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# Theory of Elastic Stability

01 Jan 1936-

About: The article was published on 1936-01-01 and is currently open access. It has received 8152 citations till now.

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2,329 citations

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24 Aug 2001TL;DR: In this paper, the authors introduce the theory of thin plates and thin shells, and apply it to the analysis of shell structures, including the moment theory of circular cylindrical shells.

Abstract: Part 1 Thin plates: introduction the fundamentals of the small-deflection plate bending theory rectangular plates circular plates bending of plates of various shapes plate bending by approximate and numerical methods advanced topics buckling of plates vibration of plates. Part 2 Thin shells: introduction to the general linear shell theory geometry of the middle surface the general linear theory of thin shells the membrane theory of shells applications of the membrane theory to the analysis of shell structures moment theory of circular cylindrical shells the moment theory of shells of revolution approximate theories of shell analysis and their application advanced topics buckling of shells vibration of shells. Appendices: some reference data Fourier series expansion verification of relations of the theory of surfaces derivation of the strain-displacement relations verification of equilibrium equations.

980 citations

### Cites methods from "Theory of Elastic Stability"

...A comprehensive analysis of linear and nonlinear buckling problems for thin plates of various shapes under various types of loads, as well as a considerable presentation of available results for critical forces and buckling modes, which can be used in engineering design, were presented by Timoshenko and Gere [38], Gerard and Becker [42], Volmir [43], Cox [44], etc....

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TL;DR: In this paper, thin gold films are made on an elastomeric substrate with built-in compressive stress to form surface waves, which function as elastic electrical conductors.

Abstract: Stripes of thin gold films are made on an elastomeric substrate with built-in compressive stress to form surface waves. Because these waves can be stretched flat they function as elastic electrical conductors. Surprisingly, we observe electrical continuity not only up to an external strain of ∼2% reached by stretching the films first flat (∼0.4%) and then to the fracture strain of free-standing gold films (∼1%), but up to ∼22%. Such large strains will permit making stretchable electric conductors that will be essential to three-dimensional electronic circuits.

939 citations

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TL;DR: In this paper, a comprehensive assessment of recent developments of nonlinear isolators in the absence of active control means is presented, which highlights resolved and unresolved problems and recommendations for future research directions.

885 citations

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TL;DR: The continuous tuning of the electronic structure of atomically thin MoS2 on flexible substrates by applying a uniaxial tensile strain demonstrates the potential of two-dimensional crystals for applications in flexible electronics and optoelectronics.

Abstract: We demonstrate the continuous tuning of the electronic structure of atomically thin MoS2 on flexible substrates by applying a uniaxial tensile strain. A redshift at a rate of ~70 meV per percent applied strain for direct gap transitions, and at a rate 1.6 times larger for indirect gap transitions, have been determined by absorption and photoluminescence spectroscopy. Our result, in excellent agreement with first principles calculations, demonstrates the potential of twodimensional crystals for applications in flexible electronics and optoelectronics.

730 citations

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TL;DR: In this paper, the authors compare the results of a singular solution of Navier's equations of motion of viscous fluid with the results obtained from many experiments, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved.

Abstract: 1. The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St. Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a co-efficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies which rendered the actual motion other than that to which the singular solution referred and not as disproving the assumption.

1,409 citations

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1,055 citations

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TL;DR: In this paper, the authors present an extension of Hooke's Law for determining the stability under stress of thin shells of isotropic elastic material, which they use to determine the equilibrium of an elementary volume of the substance by considering the forces acting upon it.

Abstract: The object of the present paper is to derive equations that are adequate to decide questions of the stability under stress of thin shells of isotropic elastic material. Equations for the same purpose have been given by R. V. Southwell, who used a method that is closely followed in a part of this paper. Such equations must contain terms that may be, and are, neglected in applications of the theory of elasticity to problems in which the stability of configurations is not considered. The truth of Kirchhoff's uniqueness theorem, which has reference to the ordinary equations of elasticity, in which powers of the displacement co-ordinates above the first are neglected, is sufficient proof of this statement. In practice it is generally sufficient to retain only the first and second order terms, and no terms of higher order are considered here. To obtain such equations an extended form of Hooke's Law is necessary; the extension made by Southwell is used in this paper. There are then two methods available for the derivation of the equations. Either we may obtain the three conditions for the equilibrium of an elementary volume of the substance by considering the forces acting upon it, or we may calculate the energy of strain correct to the third order of displacement co-ordinates, and deduce the equations by variation of this function. The first method has been used in one place here, as it would appear to be the simpler in the particular case of a plane plate, in which only one of the equations, and that the simplest, is required. However, the stability equations for a cylindrical shell are also obtained, and then all three equations are necessary. The derivation by the first method of each one of these is a laborious matter, while using the second method there is only one calculation, that of the strain energy function, to be made. Consequently, for this purpose, as in general, the second method seems to be preferable.

1,003 citations

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468 citations