Finite element formulation for implicit magnetostrictive constitutive relations
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ORIGINAL PAPER
Finite element formulation for implicit magnetostrictive constitutive relations S. Sudersan1 · U. Saravanan2 · A. Arockiarajan1 Received: 29 April 2020 / Accepted: 21 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Magnetostrictive materials that couple mechanical and magnetic domains have been widely explored for use in sensors and actuators. These materials often exhibit a nonlinear material response under applied magnetic fields, which limits the use of linear constitutive models. Furthermore, the nonlinear constitutive relations tend to be implicit in nature. Hence, a finite element scheme that can handle the implicit relationship between the mechanical (stresses and strains) and magnetic (magnetic flux density and magnetic field) quantities is proposed in order to arrive at solutions to boundary value problems. In the proposed scheme, while the physical requirements of equilibrium and strain–displacement relation are satisfied point-wise, the constitutive relations hold in a weak integral sense. A fully coupled magnetostrictive plane stress rectangular element is developed based on the proposed scheme and its efficacy in arriving at solutions to coupled field boundary value problems is illustrated by subjecting the element to standard loading conditions. Keywords Finite element method · Magnetostrictive materials · Implicit constitutive relations · Material nonlinearity · Multiphysics problems
1 Introduction Magnetostrictive materials are multiferroic materials that exhibit a coupling between ferroelastic and ferromagnetic order parameters. This results in strains developed under an applied magnetic field and magnetic fields developed under applied stresses [1]. This has fostered a plethora of research works characterizing their sensor and actuator characteristics for use in numerous applications like vibration control, energy harvesting, force measurements, etc [2,3]. These materials, however, exhibit a nonlinear response when Saravanan thanks the financial support received from Science and Engineering Research Board under Grant MTR/2017/000350.
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A. Arockiarajan [email protected] S. Sudersan [email protected] U. Saravanan [email protected]
1
Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India
2
Department of Civil Engineernig, Indian Institute of Technology Madras, Chennai 600036, India
the applied external magnetic field or stress is beyond the coercive limit. Hence it is necessary to use appropriate nonlinear constitutive relations that govern the coupled behavior of these materials. Furthermore, these materials show negligible hysteresis and hence, it is safe to assume the response is similar to that of a nonlinear elastic material. A number of constitutive relations have been proposed for the magnetostrictive behavior on the basis of experimental results obtained by Moffett et al. [4]. Some of the early phenomenological models describing the stress–strain behavior and the magnetization-field beh
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