Computational Mechanics

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

Damage-based fracture with electro-magnetic coupling

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


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.


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


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • P. Areias
    • 1
    • 2
    Email author
  • 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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