Magnetoelastic Boundary-Value Problems

  • Luis Dorfmann
  • Ray W. Ogden
Chapter

Abstract

In this chapter we first summarize the constitutive equations of a nonlinear isotropic magnetoelastic material based on the Lagrangian magnetic field and magnetic induction along with the associated governing differential equations and boundary conditions. We then apply the theory to a number of representative boundary-value problems. We examine two problems for a slab of material involving homogeneous deformation and a uniform magnetic field normal to the faces of the slab, namely pure homogeneous strain and simple shear, using the constitutive law based on the magnetic induction as the independent magnetic variable. A simple prototype energy function is used to illustrate the effect of the magnetic field on the stress in the material for pure homogeneous strain while general theoretical results for the stress and magnetic field are obtained in the case of simple shear. We then consider two non-homogeneous deformations of a thick-walled circular cylindrical tube, with the magnetic field as the independent magnetic variable. These are the extension and inflation of a tube and the helical shear of a tube with either an axial or an azimuthal magnetic field. In the first of these problems we focus on determining the effect of the magnetic field on the pressure and axial load characteristics of the material response. For the helical shear problem, we highlight the restrictions on the constitutive law for which the considered deformation is admissible and then obtain explicit results for a specific form of energy function.

Keywords

Incompressibility 

References

  1. Dorfmann A, Ogden RW (2004a) Nonlinear magnetoelastic deformations. Q J Mech Appl Math 57:599–622CrossRefMATHMathSciNetGoogle Scholar
  2. Dorfmann A, Ogden RW (2004b) Nonlinear magnetoelastic deformations of elastomers. Acta Mechanica 167:13–28CrossRefMATHGoogle Scholar
  3. Dorfmann A, Ogden RW (2005) Some problems in nonlinear magnetoelasticity. Z Angew Math Mech (ZAMP) 56:718–745CrossRefMATHMathSciNetGoogle Scholar
  4. Guan X, Dong X, Ou, J (2008) Magnetostrictive effect in magnetorheological elastomers. J Mag Mag Mater 320:158–163CrossRefGoogle Scholar
  5. Jiang X, Ogden RW (1998) On azimuthal shear of a circular cylindrical tube of compressible elastic material. Q J Mech Appl Math 51:143–158CrossRefMATHMathSciNetGoogle Scholar
  6. Jolly MR, Carlson JD, Muñoz BC (1996) A model of the behaviour of magnetorheological materials. Smart Mater Struct 5:607–614CrossRefGoogle Scholar
  7. Ogden RW (1997) Non-linear elastic deformations. Dover Publications, New YorkGoogle Scholar
  8. Ogden RW, Dorfmann A (2005) Magnetomechanical interactions in magneto-sensitive elastomers. In: Austrell P-E, Kari L (eds) Proceedings of the third European conference on constitutive models for rubber. Balkema, Rotterdam, pp 531–543Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Luis Dorfmann
    • 1
  • Ray W. Ogden
    • 2
  1. 1.Department of Civil and Environmental EngineeringTufts UniversityMedfordUSA
  2. 2.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK

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