Some problems in nonlinear magnetoelasticity

Z. angew. Math. Phys. 56 (2005) 718–745 0044-2275/05/040718-28 DOI 10.1007/s00033-004-4066-z c 2005 Birkhäuser Verlag, Basel
Zeitschrift für angewandte Mathematik und Physik ZAMP Some problems in nonlinear magnetoelasticity A. Dorfmann and R. W. Ogden∗ Abstract. In this paper we examine the influence of magnetic fields on the static response of magnetoelastic materials, such as magneto-sensitive elastomers, that are capable of large deformations. The analysis is based on a simple formulation of the mechanical equilibrium equations and constitutive law for such materials developed recently by the authors, coupled with the governing magnetic field equations. The equations are applied in the solution of some simple representative and illustrative problems, with the focus on incompressible materials. First, we consider the pure homogeneous deformation of a slab of material in the presence of a magnetic field normal to its faces. This is followed by a review of the problem of simple shear of the slab in the presence of the same magnetic field. Next we examine a problem involving non-homogeneous deformations, namely the extension and inflation of a circular cylindrical tube. In this problem the magnetic field is taken to be either axial (a uniform field) or circumferential. For each problem we give a general formulation for the case of an isotropic magnetoelastic constitutive law, and then, for illustration, specific results are derived for a prototype constitutive law. We emphasize that in general there are significant differences in the results for formulations in which the magnetic field or the magnetic induction is taken as the independent magnetic variable. This is demonstrated for one particular problem, in which restrictions are placed on the admissible class of constitutive laws if the magnetic induction is the independent variable but no restrictions if the magnetic field is the independent variable. Mathematics Subject Classification (2000). 74B20, 74F15. Keywords. Nonlinear magnetoelasticity, magnetoelastic interactions, magnetoelastic elastomers. 1. Introduction In some recent papers [1, 2, 3] we have discussed alternative formulations of the governing equilibrium equations for nonlinear magnetoelastic deformations of magneto-sensitive (MS) solids, and have applied the theory in the solution of several illustrative boundary-value problems. In the most recent of these papers [3] we have provided a rather elegant and simple formulation of the equations based on the use of a modified free-energy function with the referential magnetic induction vector as the independent magnetic variable. We also provided an alternative formulation with the magnetic field itself as the independent variable. ∗ Corresponding author, e-mail: [email protected] Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 719 The differences arising from use of one or other of these formulations are quite significant. As indicated in [3], for certain problems involving non-homogeneous deformations, restrictions are placed on the class of constitutive laws admitting particular magnetic (or magnetic induction) fields. These restrictions are generally more severe when the magnetic induction is used as the independent magnetic variable. A case in point is highlighted in the present paper for the problem of extension and inflation of a tube with an azimuthal magnetic field. The motivation for this work, which is discussed in detail in the papers cited above and not therefore repeated here, lies in the increasing use of these materials, in particular MS-elastomers, as ‘smart materials’ in electromechanical devices. Other recent related theoretical works are those by Brigadnov and Dorfmann [4], Steigmann [5] and Kankanala and Triantafyllidis [6], while the theoretical foundations for a continuum deforming in the presence of an electromagnetic field have been provided by, for example, Truesdell and Toupin [7]. Formulations of the magnetoelastic equations, based in part on the work by Brown [8], Hutter [9, 10] and Pao [11], are summarized in the monographs by Hutter [12] and Eringen and Maugin [13], for example; see also the more recent book by Kovetz [14]. In the present paper we continue the development in [3] by applying the equations to some simple deformations in order to illustrate the influence of the magnetic field on the mechanical response. In Section 2 we summarize the basic equations in general form following the formulation given in [3], together with the appropriate constitutive law for a compressible or an incompressible material. Equations based on either the magnetic induction vector or the magnetic field vector as the independent magnetic variable are included. Problems involving homogeneous deformations are considered in Section 3. First, pure homogeneous deformation of a slab of material with a magnetic field normal to its faces is discussed and this is followed by an analysis of the problem of simple shear of a slab with a magnetic field initially normal to the direction of shear and in the plane of shear. A variant of the latter problem was considered in [4] for a particular form of constitutive law but here we revisit this problem in the general setting of the constitutive framework discussed in Section 2. In Section 4 we consider the problem of inflation and extension of a circular cylindrical tube under internal pressure and axial load. It is appropriate in this problem to consider a (uniform) axial magnetic field or a circumferential magnetic field. In the latter case, with circular symmetry preserved, the field component is inversely proportional to the radial coordinate so that a singularity arises along the tube axis unless this is excluded from the domain of interest by introducing a cylindrical core concentric with the tube. The field is generated by a steady current along this core. The results in the latter problem highlight the differences in the two distinct (but in principle equivalent) formulations. If the magnetic field vector is taken as the independent magnetic variable then no restrictions are imposed by the field equations on the form of constitutive law for the considered combination of deformation and magnetic field to be admissible. By contrast, when the magnetic 720 A. Dorfmann and R. W. Ogden ZAMP induction vector is used as the independent variable the problem admits a solution only for a very restricted class of constitutive laws. In these problems we restrict attention to incompressible isotropic magnetoelastic materials, and each problem is formulated without restriction otherwise on the form of the constitutive law, but with the proviso that the restriction mentioned in the above paragraph arises naturally in one particular problem. In each case the results are illustrated by use of a prototype material model. At present there is very little experimental data available for the considered materials, in particular data that (a) show in detail how the mechanical response in different tests changes with the magnetic field and/or (b) characterize the magnetization of the material at different strain levels. Part of the objective of considering the simple problems discussed here is therefore to provide a theoretical basis for the analysis of experimental data and to encourage the generation of experimental results for comparison with the theory. 2. Basic equations 2.1. Kinematics We consider a magnetoelastic solid that is subject to a magnetic field and occupies a region B0 in a three-dimensional Euclidean space. Let the magnetic field be denoted H0 , the associated magnetic induction vector B0 and the magnetization vector M0 . Application of the magnetic field will in general have generated an initial deformation of the body from its natural (stress-free) configuration in which it is subject to neither magnetic fields nor mechanical body forces or surface tractions. This initial deformation, achieved purely by the application of the magnetic field, is the effect referred to as magnetostriction, and in general it will be associated with a (residual) stress distribution in B0 dependent on B0 . For any given magnetic field we take the geometrical configuration B0 to be the reference configuration from which subsequent deformation generated by the application of mechanical loads is measured. The geometry of B0 can be maintained, if required, when B0 is changed if appropriate mechanical loads are applied. Let a typical point of the solid be labelled by its position vector X in the reference configuration B0 relative to an arbitrarily chosen origin. When the body is deformed the point X has a new position x = χ(X) in the resulting deformed configuration, which we denote by B. Time dependence is not considered here. The vector field χ describes the deformation of the body and is defined for X ∈ B0 ∪∂B0 , where the boundary of B0 is denoted by ∂B0 . The deformation gradient tensor F relative to B0 , and its determinant, are F = Gradχ, J = det F > 0, (1) respectively, where Grad denotes the gradient operator with respect to X and wherein the notation J is defined. Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 721 2.2. Magnetic balance equations In the absence of deformation F = I and the relevant magnetic vector fields are B0 , H0 and M0 , which are related by the standard formula B0 = µ0 (H0 + M0 ), (2) where the constant µ0 is the magnetic permeability in vacuo. In the absence of material we note that M0 = 0 and (2) reduces accordingly to B0 = µ0 H0 . In the reference configuration, the vectors B0 and H0 satisfy the forms of Maxwell’s equations specialized to the considered static situation as DivB0 = 0, CurlH0 = 0, (3) where, respectively, Div and Curl are the divergence and curl operators with respect to X. A second connection between the vector fields B0 , H0 and M0 will be provided by an appropriate form of constitutive law, which will be discussed in a general setting in Section 2.5. When the material is deformed with deformation gradient F the initial vector fields B0 , H0 and M0 (which are functions of X) change, and we denote the corresponding fields in the deformed configuration B by Bf (the magnetic induction), Hf (the magnetic field) and Mf (the magnetization), respectively. These are Eulerian vector fields and may be regarded as ‘push forward’ versions of the initial fields from B0 to B. This accounts for the presence of the subscript f , signifying ‘forward’. We have the connections Bf = J −1 FB0 , −T −1 T Hf = F−T H0 , (4) T where F = (F ) and denotes the transpose of a second-order tensor. These arise naturally through the standard identities Div B0 = JdivBf , CurlH0 = JF−1 curlHf (5) (or their integral versions), which require that χ be suitably well behaved. It follows that the counterparts of (3) hold in the deformed configuration, i.e. div Bf = 0, curlHf = 0, (6) where div and curl are the usual divergence and curl operators with respect to x. Analogously to Hf we define Mf = F−T M0 and then, in terms of the Eulerian vectors, equation (2) becomes Bf = µ0 J −1 b(Hf + Mf ), (7) where b = FFT is the left Cauchy-Green deformation tensor. We point out that (7) is somewhat different in form from (2) in that it involves the multiplier J −1 b on the right-hand side, which highlights the fact that the form of equation (2) is not preserved under general deformations. Now consider the situation in which the body is in the deformed configuration B and subject to a magnetic field H with magnetic induction B and magnetization 722 A. Dorfmann and R. W. Ogden ZAMP M. These fields are related by the counterpart of equation (2), namely B = µ0 (H + M). (8) Equation (8) defines the third vector field when one vector is used as an independent variable and the other is given by a constitutive equation. Note that for a given deformation there is in general no immediate connection between B, H, M and Bf , Hf , Mf since, for example, B and Bf (equivalently, B0 ) may be chosen independently. However, when there is no deformation (8) reduces to (2) and B, H, M may then be equated to the initial fields B0 , H0 , M0 . In this Eulerian description, the relevant Maxwell field equations to be satisfied by B and H are div B = 0, curlH = 0, (9) the counterparts of (3). Just as the initial fields may be pushed forward, the fields B and H may be pulled back from B to B0 to their Lagrangian forms, denoted Bl and Hl respectively, by the identifications Bl = JF−1 B, Hl = FT H, (10) as in [3]. Since H and M occur as a sum in (8), it is appropriate to define, similarly to (10)2 , a Lagrangian form of M, denoted Ml , by Ml = FT M. (11) On use of (10) and (11) in (8) we obtain J −1 cBl = µ0 (Hl + Ml ), (12) where c is the right Cauchy-Green deformation tensor defined by c = FT F. This is another manifestation of the lack of form invariance of the relation (8). In general, there is no immediate connection between Bl , Hl , Ml and B0 , H0 , M0 . Suppose, however, that a magnetic field is applied prior to deformation and we take B0 as the independent magnetic variable. Then, H0 and M0 are determined in terms of B0 by a constitutive law and (2) together. After deformation B0 becomes Bf , as defined by (4)1 . If we define B to be Bf , this amounts to applying the field B after deformation (without changing the deformation), and then Bl = B0 . The corresponding values of H and M are obtained by means of a constitutive law together with (8). In general they are not the same as Hf and Mf . We shall elaborate on this point in connection with the constitutive laws discussed in Section 2.5. 2.3. Mechanical balance equations Let ρ0 and ρ denote the mass densities of the material in the reference and deformed configurations, B0 and B respectively. Then, for our purposes, the conservation of Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 723 mass equation is written conveniently in the form Jρ = ρ0 , (13) where, we recall, J = det F. In the absence of mechanical body forces, the equilibrium equation for a magnetoelastic solid may be written in the Eulerian form divτ = 0, (14) where τ is the total stress tensor, which, according to the balance of angular momentum, is symmetric (for detailed discussion of different stress tensors in the present context see Steigmann [5]). We note, in particular, that the influence of the magnetic field on the deforming continuum is thereby incorporated through the stress tensor and not through body force terms (which, in any case, can be written as the divergence of a second-order tensor). With τ we may associate a total ‘nominal’ stress tensor, analogous to that used in elasticity theory [15], here denoted T and given by T = JF−1 τ . (15) Then, the equilibrium equation (14) is expressed simply in Lagrangian form as DivT = 0. (16) 2.4. Boundary conditions In order to formulate boundary-value problems we need material constitutive laws in which τ and H (or M) are given in terms of the basic variables F and B. These will be discussed in the following section. In addition, appropriate boundary conditions must be satisfied by the fields B, H, τ and χ. These are summarized here. Across a surface of discontinuity within the material or across a surface bounding the considered material in the deformed configuration the vector fields B and H satisfy the standard jump conditions n · [B] = 0, n × [H] = 0, (17) where the square brackets indicate a discontinuity across the surface and n is the unit normal to the surface, which, at the material boundary, is taken as the outward pointing normal. The Lagrangian counterparts of the equations in (17) are N · [Bl ] = 0, N × [Hl ] = 0, (18) where N is the unit normal to the surface in the reference configuration that deforms into the surface with unit normal n. Provided F is continuous equations (17) and (18) are equivalent. For the reference fields B0 and H0 equations (18) 724 A. Dorfmann and R. W. Ogden ZAMP also apply with the subscript l replaced by 0, and the push forward fields Bf and Hf satisfy (17) with the subscripts f attached. Across any surface within the material the traction calculated from the total stress tensor must be continuous. This continuity condition is, in its Eulerian and Lagrangian forms, [τ ]n = 0, [T]N = 0. (19) At an external boundary of the considered body the latter, for example, may be replaced by specification of the traction in the form TN = t, (20) where t in general includes both magnetic and mechanical contributions. The mechanical part is a prescribed function of position that may depend on the deformation, as is the case for a pressure loading for example. The magnetic part is determined by the expression for the Maxwell stress exterior of the body (if the exterior is a vacuum, for example), which itself is determined by continuity of the relevant magnetic field components across the boundary. Another possible mechanical boundary condition involves spatial constraint of (part of) the boundary of the body, in which case the relevant boundary condition requires specification of x = χ(X) as a function of X on that part of the boundary. 2.5. Constitutive equations For isothermal deformations the free energy may be treated as a function of F and one of the magnetic vector field variables. In [1, 2] the variables F and B were taken as the independent variables (as in, for example, Kovetz [14]), and the free energy was denoted Ψ = Ψ(F, B). The connection (10)1 between B and Bl allows Ψ(F, B) to be regarded equally as a function of F and Bl , and then, following [2, 3], we write Φ(F, Bl ) ≡ Ψ(F, J −1 FBl ). (21) Note that since Bl is a Lagrangian vector it is indifferent to observer transformations in the deformed configuration, while the deformation gradient F changes to QF, where Q is the rotation tensor associated with the transformation. For Φ to be frame indifferent (objective) we must have Φ(QF, Bl ) = Φ(F, Bl ) (22) for all proper orthogonal Q. This requirement enables Φ to be treated as a function of c = FT F (instead of F), and Bl . Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 725 2.5.1. Compressible materials If there is no internal mechanical constraint then standard thermomechanical arguments based on use of Φ lead to 1 ∂Φ + µ−1 (23) τ = ρF 0 [B ⊗ B − (B · B)I], ∂F 2 where I is the identity tensor, and the symmetry requirement on τ is ∂Φ is symmetric, (24) F ∂F which follows automatically from the objectivity condition (22). From (23) and the definition (15) an expression may be given for the associated (total) nominal stress T, and the corresponding Lagrangian form of the magnetization is given by ∂Φ Ml = −ρ0 . (25) ∂Bl In the absence of material Φ = 0, Ml = 0 and equation (23) reduces to 1 (26) τ = µ−1 0 [B ⊗ B − (B · B)I], 2 which is the expression for the Maxwell stress in vacuo. Thus, in (23) the contributions to the stress from the interaction between the material and the magnetic field and that due to the magnetic field in the absence of material are conveniently separated out. At a boundary between the material and a vacuum it is the traction calculated from the stress (26) that is needed in any traction boundary condition (along with applied mechanical tractions) to balance the traction calculation from (23), but, of course, the vector B will in general be different on the two sides of the boundary. An alternative and compact formulation that incorporates the Maxwell stress contribution into a modified free energy, denoted Ω = Ω(F, Bl ) and defined by 1 J −1 Bl · (cBl ), (27) Ω(F, Bl ) = ρ0 Φ + µ−1 2 0 was introduced in [3]. This allows T and τ to be written in the simple forms ∂Ω ∂Ω , τ = J −1 F , ∂F ∂F which can then be used in the equilibrium equations (16) and (14). We also have, on use of (12), T= (28) ∂Ω , (29) ∂Bl and, for a given Bl , Ml is then determined from (12). Noting that in the reference configuration we set Bl = B0 , it is convenient here to define (30) Ω0 (B0 ) = Ω(I, B0 ), Hl = 726 A. Dorfmann and R. W. Ogden ZAMP and then we have H0 = ∂Ω0 , ∂B0 (31) and M0 is given by (2). 2.5.2. Incompressible materials For an incompressible magnetoelastic solid the constraint det F ≡ 1 (32) must be satisfied for all deformations. The modified free energy (27) then simplifies to 1 Bl · (cBl ), (33) Ω = ρ0 Φ + µ−1 2 0 and the total stress (28)2 is replaced by τ =F ∂Ω − pI, ∂F (34) where p is a Lagrange multiplier associated with the constraint (32). The other relevant Eulerian quantities are given by B = FBl , H = F−T ∂Ω , ∂Bl M = µ−1 0 B − H. (35) The corresponding Lagrangian expressions are T= ∂Ω − pF−1 ∂F (36) and Bl , Hl = ∂Ω , ∂Bl Ml = µ−1 0 cBl − Hl . (37) Note that for an incompressible material part of the standard Maxwell stress is incorporated into the ‘arbitrary hydrostatic pressure’ term but that outside the material (in vacuum) the Maxwell stress is given by (26). 2.6. Isotropic magnetoelastic materials We now restrict attention to isotropic magnetoelastic materials, with Φ, or equivalently Ω, regarded as a function of c and Bl . For such materials the material symmetry is similar to that associated with a transversely isotropic elastic material, for which there is a preferred direction in the reference configuration analogous to Bl . Here, however, Bl is not a unit vector so the theory involves one more invariant than is the case for transverse isotropy. Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 727 The magnetoelastic material considered here is said to be isotropic if Φ (or Ω) is an isotropic function of the two tensors c and Bl ⊗ Bl . In this case the dependence of Φ (or Ω) is reduced to dependence on the six invariants  1
(tr c)2 − tr(c2 ) , I3 (c) = det c = J 2 , (38) I1 (c) = trc, I2 (c) = 2 2 I4 = |Bl | , I5 = (cBl ) · Bl , I6 = (c2 Bl ) · Bl , (39) where tr is the trace of a second-order tensor. Note that the invariants (39) are unaffected by a reversal in the sign of Bl . A slightly more general theory could accommodate direction dependence of Bl , but here we restrict attention to the simpler case. Henceforth we concentrate on incompressible materials, so that I3 ≡ 1, and we write Ω = Ω(I1 , I2 , I4 , I5 , I6 ). In what follows the subscripts 1, 2, 4, 5, 6 on Φ or Ω signify differentiation with respect to I1 , I2 , I4 , I5 , I6 , respectively. Then, τ , for example, is given explicitly in terms of Ω as and τ = 2Ω1 b + 2Ω2 (I1 b − b2 ) − pI + 2Ω5 B ⊗ B + 2Ω6 (B ⊗ bB + bB ⊗ B), (40) H = 2(Ω4 b−1 B + Ω5 B + Ω6 bB) = µ−1 0 B − M. (41) Note that (33) may be written 1 Ω = ρ0 Φ + µ−1 I5 , (42) 2 0 the final term of which is associated with the Maxwell stress, as discussed in Section 2.5.1. 2.7. The undeformed configuration An important consideration for all subsequent analysis is the status of any (residual) magnetic field in the undeformed configuration B0 , in the absence of applied mechanical loading (or in the presence of loading applied to maintain the configuration B0 when B0 is changed). Suppose the material is subject to a magnetic (induction) field B0 in this configuration. Here this is taken to be associated with an applied external field rather than with a field ‘frozen-in’ during the curing process of the host material. Then, F = I and hence I1 = I2 = 3, I4 = I5 = I6 = B0 · B0 . (43) Let τ 0 and H0 be the values of τ and H in this configuration, calculated from the appropriate specializations of equations (34) and (35) with (33) since the material is incompressible. Then, within the material, we have τ 0 = [−p + 2(Ω1 + 2Ω2 )]I + 2(Ω5 + 2Ω6 )B0 ⊗ B0 , (44) 728 A. Dorfmann and R. W. Ogden ZAMP with Ω1 , Ω2 , Ω5 , Ω6 evaluated for the invariants given by (43). It is convenient here to adapt the notation Ω0 defined in (30) to the present situation by defining Ω0 (I4 ) = Ω(3, 3, I4 , I4 , I4 ). (45) Then, we have simply H0 = 2Ω0 B0 , M0 = µ−1 0 B0 − H0 , (46) where the prime signifies differentiation with respect to I4 . This expression for H0 needs to be accounted for when satisfying equations (3). In particular, equation (3)2 in general places a restriction on B0 for any given form of Ω. The fact that H0 is parallel to B0 is a consequence of isotropy. Outside the material (in vacuum) the (Maxwell) stress is 1 τ 0 = µ−1 0 [B0 ⊗ B0 − (B0 · B0 )I], 2 (47) subject to the continuity conditions (18) and (19). The stress τ 0 may be regarded as a residual stress in the material in configuration B0 induced by the magnetic (induction) field B0 . The equilibrium equation (14) reduces to Divτ 0 = 0, (48) with τ 0 given by (44). If there is both a ‘frozen-in’ field and an externally applied field then there would in general be two ‘preferred directions’ in B0 , the material would no longer be isotropic in the sense described above and the list of invariants would need to be extended to account for the second field, thereby considerably complicating the constitutive description of the material. This possibility is not considered here. In the following sections we apply the equations summarized in Section 2 for incompressible materials to some representative boundary-value problems. In each case we begin with a magnetic induction field B0 given in the reference configuration and we set its push forward FB0 under the deformation to be the field B. In other words, we choose the independent Lagrangian variable Bl to be the initial field B0 . The associated fields H and M are then calculated as described above. 3. Homogeneous deformations 3.1. Pure homogeneous deformation Here we consider a slab of material of uniform finite thickness, with (plane) faces normal to the X2 direction and of infinite extent in the X1 and X3 directions, where (X1 , X2 , X3 ) are Cartesian coordinates in the reference configuration B0 . The material is subject to an initial magnetic induction field B0 , which is taken Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 729 to be uniform and in the X2 direction with component B02 . Deformation associated with B0 is prevented by the application of an appropriate traction on the faces of the slab, so that B0 is the reference configuration from which the subsequent deformation is measured. Suppose that the slab is now subject to the pure homogeneous deformation x1 = λ1 X1 , x2 = λ2 X2 , x3 = λ3 X3 , (49) where λ1 , λ2 , λ3 are constants. The deformation gradient is diagonal with components (λ1 , λ2 , λ3 ), and Bf has just an x2 component, namely Bf 2 = λ2 B02 . We set B = Bf , so that B2 = Bf 2 . The associated invariants are calculated in terms of two independent principal stretches, say λ1 and λ2 , as −2 −2 2 2 I1 = λ21 + λ22 + λ−2 I2 = λ−2 1 λ2 , 1 + λ2 + λ1 λ2 , 2 2 4 I4 = B02 , I5 = λ2 I4 , I6 = λ2 I4 , (50) (51) which involve three independent quantities in all, namely λ1 and λ2 and I4 . We may therefore regard the (modified) free energy as a function of λ1 , λ2 , I4 , and we write (52) ω(λ1 , λ2 , I4 ) = Ω(I1 , I2 , I4 , I5 , I6 ). Then, on specializing (40) and (41), we obtain the component expressions τ11 − τ33 = λ1 ω1 , τ22 − τ33 = λ2 ω2 , (53) M2 = µ−1 0 B2 − H2 , (54) and H2 = 2λ−2 2 ω4 B2 , for the stress differences, magnetic field and magnetization within the slab, where the subscripts 1, 2, 4 on ω indicate differentiation with respect to λ1 , λ2 , I4 respectively. Note that the push forward of H0 has component Hf 2 = λ−1 2 H02 and that H02 = 2ω0 B02 , where ω0 (I4 ) ≡ ω(1, 1, I4 ) and the prime again denotes differentiation with respect to I4 . Clearly, H2 and Hf 2 match only if ω4 = ω0 , i.e. only if ω4 is independent of the deformation (but this is not a requirement). Since the considered deformation is homogeneous and the magnetic field uniform all the governing differential equations are satisfied trivially. Outside the slab, assumed to be a vacuum, the magnetic induction is, by continuity, the same as that within the slab, while, from (26) and (51)2 , the Maxwell stress has components 1 I5 = −τ22 . (55) τ11 = τ33 = − µ−1 2 0 If no mechanical traction is supplied to the plane faces of the slab then the normal stress τ22 is continuous, and hence (53) and (55) yield τ33 = 1 −1 2 µ λ I4 − λ2 ω2 , 2 0 2 τ11 = τ33 + λ1 ω1 , (56) 730 A. Dorfmann and R. W. Ogden ZAMP which provide expressions for the stresses τ11 and τ33 required to produce the deformation in the presence of the magnetic field. More particularly, if we specialize to the case of equibiaxial deformation, with λ1 = λ3 and λ2 = λ−2 1 , we obtain 1 −1 −4 −2 µ λ I4 − λ−2 (57) 1 ω2 (λ1 , λ1 , I4 ), 2 0 1 where the latter provides an expression for the stress τ11 as a function of λ1 and I4 . In the reference configuration ω1 (λ1 , λ−2 1 , I4 ) = 0, τ11 = 1 −1 µ I4 − ω2 (1, 1, I4 ), (58) 2 0 which is the initial lateral stress needed to prevent deformation due to the magnetic field. τ11 = 3.1.1. Illustration For purposes of illustration we now choose a specific form of ω, namely ω(λ1 , λ2 , I4 ) = 1 1 −1 −2 µ(I4 )(λ21 + λ22 + λ−2 1 λ2 − 3) + ν(I4 ) + µ0 I5 , 2 4 (59) where µ and ν are functions of I4 . Note that for simplicity the factor 1/4 in the final term has been chosen so that τ11 in (58) vanishes, but this can be modified as necessary. Equally, I4 may be replaced by I5 in the argument of the function ν to provide a slightly different model. In the absence of a magnetic field I4 = 0, and then µ(0) is just the shear modulus of the material in the reference configuration and (59) is the classical neo-Hookean strain-energy function provided, for compatibility, we set ν(0) = 0. The term in ν, combined with the final term, accounts for the magnetic energy in the material in the reference configuration. This yields λ1 ω1 = µ(I4 )(λ21 − λ23 ), and ω4 = 1 λ2 ω2 = µ(I4 )(λ22 − λ23 ) + µ−1 I5 , 2 0 1  1 µ (I4 )(I1 − 3) + ν  (I4 ) + µ−1 λ2 . 2 4 0 2 (60) (61) For the considered equibiaxial stress we then have, from (57)2 , τ11 = µ(I4 )(λ21 − λ−4 1 ). (62) The corresponding expression for the magnetization is M2 = 1 −1 2 [µ λ − 2µ (I4 )(I1 − 3) − 4ν  (I4 )]λ−1 2 B02 . 2 0 2 (63) For definiteness we choose µ(I4 ) = µ1 + µ−1 0 I4 , where µ1 = µ(0) and  is a dimensionless material parameter. Then (62) reduces (in dimensionless form) to ∗ τ11 ≡ τ11 /µ1 = (1 + m)(λ21 − λ−4 1 ), (64) Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 731 where m is another dimensionless parameter, defined by m = µ−1 0 I4 /µ1 and reflecting the magnitude of the initial magnetic (induction) field. Note that this expression is not affected by the function ν. The corresponding dimensionless magnetization is defined and given by 1 −2 4 2  2 (65) M2∗ ≡ M2 B02 /µ1 = mλ−2 1 − m(2λ1 + λ1 − 3λ1 ) − 2µ0 mν (I4 )λ1 . 2 In the reference configuration this reduces to 1 ∗ M02 = m − 2µ0 mν  (I4 ). (66) 2 Thus, we can interpret the function ν  (I4 ) as characterizing the magnetization in the absence of deformation. ∗ against λ1 for different values of the parameter m with  In Figure 1 we plot τ11 taken to be 0.2. Clearly, the stiffness of the material increases with the value of the magnetic field strength in the small strain region. We emphasize that this result is for a very simple prototype example of constitutive law but it does nevertheless reflect the limited available data for MS-elastomers (for references and discussion, see [1, 2]). There is some limited evidence to suggest [16] that the stiffness of the material reaches a maximum as the magnetic field strength increases, but, in using a linear form for µ(I4 ), we have not attempted to model this possible effect although it could easily be accommodated if more comprehensive data indicate that this is indeed an important characteristic of the considered materials. Although data for magnetization of MS-elastomers under deformation do not appear to be available it is of interest to examine the predictions of the magnetization in respect of the same simple model. Therefore, in Figure 2, we show the dimensionless magnetization (65) plotted against λ1 for a series of values of the dimensionless parameter m (in this case 0.5, 1, 2, 3, 3.5), again for  = 0.2. We choose a form of ν  (I4 ) that leads to magnetic saturation in the reference configuration. Typically, this means that |B0 |, and hence I4 , tends to a finite limit as |H02 | increases. This behaviour is accommodated by taking   m α 1 tanh−1 (67) µ0 ν  (I4 ) = − , 2m β 4 where α and β are dimensionless material constants. Then, for λ1 = 1 we have ∗ ≡ H02 B02 /µ1 = α tanh−1 (m/β). The limiting value of m is β. In the absence H02 of the applied magnetic field (m = 0) there is, of course, no magnetization. For illustrative purposes, we have taken α = 2.8, β = 4. Note that the sign of the magnetization reverses with that of B02 but that the dimensionless magnetization defined here does not similarly reverse sign since it depends on B02 only through I4 . The (dimensionless) magnetization is positive at λ1 = 1 for the smaller values of m > 0 but negative for larger values (m greater than approximately 3.3) and in each case it decreases as λ1 increases. Of course, for different forms of ν  (I4 ) very different behaviour can be predicted, as is also the case for different values of α and β in (67). Note that because of the connection 732 A. Dorfmann and R. W. Ogden ZAMP 5 1.5 1@ 0.5 @@ 0 @@@@ @@@R R @@@ R @@ R @ 4 ∗ τ11 3 2 1 0 1 1.2 1.4 1.6 1.8 2 λ1 ∗ from equation (64) against stretch λ for the Figure 1. Plot of the dimensionless stress τ11 1 following values of the dimensionless parameter m: 0, 0.5, 1, 1.5. The gradient increases with m. B2 = λ−2 1 B02 , for a given B02 , the magnetic induction B2 becomes smaller and smaller as λ1 increases, but for the simple model considered here the corresponding magnetic field behaves as 2λ21 ω4 B02 , which, in dimensionless form, tends to 2mλ41 as λ1 → ∞. The equibiaxial extension of a thin plate in the presence of a magnetic field normal to its plane provides a possible means of testing the above predictions experimentally and informing the construction of more realistic material models. 3.2. Simple shear Consider the slab of material discussed in Section 3.1 but now, instead of pure homogeneous deformation, subject to a simple shear deformation in the X1 direction in the (X1 , X2 ) plane with amount of shear γ (which is uniform). The matrix of Cartesian components F of the deformation gradient tensor F is   1 γ 0 F = 0 1 0. 0 0 1 (68) Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 733 2 0 0.5 −2 M2∗ 1 −4 −6 2 −8 −10 −12 3 −14 −16 −18 1 3.5 1.2 1.4 1.6 1.8 2 λ1 Figure 2. Plot of the dimensionless magnetization M2∗ from equation (65) against stretch λ1 for the following values of the dimensionless parameter m: 0.5, 1, 2, 3, 3.5. For m = 0 we have M2∗ ≡ 0. For the considered model M2∗ is positive at λ1 = 1 for the smaller values of m but negative for m larger than about 3.3. As λ1 increases M2∗ decreases in each case. The corresponding matrices of the left and right Cauchy-Green deformation tensors b = FFT and c = FT F, written b and c, are     1 + γ2 γ 0 1 γ 0 1 0  , c =  γ 1 + γ2 0  , b= γ (69) 0 0 1 0 0 1 while the components of b2 and b−1 , required in equations (40) and (41), are given by the matrices     1 + 3γ 2 + γ 4 γ(2 + γ 2 ) 0 1 −γ 0 1 + γ 2 0  , b−1 =  −γ 1 + γ 2 0  . b2 =  γ(2 + γ 2 ) (70) 0 0 1 0 0 1 The principal invariants I1 , I2 defined in (38) simplify to I1 = I2 = 3 + γ 2 . (71) For this problem we again take B0 to be in the X2 direction with component B02 . The corresponding magnetic field H0 similarly has a single component H02 . Then, the components (Bf 1 , Bf 2 , Bf 3 ) of Bf in the deformed configuration follow 734 A. Dorfmann and R. W. Ogden ZAMP from the component form of (35)1 with Bf = B = FBl = FB0 on use of (68). Thus, (72) Bf 1 = γB02 , Bf 2 = B02 , Bf 3 = 0, which shows that the magnetic induction vector changes direction as the amount of shear changes. From (39) we calculate, for the considered deformation, 2 I4 = B02 , I5 = I4 (1 + γ 2 ), I6 = I4 (1 + 3γ 2 + γ 4 ). (73) We now set B = Bf . The components of bB, which are required in the expressions (40) for τ and (41) for H, are given by
T bB = 2γ + γ 3 , 1 + γ 2 , 0 B02 , (74) and b−1 B, which is needed to determine H, has the component form T b−1 B = [0, 1, 0] B02 . (75) The resulting components of τ , obtained from equation (40), are
 τ11 = −p + 2Ω1 (1 + γ 2 ) + 2Ω2 (2 + γ 2 ) + 2I4 γ 2 Ω5 + 2Ω6 (2 + γ 2 ) ,
 (76) τ22 = −p + 2Ω1 + 4Ω2 + 2I4 Ω5 + 2Ω6 (1 + γ 2 ) , τ33 = −p
+ 2Ω1 + 2Ω2 (2 + γ 2 ),  (77) τ12 = 2γ Ω1 + Ω2 + I4 [Ω5 + Ω6 (3 + 2γ 2 )] , and τ13 = τ23 = 0. The components of the magnetic field H and magnetization vector M are, from (41),
 H1 = 2γB02 Ω5 + Ω6 (2 + γ 2 ) , M1 = γµ−1 (78) 0 B02 − H1 ,
 H2 = 2B02 Ω4 + Ω5 + Ω6 (1 + γ 2 ) , M2 = µ−1 B − H , (79) 02 2 0 with H3 = M3 = 0. Note that for this problem Hf = F−T H0 = H0 . In view of (71) and (73) there remain two independent variables, namely γ and I4 . It is again convenient to use a reduced form of the modified energy Ω, as a function of these two variables only. We define the appropriate specialization, denoted ω, by (80) ω(γ, I4 ) = Ω(I1 , I2 , I4 , I5 , I6 ), with (71) and (73). It follows that
 ωγ = 2γ Ω1 + Ω2 + I4 (Ω5 + Ω6 (3 + 2γ 2 )) , ω4 = Ω4 + Ω5 (1 + γ 2 ) + Ω6 (1 + 3γ 2 + γ 4 ), (81) where the subscripts γ and 4 on ω indicate partial differentiation with respect to γ and I4 . This allows, in particular, the shear stress to be expressed in a very simple form, namely (82) τ12 = ωγ , just as in standard nonlinear elasticity theory. Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 735 The components of the magnetic field in the deformed configuration can be combined simply in the forms γH1 + H2 = 2ω4 B02 , the first of which reduces to H1 − γH2 = −2γ(Ω4 − Ω6 )B02 , H02 = 2ω0 B02 (83) (84) in the undeformed configuration. Outside the material the relevant in-plane components of the Maxwell stress are calculated by using the continuity of B2 and H1 . Inside the material, H1 is given by (78)1 and outside the material B1 = µ0 H1 . The components of the Maxwell stress are then calculated from (26) as τ22 = −τ11 = and 1 −1 µ I4 − 2µ0 γ 2 I4 [Ω5 + (2 + γ 2 )Ω6 ]2 2 0 τ12 = 2γI4 [Ω5 + (2 + γ 2 )Ω6 ]. (85) (86) The differences between the components of τ22 and τ12 inside and outside the material give the mechanical traction components required to achieve the considered deformation. 3.2.1. Illustration For the model used in Section 3.1.1 and for the considered deformation, we have 1 1 I4 (1 + γ 2 ), (87) ω = µ(I4 )γ 2 + ν(I4 ) + µ−1 2 4 0 and hence, from (82), 1 τ12 = [µ(I4 ) + µ−1 I4 ]γ. (88) 2 0 This is linear in γ with a gradient that is a function of the magnetic field strength through I4 . The linearity is a consequence of the adoption of the neo-Hookean model for the underlying elastic response. The precise details of the function µ(I4 ) remain to be characterized when sufficient data become available. Note that this expression is independent of ν(I4 ), as in the corresponding stresses for the example considered in Section 3.1.1. The components (78)2 and (79)2 of the magnetization are given by 1 1 γB02 , M2 = µ−1 B02 − [µ (I4 )γ 2 + 2ν  (I4 )]B02 , (89) M1 = µ−1 2 0 2 0 and we note that M2 does depend on ν(I4 ) and exhibits quadratic dependence on γ. We observe that while µ(I4 ) characterizes the dependence of the (mechanical) stiffness on the magnetic field the function ν(I4 ), through its derivative, characterizes the magnetization M02 in the reference configuration (where γ = 0), as discussed in Section 3.1.1. 736 A. Dorfmann and R. W. Ogden ZAMP In the simple shear problem examined by Brigadnov and Dorfmann [4] a magnetic field was applied normal to the faces of the slab after deformation. If we do this here then the results are similar to those obtained above with a few differences. If we take B2 as the only component of the magnetic induction then the invariants defined by (39) are given by I4 = (1 + γ 2 )I5 , I5 = I6 = B22 . Then, defining ω by ω(γ, I5 ) = Ω(I1 , I2 , I4 , I5 , I6 ) with (71), we obtain, instead of (78)1 and (79)1 , H1 = 2γ(Ω6 − Ω4 )B2 , H2 = 2ω5 B2 . (90) If, instead of using B as the independent variable, H is used, with an appropriate recasting of the constitutive law, as described in [3] and in Section 4.4 of the present paper, the results are very similar again. In particular, if H has only a component H2 then B will have components B1 and B2 . 4. Extension and inflation of a tube For any problem involving non-uniform fields the relevant magnetic governing equations to be solved for the initial magnetic vectors are DivB0 = 0, CurlH0 = 0, (91) together with H0 = 2Ω0 B0 , (92) in which the factor on the right-hand side depends on I4 = B0 · B0 . Equations (91) guarantee that the corresponding equations for Bf = FB0 and Hf = F−T H0 in the deformed configuration are automatically satisfied. However, if we take B = Bf and calculate H from (41) with Bl = B0 then curlH = 0 does not in general follow and needs to be checked. In general, this equation will impose restrictions on the class of constitutive laws for which the considered combination of deformation and magnetic induction is admissible. This is the case whether or not B is chosen as above since, for any given B, H has to be calculated from the constitutive law (41). The analysis is based on equations (40) and (41) for τ and H for an incompressible material, with Ω given by (42) as a function of the invariants I1 , I2 , I4 , I5 , I6 defined in (38) and (39), together with the equilibrium equation divτ = 0 (93) in the absence of mechanical body forces, coupled with suitable boundary conditions. Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 737 4.1. Kinematics We now apply these equations to the problem of an infinitely long circular cylindrical tube whose reference cross-sectional geometry is defined by A ≤ R ≤ B, 0 ≤ Θ ≤ 2π (94) with respect to cylindrical polar coordinates (R, Θ, Z). Note that a tube of finite length would present difficulties associated with compatibility of the magnetic boundary conditions on the ends of the tube and on the lateral surfaces and would not admit the simple nature of the fields arising for a tube of infinite length. The tube is deformed by combined uniform axial extension and radial expansion so that the deformed geometrical cross-section is described by the equations a ≤ r ≤ b, 0 ≤ θ ≤ 2π (95) in terms of cylindrical polar coordinates (r, θ, z). Since the material is incompressible the deformation has the form 2 2 r2 = a2 + λ−1 z (R − A ), θ = Θ, z = λz Z, (96) where λz (constant) is the axial stretch. The deformation gradient is diagonal with respect to the cylindrical coordinate axes and the associated principal stretches are written r ≡ λ, λ3 = λz , λ2 = (97) λ1 = λ−1 λ−1 z , R wherein the notation λ is defined. We regard λ and λz as the two independent stretches. 4.2. The magnetic field In cylindrical polar coordinates (R, Θ, Z) equations (91) have the explicit forms ∂B0R 1 ∂B0Z 1 ∂B0Θ + B0R + + = 0, ∂R R R ∂Θ ∂Z (98) and 1 ∂H0Z ∂H0Θ ∂H0R ∂H0Z 1 ∂(RH0Θ ) 1 ∂H0R − = 0, − = 0, − = 0, (99) R ∂Θ ∂Z ∂Z ∂R R ∂R R ∂Θ where (B0R , B0Θ , B0Z ) are the components of B0 and (H0R , H0Θ , H0Z ) those of H0 . We consider only magnetic fields that preserve the circular cylindrical symmetry of the problem, so that these components are functions of R only and equations (98) and (99) simplify accordingly. If the field is purely radial, then we deduce that B0R = C , R H0R = 2ω0 B0R , (100) 738 A. Dorfmann and R. W. Ogden ZAMP 2 where C is a constant, I4 = B0R and ω0 = ω(1, 1, I4 ) is defined similarly to the corresponding expression in Section 3.1. (For the present deformation ω is defined below.) However, since B0R must be continuous across a circular cylindrical surface its singularity at R = 0 cannot be avoided and we conclude that C = 0. (We remark that in [1, 2] a radial field was used and the singularity was overlooked.) Thus, it suffices to consider either an axial or an azimuthal field. If the magnetic field is purely axial then we deduce from (99) that H0Z (and hence B0Z ) is constant and we have H0Z = C, H0Z = 2ω0 B0Z , (101) where C is again a constant, and 2 , I4 = B0Z I5 = λ2z I4 , I6 = λ4z I4 . (102) The deformed fields are given by Hf z = λ−1 z H0Z , Bz = Bf z = λz B0Z . (103) If the magnetic field is purely circumferential then the solution of equations (98) and (99) is given by H0Θ = C , R H0Θ = 2ω0 B0Θ , (104) C again being a constant and 2 I4 = B0Θ , I5 = λ2 I4 , I6 = λ4 I4 . (105) The deformed fields are given by Hf θ = λ−1 H0Θ , Bθ = Bf θ = λB0Θ . (106) In the case of H0Θ the singularity at R = 0 can be excluded by use of a concentric circular cylindrical core across the boundary of which H0Θ satisfies an appropriate jump condition associated with an axial steady current (such a jump condition requires a generalization of the continuity condition (18)2 ). The deformation is locally a pure homogeneous strain and we may take the (modified) free energy to depend on only λ, λz , I4 , and, whichever of the axial or azimuthal magnetic fields is used, write ω(λ, λz , I4 ) = Ω(I1 , I2 , I4 , I5 , I6 ). (107) This is similar to the definition (52), except for the notation used for the stretches and the fact that here there is dependence on the radial coordinate. Whether the magnetic field is axial or azimuthal the stress differences are given by the simple formulas τθθ − τrr = λωλ , τzz − τrr = λz ωλz . (108) The magnetic field H and magnetization M are axial in the first case with components Mz = µ−1 (109) Hz = 2λ−1 z ω4 B0Z , 0 λz B0Z − Hz . Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 739 In the second case they are circumferential with Mθ = µ−1 0 λB0Θ − Hθ . Hθ = 2λ−1 ω4 B0Θ , (110) In the above the subscripts λ, λz , 4 signify partial derivatives with respect to λ, λz , I4 , respectively. In the reference configuration equations (108), (109) and (110) reduce to, respectively, τ0ΘΘ − τ0RR = ωλ (1, 1, I4 ), H0Z = 2ω0 B0Z , H0Θ = 2ω0 B0Θ , τ0ZZ − τ0RR = ωλz (1, 1, I4 ), µ−1 0 B0Z (111) − H0Z , (112) M0Θ = µ−1 0 B0Θ − H0Θ . (113) M0Z = In either the problem with an axial or a circumferential magnetic field the only component of the equilibrium equation not satisfied trivially is the radial equation 1 dτrr + (τrr − τθθ ) = 0 dr r (114) in the deformed configuration. The expression for the stress difference in (114) is obtained from (108)1 , and this equation then serves to determine τrr (or equivalently p) subject to appropriate boundary conditions. In the reference configuration the (residual) stresses must satisfy the equation 1 dτ0RR + (τ0RR − τ0ΘΘ ) = 0, dR R (115) in which (111)1 should be used. 4.3. Solutions 4.3.1. Axial magnetic field In this case B0Z is constant and connected to (constant) H0Z through (101)2 . The corresponding value of Hf z is λ−1 z H0Z . Since, in the absence of surface currents, the tangential component of H must be continuous across an interface it follows that Hz is continuous across the surfaces r = a and r = b and is therefore spatially uniform in the present problem. Outside the material (assumed to be vacuum) the magnetic induction is given by µ0 Hz , and there the components of the Maxwell stress are given by 1 (116) τrr = τθθ = − µ0 Hz2 = −τzz . 2 In particular, τrr contributes to the (radial) traction on the boundaries r = a and r = b, but has the same value on each boundary. Suppose that inflation of the tube is achieved by application of an internal pressure P with no mechanical load applied on the exterior boundary. Then we 740 A. Dorfmann and R. W. Ogden ZAMP can apply boundary conditions in the form 1 1 τrr = − µ0 Hz2 on r = b. (117) τrr = −P − µ0 Hz2 on r = a, 2 2 Integration of equation (114), use of the relation (108)1 and application of the boundary conditions (117) leads to b P = a λωλ dr , r (118) and we note, in particular, that this does not involve the exterior Maxwell stress. The corresponding expression for the resultant axial load N on a cross-section of the cylinder is calculated from N = 2π b a τzz rdr, (119) using (108), (114) and (118), as N =π b a 1 (2λz ωλz − λωλ )rdr + πa2 P − π(b2 − a2 )µ0 Hz2 . 2 (120) The expression for P is identical in form to the corresponding equation in the purely elastic theory, while that for N differs by the inclusion of the final term involving Hz (see, for example, Ogden [15], pp. 289–291), but now ω includes the influence of an axial magnetic field through I4 . Note that Hz is given by (109)1 . For the model used in Section 3.1.1 the expression for P takes the form b P = µ(I4 ) a (λ2 − λ−2 λ−2 z ) dr . r (121) This differs from the corresponding expression in the purely elastic case (with the neo-Hookean strain-energy function) only by virtue of the dependence of µ on I4 . Thus, if, for example, µ is an increasing function of I4 then the pressure-radius response stiffens as a result of the presence of the axial magnetic field. A similar comment applies to N in that it is influenced by the dependence of µ on I4 , but there is also an additional contribution from the ‘Maxwell’ term in the energy function. Thus, we can write 1 2 2 (122) N = Nnh + π(b2 − a2 )(µ−1 0 λz I4 − µ0 Hz ), 2 where Nnh is the expression for N obtained for the neo-Hookean material but with µ dependent on I4 . Consider next the residual stress field governed by (115), which is now written 1 dτ0RR = ωλ (1, 1, I4 ). (123) dR R Since I4 is constant this may be integrated immediately. Use of the boundary conditions (117) specialized to the reference configuration with P = 0 leads to the Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 741 solution τ0RR = constant, the constant being equal to the value of the Maxwell 2 /2, with H0Z stress on each of the boundaries R = A and R = B, i.e. −µ0 H0Z given by (104). We also deduce that ωλ (1, 1, I4 ) = 0, which is consistent with P = 0 in equation (118) for this specialization. Then, from (111), we have, within the material, (124) τ0ZZ = ωλz (1, 1, I4 ) − 2µ0 I4 (ω0 )2 , which is independent of R. The sign of this term therefore determines whether the load associated with the axial magnetic field required to maintain the initial geometry is tensile or compressive. Depending on the form of ω(λ, λz , I4 ) this can be positive or negative. 4.3.2. Circumferential magnetic field This is similar to the above except that in ω the influence of the circumferential 2 , which depends on the radial magnetic field is incorporated through I4 = B0Θ coordinate. We also have Hf θ = λ−1 H0Θ . Here we must have continuity of Hθ at the boundaries r = a and r = b, with C = 2ω0 B0Θ (125) R within the material. Suppose that there is a central core of radius c < a carrying a steady current I. Then C = I/2π. This core is excluded from consideration. In principle, the second equation in (125) determines I4 as a function of R when I is given. In order to satisfy the equation curlH = 0 it is necessary (under the assumed cylindrical symmetry) that rHθ is constant. When taken together with the first equation in (125) this leads to the conclusion that ω4 must depend only on I4 for this deformation and therefore puts a restriction on the class of material models admitting extension and inflation in the presence of an azimuthal magnetic field. Indeed, this restriction is quite severe since it implies that ω is decoupled in the form ω (1) (λ, λz ) + ω (2) (I4 ), where ω (1) depends only on the deformation and ω (2) only on the initial magnetic field, through I4 . Outside the material (in the regions c < r < a and r > b) the components of the Maxwell stress are 1 (126) τrr = τzz = −τθθ = − µ0 Hθ2 ≡ −τm (r), 2 wherein the notation τm (r) is defined. Since this depends on the radius the radial components τm (a) and τm (b) now influence the pressure-radius relation, which becomes b dr (127) λωλ . P + τm (a) − τm (b) = r a Hθ = 2λ−1 ω4 B0Θ , H0Θ = The corresponding expression for N in this case is N =π b a (2λz ωλz − λωλ )rdr + πa2 P − π[b2 τm (b) − a2 τm (a)]. (128) 742 A. Dorfmann and R. W. Ogden ZAMP However, since we must have rHθ = constant, the term b2 τm (b)−a2 τm (a) vanishes. In the absence of internal pressure in the reference configuration, equation (128) reduces to N =π B A [2ωλz (1, 1, I4 ) − ωλ (1, 1, I4 )]RdR, (129) which gives the axial load required to prevent deformation due to the applied magnetic field. For the specific model (59) this reduces further to 1 N = − πµ−1 2 0 B A I4 RdR, (130) which is negative. Thus, for this model, initial application of the circumferential magnetic field requires a compressive axial load to prevent axial extension of the tube. 4.4. An alternative formulation In view of the restriction on the class of constitutive laws imposed by the problem considered in Section 4.3 in the case of an azimuthal magnetic induction field we now consider an alternative formulation of the governing equations based on use of H (or, equivalently, Hl ) as the independent magnetic variable. We make use of the Legendre transformation Ω∗ (F, Hl ) = Ω(F, Bl ) − Hl · Bl , (131) which defines Ω∗ (F, Hl ) as a function of F and Hl . Then, it follows that T= with Ml given by ∂Ω∗ , ∂F Bl = − ∂Ω∗ , ∂Hl Ml = µ−1 0 cBl − Hl . (132) (133) Alternatively, instead of defining Ω∗ (F, Hl ) via (131), one could begin with an energy defined as a function of F and H or F and Hl and construct the constitutive law ab initio in terms of Hl , as was done in Section 2.5 for Bl . In the same way as we may choose Bl = B0 , we may now set, instead, Hl = H0 , but then, in general, Bl = B0 . Here, we use the notation K4 , K5 , K6 in respect of an isotropic material for the counterparts of I4 , I5 , I6 . These are defined by K1 = |H0 |2 , K5 = (cH0 ) · H0 , K6 = (c2 H0 ) · H0 . (134) For an incompressible material, for example, the total stress is τ = 2Ω∗1 b + 2Ω∗2 (I1 b − b2 ) − pI + 2Ω∗5 bH ⊗ bH + 2Ω∗6 (bH ⊗ b2 H + b2 H ⊗ bH), (135) Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 743 and the magnetic induction B = −2(Ω∗4 bH + Ω∗5 b2 H + Ω∗6 b3 H), (136) where now Ω∗ = Ω∗ (I1 , I2 , K4 , K5 , K6 ) and Ω∗i is defined as ∂Ω∗ /∂Ii for i = 1, 2, and ∂Ω∗ /∂Ki for i = 4, 5, 6. In the reference configuration, we have I1 = I2 = 3, K4 = K5 = K6 = |H0 |2 , (137) and we define Ω∗0 (K4 ) = Ω∗ (3, 3, K4 , K4 , K4 ). (138) τ 0 = [−p + 2(Ω∗1 + 2Ω∗2 )]I + 2(Ω∗5 + 2Ω∗6 )H0 ⊗ H0 , (139) B0 = −2(Ω∗4 + Ω∗5 + Ω∗6 )H0 ≡ −2Ω∗0  (K4 )H0 , (140) Then, and wherein the prime indicates differentiation with respect to K4 and Ω∗i , i = 1, 2, 4, 5, 6, is evaluated for (137). 4.4.1. The circumferential magnetic field revisited We now review briefly the problem considered in Section 4.3.2 on the basis of this alternative formulation. In order to satisfy curlH = 0 the azimuthal field is given by C C = λ−1 H0Θ , H0Θ = , Hθ = Hf θ = (141) R r where C is a constant, and it follows that 2 K4 = H0Θ , K5 = λ2 K4 , K6 = λ4 K4 . (142) Analogously to (107) we define ω ∗ (λ, λz , K4 ) = Ω∗ (I1 , I2 , K4 , K5 , K6 ), (143) with ω0∗ (K4 ) = ω ∗ (1, 1, K4 ). Then, equations (108) are replaced by τθθ − τrr = λωλ∗ , τzz − τrr = λz ωλ∗z , (144) and (110) by Bθ = −2λω4∗ H0Θ . (145) Equations (127) and (128) are unchanged except that ω is replaced by ω ∗ . With the present formulation the equation divB = 0 is automatically satisfied and places no restriction on the form of constitutive law, unlike the situation in Section 4.3.2. 744 A. Dorfmann and R. W. Ogden ZAMP 5. Conclusions In this paper we have analyzed some basic problems in the theory of nonlinear isotropic magnetoelasticity in order to provide a framework within which the theory can be compared with experiment data for, in particular, magneto-sensitive elastomers. This requires development of experimental tests that can provide appropriate data since extensive data of the required type are not currently available. In particular, more information on the change in material constants with magnetic field is needed and on the deformation dependence of the magnetization of these materials. Once such data become available, a key objective will be to characterize these materials through use of specific forms of constitutive law based on a free energy function or a variant of this function. As a starting point for this process the simple prototype model introduced in Section 3.1.1 includes two constitutive functions, one related to the material stiffness and the other to the magnetization. An important consideration is whether it is appropriate to use the magnetic field H or the magnetic induction B as the independent variable. While, in principle, the associated formulations of the constitutive law and the governing field equations are equivalent, in practice, at least for the problem considered in Section 4.3.2 and for several problems considered in [3], more restrictions are placed on the admissible class of constitutive laws if B is used rather than H. This problem would be compounded if the magnetization M were used as the independent variable since the fields H and B then derived via the constitutive law and the connection B = µ0 (H + M) have to satisfy curlH = 0 and divB = 0, respectively. Acknowledgements This research was partially supported by the Research Directorates General of the European Commission (through project GRD1-1999-11095). The authors gratefully acknowledge this support. References [1] A. Dorfmann and R.W. Ogden, Magnetoelastic modelling of elastomers, European J. Mech. A/Solids 22 (2003), 497–507. [2] A. Dorfmann and R.W. Ogden, Nonlinear magnetoelastic deformations of elastomers, Acta Mechanica 167 (2003), 13–28. [3] A. Dorfmann and R.W. Ogden, Nonlinear magnetoelastic deformations, Quarterly Journal of Mechanics and Applied Mathematics, 57 (2004), 599–622. [4] I.A. Brigadnov and A. Dorfmann, Mathematical modeling of magneto-sensitive elastomers, Int. J. Solids Structures 40 (2003), 4659–4674. [5] D.J. Steigmann, Equilibrium theory for magnetic elastomers and magnetoelastic membranes, Int. J. Nonlinear Mech. 39 (2004), 1193–1216. [6] S.V. Kankanala and N. Triantafyllidis, On finitely strained magnetorheological elastomers, J. Mech. Phys. Solids 52 (2004), 2869–2908. Vol. 56 (2005) Some problems in nonlinear magnetoelasticity 745 [7] C. Truesdell and R. Toupin The Classical Field Theories. In: Handbuch der Physik, Vol. III/1 (S. Flügge, ed.), (Springer, Berlin 1960). [8] W.F. Brown, Magnetoelastic Interactions (Springer, Berlin 1966). [9] K. Hutter, On thermodynamics and thermostatics of viscous thermoelastic solids in the electromagnetic fields. A Lagrangian formulation, Arch. Rat. Mech. Anal. 54 (1975), 339– 366. [10] K. Hutter, A thermodynamic theory of fluids and solids in the electromagnetic fields, Arch. Rat. Mech. Anal. 64 (1977), 269–289. [11] Y.H. Pao, Electromagnetic forces in deformable continua, Mechanics Today, Vol. 4 (ed. S. Nemat-Nasser) Pergamon Press, Oxford 1978, pp. 209–306. [12] K. Hutter and A.A.F. van de Ven, Field Matter Interactions in Thermoelastic Solids, Lecture Notes in Physics, vol. 88 (Springer, Berlin 1978). [13] A.C. Eringen and G.A. Maugin Electrodynamics of Continua I. Springer Verlag, New York 1990. [14] A. Kovetz, Electromagnetic Theory (University Press, Oxford 2000). [15] R.W. Ogden, Non-linear Elastic Deformations. Dover, New York 1997. [16] M.R. Jolly, J.D. Carlson and B.C. Muñoz, A model of the behaviour of magnetorheological materials, Smart Mater. Struct. 5 (1996), 607–614. A. Dorfmann Center of Mechanics and Structural Dynamics Vienna University of Technology Wiedner Hauptstrasse 8–10 1040 Vienna Austria R. W. Ogden Department of Mathematics University of Glasgow Glasgow G12 8QW UK (Received: May 17, 2004)