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Saint-Venant problem for solids with helical anisotropy

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We discuss the solution of Saint-Venant’s problem for solids with helical anisotropy. Here the governing relations of the theory of elasticity in terms of displacements were presented using the helical coordinate system. We proposed an approach to construct elementary Saint-Venant solutions using integration of ordinary differential equations with variable coefficients in the case of a circular cylinder with helical anisotropy. Elementary solutions correspond to problems of extension, of torsion, of pure bending and of bending of shear force. The solution of the problem is obtained using small parameter method for small values of twist angle and numerically for arbitrary values. Numeric results correspond to problems of extension–torsion. Dependencies of the stiffness matrix (in dimensionless form) on angle between the tangent to the helical coil and the axis of the cylinder corresponding to stiffness on stretching and torsion are illustrated graphically for different values of material and geometrical parameters.

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Correspondence to Natalia V. Kurbatova.

Additional information

The first author acknowledges the supports by the Ministry of Education and Science of the Russian Federation, Project No. 1334 the base part of the Job No. 2014/174—public works in the field of scientific activity.

Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell’Isola.

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Kurbatova, N.V., Ustinov, Y.A. Saint-Venant problem for solids with helical anisotropy. Continuum Mech. Thermodyn. 28, 465–476 (2016).

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  • Helical anisotropy
  • Saint-Venant problem
  • Elastic cylinder