Universal relations in coupled electro-magneto-elasticity
Introduction
In general, an incompressible isotropic electro-magneto-elastic (EME) material is an important class of the man-made material. In line with that, the material is referred to as EME or smart material whose mechanical response is coupled with an electromagnetic effect (Bardzokas, Filshtinsky, Filshtinsky, 2007, Alblas, 1974, Dai, Fu, Yang, 2007). Electrostriction and magnetostriction are the fundamental examples of electro-magneto-mechanical phenomenon shown by smart materials with an application of electromagnetic field (Shkel, Klingenberg, 1996, Pelrine, Kornbluh, Joseph, 1998, Lee, 1955, Callen, 1968). In current scenario, EME coupling effect got the current industrial attention due to inherent applications like electronic packaging, medical ultrasonic imaging, sensors, and actuators (Aboudi, 2001, Van den Boomgaard, Terrell, Born, Giller, 1974, Li, Dunn, 1998) etc. These applications recently demand the development of the constitutive relationships to study their EME deformations, effectively. The constitutive relationships of smart materials may be directly used to obtain the general solutions of EME coupled problems.
In particular, the coupling behavior of smart materials in presence of an electromagnetic field is a challenging task to describe accurately, because their elastic properties quickly get affected with external fields (Bustamante, Dorfmann, Ogden, 2009, Carlson, Jolly, 2000). But, this can be studied through the development of the coupling type of universal relations. In general, constitutive modeling provides the fundamental steps to model the exact EME coupled behaviors of smart materials up to a certain level. These fundamentals steps to model the EME coupled behaviors of smart materials may be seen in Fig. 1. These steps are based on the fundamental laws of physics alongside with thermodynamics. Further, the constitutive modeling provides important relationships known as constitutive relations through which we may formulate the connections between the stress tensor to the deformation as well as an applied electromagnetic field. The constitutive modeling specifically tries to answer the question that in what manner the material will behave with an applied stimulus. In addition, some mathematical equations that hold for every material in a specified class are known as the universal relations. And, these relations also correlate the components of total Cauchy stress tensor with the deformation and applied electromagnetic field variables (Beatty, 1987, Saccomandi, 2001, Saccomandi) similar to the constitutive relations. A key objective of the universal relations is to help the material experimentalist to specify the material in a particular constitutive class.
In the literature, Van den Boomgaard et al. (1974) proposed a new class of material behavior known as electro-magneto-elastic coupling effect that may be observed with the combination of piezo-electric and piezo-magnetic composites. Then, the micro-structural properties and the relationships between them for these class of smart materials, are studied by different authors (Li, Dunn, 1998, Bichurin, Filippov, Petrov, Laletsin, Paddubnaya, Srinivasan, 2003, Petrov, Srinivasan, Laletsin, Bichurin, Tuskov, Paddubnaya, 2007). Next, the additional coupled properties with the introduction of discontinuous reinforcement in these materials, are addressed by Bracke and Van Vliet (1981), Nan (1994), Benveniste (1995) and Tong et al. (2008). Further, the first standardised approach to develop the constitutive model for smart materials were presented by Hayes and Knops (1966), and most recently by Dorfmann and Ogden (2005) and Bustamante et al. (2006). Specifically, very few number of short-notes (Beatty, 1987, Rivlin, 1947, Beatty, 1987, Bustamante, Ogden, 2006, Dorfmann, Ogden, Saccomandi, 2004) on the EME coupled universal relations are available in the literature for incompressible isotropic EME materials starting from the original incompressible isotropic elastic deformation work by Rivlin (1947), Beatty, 1987, Beatty, 1987 and isotropic electro-elastic and magneto-elastic deformation works by Bustamante and Ogden (2006) and Dorfmann et al. (2004).
From the literature, we may conclude that limited research have been performed to model the coupled behavior of EME materials. However, most of them lack of simplicity in their formulation and a combined analysis that considers the general electro-magneto-elastic coupling effect lacks in the literature. Therefore, the research area of EME coupled constitutive modeling is broad and still open to researchers for the accurate prediction of the electro-magneto-mechanical coupling behavior of smart materials.
The primary aim of the current study is to develop the universal relations in coupled electro-magneto-elasticity. In line with that, we first introduce a new inequality on non-coaxiality of tensors T and b parallel to an equation on coaxiality one existing in the literature. Wherein, T and b denote the total Cauchy stress tensor and left Cauchy green deformation tensor for the considered electro-magneto-elastic deformation of continua, respectively. As a first step for the same, we adopt a unification of electro-magnetic theory and continuum mechanics in line with Kumar and Sarangi (2019), Jordan and Eringen (1964), Bustamante and Ogden (2006) and Dorfmann et al. (2004). In addition, a new class of the possible deformation families are also proposed through the formulated coupled universal relations. These deformation families provide the direct connections between the physical coupling components of electro-elastic and magneto-elastic coupling parameters of smart materials. And, these deformation families may also be counted as an additional class of controllable deformation families proposed by Beatty, 1987, Beatty, 1987 under external electromagnetic field in every homogeneous incompressible isotropic EME material.
Section snippets
Field equations
In current section, we summarize the electro-magneto-elastic deformation-based physical equations alongside with the principles of thermodynamics for smart materials (Eringen and Maugin, 2012).
Constitutive theories
In current section, an EME deformation of a continua is formulated followed by the fundamental laws of thermodynamics. Additionally, a new concept of an amended form of energy density function (Kumar and Sarangi, 2019) in electro-magneto-elasticity parallel to Dorfmann and Ogden (2005) in electro-elasticity is adopted, which profitably undertakes the physical insight of the Maxwell stress tensor under large deformations.
In general, the constitutive relationships for an isotropic EME material
Universal relations
In current section, we develop the EME coupling type of universal relations through a new inequality for a class of an EME material parallel to an equation for a class of an elastic material existing in the literature.
In general, a class of universal relations for isotropic electro-magneto-elastic coupling are fully depend on the total Cauchy stress tensor T and the left Cauchy Green tensor b. The symmetry and skew-symmetry of the total Cauchy stress tensor decide the different
Electro-magneto-elastic deformation families
In current section, we propose the possible electro-magneto-elastic deformation families based on the developed coupled universal relations (18).
In order to define the deformation families for an electro-elastic deformation of continua, we consider a Beatty (1987a) type deformation given as followsand
Now, the coupled universal relation (18) for an above Beatty (1987a) type deformation may be rewritten in a
Application to a magnetostriction phenomenon
In current section, we apply the developed coupled universal relations (18) to determine the possible electro-elastic and magneto-elastic coupled forces generated on the system of an incompressible isotropic EME material. Additionally, we also checked the possible deformation family for the same.
For that, we may consider a magnetostriction phenomenon, which was experimentally analysed by López-López et al. (2010). In that experiment, a circular plate-plate geometry of the rheometer shown in
Concluding remarks
In the present paper, the modeling of coupled electro-magneto-elastic (EME) behavior of smart materials is formulated through the finite deformation of EME continua. A new concept of amended energy density function (Kumar, Sarangi, 2019, Dorfmann, Ogden, 2005) that depends on nine invariants (12) is adopted to obtain the total Cauchy stress tensor, electric displacement vector and the magnetic induction vector for an EME material. The developed coupled universal relations ((18), (22)) and their
Conflict of Competing Interest
The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript
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