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Computational Mechanics

, Volume 51, Issue 5, pp 629–640 | Cite as

Damage-based fracture with electro-magnetic coupling

  • P. Areias
  • H. G. Silva
  • N. Van Goethem
  • M. Bezzeghoud
Original Paper

Abstract

A coupled elastic and electro-magnetic analysis is proposed including finite displacements and damage-based fracture. Piezo-electric terms are considered and resulting partial differential equations include a non-classical wave equation due to the specific constitutive law. The resulting wave equation is constrained and, in contrast with the traditional solutions of the decoupled classical electro-magnetic wave equations, the constraint is directly included in the analysis. The absence of free current density allows the expression of the magnetic field rate as a function of the electric field and therefore, under specific circumstances, removal of the corresponding magnetic degrees-of-freedom. A Lagrange multiplier field is introduced to exactly enforce the divergence constraint, forming a three-field variational formulation (required to include the wave constraint). No vector-potential is required or mentioned, eliminating the need for gauges. The classical boundary conditions of electromagnetism are specialized and a boundary condition involving the electric field is obtained. The spatial discretization makes use of mixed bubble-based (of the MINI type) finite elements with displacement, electric field and Lagrange multiplier degrees-of-freedom. Three verification examples are presented with very good qualitative conclusions and mesh-independence.

Keywords

Electro-magnetism Maxwell’s equations Elasticity Piezo-electricity Mixed finite element methods 

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References

  1. 1.
    Lax M, Nelson DF (1976) Maxwell equations in material form. Phys Rev B 13(4): 1777–1784MathSciNetCrossRefGoogle Scholar
  2. 2.
    Maugin GA (1988) Continuum mechanics of electromagnetic solids. Applied Mathematics and Mechanics, vol 33. North-Holland, AmsterdamGoogle Scholar
  3. 3.
    Ericksen JL (2007) On formulating and assessing continuum theories of electromagnetic fields in elastic materials. J Elast 87: 95–108MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ericksen JL (2007) Theory of elastic dielectrics revisited. Arch Ration Mech Anal 183: 299–313MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dorfmann A, Ogden RW (2006) Nonlinear electroelastic deformations. J Elast 82(2): 99–127MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Belahcen A, Fonteyn K (2008) On numerical modeling of coupled magnetoelastic problem. In: Kvamsdal T, Mathisen KM, Pettersen B (eds) 21st nordic seminar on computational mechanics. NSCM, Barcelona, CIMNEGoogle Scholar
  7. 7.
    Kuna M (2010) Fracture mechanics of piezoelectric materials—where are we right now. Eng Fract Mech 77: 309–326CrossRefGoogle Scholar
  8. 8.
    Kuna M (2006) Finite element analyses of cracks in piezoelectric structures: a survey. Arch Appl Mech 76: 725–745zbMATHCrossRefGoogle Scholar
  9. 9.
    Bathe K-J (1996) Finite element procedures. Prentice-Hall, Englewood CliffsGoogle Scholar
  10. 10.
    Ogden RW (1997) Nonlinear elastic deformations. Dover Publications, Mineola, NYGoogle Scholar
  11. 11.
    Marsden JE, Hughes TJR (1994) Mathematical foundations of elasticity. Dover Publications, New YorkGoogle Scholar
  12. 12.
    Haus HA, Melcher JR (1989) Electromagnetic fields and energy. Prentice-Hall, Englewood CliffsGoogle Scholar
  13. 13.
    Jackson JD (1999) Classical electrodynamics, 3rd edn. Wiley, New YorkGoogle Scholar
  14. 14.
    Mota A, Zimmerman JA (2011) A variational, finite-deformation constitutive model for piezoelectric materials. Int J Numer Methods Eng 85: 752–767MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Bustamante R, Ogden RW (2006) Universal relations for nonlinear electroelastic solids. Acta Mech 182: 125–140zbMATHCrossRefGoogle Scholar
  16. 16.
    Bustamante R, Dorfmann A, Ogden RW (2006) Universal relations in isotropic nonlinear magnetoelasticity. Q J Mech Appl Math 59(3): 435–450MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Linder C, Rosato D, Miehe C (2011) New finite elements with embedded strong discontinuities for the modeling of failure in electromechanical coupled solids. Comp Method Appl Mech Eng 200: 141–161MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bustamante R, Dorfmann A, Ogden RW (2011) Numerical solution of finite geometry boundary-value problems in nonlinear magnetoelasticity. Int J Solids Struct 48: 874–883zbMATHCrossRefGoogle Scholar
  19. 19.
    Bonet J, Wood RD (2008) Nonlinear continuum mechanics for finite element analysis, Second edition. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  20. 20.
    Vu DK, Steinmann P, Possart G (2007) Numerical modelling of non-linear electroelasticity. Int J Numer Methods Eng 70: 685–704MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lemaitre J, Chaboche J-L (1990) Mechanics of solid materials. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  22. 22.
    Lemaitre J (1996) A course on damage mechanics, Second edition. Springer, BerlinCrossRefGoogle Scholar
  23. 23.
    Mullins L (1969) Softening of rubber by deformation. Rubber Chem Technol 42: 339–362CrossRefGoogle Scholar
  24. 24.
    Oliver J (1989) A consistent characteristic length for smeared cracking models. Int J Numer Methods Eng 28: 461–474zbMATHCrossRefGoogle Scholar
  25. 25.
    Truesdell C, Noll W (2004) The non-linear field theories of mechanics, Third edition. Springer, BerlinCrossRefGoogle Scholar
  26. 26.
    Hughes TJR (2000) The finite element method. Linear static and dynamic finite element analysis. Dover Publications, New York (reprint of Prentice-Hall edition, 1987)Google Scholar
  27. 27.
    Korelc J (2002) Multi-language and multi-environment generation of nonlinear finite element codes. Eng Comput 18(4): 312–327CrossRefGoogle Scholar
  28. 28.
    Wolfram Research Inc. (2008) Mathematica, Version 7.0, Champaign, ILGoogle Scholar
  29. 29.
    Arnold DN, Brezzi F, Fortin M (1984) A stable finite element for the Stokes equations. Calcolo XXI(IV): 337–344MathSciNetCrossRefGoogle Scholar
  30. 30.
    Areias P, Dias-da-Costa D, Alfaiate J, Júlio E (2009) Arbitrary bi-dimensional finite strain cohesive crack propagation. Comput Mech 45(1): 61–75MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Areias P, Van Goethem N, Pires EB (2011) A damage model for ductile crack initiation and propagation. Comput Mech 47(6): 641–656zbMATHCrossRefGoogle Scholar
  32. 32.
    Areias P, Van Goethem N, Pires EB (2011) Constrained ale-based discrete fracture in shells with quasi-brittle and ductile materials. In: CFRAC 2011 international conference, Barcelona, Spain, June 2011. CIMNEGoogle Scholar
  33. 33.
    Areias P. Simplas. https://ssm7.ae.uiuc.edu:80/simplas. Accessed 15 June 2012

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • P. Areias
    • 1
    • 2
  • H. G. Silva
    • 1
    • 3
  • N. Van Goethem
    • 4
  • M. Bezzeghoud
    • 1
    • 3
  1. 1.Departamento de Física, Escola de Ciências e TecnologiaUniversidade de Évora, Colégio Luís António VerneyÉvoraPortugal
  2. 2.ICISTLisbonPortugal
  3. 3.CGEÉvoraPortugal
  4. 4.Departamento de Matemática, Centro de Matemática e Aplicacações FundamentaisCMAF, Universidade de Lisboa, Faculdade de CiênciasLisbonPortugal

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