Advertisement

Two- and three-dimensional modeling approaches in magneto-mechanics: a quantitative comparison

  • P. Metsch
  • K. A. Kalina
  • J. Brummund
  • M. Kästner
SPECIAL
  • 71 Downloads

Abstract

In this contribution, we present a qualitative and quantitative comparison of two- and three-dimensional finite-element simulations for magneto-rheological elastomers. Based on a general continuum formulation of the coupled magneto-mechanical boundary value problem, a microscopic modeling approach is applied. The merit of this strategy is a full resolution of the local magnetic and mechanical fields within the heterogeneous microstructure of magneto-rheological elastomers—it allows to account for systems with high particle-volume fractions and small inter-particle distances. In order to understand basic deformation mechanisms as well as local magneto-mechanical interactions of the spherical inclusions, the differences between simplified two-dimensional and realistic three-dimensional simulations are initially shown for the example of chain-like structures with varying arrangements of the particles. Afterwards, an appropriate scale transition scheme is used to connect the microscopic and macroscopic quantities: Different two- and three-dimensional, ideal and random microstructures are analyzed with regard to their effective magneto-mechanical behavior.

Keywords

Magneto-rheological elastomers Magneto-mechanical coupling Nonlinear finite-element method 

Notes

Acknowledgements

The present study is funded by the German Research Foundation, Priority Programmes (SPP) 1681 and 1713: Grants KA 3309/2-3 and KA 3309/5-2. This support is gratefully acknowledged. The computations were performed on a PC-Cluster at the Center for Information Services and High Performance Computing (ZIH) at TU Dresden. The authors thank the ZIH for generous allocations of computer time.

References

  1. 1.
    Ahrens, J., Geveci, B., Law, C.: ParaView: an end-user tool for large-data visualization. In: Hansen, C.D., Johnson, C.R. (eds.) Visualization Handbook, pp. 717–731. Butterworth-Heinemann, Burlington (2005)CrossRefGoogle Scholar
  2. 2.
    Becker, T., Böhm, V., Vega, J.C., Borin, D.Y., Odenbach, S., Raikher, Y., Stepanov, G., Zimmermann, K.: Studies on the dynamical behavior of magneto-sensitive elastomers in application for magnetic field controlled actuator and sensor systems. Arch. Appl. Mech., submitted (2018)Google Scholar
  3. 3.
    Biller, A.M., Stolbov, O.V., Raikher, Y.L.: Modeling of particle interactions in magnetorheological elastomers. J. Appl. Phys. 116(11), 114904 (2014)CrossRefGoogle Scholar
  4. 4.
    Bíró, O., Preis, K.: On the use of the magnetic vector potential in the finite element analysis of three-dimensional eddy currents. IEEE Trans. Magn. 25(4), 3145–3159 (1989)CrossRefGoogle Scholar
  5. 5.
    Borin, D., Stepanov, G.V.: Soft magnetoactive elastomers: synthesis and magnetomechanical characterization. Arch. Appl. Mech., submitted (2018)Google Scholar
  6. 6.
    Böse, H., Rabindranath, R., Ehrlich, J.: Soft magnetorheological elastomers as new actuators for valves. J. Intell. Mater. Syst. Struct. 23(9), 989–994 (2012)CrossRefGoogle Scholar
  7. 7.
    Bustamante, R.: Transversely isotropic nonlinear magneto-active elastomers. Acta Mech. 210(3), 183–214 (2010)CrossRefzbMATHGoogle Scholar
  8. 8.
    Carlson, J.D., Jolly, M.R.: MR fluid, foam and elastomer devices. Mechatronics 10(4–5), 555–569 (2000)CrossRefGoogle Scholar
  9. 9.
    Chatzigeorgiou, G., Javili, A., Steinmann, P.: Unified magnetomechanical homogenization framework with application to magnetorheological elastomers. Math. Mech. Solids 19(2), 193–211 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ciarlet, P.G.: Three-Dimensional Elasticity, vol. 20. Elsevier, New York (1988)zbMATHGoogle Scholar
  11. 11.
    Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963).  https://doi.org/10.1007/BF01262690 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cremer, P., Löwen, H., Menzel, A.M.: Tailoring superelasticity of soft magnetic materials. Appl. Phys. Lett. 107(17), 171903 (2015)CrossRefGoogle Scholar
  13. 13.
    Danas, K.: Effective response of classical, auxetic and chiral magnetoelastic materials by use of a new variational principle. J. Mech. Phys. Solids 105, 25–53 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Danas, K., Kankanala, S.V., Triantafyllidis, N.: Experiments and modeling of iron-particle-filled magnetorheological elastomers. J. Mech. Phys. Solids 60(1), 120–138 (2012)CrossRefGoogle Scholar
  15. 15.
    de Groot, S.R., Suttorp, L.G.: Foundations of Electrodynamics. North-Holland, Amsterdam (1972)Google Scholar
  16. 16.
    Dorfmann, A., Ogden, R.W.: Nonlinear magnetoelastic deformations of elastomers. Acta Mech. 167(1), 13–28 (2004)CrossRefzbMATHGoogle Scholar
  17. 17.
    Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua I: Foundations and Solid Media. Springer, New York (1990)CrossRefGoogle Scholar
  18. 18.
    Fetzer, J., Haas, M., Kurz, S.: Numerische Berechnung elektromagnetischer Felder, volume 627 of Kontakt & Studium. expert-Verlag, Renningen-Malmsheim (2002)Google Scholar
  19. 19.
    Galipeau, E., Ponte Castañeda, P.: The effect of particle shape and distribution on the macroscopic behavior of magnetoelastic composites. Int. J. Solids Struct. 49(1), 1–17 (2012)CrossRefGoogle Scholar
  20. 20.
    Galipeau, E., Ponte Castañeda, P.: A finite-strain constitutive model for magnetorheological elastomers: magnetic torques and fiber rotations. J. Mech. Phys. Solids 61(4), 1065–1090 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Galipeau, E., Rudykh, S., deBotton, G., Ponte Castañeda, P.: Magnetoactive elastomers with periodic and random microstructures. Int. J. Solids Struct. 51(18), 3012–3024 (2014)CrossRefGoogle Scholar
  22. 22.
    Geuzaine, C., Remacle, J.-F.: Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Eng. 79(11), 1309–1331 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Goshkoderia, A., Rudykh, S.: Stability of magnetoactive composites with periodic microstructures undergoing finite strains in the presence of a magnetic field. Compos. B Eng. 128, 19–29 (2017)CrossRefGoogle Scholar
  24. 24.
    Guan, X., Dong, X., Ou, J.: Magnetostrictive effect of magnetorheological elastomer. J. Magn. Magn. Mater. 320(3–4), 158–163 (2008)CrossRefGoogle Scholar
  25. 25.
    Haldar, K., Kiefer, B., Menzel, A.: Finite element simulation of rate-dependent magneto-active polymer response. Smart Mater. Struct. 25(10), 104003 (2016)CrossRefGoogle Scholar
  26. 26.
    Han, Y., Hong, W., Faidley, L.E.: Field-stiffening effect of magneto-rheological elastomers. Int. J. Solids Struct. 50(14–15), 2281–2288 (2013)CrossRefGoogle Scholar
  27. 27.
    Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11(5), 357–372 (1963)CrossRefzbMATHGoogle Scholar
  28. 28.
    Jackson, J.D.: Klassische Elektrodynamik, 4th edn. Walter de Gruyter, Berlin (2006)CrossRefGoogle Scholar
  29. 29.
    Javili, A., Chatzigeorgiou, G., Steinmann, P.: Computational homogenization in magneto-mechanics. Int. J. Solids Struct. 50(25–26), 4197–4216 (2013)CrossRefGoogle Scholar
  30. 30.
    Kalina, K.A., Brummund, J., Kästner, M.P.M., Borin, D.Y., Linke, J.M., Odenbach, S.: Modeling of magnetic hystereses in soft mres filled with ndfeb particles. Smart Mater. Struct. 26, 105019–105031 (2017)CrossRefGoogle Scholar
  31. 31.
    Kalina, K.A., Metsch, P., Kästner, M.: Microscale modeling and simulation of magnetorheological elastomers at finite strains: a study on the influence of mechanical preloads. Int. J. Solids Struct. 102–103, 286–296 (2016)CrossRefGoogle Scholar
  32. 32.
    Kankanala, S.V., Triantafyllidis, N.: On finitely strained magnetorheological elastomers. J. Mech. Phys. Solids 52(12), 2869–2908 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kästner, M., Obst, M., Brummund, J., Thielsch, K., Ulbricht, V.: Inelastic material behavior of polymers–experimental characterization, formulation and implementation of a material model. Mech. Mater. 52, 40–57 (2012)CrossRefGoogle Scholar
  34. 34.
    Keip, M.-A., Rambausek, M.: A multiscale approach to the computational characterization of magnetorheological elastomers. Int. J. Numer. Methods Eng. 107(4), 338–360 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Keip, M.-A., Rambausek, M.: Computational and analytical investigations of shape effects in the experimental characterization of magnetorheological elastomers. Int. J. Solids Struct. 121, 1–20 (2017)CrossRefGoogle Scholar
  36. 36.
    Linke, J.M., Borin, D.Y., Odenbach, S.: First-order reversal curve analysis of magnetoactive elastomers. RSC Adv. 6(102), 100407–100416 (2016)CrossRefGoogle Scholar
  37. 37.
    Menzel, A.: Mesoscopic modeling of magnetic gels and elastomers on the discrete particle level and links to the macroscale. Arch. Appl. Mech. submitted (2018)Google Scholar
  38. 38.
    Metsch, P., Kalina, K.A., Spieler, C., Kästner, M.: A numerical study on magnetostrictive phenomena in magnetorheological elastomers. Comput. Mater. Sci. 124, 364–374 (2016)CrossRefGoogle Scholar
  39. 39.
    Miehe, C., Rosato, D., Kiefer, B.: Variational principles in dissipative electro-magneto-mechanics: a framework for the macro-modeling of functional materials. Int. J. Numer. Meth. Eng. 86(10), 1225–1276 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Pao, Y.-H.: Electromagnetic forces in deformable continua. In: Nemat-Nasser, S. (ed.) Mechanics Today, vol. 4, chapter IV, pp. 209–305. Pergamon Press (1978)Google Scholar
  41. 41.
    Petra, C.G., Schenk, O., Anitescu, M.: Real-time stochastic optimization of complex energy systems on high-performance computers. IEEE Comput. Sci. Eng. 16(5), 32–42 (2014)CrossRefGoogle Scholar
  42. 42.
    Petra, C.G., Schenk, O., Lubin, M., Gärtner, K.: An augmented incomplete factorization approach for computing the Schur complement in stochastic optimization. SIAM J. Sci. Comput. 36(2), C139–C162 (2014)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Ponte Castañeda, P., Galipeau, E.: Homogenization-based constitutive models for magnetorheological elastomers at finite strain. J. Mech. Phys. Solids 59(2), 194–215 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Romeis, D., Metsch, P., Kästner, M., Saphiannikova, M.: Theoretical models for magneto-sensitive elastomers: a comparison between continuum and dipole approaches. Phys. Rev. E 95, 042501 (2017)CrossRefGoogle Scholar
  45. 45.
    Romeis, D., Toshchevikov, V., Saphiannikova, M.: Elongated micro-structures in magneto-sensitive elastomers: a dipolar mean field model. Soft Matter 12, 9364–9376 (2016)CrossRefGoogle Scholar
  46. 46.
    Rudykh, S., Bertoldi, K.: Stability of anisotropic magnetorheological elastomers in finite deformations: a micromechanical approach. J. Mech. Phys. Solids 61(4), 949–967 (2013)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Saxena, P., Hossain, M., Steinmann, P.: A theory of finite deformation magneto-viscoelasticity. Int. J. Solids Struct. 50(24), 3886–3897 (2013)CrossRefGoogle Scholar
  48. 48.
    Saxena, P., Hossain, M., Steinmann, P.: Nonlinear magneto-viscoelasticity of transversally isotropic magneto-active polymers. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 470(2166), 20140082 (2014)CrossRefGoogle Scholar
  49. 49.
    Schubert, G.: Manufacture, Characterisation and Modelling of Magneto-rheological Elastomers. PhD thesis, University of Glasgow (2014)Google Scholar
  50. 50.
    Schümann, M., Borin, D.Y., Huang, S., Auernhammer, G.K., Müller, R., Odenbach, S.: A characterisation of the magnetically induced movement of ndfeb-particles in magnetorheological elastomers. Smart Mater. Struct. 26(9), 095018 (2017)CrossRefGoogle Scholar
  51. 51.
    Schümann, M., Gundermann, T., Odenbach, S.: Microscopic investigation of the reasons for field dependent changes of the properties of magnetic hybrid materials using x-ray micro tomography. Arch. Appl. Mech., submitted (2018)Google Scholar
  52. 52.
    Spieler, C., Kästner, M., Goldmann, J., Brummund, J., Ulbricht, V.: XFEM modeling and homogenization of magnetoactive composites. Acta Mech. 224(11), 2453–2469 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Stark, S., Semenov, A.S., Balke, H.: On the boundary conditions for the vector potential formulation in electrostatics. Int. J. Numer. Meth. Eng. 102(11), 1704–1732 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Stepanov, G., Borin, D.Y., Raikher, Y.L., Melenev, P., Perov, N.: Motion of ferroparticles inside the polymeric matrix in magnetoactive elastomers. J. Phys.: Condens. Matter 20(20), 204121 (2008)Google Scholar
  55. 55.
    Stepanov, G.V., Borin, D.Y., Kramarenko, E.Y., Bogdanov, V.V., Semerenko, D.A., Storozhenko, P.A.: Magnetoactive elastomer based on magnetically hard filler: synthesis and study of viscoelastic and damping properties. Polym. Sci. Ser. A 56(5), 603–613 (2014)CrossRefGoogle Scholar
  56. 56.
    Tian, T.F., Li, W.H., Deng, Y.M.: Sensing capabilities of graphite based MR elastomers. Smart Mater. Struct. 20(2), 025022 (2011)CrossRefGoogle Scholar
  57. 57.
    Vogel, F., Bustamante, R., Steinmann, P.: On some mixed variational principles in magneto-elastostatics. Int. J. Non-Linear Mech. 51, 157–169 (2013)CrossRefGoogle Scholar
  58. 58.
    Volkova, T., Böhm, V., Kaufhold, T., Popp, J., Becker, F., Borin, D., Stepanov, G., Zimmermann, K.: Motion behaviour of magneto-sensitive elastomers controlled by an external magnetic field for sensor applications. J. Magn. Magn. Mater.  https://doi.org/10.1016/j.jmmm.2016.10.009 (2016)
  59. 59.
    Weeber, R., Kreissl, P., Holm, C.: Studying the field-controlled change of shape and elasticity of magnetic gels using particle-based simulations. Arch. Appl. Mech.  https://doi.org/10.1007/s00419-018-1396-4 (2018)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • P. Metsch
    • 1
  • K. A. Kalina
    • 1
  • J. Brummund
    • 1
  • M. Kästner
    • 1
  1. 1.Institute of Solid MechanicsTechnische Universität DresdenDresdenGermany

Personalised recommendations