Journal of the Mechanics and Physics of Solids 56 (2008) 1147–1169 Magnetoelastic buckling of a rectangular block in plane strain S.V. Kankanala à , N. Triantafyllidis Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109-2140, USA Received 13 June 2007; received in revised form 15 October 2007; accepted 21 October 2007 Abstract Of interest here is the stability of a rectangular block subjected to a uniform magnetic field perpendicular to its longitudinal axis. The two ends of the block are frictionless and kept parallel to each other. This boundary value problem is motivated by the classical problem of magnetoelastic buckling in which a cantilever beam subjected to a transverse magnetic field buckles when the applied field reaches a critical value. This work presents a finite strain continuum mechanics formulation of the stability problem of a homogeneous, compressible, magnetoelastic rectangular block in plane strain subjected to a uniform transverse magnetic field. The applied variational approach employs an unconstrained energy minimization recently proposed by the authors. The analytical solution for the critical buckling fields for both the antisymmetric and symmetric modes are obtained for three different constitutive laws. The corresponding result for thin beams is extracted asymptotically for a special material and the solution is compared to previously published results. The critical magnetic field is shown to increase monotonically with the block’s aspect ratio for each material and mode type. Antisymmetric modes are always the critical buckling modes for stress saturated and neo-Hookean materials, except for a narrow range of moderate aspect ratios (about 0:25) where symmetric modes become critical. For strain-saturated solids no buckling is possible above a maximum aspect ratio. r 2007 Elsevier Ltd. All rights reserved. Keywords: Electromechanical processes; Finite strain; Particulate reinforced material; Energy methods; Stability 1. Introduction and motivation Magnetoelastic solids exhibit coupling between their mechanical and magnetic responses. Their study in the context of continuum mechanics goes back a few decades to Truesdell and Toupin (1960), Tiersten (1964), Brown (1966), and Maugin and Eringen (1972). Due to novel technological applications, such as magnetoelastic elastomers, there has been a renewed interest in these materials (e.g. DeSimone and James, 2002; Dorfmann and Ogden, 2003; Kankanala and Triantafyllidis, 2004; and Ericksen, 2006). The solution of basic nontrivial boundary value problems is the obvious next step in further examining the nature of the underlying coupling between magnetic and elastic effects. As such, attention is here focused on the classical magnetoelastic buckling problem in which a bar in a transverse magnetic field buckles when the ARTICLE IN PRESS www.elsevier.com/locate/jmps 0022-5096/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2007.10.008 à Corresponding author. Research and Innovation Center, 2101 Village Rd., MD-RIC 2115, Ford Motor Company, Dearborn, MI 48121, USA. Tel.: +1 313 594 0691; fax: +1 313 248 9051. E-mail address: skankana@ford.com (S.V. Kankanala).