Universal relations in nonlinear electro-magneto-elasticity
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O R I G I NA L
Deepak Kumar · Somnath Sarangi · Ranjan Bhattacharyya
Universal relations in nonlinear electro-magneto-elasticity
Received: 25 July 2019 / Accepted: 28 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The purpose of this article is to develop a class of universal relations for an incompressible isotropic electro-magneto-elastic (hereafter EME) material in order to generalize the continuum concept to electromagneto-elasticity. In line with that, we adopt an electro-magneto-elasticity theory following the second law of thermodynamics-based approach. More precisely, we first extend a thermodynamically consistent deformation of a continua to a coupled EME interaction through a new amended energy function (hereafter AEF). This AEF succeeds the physical insight of the Maxwell stress tensor (hereafter MST) under large deformations. Next, we introduce a new inequality Tb−bT = 0 for a class of an EME material parallel to an equation Tb−bT = 0 for a class of an elastic material existing in the literature. At last, the formulated universal relations are applied to some homogeneous and non-homogeneous deformations to exemplify the consequences of an electromagnetic field on the mechanical deformation. Additionally, the validity of the proposed universal relations in electromagneto-elasticity is also checked by obtaining an existing universal relation in nonlinear elasticity in the absence of an applied electromagnetic field. Keywords Electro-magneto-elasticity · Electro-magneto-elastic materials · Universal relations
1 Introduction A material, whose elastic property and elastic state vary with an electromagnetic field application, is known as an EME material. In addition, an EME material also recognized as a smart material, which may establish the juncture between electric, magnetic and mechanical fields [1–4]. By stuffing the electro-magneto-active particles into rubber matrix of soft elastomers, a large deformations with punctual response may be capture in the company of an electromagnetic field. The large deformation and rapid response make EME materials specific to the vibration, noise-suppression, and other engineering and medical applications. Recently, increasing applications of smart materials demand an advancement in the constitutive relationships, which may be utilized in solving the different boundary-value problems with different types of commercial finite element software. In general, the behaviour of smart materials is quite complex with the application of external fields [5–7]. However, these complex behaviours of smart materials may be easily described with the universal constitutive D. Kumar (B) Department of Applied Mechanics, Indian Institute of Technology, Delhi 110016, India E-mail: [email protected] D. Kumar · S. Sarangi Department of Mechanical Engineering, Indian Institute of Technology, Patna 801103, India R. Bhattacharyya Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India
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