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Journal of Elasticity
Erschienen in: Journal of Elasticity 1-4/2023

25.04.2023 | Research

Mechanical Response of Metal Solenoids Subjected to Electric Currents

verfasst von: R. S. Elliott, N. Triantafyllidis

Erschienen in: Journal of Elasticity | Ausgabe 1-4/2023

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Abstract

Solenoids, ubiquitous in electrical engineering applications, are devices formed from a coil of wire that use electric current to produce a magnetic field. In contrast to typical electrical engineering applications that pertain to their magnetic field, of interest here is their use as actuators by studying their mechanical deformation. An analytically tractable model of parallel, coaxial circular rings is used to find the solenoid’s axial deformation when subjected to a combined electrical (current) and mechanical (axial force) loading. Both finite and infinite solenoids are considered and their equilibrium configurations as well as their stability are investigated as functions of their geometry and applied current intensity.
Vorheriger Artikel Dislocation-Free Zone at a Mode II Edge Crack Tip
Nächster Artikel Selecting Generalized Continuum Theories for Nonlinear Periodic Solids Based on the Instabilities of the Underlying Microstructure

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Fußnoten
1
It should be noted here that a coil with parallel rings can also be realized experimentally (see [7]).
 
2
Due to the small pitch angle of the solenoid coils considered here, even for relatively large relative displacements \(e_{i}\) between turns \((e_{i} \equiv \lambda _{i} - 1)\) — in Sect. 2.2 we consider up to \(|e_{i}|= |e| =0.5\) — the force-displacement relation for the coil is assumed to remain linear. Large strain nonlinear effects are taken into account in the hyperelastic encasing tube which provides a stiffening response with increasing \(e\).
 
3
The tube is assumed incompressible neo-Hookean and in a uniaxial stress state; different constitutive laws could easily be considered.
 
4
The self-inductance of a circular ring of radius \(a\) is a constant \(L_{ii} = c_{i}\) that depends only on the coil wire’s cross-sectional geometry; \(c_{i}=a[\ln (8a/\rho )-7/4]\) for a circular section of radius \(\rho \) (e.g., see [5, 10]).
 
5
For simplicity the naked coil ring spacing is given but the uniform strain solution \(e\) in the presence of an encasing hyperelastic tube is a straightforward solution of (2.6).
 
6
An externally applied reaction force \(f_{ext} = f \) is needed for maintaining the applied strain \(e\).
 
7
As expected from translational invariance, an eigenvalue \(\beta =0\) always exists and corresponds to \(\omega =0\) (see also (2.11)).
 
8
Standard MKSA system of units is used in this subsection.
 
9
In Sect. 3.2 a coil of diameter \(2a = 10^{-1}\) m is considered with a wire of thickness \(\rho = 10^{-3}\) m.
 
Literatur
1.
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1965) MATH
2.
Gourdin, W.H.: Analysis and assessment of electromagnetic ring expansion at a high-strain-rate test. J. Appl. Phys. 65, 411–422 (1989) CrossRef
3.
Griffiths, D.: Introduction to Electrodynamics. Pearson Education, Upper Saddle River (2014)
4.
Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1999) MATH
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Kovetz, A.: Electromagnetic Theory. Oxford University Press, London (2000) MATH
6.
Love, A.E.H.: A Theasise on the Mathematical Theory of Elasticity. Dover, New York (1944)
7.
Mainy, A.: Dynamic buckling of thin metallic rings under external pressure. Master’s thesis, University of Texas (2012)
8.
Thomas, J., Triantafyllidis, N.: On electromagnetic forming processes in finitely strained solids: theory and examples. J. Mech. Phys. Solids 57(8), 1391–1416 (2009) MathSciNetCrossRefMATH
9.
Thomas, J.D., Seth, M., Bradley, J., Daehn, G., Triantafyllidis, N.: Forming limits for electromagnetically expanded aluminum alloy tubes: theory and experiment forming limits for electromagnetically expanded aluminum alloy tubes: theory and experiment. Acta Mater. 55, 2863–2873 (2007) CrossRef
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Triantafyllidis, N., Waldenmyer, J.: Onset of necking in electro-magnetically formed rings. J. Mech. Phys. Solids 52, 2127–2148 (2004) CrossRefMATH
11.
Zhang, H., Ravi-Chandar, K.: On the dynamics of necking and fragmentation – I. Real-time and post-mortem observations in Al 6061-O. Int. J. Fract. 142, 183–217 (2006) CrossRef
12.
Zhang, H., Ravi-Chandar, K.: On the dynamics of necking and fragmentation – IV. Expansion of Al 6061-O tubes. Int. J. Fract. 163, 41–65 (2010) CrossRefMATH
Metadaten
Titel
Mechanical Response of Metal Solenoids Subjected to Electric Currents
verfasst von
R. S. Elliott
N. Triantafyllidis
Publikationsdatum
25.04.2023
Verlag
Springer Netherlands
Erschienen in
Journal of Elasticity / Ausgabe 1-4/2023
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI
https://doi.org/10.1007/s10659-023-10015-y

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