Abstract
In the plane problem of the theory of elasticity about a cantilever strip bending, we study the stress state near its fixed end. It is found that the solution singularity at the corner points does not have any physical nature and is generated by specific characteristics of the statement of the problem in which it is assumed that the stress tensor symmetry is violated at these points.
We consider Reissner’s problem about the stress state of the corner region one of whose sides is loaded by constant tangential stresses. We found that the statement of this problem in polar coordinates, which does not violate the stress tensor symmetry for a given loading, results in a regular solution. But the statement of the problem in Cartesian coordinates, which contradicts the stress tensor symmetry, results in a solution with singularities.
On the basis of the obtained results, we conclude that the singularity of solutions of elasticity problems is related to the statement of these problems, which explicitly or implicitly assumes violations of the stress tensor symmetry.
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References
S. P. Timoshenko and J. N. Goodyear, Theory of Elasticity (McGraw-Hill, New York, 1951; Nauka, Moscow, 1975).
V. V. Vasil’ev and S. A. Lur’e, “AVariant of the Refined Theory of Bending for a Laminar Plastic Beam,” Mekh. Polimer., No. 4, 674–681 (1972) [PolymerMech. (Engl. Transl.) 8 (4), 582–588 (1972)].
S. A. Lurie and V. V. Vasiliev, The Biharmonic Problem in the Theory of Elasticity (Gordon and Breach, Australia etc., 1995).
E. Reissner, “Note on the Theorem of the Symmetry of the Stress Tensor,” in Proc. of the 4th Midwestern Conf. on Solid Mechanics (University of Texas Press, Austin, 1959), pp. 192–194.
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Original Russian Text © V.V. Vasil’ev, 2010, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2010, No. 2, pp. 62–72.
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Vasil’ev, V.V. Stress tensor symmetry and singular solutions in the theory of elasticity. Mech. Solids 45, 205–213 (2010). https://doi.org/10.3103/S0025654410020068
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DOI: https://doi.org/10.3103/S0025654410020068