Damage-based fracture with electro-magnetic coupling
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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.
KeywordsElectro-magnetism Maxwell’s equations Elasticity Piezo-electricity Mixed finite element methods
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- 2.Maugin GA (1988) Continuum mechanics of electromagnetic solids. Applied Mathematics and Mechanics, vol 33. North-Holland, AmsterdamGoogle Scholar
- 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
- 9.Bathe K-J (1996) Finite element procedures. Prentice-Hall, Englewood CliffsGoogle Scholar
- 10.Ogden RW (1997) Nonlinear elastic deformations. Dover Publications, Mineola, NYGoogle Scholar
- 11.Marsden JE, Hughes TJR (1994) Mathematical foundations of elasticity. Dover Publications, New YorkGoogle Scholar
- 12.Haus HA, Melcher JR (1989) Electromagnetic fields and energy. Prentice-Hall, Englewood CliffsGoogle Scholar
- 13.Jackson JD (1999) Classical electrodynamics, 3rd edn. Wiley, New YorkGoogle Scholar
- 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
- 28.Wolfram Research Inc. (2008) Mathematica, Version 7.0, Champaign, ILGoogle Scholar
- 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.Areias P. Simplas. https://ssm7.ae.uiuc.edu:80/simplas. Accessed 15 June 2012