Mechanics of Solids (about journal) Mechanics of Solids
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IssuesArchive of Issues2014-2pp.225-236

Archive of Issues

Total articles in the database: 10864
In Russian (. . ): 8009
In English (Mech. Solids): 2855

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A.D. Chernyshov, "Torsion of an Elastic Rod Whose Cross-Section Is a Parallelogram, Trapezoid, or Triangle, or Has an Arbitrary Shape by the Method of Transformation to a Rectangular Domain," Mech. Solids. 49 (2), 225-236 (2014)
Year 2014 Volume 49 Number 2 Pages 225-236
DOI 10.3103/S0025654414020125
Title Torsion of an Elastic Rod Whose Cross-Section Is a Parallelogram, Trapezoid, or Triangle, or Has an Arbitrary Shape by the Method of Transformation to a Rectangular Domain
Author(s) A.D. Chernyshov (Voronezh State Technological Academy, pr-t Revolyutsii 19, Voronezh, 394000 Russia, chernyshovad@mail.ru)
Abstract A coordinate transformation is used to take the domain of the rod cross-section to a rectangular domain for which the spectra of eigenfunctions and eigenvalues are known. The torsion function is represented as a generalized Fourier series to reduce the problem to solving a closed linear system of algebraic equations for the expansion coefficients. It is shown that these Fourier series converge absolutely, because the expansion coefficients decrease by a cubic law depending on the term number. We prove that the approximate solution in the form of a finite sum of the Fourier series converges to the exact solution. This theorem is generalized to the case of a rod cross-section of arbitrary shape.
Keywords elastic rod, cross-section, parallelogram, trapezoid, triangle, fast Fourier series
References
1.  A. I. Lurie, The Theory of Elasticity (Nauka, Moscow, 1970) [in Russian].
2.  S. P. Timoshenko and J. N. Goodyear, Theory of Elasticity (McGraw-Hill, New York, 1951; Nauka, Moscow, 1975).
3.  N. I. Muskhelishvili, Several Fundamental Problems of Mathematical Theory of Elasticity (Izd-vo AN SSSR, Moscow, 1954) [in Russian].
4.  N. Kh. Arutyunyan and B. L. Abramyan, Torsion of Elastic Bodies (Fizmatgiz, Moscow, 1963) [in Russian].
5.  V. A. Lomakin, "Torsion of Rods with Elastic Properties Depending on the Stress State Form," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 30-39 (2002) [Mech. Solids (Engl. Transl.) 37 (4), 23-30 (2002)].
6.  A. V. Konovalov, "Torsion of Cylindrical Rods and Pipes with Large Plastic Strains," with Large Plastic Strains," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 102-111 (2001) [Mech. Solids (Engl. Transl.) 36 (3), 86-94 (2001)].
7.  L. M. Zubov, "The Non-Linear Saint-Venant Problem of the Torsion, Stretching and Bending of a Naturally Twisted Rod," Prikl. Mat. Mekh. 70 (2), 332-343 (2006) [J. Appl. Math. Mech. (Engl. Transl.) 70 (2), 300-310 (2006)].
8.  G. I. Bykovtsev and D. D. Ivlev, Theory of Plasticity (Dal'nauka, Vladivostok, 1998) [in Russian].
9.  V. A. Il'in, Spectral Theory of Differential Operators. Self-Adjoint Differential Operators (Nauka, Moscow, 1991) [in Russian].
10.  E. M. Kartashov, Analytical Methods in Heat Conductivity of Solids (Vysshaya Shkola, Moscow, 2001) [in Russian].
11.  A. D. Chernyshov, "Solving Nonlinear Boundary Value Problems by the Spectral Decomposition Method," Dokl. Ross. Akad. Nauk 411 (6), 775-778 (2006) [Dokl. Phys. (Engl. Transl.) 51 (12), 697-700 (2006)].
12.  G. P. Tolstov, Fourier Series (Dover, New York, 1976; Nauka, Moscow, 1980).
13.  A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1981; Dover, New York, 1999).
Received 04 April 2011
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