Continuum Mechanics and Thermodynamics

, Volume 29, Issue 2, pp 535–557

# Magnetostriction of a sphere: stress development during magnetization and residual stresses due to the remanent field

• Felix A. Reich
• Wilhelm Rickert
• Oliver Stahn
• Wolfgang H. Müller
Original Article

## Abstract

Based on the principles of rational continuum mechanics and electrodynamics (see Truesdell and Toupin in Handbuch der Physik, Springer, Berlin, 1960 or Kovetz in Electromagnetic theory, Oxford University Press, Oxford, 2000), we present closed-form solutions for the mechanical displacements and stresses of two different magnets. Both magnets are initially of spherical shape. The first (hard) magnet is uniformly magnetized and deforms due to the field induced by the magnetization. In the second problem of a (soft) linear-magnetic sphere, the deformation is caused by an applied external field, giving rise to magnetization. Both problems can be used for modeling parts of general magnetization processes. We will address the similarities between both settings in context with the solutions for the stresses and displacements. In both problems, the volumetric Lorentz force density vanishes. However, a Lorentz surface traction is present. This traction is determined from the magnetic flux density. Since the obtained displacements and stresses are small in magnitude, we may use Hooke’s law with a small-strain approximation, resulting in the Lamé-Navier equations of linear elasticity theory. If gravity is neglected and azimuthal symmetry is assumed, these equations can be solved in terms of a series. This has been done by Hiramatsu and Oka (Int J Rock Mech Min Sci Geomech Abstr 3(2):89–90, 1966) before. We make use of their series solution for the displacements and the stresses and expand the Lorentz tractions of the analyzed problems suitably in order to find the expansion coefficients. The resulting algebraic system yields finite numbers of nonvanishing coefficients. Finally, the resulting stresses, displacements, principal strains and the Lorentz tractions are illustrated and discussed.

## Keywords

Magnetostriction Closed-form solution Spherical magnets Linear-elastic

## General quantities

$$\varvec{v}$$

Barycentric velocity (m/s)

$$\varvec{v}_{\text {I}}$$

Barycentric surface velocity (m/s)

$$\varvec{w}$$

Velocity of a singular surface (m/s)

$$w_\bot$$

Normal velocity of a singular surface (m/s)

$$\varvec{n}$$

Normal vector (1)

$$P_n$$

nth Legendre polynomial (1)

R

Radius of the magnetic sphere (m)

r

$$\tilde{r}$$

$$\vartheta$$

Polar spherical angle, $$\vartheta \in [0, \uppi ]$$ (1)

x

Cosine of polar spherical angle (1)

$$\xi$$

$$\tilde{\xi }$$

z

Axial cylindrical coordinate (m)

$$\tilde{z}$$

Dimensionless axial cylindrical coordinate, (1)

$$\varphi$$

Azimuthal angle, $$\varphi \in [0, 2 \uppi )$$ (1)

$$\varvec{u}$$

Displacement vector (m)

$$u_r$$

Radial displacement component w.r.t. $$\varvec{e}_r$$ (m)

$$u_\vartheta$$

Polar displacement component w.r.t. $$\varvec{e}_\vartheta$$ (m)

$$u_\varphi$$

Azimuthal displacement component w.r.t. $$\varvec{e}_\varphi$$ (m)

$$(\cdot )^\mathrm {I}$$

Interior of magnet

$$(\cdot )^\mathrm {E}$$

Exterior of magnet

$$(\cdot )^{\text { (I)}}$$

Problem of uniformly magnetized sphere

$$(\cdot )^{\text { (II)}}$$

Problem of linear-magnetic sphere

$$\tilde{(\cdot )}$$

A normalized dimensionless function (1)

$$\{\varvec{e}_i\}$$

Local orthonormal basis, $$i \in \{r, \vartheta , \varphi \}$$ (1)

$$\varvec{e}_z$$

Cylindrical axial unit vector (1)

$$\varvec{e}_\xi$$

$$\varvec{1}$$

Unit tensor, $$\varvec{1} = \sum _{i=1}^3 \varvec{e}_i \otimes \varvec{e}_i$$ (1)

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Interface projector, $$\varvec{1}_{\text {I}}= \varvec{1} - \varvec{n} \otimes \varvec{n}$$ (1)

$$\delta _{ij}$$

Kronecker delta (1)

$$\nabla$$

Nabla operator, $$\nabla =\varvec{e}_r \frac{\partial }{\partial r} + \varvec{e}_\vartheta \frac{1}{r} \frac{\partial }{\partial \vartheta } + \varvec{e}_\varphi \frac{1}{r\sin \vartheta } \frac{\partial }{\partial \varphi }$$ (1/m)

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Surface nabla, $$\nabla _{\text {I}}= \varvec{1}_{\text {I}}\cdot \nabla$$ (1/m)

H

Mean curvature, $$H=-\tfrac{1}{2}\nabla _{\text {I}}\cdot \varvec{n}$$ (1/m)

## Mechanics

$$\varvec{\sigma }$$

Cauchy stress tensor (N/m$$^2$$)

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Cauchy surface stress tensor (N/m)

$$\sigma _{ij}$$

Components of $$\varvec{\sigma }$$ w.r.t. $$\{\varvec{e}_i \otimes \varvec{e}_j \}$$, $$i, j \in \{r, \vartheta , \varphi \}$$ (N/m$$^2$$)

$$\sigma _{\mathrm {vM}}$$

Equivalent von Mises stress (N/m$$^2$$)

$$\varvec{\varepsilon }$$

Linear strain tensor (1)

$$\varepsilon _{ij}$$

Components of $$\varvec{\varepsilon }$$ w.r.t. $$\{\varvec{e}_i \otimes \varvec{e}_j \}$$, $$i, j \in \{r, \vartheta , \varphi \}$$ (1)

$$\varLambda _i$$

A principal strain value, $$i \in \{1, 2, 3\}$$ (1)

$$\rho$$

Mass density (kg/m$$^3$$)

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Surface mass density (kg/m$$^2$$)

$$\lambda$$

Lamé’s first parameter (N/m$$^2$$)

$$\mu$$

Lamé’s second parameter (N/m$$^2$$)

## Electrodynamics

$$\varvec{B}$$

Magnetic flux density (T)

$$\varvec{\mathfrak {H}}$$

Potential of free electric current (A/m)

$$\mathfrak {H}_0$$

Amplitude of an external field $$\varvec{\mathfrak {H}}$$ (A/m)

$$V_\mathrm {m}$$

A potential of $$\varvec{\mathfrak {H}}$$, where $$\varvec{\mathfrak {H}}= - \nabla V_\mathrm {m}$$ (A)

$$\varvec{\mathfrak {D}}$$

Potential of free electric charge (C/m$$^2$$)

$$\varvec{E}$$

Electric field (N/C)

$$\varvec{M}$$

Minkowski magnetization (A/m)

$$M_0$$

Strength of the magnet’s uniform magnetization (A/m)

$$\varvec{P}$$

Polarization (C/m$$^2$$)

$$\mu _0$$

Vacuum permeability (N/A$$^2$$)

$$\mu _\mathrm {r}$$

Relative permeability (1)

$$\epsilon _0$$

Vacuum permittivity $$({\mathrm {A}^2 \mathrm {s}^2}/{({\mathrm {N}\, \mathrm {m}^2})})$$

q

Total electric charge density (C/m$$^3$$)

$$q^{\text {f}}$$

Free electric charge density (C/m$$^3$$)

$$q^{\text {r}}$$

Bound electric charge density (C/m$$^3$$)

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Singular free electric charge density (C/m$$^2$$)

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Singular bound electric charge density (C/m$$^2$$)

$$\varvec{J}$$

Total electric current density (A/m$$^2$$)

$$\varvec{J}^{\text {f}}$$

Free electric current density (A/m$$^2$$)

$$\varvec{J}^{\text {r}}$$

Bound electric current density (A/m$$^2$$)

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Singular total electric current density (A/m)

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Singular free electric current density (A/m)

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Singular bound electric current density (A/m)

$$\varvec{j}^{\text {f}}$$

Free diffusive electric current density (A/m$$^2$$)

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Singular free diffusive electric current density (A/m)

## Mixed mechanics and electrodynamics

$$\varvec{f}^\mathrm {L}$$

Volumetric Lorentz force density (N/m$$^3$$)

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Surface Lorentz force density (N/m$$^2$$)

## References

1. 1.
Brown, W.F.: Magnetoelastic Interactions, Springer Tracts in Natural Philosophy, vol. 9. Springer, Berlin (1966)Google Scholar
2. 2.
Dziubek, A.: Equations for two-phase flows: a primer. arXiv:1101.5732 (2011)
3. 3.
Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua I: Foundations and Solid Media. Springer, Berlin (2012)Google Scholar
4. 4.
Fitzpatrick R.: Leture notes: Classical electromagnetism: an intermediate level course. University of Texas, Austin (2014)Google Scholar
5. 5.
Guhlke, C.: Theorie der elektrochemischen Grenzfläche. Ph.D. thesis, Technische Universität Berlin (2015)Google Scholar
6. 6.
Hiramatsu, Y., Oka, Y.: Determination of the tensile strength of rock by a compression test of an irregular test piece. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 3(2), 89–90 (1966)
7. 7.
Jackson, J.D.: Classical Electrodynamics, 2nd edn. Wiley, Hoboken (1975)
8. 8.
Kovetz, A.: Electromagnetic Theory. Oxford University Press, Oxford (2000)
9. 9.
Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover Books on Engineering and Engineering Physics. Dover Publications, Mineola (1944)Google Scholar
10. 10.
Müller, I.: Thermodynamics. Interaction of Mechanics and Mathematics Series. Pitman, Boston (1985)Google Scholar
11. 11.
Müller, W.H.: An Expedition to Continuum Theory. Solid Mechanics and Its Applications Series. Springer, Berlin (2014)
12. 12.
Raikher, Y.L., Stolbov, O.V.: Deformation of an ellipsoidal ferrogel sample in a uniform magnetic field. J. Appl. Mech. Tech. Phys. 46(3), 434–443 (2005)
13. 13.
Rinaldi, C., Brenner, H.: Body versus surface forces in continuum mechanics: is the Maxwell stress tensor a physically objective Cauchy stress? Phys. Rev. E 65(3), 036615 (2002)
14. 14.
Slattery, J.C., Sagis, L., Oh, E.S.: Interfacial Transport Phenomena. Springer, Berlin (2007)
15. 15.
Steinmann, P.: On boundary potential energies in deformational and configurational mechanics. J. Mech. Phys. Solids 56(3), 772–800 (2008)
16. 16.
Stratton, J.A.: Electromagnetic Theory. McGraw-Hill, New York (1941)
17. 17.
Truesdell, C.A., Toupin, R.: The classical field theories. In: Handbuch der Physik, Bd. III/1, pp. 226–793; appendix, pp. 794–858. Springer, Berlin (1960). With an appendix on tensor fields by J. L. EricksenGoogle Scholar
18. 18.
Wei, X.X., Wang, Z.M., Xiong, J.: The analytical solutions for the stress distributions within elastic hollow spheres under the diametrical point loads. Arch. Appl. Mech. 85(6), 817–830 (2015)

## Authors and Affiliations

• Felix A. Reich
• 1
Email author
• Wilhelm Rickert
• 1
• Oliver Stahn
• 1
• Wolfgang H. Müller
• 1
1. 1.Institut für Mechanik, Kontinuumsmechanik und MaterialtheorieTechnische Universität BerlinBerlinGermany