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Deformations of helical spring and cuboid into hollow cylinders

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Abstract

We study finite inhomogeneous deformations of a helical spring with a rectangular cross-section and a long cuboid. Two surfaces of the spring or the cuboid are joined to obtain a hollow cylinder. When body forces are absent the equilibrium equations reduce to ordinary differential equations. The stress-strain states are the same in each cross-section. The proposed deformations correspond to an inflation, an extension and a torsion of the obtained hollow cylinders. If the obtained cylinders are free of external applied loads, then they have residual stresses.

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Abbreviations

r, \(\phi\), z :

Cylindrical coordinates in reference configuration

R, \({\varPhi }\), Z :

Cylindrical coordinates in deformed configuration

\(\mathbf {e}_r\), \(\mathbf {e}_\phi\), \(\mathbf {e}_z\) :

Cylindrical bases in reference configuration

\(\mathbf {e}_R\), \(\mathbf {e}_{\varPhi }\), \(\mathbf {e}_Z\) :

Cylindrical bases in deformed configuration

\(\mathbf {r}\), \(\mathbf {R}\) :

Position vectors of point of body in reference and deformed configurations

\(\chi\), \(\upsilon\), \(\zeta\) :

Lagrangian coordinates

\(\chi _1\), \(\chi _2\), \(\upsilon _1\), \(\upsilon _2\), h, \(\alpha\) :

Initial geometrical parameters

a, b, c, d, \(\tau\) :

Deformation parameters

\(\mathbf {E}\) :

Unit tensor

\(\mathbf {C}\) :

Deformation gradient

\(\mathbf {G}\), \(\mathbf {g}\), \(\mathbf {g}^{-1}\) :

Cauchy–Green, Almansi, Finger strain measures

\(I_1\), \(I_2\), \(I_3\) :

Invariants of Cauchy–Green strain measure

\(\mathbf {T}\) :

Cauchy stress tensor

W :

Strain-energy function

\(\mathbf {F}\), \(\mathbf {M}\) :

Resultant force and resultant couple on cross-section

\(q_1\), \(q_2\) :

Pressures on inner and outer surfaces of cylinder

\(R_1\), \(R_2\) :

Inner and outer radii of cylinder

p :

Hydrostatic pressure for incompressible material

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Acknowledgements

The research was supported by Russian Foundation for Basic Research (Grant 15-01-01492 A).

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  1. Department of Elasticity Theory, Institute of Mathematics, Mechanics and Computer Science Named After of I.I. Vorovich, Southern Federal University, ul. Milchakova 8a, Rostov-on-Don, Russian Federation, 344090

    Alexey M. Kolesnikov

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Kolesnikov, A.M. Deformations of helical spring and cuboid into hollow cylinders. Meccanica 53, 2161–2170 (2018). https://doi.org/10.1007/s11012-017-0805-z

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