Continuum Mechanics and Thermodynamics

, Volume 28, Issue 5, pp 1421–1433 | Cite as

Inelastic deformation of conductive bodies in electromagnetic fields

  • Holm Altenbach
  • Oleg Morachkovsky
  • Konstantin NaumenkoEmail author
  • Denis Lavinsky
Original Article


Inelastic deformation of conductive bodies under the action of electromagnetic fields is analyzed. Governing equations for non-stationary electromagnetic field propagation and elastic–plastic deformation are presented. The variational principle of minimum of the total energy is applied to formulate the numerical solution procedure by the finite element method. With the proposed method, distributions of vector characteristics of the electromagnetic field and tensor characteristics of the deformation process are illustrated for the inductor–workpiece system within a realistic electromagnetic forming process.


Electromagnetic field Conductive body Inelastic deformation 


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  1. 1.
    Alessandroni S., Andreaus U., Dell’Isola F., Porfiri M.: Piezo-electromechanical (PEM) Kirchhoff–Love plates. Eur. J. Mech. A Solids 23(4), 689–702 (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Altenbach H., Brigadnov I., Eremeyev V.: Oscillations of a magneto-sensitive elastic sphere. ZAMM 88(6), 497–506 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambarcumyan S.A., Bagdarsanyan G.E., Belubekyan M.B.: Magnetoelasticity of Plates and Shells. Nauka, Moscow (1977) (in Russ.)Google Scholar
  4. 4.
    Andreaus U., Dell’Isola F., Porfiri M.: Piezoelectric passive distributed controllers for beam flexural vibrations. J. Vib. Control 10(5), 625–659 (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bay F., Labbé V., Favennec Y., Chenot J.: A numerical model for induction heating processes coupling electromagnetism and thermomechanics. Int. J. Numer. Methods Eng. 58(6), 839–867 (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Belokon’ A.V., Eremeyev V.A., Nasedkin A.V., Solov’yev A.N.: Partitioned schemes of the finite-element method for dynamic problems of acoustoelectroelasticity. J. Appl. Math. Mech. 64(3), 367–377 (2000)CrossRefGoogle Scholar
  7. 7.
    Bertram, A.: Finite gradient elasticity and plasticity: a constitutive thermodynamical framework. Contin. Mech. Thermodyn. 27(6), 1039–1058 (2015)Google Scholar
  8. 8.
    Cazzani A., Atluri S.N.: Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes. Comput. Mech. 11(4), 229–251 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cazzani A., Garusi E., Tralli A., Atluri S.N.: A four-node hybrid assumed-strain finite element for laminated composite plates. CMC Comput. Mater. Contin. 2(1), 23–38 (2005)zbMATHGoogle Scholar
  10. 10.
    Chadwick, P.: Elastic wave propagation in a magnetic field. In: Actes IX Congress International of Applied Mechanics, vol. 7, pp. 143–158. Univ. Bruxelles, Bruxelles (1957)Google Scholar
  11. 11.
    Dorfmann A., Ogden R.: Magnetoelastic modelling of elastomers. Eur. J. Mech. A Solids 22(4), 497–507 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dorfmann A., Ogden R.: Some problems in nonlinear magnetoelasticity. ZAMP 56(4), 718–745 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dorfmann L., Ogden R.W.: Nonlinear Theory of Electroelastic and Magnetoelastic Interactions. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  14. 14.
    Dunkin J.W., Eringen A.C.: Propagation of waves in an electromagnetic elastic solid. Int. J. Eng. Sci. 1, 461–495 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Eringen A.C.: Mechanics of Continuum. Wiley, New York (1967)zbMATHGoogle Scholar
  16. 16.
    Eringen A.C., Maugin G.A.: Electrodynamics of Continua, vol. 2. Springer, New York (1989)Google Scholar
  17. 17.
    Favennec Y., Labbé V., Bay F.: Induction heating processes optimization a general optimal control approach. J. Comput. Phys. 187(1), 68–94 (2003)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Giorgio I., Galantucci L., Della Corte A., Del Vescovo D.: Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications. Int. J. Appl. Electromagn. Mech. 47(4), 1051–1084 (2015)CrossRefGoogle Scholar
  19. 19.
    Hutter K., van de Ven A.A.: Field–Matter Interaction in Thermoelastic Solids. Lecture Notes in Physics, vol. 88. Springer, Berlin (1978)CrossRefGoogle Scholar
  20. 20.
    Kaliski S., Petykiewicz J.: Dynamical equations of motion coupled with the field of temperatures and resolving functions for elastic and inelastic anisotropic bodies in the magnetic field. Proc. Vibr. Probl. 1(3), 81–94 (1960)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Knopoff L.: The interactions between elastic waves motions and a magnetic field in electric conductor. J. Geophys. Res. 60, 441–456 (1955)ADSCrossRefGoogle Scholar
  22. 22.
    Lange K.: Handbook of Metal Forming, pp. 27.32–27.39. McGraw-Hill, New York (1985)Google Scholar
  23. 23.
    Längler F., Naumenko K., Altenbach H., Ievdokymov M.: A constitutive model for inelastic behavior of casting materials under thermo-mechanical loading. J. Strain Anal. Eng. Des. 49, 421–428 (2014)CrossRefGoogle Scholar
  24. 24.
    Livshitz, Y., Gafri, O.: Technology and equipment for industrial use of pulse magnetic fields. In: Ren, Z., Besbes, M., Boukhtache, S. (eds.) IEEE International Pulsed Power Conference, vol. 1, pp. 475–478 (1999)Google Scholar
  25. 25.
    Maugin C.A.: Electromagnetic internal variables in electromagnetic continua. Arch. Mech. 30(1), 927–936 (1981)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Maugin G.A.: Continuum Mechanics of Electromagnetic Solids. Elsevier, New York (1988)zbMATHGoogle Scholar
  27. 27.
    Maxwell J.C.: A Treatise on Electricity and Magnetism. Clarendon Press, Oxford (1873)zbMATHGoogle Scholar
  28. 28.
    Miehe C., Schänzel L.M., Ulmer H.: Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput. Methods Appl. Mech. Eng. 294, 449–485 (2015)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Nasedkin A.V., Eremeyev V.A.: Harmonic vibrations of nanosized piezoelectric bodies with surface effects. ZAMM 94(10), 878–892 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Naumenko, K., Altenbach, H.: Analysis of inelastic behavior for high temperature materials and structures. In: Altenbach, H., Matsuda, T., Okumura, D. (eds.) From Creep Damage Mechanics to Homogenization Methods, Advanced Structured Materials, vol. 64., pp. 241–298. Springer International Publishing (2015)Google Scholar
  31. 31.
    Naumenko K., Altenbach H., Kutschke A.: A combined model for hardening, softening and damage processes in advanced heat resistant steels at elevated temperature. Int. J. Damage Mech. 20, 578–597 (2011)CrossRefGoogle Scholar
  32. 32.
    Naumenko K., Gariboldi E.: A phase mixture model for anisotropic creep of forged Al–Cu–Mg–Si alloy. Mater. Sci. Eng. A 618, 368–376 (2014)CrossRefGoogle Scholar
  33. 33.
    Nemkov V., Goldstein R.: Computer simulation for fundamental study and practical solutions to induction heating problems. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. 22(1), 181–191 (2003)CrossRefGoogle Scholar
  34. 34.
    Nowacki W.: Efekty Elektromagnetyczne w Stałych Ciałach Odkształcalnych. Państwowe Wydawnictwo Naukowe, Warsaw (1983)Google Scholar
  35. 35.
    Podstrigach Y.S., Burak Y.I.: Some peculiarity in development models in solid mechanics including electronic processes. Dokl. Ukr. Acad. Sci. 12, 18–31 (1970) (in Russ.)Google Scholar
  36. 36.
    Podstrigach Y.S., Burak Y.I., Kondrat V.F.: Magnetothermoelasticity in Conducting Solids. Naukv Dumka, Kiev (1982) (in Russ.)Google Scholar
  37. 37.
    Sedov L.L.: A Course in Continuum Mechanics, vol. 2: Physical Foundations and Formulations of Problems. Wolters-Noordhoff Publishing, Groningen (1972)zbMATHGoogle Scholar
  38. 38.
    Simo J., Hughes T.: Computational Inelasticity. Springer, New York, Berlin, Heidelberg (1998)zbMATHGoogle Scholar
  39. 39.
    Truesdell C., Toupin R.: The classical field theories. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/1, pp. 226–793. Springer, Berlin (1960)Google Scholar
  40. 40.
    Turenko A.N.: Pulse magnetic fields to progressive technologies. In: Turenko, A.N., Batygin, Y.V., Gnatov, A.V. (eds.) Theory and Experiment of Thin-Walled Metal Attraction by Pulsed Magnetic Fields, KhNADU, Kharkov (2009) (in Russ.)Google Scholar
  41. 41.
    Vovk, A., Vovk, V., Sabelkin, V., Taran, V.: Mathematical modeling of impulsive forming processes using various energy sources and transmitting medium. In: ICHSF 2006, pp. 95–105. Dortmund (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Holm Altenbach
    • 1
  • Oleg Morachkovsky
    • 2
  • Konstantin Naumenko
    • 1
    Email author
  • Denis Lavinsky
    • 2
  1. 1.Institut für Mechanik, Fakultät für MaschinenbauOtto-von-Guericke-Universität MagdeburgMagdeburgGermany
  2. 2.National Technical University “Kharkiv Polytechnic Institute”KharkivUkraine

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