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On the lower-order theories of continua with application to incremental motions, stability and vibrations of rods

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Abstract

The relative merit of lower-order theories, which have been deduced from the three-dimensional theories of continua, is evaluated with respect to the quantified and un-quantified errors in mathematically modeling the physical response of structural elements. Then, the one-dimensional theories are derived with high accuracy, internal consistency and flexibility from the three-dimensional theory of elasticity in order to govern the nonlinear and incremental motions and stability of a functionally graded rod. First, a kinematic-based method of separation of variables is introduced as a method of reduction, which may lead to the lower-order theories with the same order of errors of the three-dimensional theories, and the nonlinear theories of the rod are derived under Leibnitz’s postulate of structural elements by use of Hamilton’s principle. A theorem of uniqueness is proved in solutions of the linear equations of the rod by means of the logarithmic convexity argument. Next, the kinematic basis is expressed by the power series expansion in the cross-sectional coordinates using Weierstrass’s theorem. Mindlin’s method is used so as to derive the equations in an invariant and fully variational form for the small motions superposed on a static finite deformation, the stability analysis and the high-frequency vibrations of the rod. Moreover, the free vibrations of the rod are considered, the basic properties of eigenvalues are examined, and Rayleigh’s quotient is obtained. The invariant equations of the rod, which are expressible in any system of orthogonal coordinates, may provide simultaneous approximations on all the field variables in a direct method of solutions. The equations are indicated to contain some of earlier equations of rods, as special cases, and also, the numerical elasticity solution of a sample application is presented.

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Authors and Affiliations

  1. Department of Civil Engineering, Boğaziçi University, 34342, Bebek, Istanbul, Turkey

    G. Altay

  2. Istanbul Technical University, Teknik Üniversite PTT, Gümüşsuyu, P.K.9, 34437, Istanbul, Turkey

    M. C. Dökmeci

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Correspondence to M. C. Dökmeci.

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Communicated by Andreas Öchsner.

Dedicated to late Prof. Dipl.-Ing. (ITU) & Dr.-Ing. (ETH, Zürich) Mustafa İnan, with the greatest respect to the highest integrity, and the outstanding teaching of mechanics of materials and scientific achievement, in honour of the centenary of his birth.

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Altay, G., Dökmeci, M.C. On the lower-order theories of continua with application to incremental motions, stability and vibrations of rods. Continuum Mech. Thermodyn. 26, 715–751 (2014). https://doi.org/10.1007/s00161-013-0324-7

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  • DOI: https://doi.org/10.1007/s00161-013-0324-7

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