Magnetoelastic deformation of conductive semilinear hyperelastic solids

. This work investigates the radial deformation of conductive magneto-hyperelastic solid cylinder subject to azimuthal magnetic field. It shows effect of the current density on the radial deformation of the solid. A simple magnetoelastic energy function is proposed for the cylinder under consideration such that its purely elastic part corresponds to the strain energy of the well-known semilienar hyperelastic materials. The consequent magnetoelasticity field equations, in conjunction with the accompanying boundary conditions, are specialized for application to the problem of radial deformation of solid cylinder. The obtained magnetoelastic constitutive model shows that the stress distribution in the solids is sensitive to the magnetic induction while the associated magnetic field at point within the cylinder is deformation-dependent. Furthermore, it is revealed that the azimuthal magnetic induction produced by steady current within the solid cylinder increases along its radius. Finally, and among other things, the graphical illustration shows that the effect of steady axial current density on the magnitude of the displacement function at points within the cylinder is significantly pronounced.


Introduction
Conductive magnetoactive elastomers (CMEs) are magneto-sensitive materials that conduct electricity.They are manufactured by mixing micron/nano -size magnetic and conductive particles into nonmagnetic rubber-like matrices.CMEs exhibit change in mechanical response when subject to applied magnetic field and/ or current of electricity.The widespread applications of these materials have continued to instigate the needs for the development of new magnetoelasticity theories.In the fundamental formulation, magnetoelasticity field equations govern magnetomechanical interaction of solids.These equations consist of magnetostatic and elasticity fields equations, and are used to construct solutions to problems involving magnetoelastic deformation.
Finite magnetoelastic interaction of solids has long been a subject of interest since the classic studies of Maugin [1], Eringen and Maugin [2] and Pao [3].Recently, Pei et al. [4] investigated nonlinear magnetoelastic deformation of porous solids; Reddy and Saxena [5] studied instabilities in axisymmetric magnetoelastic deformation of a cylindrical membrane; Garcia-Gonzalez and Hossain [6] proposed a microstructural-based approach to model magneto-viscoelasticity materials at finite strain; Ren et al. [7] studied multi-functional soft-bodied jellyfish-like swimming; Bostola and Hossain [8] gave a review on magneto-mechanical charactirizations of magnetorheological elastomers; Dorfman and Ogden [9] studied nonlinear theory of electroelastic and magnetoelastic interactions; Nedjar [10] proposed a modelling framework for finite strain magnetoviscoelasticity; and Saxena et al. [11] developed a finite deformation theory for magneto viscoelasticity.
In view of Fadodun et al. [12], this work proposes a simple magnetoelastic energy function for conductive semilinear magnetohyperelastic solids.Using the laws of thermodynamic, Coleman-Noll procedure and tensor calculus, the study develops a magnetoelastic constitutive model for the solids under consideration.The consequent magnetoelasticity field equations together with the accompanying boundary conditions are specialized for applications to the problem of radial deformation of a conductive magnetoelastic cylinder subject to steady current of electricity.The rest of the paper is as follow: the first sections present magnetoelasticity field equations and constitutive relations while the remaining sections detail the solution to the radial deformation problem of magneto-sensitive solid cylinder subject to steady axial current density.

Kinematics
Consider a stress-free conductive magnetohyperelastic solid occupying the reference configuration Ω 0 ⊂ ℝ 3 with smooth boundary Ω 0 and surface outward unit normal vector  � �⃗ .When subject to magnetic field and /or mechanical surface load the body deforms onto deformed configuration Ω with boundary Ω and surface outward unit normal vector  �⃗.The deformation of the body is defined by vector function  �⃗ such that  ⃗ =  �⃗( ⃗ ) where  ⃗ denotes position vector of material points in Ω 0 and  ⃗ represents position vector of the corresponding material points in Ω.The closures Ω 0 and Ω in Eq. ( 1) are defined by The deformation gradient  is defined by where  is the gradient operator with respect to Ω 0 .At an arbitrary point  ⃗ , the determinant det() > 0 measures the local volume change.
Applying the polar decomposition theorem, the deformation gradient  is decomposed into product of second-rank tensors   and U where   is the orthogonal rotation tensor and  is the right stretch symmetric tensor.The tensors  and   are obtained by the relations where   is the transpose of ,  −1 is the inverse of  and  =    is the right Cauchy-Green deformation tensor [12].

Eulerian Form/Description: Magneto-Mechanical Field Equations
Let  � �⃗ ,  �⃗   ��⃗ denote the Eulerian forms of the magnetic field, magnetic induction and effective magnetization vectors respectively.For a purely magnetostatic field produced by steady current, and in the absence of electric interaction and surface current the Maxwell's field equations read where the operators    are defined in the deformed configuration Ω and  ⃗ is the current density in Eulerian form.The current density  ⃗ satisfies the equation In magnetic materials, the vectors  � �⃗ ,  �⃗   ��⃗ are related by the constitutive law where  is the magnetic permeability of the material and  0 is the magnetic permeability of free space.
In free space exterior to the body, the corresponding magnetostatic fields are denoted by vectors  � �⃗ * and  �⃗ * , which are governed by the equations At the bounding surface of the considered material in the deformed configuration Ω, the standard boundary conditions associated with Eq. ( 5) are where  �⃗ is the unit outward normal vector on Ω.
Let  denote the total stress tensor which incorporates magnetostatic body forces.In the absence of mechanical body forces the mechanical equilibrium equation reads The standard boundary condition accompanying the equilibrium equation is where  ⃗  is the mechanical traction on Ω per unit area,  ⃗  =  *  �⃗ is the load due to the Maxwell stress and  is the second-order unit tensor in Ω [9,12].
The magnetostatic field equations in Lagrangian forms read where the operators    are defined in the reference configuration Ω 0 , and  ⃗  = det()  −1  ⃗ is the Lagrangian current density satisfying the equation Similarly, the vectors Let  � �⃗  ,  �⃗    ��⃗  are related by In addition, the vectors  � �⃗    �⃗  satisfy the standard boundary conditions Let  denote the total first Piola-Kirchhoff's stress tensor.The total stress tensor  and first Piola-Kirchhoff's stress tensor  are related by where  − is the inverse of   .In term of  the equilibrium equation assumes the equivalent form The corresponding boundary condition reads where  ⃗  is the mechanical traction on Ω 0 per unit area,  ⃗  =  *  � �⃗ and  * =  = det()  *  − is the pull back version of the Maxwell stress  * [12].

Magnetoelastic Energy Function and Constitutive Model
In order to complete the mathematical equations formulation for the study, we choose the deformation gradient  and magnetic induction vector  �⃗  as independent variables; and model magnetoelastic constitutive laws that give first Piola-Kirchhoff stress tensor  and magnetic field vector  � �⃗  in terms of  and  �⃗  .Consequently, we take the magnetoelastic Helmholtz free energy function Φ = Φ(,  �⃗  ) to depend on  and  �⃗  , and ensure the objectivity condition is satisfied for all proper orthogonal second-rank tensor .
Using the laws of thermodynamics and Coleman-Noll procedure, the first Piola-Kirchhoff stress tensor  and the Lagrangian magnetic field vector  � �⃗  are obtained through the relations for an isotropic semilinear hyperelastic solid, where  1 ( −  0 ) is the first invariant of the secondrank tensor ( −  0 ),  2 ( −  0 ) =  1 ( −  0 ) 2 ,   ,   are the Lame's constants and  0 is the second-rank unit tensor in the reference configuration [12] Following Fadodun el al. [12], Melnikov and Ogden [13] and Dorfmann and Ogden [14], we generalize and consider a simple energy function of the form for the magnetoelastomeric solid under consideration such that its purely elastic part corresponds to the semilinear hyperelastic energy function in Eq. ( 22), where the scalar µ is the permeability of the solid.
The Frechet derivatives of invariants  1 ( −) 2 and  1 2 ( −  0 ) with respect to  are [12] and as the magnetoelastic constitutive model for the magnetoelastic solids under consideration, where ⨂ denotes the tensor product.
In view of Eq. ( 18), the corresponding Eulerian total stress tensor  is

Remark 1:
The obtained magnetoelastic constitutive model in Eqs. ( 27) and (28) shows that the stress distribution is sensitive to the magnetic induction generated while the magnetic field at point within the body is deformation-dependent.

Application: Magnetoelastic Deformation of Conductive Hyperelastic Cylinder
In view of the constitutive Eqs. ( 27) and ( 28), it is convenient to solve the problem of the radial deformation of conductive semilinear magneto-hyperelastic solid cylinder in the Lagrangian frame of reference.The theory of magnetoelastcity presented in the previous sections is now specialized for application to the problem of magnetoelastic deformation of a solid cylinder.The cylinder under consideration has radius  and is subject to uniform axial current density.The geometry of the cylinder is assumed to be sufficiently long/thin such that the edge effect is neglected.Let the cylindrical coordinates (, Θ, ) with associated unit basis vectors  �⃗  ,  �⃗ Θ ,  �⃗  describe the position vector  �⃗ =  �⃗  +  �⃗  of material point of the cylinder in the reference configuration defined by where  and  are the radius and length of the cylinder respectively.Invoking the constraint of circular symmetry, and let the cylindrical coordinates (, , ) with unit basis vectors  ⃗  ,  ⃗  ,  ⃗  give the position vector  ⃗ =  ⃗  +  ⃗  of the corresponding material point in the deformed configuration, the deformation of the cylinder is defined by where () is a function of  only and   is the uniform axial stretch.Using Eqs. ( 2) and (30), the deformation gradient  is where  ⃗  ,  ⃗  ,  ⃗  and  �⃗  ,  �⃗ Θ ,  �⃗  are the orthonormal basis vectors in Ω and Ω 0 respectively.Using Eqs.(3), ( 4) and (31) gives where for any second-rank tensor , ()  denotes the components of .

Solution of magnetostatic field equations
Let the axis of the solid cylinder of radius  and constant conductivity  be taken along the  axis and let  ⃗  =    �⃗  be the uniform axial current density along the axis of the cylinder, where   is the magnitude of  ⃗  and  �⃗  is the unit vector along the axis of the cylinder.For this problem, Eq. ( 15) is satisfied for uniform  ⃗  .The solution of Eq. 14(b) is obtained by introducing a uniquely defined vector (magnetic vector potential)  ⃗  such that Using Eq. 33(b) and  �⃗  = µ � �⃗  in Eq. 14(a) gives The form of Eq. ( 34) suggests that  ⃗  = (, Θ, ) �⃗  where (, Θ, ) is a function of cylindrical coordinates , Θ  .Meanwhile Eq. 33(b) shows that G is independent of  and by symmetry,  is independent of Θ, thus,  ⃗  = () �⃗  is a function of  only [15].
The solution of Eq. (36) yields where   ,  = 1,2,3,4 are constants to be determined.Since () must be finite along the axis of the tube ( = 0),  1 = 0. Thus, The constant  3 is obtained by using the Maxwell's first circuital relation where  is the current flowing in the tube.Hence, Substituting Eq. (40) into Eq.(38) gives Recall that the magnetic vector potential is continuous at the surface of seperation, thus, and setting  4 = 0 (without loss of generality) gives Substituting Eq. (43) into Eq.( 41) gives the solution

Conclusion
The study develops a new magnetoelastic constitutive theory for modelling magneto-mechanical interaction of solids.The theory is specialized for application to the problem of radial deformation of a solid circular cylindrical made of conductive semilinear magnetohyperelastic materials.It is obtained that the stress propagation in the solid cylinder is sensitive to the magnetic induction produced by uniform axial current density while the associated magnetic field is deformationdependent. Furthermore, it is shown that the effect of uniform axial current density on the deformation of the tube is significantly pronounced.Finally, the results in this study find applications in design of soft actuators, sensors and energy harvesters to mention a few.

)= 1 µFig. 1 :
Fig. 1: This plot shows that the effect of steady current density on the magnitude of displacement function at point within solid cylinder is significantly pronounced.

Fig. 2 :
Fig. 2: This plot shows that the azimuthal magnetic induction produced by the steady current within the solid cylinder increases linearly along the radius of the cylinder.