown as early as 1900 by Verschaffelt. 

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