Incremental Magnetoelastic Deformations, with Application to Surface Instability
aa r X i v : . [ phy s i c s . c l a ss - ph ] D ec Incremental magnetoelastic deformations,with application to surface instability
M. Ott´enio, M. Destrade, R.W. Ogden,2007
Abstract
In this paper the equations governing the deformations of infinitesi-mal (incremental) disturbances superimposed on finite static deforma-tion fields involving magnetic and elastic interactions are presented.The coupling between the equations of mechanical equilibrium andMaxwell’s equations complicates the incremental formulation and par-ticular attention is therefore paid to the derivation of the incrementalequations, of the tensors of magnetoelastic moduli and of the incre-mental boundary conditions at a magnetoelastic/vacuum interface.The problem of surface stability for a solid half-space under planestrain with a magnetic field normal to its surface is used to illustratethe general results. The analysis involved leads to the simultaneousresolution of a bicubic and vanishing of a 7 × One of the main reasons for industrial interest in rubber-like materials re-sides in their ability to dampen vibrations and to absorb shocks. This paperis concerned with an extension of the nonlinear elasticity theory adopted fordescribing the properties of these materials to incorporate nonlinear mag-netoelastic effects so as to embrace a class of solids referred to as magneto-sensitive (MS) elastomers. These “smart” elastomers typically consist ofan elastomeric matrix (rubber, silicon, for example) with a distribution of1errous particles (with a diameter of the order of 1–5 micrometers) withintheir bulk. They are sensitive to magnetic fields in that they can deformsignificantly under the action of magnetic fields alone without mechanicalloading, a phenomenon known as magnetostriction . As a result, their me-chanical damping abilities can be controlled by applying suitable magneticfields. This coupling between elasticity and magnetism was probably firstobserved by Joule in 1847 when he noticed that a sample of iron changed itslength when magnetized.In general, the physical properties of magnetoelastic materials depend onfactors such as the choice of magnetizable particles, their volume fractionwithin the bulk, the choice of the matrix material, the chemical processes ofcuring, etc.; see [1] for details, and also [2] for an experimental study on amagneto-sensitive elastomer.The coupling between magnetism and nonlinear elasticity has generatedmuch interest over the last 50 years or so, as illustrated by the works ofTruesdell and Toupin [3], Brown [4], Yu and Tang [5], Maugin [6], Eringenand Maugin [7], Kovetz [8], and others. The corresponding engineering ap-plications are more recent (see Jolly et al. [9], or Dapino [10], for instance)and have generated renewed impetus in theoretical modelling (see, for exam-ple, Dorfmann and Brigadnov [11]; Dorfmann and Ogden [12]); Kankanalaand Triantafyllidis [13]. Here, we derive the (linearized) equations governingincremental effects in a magnetoelastic solid subject to finite deformation inthe presence of a magnetic field. These equations are then used to examinethe problem of surface stability of a homogeneously pre-strained half-spacesubject to a magnetic field normal to its (plane) boundary. Related works onthis subject include the studies of McCarthy [14], van de Ven [15], Boulanger[16, 17], Maugin [18], Carroll and McCarthy [19] and Das et al. [20].We adopt the formulation of Dorfmann and Ogden [12] as the startingpoint for the derivation of the incremental equations. This involves a totalstress tensor and a modified strain energy function or total energy function ,which enable the constitutive law for the stress to be written in a form verysimilar to that in standard nonlinear elasticity theory. The coupled governingequations then have a simple structure. We summarize these equations inSection 2. For incompressible isotropic magnetoelastic materials the energydensity is a function of five invariants, which we denote here by I and I ,the first two principal invariants of the Cauchy-Green deformation tensors,and I , I , I , three invariants involving a Cauchy-Green tensor and themagnetic induction vector. This formulation is similar in structure to thatassociated with transversely isotropic elastic solids (see Spencer [21]). Thegeneral incremental equations of nonlinear magnetoelasticity are then derivedin Section 3. Therein we define the various magnetoelastic ‘moduli’ tensors2nd provide general incremental boundary conditions. Care is needed inderiving the boundary equations since the Lagrangian fields in the solid andthe Eulerian fields in the vacuum must be reconciled.Section 4 provides a brief summary of the basic equations associated withthe pure homogeneous plane strain of a half-space of magnetoelastic mate-rial with a magnetic field normal to its boundary. In Section 5, the generalincremental equations are applied to the analysis of surface stability. Not sur-prisingly, the resulting bifurcation criterion is a complicated equation, evenwhen the pre-stress corresponds to plane strain and the magnetic inductionvector is aligned with a principal direction of strain, as is the case here. Thebifurcation equation comes from the vanishing of the determinant of a 7 × I , I , I , and I . Of course, these invariants are nonlinear in the deforma-tion and the theory remains highly nonlinear. The bicubic then factorizesand a complete analytical resolution follows. In addition to the two elasticMooney-Rivlin parameters (material constants), the material model involvestwo magnetoelastic coupling parameters. The stability behaviour of the half-space depends crucially on the values of these coupling parameters and alsoon the magnitude of the magnetic field. In particular, a judicious choice ofparameters can stabilize the half-space relative to the situation in the ab-sence of a magnetic field. Equally, the half-space can become de-stabilizedfor different choices of the parameters. Thus, even this very simple modelillustrates the possible complicated nature of the magnetoelastic coupling inthe nonlinear regime. In this section the equations for nonlinear magnetoelastic deformations, asdeveloped by Dorfmann and Ogden [12, 22, 23, 24], are summarized for sub-sequent use in the derivation of the incremental equations.We consider a magnetoelastic body in an undeformed configuration B ,with boundary ∂ B . A material point within the body in that configurationis identified by its position vector X . By the combined action of appliedmechanical loads and magnetic fields, the material is then deformed from B to the configuration B , with boundary ∂ B , so that the particle located at X in B now occupies the position x = χ ( X ) in the deformed configuration B .3he function χ describes the static deformation of the body and is a one-to-one, orientation-preserving mapping with suitable regularity properties.The deformation gradient tensor F relative to B is defined by F = Grad χ , F iα = ∂x i /∂X α , Grad being the gradient operator in B . The magnetic fieldvector in B is denoted H , the associated magnetic induction vector by B and the magnetization vector by M .To avoid a conflict of standard notations, the Cauchy-Green tensors arerepresented here by lower case characters; thus, the left and right Cauchy-Green tensors are b = F F t and c = F t F , respectively, where t denotes thetranspose. The Jacobian of the deformation gradient is J = det F , and theusual convention J >
Conservation of the mass for the material is here expressed as
J ρ = ρ , (2.1)where ρ and ρ are the mass densities in the configurations B and B , respec-tively. For an incompressible material, J = 1 is enforced so that ρ = ρ .The equilibrium equation in the absence of mechanical body forces, isgiven in Eulerian form by div τ = , (2.2)where τ is the total Cauchy stress tensor , which is symmetric , and div is thedivergence operator in B . The total nominal stress tensor T is then definedby T = J F − τ , (2.3)so that the Lagrangian counterpart of the equilibrium equation (2.2) isDiv T = , (2.4)Div being the divergence operator in B .Let N denote the unit outward normal vector to ∂ B and n the cor-responding unit normal to ∂ B . These are related by Nanson’s formula n d a = J F − t N d A , where d A and d a are the associated area elements. Thetraction on the area element in ∂ B may be written τ n d a or as T t N d A . Atraction boundary condition might therefore be expressed in the form T t N = t a , (2.5)where t a is the applied traction per unit reference area. If this is independentof the deformation then the traction is said to be a dead load .4 .2 Magnetic balance laws In the Eulerian description, Maxwell’s equations in the absence of time de-pendence, free charges and free currents reduce todiv B = 0 , curl H = , (2.6)which hold both inside and outside a magnetic material, where curl relatesto B . Thus, B and H can be regarded as fundamental field variables. Athird vector field, the magnetization, when required, can be defined by thestandard relation B = µ ( H + M ) . (2.7)We shall not need to make explicit use of the magnetization in this paper.Associated with the equations (2.6) are the boundary continuity condi-tions ( B − B ⋆ ) · n = 0 , ( H − H ⋆ ) × n = , (2.8)wherein B and H are the fields in the material and B ⋆ and H ⋆ the corre-sponding fields exterior to the material, but in each case evaluated on theboundary ∂ B .Lagrangian counterparts of B and H , denoted B l and H l , respectively,are defined by B l = J F − B , H l = F t H , (2.9)and in terms of these quantities equations (2.6) becomeDiv B l = 0 , Curl H l = , (2.10)where Curl is the curl operator in B . We note in passing that a Lagrangiancounterpart of M may also be defined, one possibility being M l = F t M .The boundary conditions (2.8) can also be expressed in Lagrangian form,namely ( B l − J F − B ⋆ ) · N = 0 , ( H l − F t H ⋆ ) × N = , (2.11)evaluated on the boundary ∂ B . There are many possible ways to formulate constitutive laws for magnetoelas-tic materials based on different choices of the independent magnetic variableand the form of energy function. For present purposes it is convenient touse a formulation involving a ‘total energy function’, or ‘modified free energyfunction’, which is denoted here by Ω, following Dorfmann and Ogden [12].5his is defined per unit reference volume and is a function of F and B l :Ω( F , B l ). This leads to the very simple expressions T = ∂ Ω ∂ F , H l = ∂ Ω ∂ B l (2.12)for a magnetoelastic material without internal mechanical constraints, and T = ∂ Ω ∂ F − p F − , H l = ∂ Ω ∂ B l (2.13)for an incompressible material, where p is a Lagrange multiplier associatedwith the constraint det F = 1. Note that the expression for H l is unchangedexcept that now det F = 1 in Ω.The Eulerian counterparts of the above equations are τ = J − F ∂ Ω ∂ F , H = F − t ∂ Ω ∂ B l (2.14)for an unconstrained material, where F − t = ( F − ) t , and τ = F ∂ Ω ∂ F − p I , H = F − t ∂ Ω ∂ B l , (2.15)where I is the identity tensor. We emphasize that the first equation in eachof (2.12)–(2.15) has exactly the same form as for a purely elastic material inthe absence of a magnetic field. In general the mechanical properties of magnetoelastic elastomers have fea-tures that are similar to those of transversely isotropic materials. Duringthe curing process a preferred direction is ’frozen in’ to the material if thecuring is done in the presence of a magnetic field, which aligns the magneticparticles. If cured without a magnetic field then the distribution of particlesis essentially random and the resulting magnetoelastic response is isotropic.We focus on the latter case here for simplicity, but the corresponding analysisfor the more general case follows the same pattern, albeit more complicatedalgebraically. A general constitutive theory for the former situation has beendeveloped by Bustamante and Ogden [25] and applied to some simple prob-lems. For isotropic materials, the energy function Ω depends only on c and B l ⊗ B l , through the six invariants I = tr c , I = (cid:2) (tr c ) − (tr c ) (cid:3) , I = det c = J ,I = B l · B l , I = ( c B l ) · B l , I = ( c B l ) · B l . (2.16)6or incompressible materials, I = 1 and only the five invariants I , I , I , I , and I remain. The total stress tensor τ is then expressed as τ = − p I +2Ω b +2Ω ( I b − b )+2Ω B ⊗ B +2Ω ( B ⊗ bB + bB ⊗ B ) , (2.17)where Ω i = ∂ Ω /∂I i , and the total nominal stress tensor T as T = − p F − + 2Ω F t + 2Ω ( I F t − F t b )+ 2Ω B l ⊗ B + 2Ω ( B l ⊗ bB + F t B ⊗ B ) . (2.18)Finally, the magnetic field vector H is found from (2.15) as H = 2(Ω b − B + Ω B + Ω bB ) , (2.19)and its Lagrangian counterpart is H l = 2(Ω B l + Ω cB l + Ω c B l ) . (2.20) In vacuum, there is no magnetization and the standard relation (2.7) reducesto B ⋆ = µ H ⋆ , (2.21)where the star is again used to denote a quantity exterior to the material.Also, the stress tensor τ is now the Maxwell stress τ ⋆ , given by τ ⋆ = µ − [ B ⋆ ⊗ B ⋆ − ( B ⋆ · B ⋆ ) I ] , (2.22)which, since div B ⋆ = 0 and curl B ⋆ = , satisfies div τ ∗ = . Suppose now that both the magnetic field and, within the material, thedeformation undergo incremental changes (which are denoted by superposeddots). Let ˙ F and ˙ B l be the increments in the independent variables F and B l . It follows from (2.12) that the increment ˙ T in T and the increment ˙ H l in H l are given in the form ˙ T = A ˙ F + Γ ˙ B l , ˙ H l = Γ ˙ F + K ˙ B l , (3.1)7here A , Γ and K are, respectively, fourth-, third- and second-order tensors,with components defined by A αiβj = ∂ Ω ∂F iα ∂F jβ , Γ αiβ = ∂ Ω ∂F iα ∂B l β = ∂ Ω ∂B l β ∂F iα , K αβ = ∂ Ω ∂B l α ∂B l β . (3.2)We refer to these tensors as magnetoelastic moduli tensors . We note thesymmetries A αiβj = A βjαi , K αβ = K βα , (3.3)and observe that Γ has no such indicial symmetry. The products in (3.1) aredefined so that, in component form, we have˙ T αi = A αiβj ˙ F jβ + Γ αiβ ˙ B l β , ˙ H l α = Γ βiα ˙ F iβ + K αβ ˙ B l β . (3.4)For an unconstrained isotropic material, Ω is a function of the six in-variants I , I , I , I , I , I , and the expressions (3.2) can be expanded in theforms A αiβj = X m =1 , m =4 6 X n =1 , n =4 Ω mn ∂I n ∂F iα ∂I m ∂F jβ + X n =1 , n =4 Ω n ∂ I n ∂F iα ∂F jβ , Γ αiβ = X m =4 6 X n =1 , n =4 Ω mn ∂I m ∂B l β ∂I n ∂F iα + X n =5 Ω n ∂ I n ∂F iα ∂B l β , K αβ = X m =4 6 X n =4 Ω mn ∂I m ∂B l α ∂I n ∂B l β + X n =4 Ω n ∂ I n ∂B l α ∂B l β , (3.5)where Ω n = ∂ Ω /∂I n , Ω mn = ∂ Ω /∂I m ∂I n . Expressions for the first andsecond derivatives of I n , n = 1 , . . . ,
6, are given in the Appendix.For an incompressible material, T is given by (2.13) and its incrementis then ˙ T = A ˙ F + Γ ˙ B l − ˙ p F − + p F − ˙ F F − , (3.6)which replaces (3.1) in this case. On the other hand, H l is still given by(2.12) and its increment is unaffected by the constraint of incompressibility,except, of course, since Ω is now independent of I = 1, the summations inequations (3.5) omit m = 3 and n = 3.It is now a simple matter to obtain the incremental forms of the (La-grangian) governing equations. We haveDiv ˙ T = , Div ˙ B l = 0 , Curl ˙ H l = . (3.7)8hese equations can be transformed into their Eulerian counterparts (indi-cated by a zero subscript) by means of the transformations ˙ T = J − F ˙ T , ˙ B l = J − F ˙ B l , ˙ H l = F − t ˙ H l (3.8)(with J = 1 for an incompressible material), leading todiv ˙ T = , div ˙ B l = 0 , curl ˙ H l = . (3.9)Now let u denote the incremental displacement vector x − X . Then, ˙ F = Grad u = (grad u ) F , where grad is the gradient operator with respect to x . We use the notation d for the displacement gradient grad u , in components d ij = ∂u i /∂x j . From (3.8) and (3.1) we then have ˙ T = A d + Γ ˙ B l , ˙ H l = Γ d + K ˙ B l , (3.10)where, in index notation, the tensors A , Γ , and K are defined by A jisk = J − F jα F sβ A αiβk , Γ jik = F jα F − βk Γ αiβ , K ij = J F − αi F − βj K αβ (3.11)for an unconstrained material. For an incompressible material J = 1 in theabove and (3.10) is replaced by ˙ T = A d + Γ ˙ B l + p d − ˙ p I , ˙ H l = Γ d + K ˙ B l , (3.12)and the incremental incompressibility condition isdiv u = 0 . (3.13)Notice that A and K inherit the symmetries of A and K , respectively,so that A jisk = A skji , K ij = K ji . (3.14)Finally, using the incremental form of the rotational balance condition F T = (
F T ) t , we find that Γ has the symmetryΓ ijk = Γ jik , (3.15)and we uncover the connections A jisk − A ijsk = τ js δ ik − τ is δ jk (3.16)between the components of the tensors A and τ for an unconstrained mate-rial (see, for example, Ogden [29] for the specialization of these in the purelyelastic case), and A jisk − A ijsk = ( τ js + pδ js ) δ ik − ( τ is + pδ is ) δ jk (3.17)9or incompressible materials (see Chadwick [30] for the elastic specialization).Following Prikazchikov [31], we decompose the tensor A into the sum A = A (0)0 + A (5)0 + A (6)0 . (3.18)The first term A (0)0 does not involve any derivatives with respect to I , I , and I . Clearly, this term is very similar to the tensor of elastic moduli associatedwith isotropic elasticity in the absence of magnetic fields. In component formit is given by J A (0)0 jisk = 4 b ij b ks Ω + 4 N ij N ks Ω + 4 J δ ij δ ks Ω + 4( b ks N ij + b ij N ks )Ω +4( b ks δ ij + b ij δ ks )Ω + 4 J ( N ks δ ij + N ij δ ks )Ω +2 δ ik b js Ω + 2(2 b ij b ks + δ ik N js − b jk b is − b ik b js )Ω +2 J (2 δ ij δ ks − δ is δ jk )Ω , (3.19)where N ij = b kk b ij − b ik b kj (3.20)and b ij are the components of b .The terms A (5)0 jisk and A (6)0 jisk may be expressed in the forms A (5)0 jisk = A jisk Ω + X m =1 , m =4 A m (5)0 jisk Ω m , A (6)0 αiβj = A jisk Ω + X m =1 , m =4 A m (6)0 jisk Ω m , (3.21)where A jisk = 0 and A jisk = 2 J − a j a s δ ik , A jisk = 4 J − ( a k a s b ij + a i a j b ks ) , A jisk = 4 J − ( a k a s N ij + a i a j N ks ) , A jisk = 4 J ( a k a s δ ij + a i a j δ ks ) , A jisk = 4 J − a j a i a s a k , A jisk = 2 J − ( a j a i H ks + a k a s H ij ) , (3.22)with H ij = a j a k b ik + a i a k b jk , a i = F iα B l α . (3.23)Similarly, A jisk = 0 and A jisk = 2 J − ( δ ik H js + a i a s b jk + a j a k b is + a j a s b ik + a i a k b js ) , A jisk = 4 J − ( b ks H ij + b ij H ks ) , A jisk = 4 J − ( H ij N ks + H ks N ij ) , A jisk = 4 J ( H ks δ ij + H ij δ ks ) , A jisk = 2 J − ( a i a j H ks + a k a s H ij ) , A jisk = 4 J − H ij H ks . (3.24)10he tensor Γ is decomposed as Γ = Γ (1)0 + Γ (2)0 + Γ (3)0 + Γ (5)0 + Γ (6)0 , (3.25)with components given byΓ (1)0 jik = 4 b ij M k , Γ (2)0 jik = 4 N ij M k , Γ (3)0 jik = 4 J δ ij M k , Γ (5)0 jik = 4 a j a i M k + 2( a j δ ik + a i δ jk )Ω , Γ (6)0 jik = 4 H ij M k + 2( δ ik a s b js + a i b jk + δ jk a s b is + a j b ik )Ω , (3.26)where M ik = F − αk B l α Ω i + a k Ω i + a j b jk Ω i . (3.27)Finally, we represent K in the form K = K (4)0 + K (5)0 + K (6)0 , (3.28)with components K (4)0 ij = 2 J F − αi (2 B l α M j + F − αj )Ω , K (5)0 ij = 2 J (2 a i M j + δ ij Ω ) , K (6)0 ij = 2 J (2 a k b ik M j + b ij Ω ) . (3.29)For an incompressible material, the above expressions are unaltered ex-cept that J = 1 and all the terms Ω and Ω n , n = 1 , . . . , A ijkl , Γ ijk , K ij are omitted. The standard relation B = µ H in vacuum is incremented to ˙ B ⋆ = µ ˙ H ⋆ , (3.30)where ˙ B ⋆ and ˙ H ⋆ are the increments of B ⋆ and H ⋆ , respectively. Thesefields satisfy Maxwell’s equationsdiv ˙ B ⋆ = 0 , curl ˙ H ⋆ = . (3.31)Finally, we increment the Maxwell stress of (2.22) to ˙ τ ⋆ = µ − [ ˙ B ⋆ ⊗ B ⋆ + B ⋆ ⊗ ˙ B ⋆ − ( B ⋆ · ˙ B ⋆ ) I ] , (3.32)noting that div ˙ τ ⋆ = . 11 .3 Incremental boundary conditions At the boundary of the material, in addition to any applied traction t a (de-fined per unit reference area), there will in general be a contribution fromthe Maxwell stress exterior to the material. This is a traction τ ⋆ n per unitcurrent area and can be ‘pulled back’ to the reference configuration to givea traction J τ ⋆ F − t N per unit reference area, in which case the boundarycondition (2.5) is modified to T t N = J τ ⋆ F − t N + t a . (3.33)On taking the increment of this equation, we obtain ˙ T t N = J ˙ τ ⋆ F − t N − J τ ⋆ F − t ˙ F t F − t + ˙ J τ ⋆ F − t N + ˙ t a , (3.34)and hence, on updating this from the reference configuration to the currentconfiguration, ˙ T t n = ˙ τ ⋆ n − τ ⋆ d t n + (div u ) τ ⋆ n + ˙ t a . (3.35)Proceeding in a similar fashion for the other fields, we increment themagnetic boundary conditions (2.11) to give, again after updating,( ˙ B l + dB ⋆ − (div u ) B ⋆ − ˙ B ⋆ ) · n = 0 (3.36)and ( ˙ H l − d t H ⋆ − ˙ H ⋆ ) × n = . (3.37) Here we summarize the basic equations for the pure homogeneous deforma-tion of a half-space in the presence of a magnetic field normal to its boundaryprior to considering a superimposed incremental deformation in Section 5.
Let X , X , X be rectangular Cartesian coordinates in the undeformedhalf-space B and take X = 0 to be the boundary ∂ B , with the materialoccupying the domain X ≥
0. In order to minimize the number of param-eters, we consider the material to be incompressible and subject to a plane12train in the ( X , X ) plane. With respect to the Cartesian axes, the defor-mation is then defined by x = λX , x = λ − X , x = X . The componentsof the deformation gradient tensor F and the right Cauchy-Green tensor c are written F and c , respectively, and are given by F = λ λ −
00 0 1 , c = λ λ −
00 0 1 , (4.1)where λ is the principal stretch in the X direction. The invariants I and I are therefore I = I = 1 + λ + λ − . (4.2)We take the magnetic induction vector B to be in the x direction andto be independent of x and x . It then follows from div B = 0 that itscomponent B is constant. Thus, B = 0 , B = 0 , B = 0 . (4.3)The associated Lagrangian field B l = F − B then has components B l = 0 , B l = λB , B l = 0 , (4.4)and the invariants involving the magnetic field are I = B l , I = λ − I , I = λ − I . (4.5)We may now compute the stress field using (2.17), (4.1) and (4.4). Theresulting non-zero components of τ are τ = 2Ω λ + 2Ω ( λ + 1) − p,τ = 2Ω λ − + 2Ω (1 + λ − ) − p + 2Ω λ − I + 4Ω λ − I ,τ = 2Ω + 2Ω ( λ + λ − ) − p. (4.6)The magnetic field H has components given by (2.19) as H = 0 , H = 2(Ω + λ − Ω + λ − Ω ) λB l , H = 0 . (4.7)Since B l and λ are constant, all the fields are uniform and the equilibriumequations and Maxwell’s equations are satisfied.In view of (4.2) and (4.5), there are only two independent variables, λ and I . We thus introduce a specialization ω ( λ, I ) of the total energy Ω, bythe definition ω ( λ, I ) = Ω(1 + λ + λ − , λ + λ − , I , λ − I , λ − I ) , (4.8)13rom which it follows that ω λ = 2 λ − [( λ − λ − )(Ω + Ω ) − λ − I Ω − λ − I Ω ] ,ω = Ω + λ − Ω + λ − Ω , (4.9)where ω λ = ∂ω/∂λ, ω = ∂ω/∂I . Hence, τ − τ = λω λ , H = 2 λB l ω . (4.10) From the boundary conditions (2.8) applied at the interface x = X = 0,we have B ⋆ = B and H ⋆ = H ⋆ = 0, while from (2.21) it follows that B ⋆ = B ⋆ = 0 and H ⋆ = µ − B ⋆ = µ − B . Outside the material we takethe magnetic field to be uniform and equal to its interface value, Maxwell’sequations are then satisfied identically, B ⋆ therefore has components B ⋆ = 0 , B ⋆ = B = λ − B l , B ⋆ = 0 , (4.11)and H ⋆ has components H ⋆ = 0 , H ⋆ = µ − B = µ − λ − B l , H ⋆ = 0 . (4.12)From these expressions, we deduce that the non-zero components of theMaxwell stress (2.22) are given by τ ⋆ = − τ ⋆ = − µ − B = − µ − λ − I = τ ⋆ . (4.13)The applied mechanical traction on x = 0 required to maintain the planestrain deformation has a single non-zero component τ − τ ⋆ . We now address the question of surface stability for the deformed half-spaceby establishing a bifurcation criterion based on the incremental static solutionof the boundary-value problem. Biot [32] initiated this approach, which hassince been successfully applied to a great variety of boundary-value problems;see Ogden [33] for pointers to the vast literature on the subject.14 .1 Magnetoelastic moduli
First we note that since F ij = 0 for i = j and B l = B l = 0 several simpli-fications occur in the expressions for the components of the magnetoelasticmoduli tensors A , Γ , K . In particular, we have A iijk = 0 , K ij = 0 , for j = k, Γ ii = Γ ii = Γ ii = Γ ii = 0 , Γ ijk = 0 , for i = j = k = i. (5.1)For subsequent use we compute the quantities a = A , b = A + A − A − A , c = A ,d = Γ , e = Γ − Γ , f = K , g = K . (5.2)Explicitly, we obtain a = 2 λ (Ω + Ω ) + 2 I Ω ,b = ( λ + λ − )(Ω + Ω ) + I [ λ − Ω + (6 λ − − ]+ 2( λ + λ − − + 2Ω + Ω )+ 4 I ( λ − − + Ω + 2 λ − (Ω + Ω )]+ 2 I λ − (Ω + 4 λ − Ω + 4 λ − Ω ) ,c = 2 λ − (Ω + Ω ) + 2 I [ λ − Ω + (2 λ − + 1)Ω ] ,d = 2 B l λ [ λ − Ω + ( λ − + 1)Ω ] ,e = 4 B l λ − [Ω + 2 λ − Ω + (1 − λ )(Ω + Ω )+ ( λ − − λ )(Ω + Ω ) + ( λ − − + Ω )+ I (Ω + λ − Ω + 2 λ − Ω + 3 λ − Ω + 2 λ − Ω )] ,f = 2( λ − Ω + Ω + λ Ω ) g = 2( λ Ω + Ω + λ − Ω ) + 4 I ( λ Ω + 2Ω + 2 λ − Ω + λ − Ω + 2 λ − Ω + λ − Ω ) . (5.3)In terms of the energy density ω ( λ, I ) we have the connections a − c = λω λ , b + c ) = λ ω λλ , e = − B l λ ω λ , g = 2 λ ( ω + 2 I ω ) , (5.4)where ω λλ = ∂ ω/∂λ , ω λ = ∂ ω/∂λ∂I and ω = ∂ ω/∂I . We seek incremental solutions depending only on the in-plane variables x and x such that u = 0 and ˙ B l = 0. Hence u i = u i ( x , x ) and ˙ B l i =15 B l i ( x , x ) for i = 1 , p = ˙ p ( x , x ). In the following, a subscriptedcomma followed by an index i signifies partial differentiation with respect to x i , i = 1 , u , + u , = 0 , (5.5)and hence there exists a function ψ = ψ ( x , x ) such that u = ψ , , u = − ψ , . (5.6)Similarly, equation (3.9) reduces to˙ B l , + ˙ B l , = 0 , (5.7)and the function φ = φ ( x , x ) is introduced such that˙ B l = φ , , ˙ B l = − φ , . (5.8)The incremental equations of equilibrium (3.9) simplify to˙ T , + ˙ T , = 0 , ˙ T , + ˙ T , = 0 . (5.9)From the identities (5.1), the only non-zero components of the incrementalstress ˙ T are found to be˙ T = ( A + p ) u , + A u , + ˙ B Γ − ˙ p, ˙ T = ( A + p ) u , + A u , + ˙ B Γ , ˙ T = ( A + p ) u , + A u , + ˙ B Γ , ˙ T = ( A + p ) u , + A u , + ˙ B Γ − ˙ p. (5.10)Also, equation (3.9) reduces to˙ H l , − ˙ H l , = 0 , (5.11)wherein are the only non-zero components of ˙ H l , which, from (5.1), are givenby˙ H l = Γ ( u , + u , ) + K ˙ B l , ˙ H l = Γ u , + Γ u , + K ˙ B l . (5.12)In terms of the functions ψ and φ equations (5.9) and (5.11) become( A − A − A ) ψ , + A ψ , − Γ φ , + Γ φ , = ˙ p , , ( A − A − A ) ψ , + A ψ , − (Γ − Γ ) φ , = − ˙ p , , (Γ − Γ − Γ ) ψ , + Γ ψ , + K φ , + K φ , = 0 . (5.13)16e eliminate ˙ p from the first two equations by cross-differentiation and ad-dition and obtain finally the coupled equations aψ , + 2 bψ , + cψ , + ( e − d ) φ , + dφ , = 0 (5.14)and dψ , + ( e − d ) ψ , + f φ , + gφ , = 0 (5.15)for ψ and φ . In vacuum, Maxwell’s equations (3.31) hold for ˙ B and ˙ H . From the secondequation, and the assumption that all fields depend only on x and x , wededuce the existence of a scalar function φ ⋆ = φ ⋆ ( x , x ) such that˙ H ⋆ = − φ ⋆, , ˙ H ⋆ = − φ ⋆, , ˙ H ⋆ = 0 . (5.16)Equation (3.30) then gives˙ B ⋆ = − µ φ ⋆, , ˙ B ⋆ = − µ φ ⋆, , ˙ B ⋆ = 0 , (5.17)and from (3.31) we obtain the equation φ ⋆, + φ ⋆, = 0 (5.18)for φ ⋆ . Finally, the incremental Maxwell stress tensor (3.32) has non-zerocomponents˙ τ ⋆ = λ − B l φ ⋆, = ˙ τ ⋆ = − ˙ τ ⋆ , ˙ τ ⋆ = − λ − B l φ ⋆, = ˙ τ ⋆ . (5.19) We now specialize the general incremental boundary conditions of Section 3.3to the present deformed semi-infinite solid. First, for ˙ t a = , the incrementaltraction boundary conditions (3.35) reduce to˙ T + τ ⋆ u , − ˙ τ ⋆ = 0 , ˙ T + τ ⋆ u , − ˙ τ ⋆ = 0 , (5.20)on x = 0. Putting together the results of this section, using (4.13), (5.2),(5.6), (5.8), (5.10), (5.16) and (5.19), we express the two equations (5.20) as( τ + µ − λ − I − c ) ψ , + cψ , + dφ , + λ − B l φ ⋆, = 0 , (5.21)17nd(2 b + c − τ + µ − λ − I ) ψ , + cψ , + eφ , + dφ , − λ − B l φ ⋆, = 0 , (5.22)which apply on x = 0. In obtaining the latter we have differentiated (5.20) with respect to x and made use of (5.13) .Next, the incremental magnetic boundary conditions (3.36) and (3.37)reduce to ˙ B l + B ⋆ u , − ˙ B ⋆ = 0 , ˙ H l − H ⋆ u , − ˙ H ⋆ = 0 (5.23)on x = 0. Using again the results of the preceding sections, we write theseas λ − B l ψ , + φ , − µ φ ⋆, = 0 , (5.24)and ( µ − λ − B l − d ) ψ , + dψ , + f φ , + φ ⋆, = 0 (5.25)on x = 0. We are now in a position to solve the incremental boundary value problem.We seek small-amplitude solutions, localized near the interface x = 0. Hencewe take solutions in the solid ( x ≥
0) to be of the form ψ = Ae − ksx e i kx , φ = kDe − ksx e i kx , (5.26)where k > π/k is the wavelength of the perturbation) and s is such that ℜ ( s ) > x ( > cs − bs + a ) A − s ( ds + d − e ) D = 0 ,s ( ds + d − e ) A − ( f s − g ) D = 0 . (5.28)For non-trivial solutions to exist, the determinant of coefficients of A and D must vanish, which yields a cubic in s , namely( cf − d ) s − [2 bf + cg + 2( d − e ) d ] s + [2 bg + af − ( d − e ) ] s − ag = 0 . (5.29)18rom the six possible roots we select s , s , s to be the three roots satisfying(5.27). We then construct the general solution for the solid as ψ = X j =1 A j e − ks j x e i kx , φ = k X j =1 D j e − ks j x e i kx , (5.30)where A j , D j , j = 1 , ,
3, are constants.For the half-space x ≤ φ ⋆ to (5.18) thatis localized near the interface x = 0. Specifically, we write this as φ ⋆ = i kC ⋆ e kx e i kx , (5.31)where C ⋆ is a constant.The constants A j and D j are related through either equation in (5.28).From the second equation, for instance, we obtain s j ( ds j + d − e ) A j + ( f s j − g ) D j = 0 , j = 1 , ,
3; no summation . (5.32)We also have the two traction boundary conditions (5.21) and (5.22), whichread( c − τ − µ − λ − I )( A + A + A ) + c ( s A + s A + s A ) − d ( s D + s D + s D ) − λ − B l C ⋆ = 0 , (5.33)and( τ − µ − λ − I − b − c )( s A + s A + s A )+ c ( s A + s A + s A ) + ( e − ds ) D + ( e − ds ) D + ( e − ds ) D − λ − B l C ⋆ = 0 . (5.34)Finally, the two magnetic boundary conditions (5.24) and (5.25) become λ − B l ( s A + s A + s A ) − ( D + D + D ) + µ C ⋆ = 0 , (5.35)and( d − µ − λ − B l )( A + A + A ) + d ( s A + s A + s A ) − f ( s D + s D + s D ) − C ⋆ = 0 . (5.36)In total, there are seven homogeneous linear equations for the seven un-knowns A j , D j , j = 1 , ,
3, and C ⋆ . The resulting determinant of coefficientsmust vanish and this equation is rather formidable to solve, particularly since19t must be solved in conjunction with the bicubic (5.29). It is in principle pos-sible to express the determinant in terms of the sums and products s + s + s , s s + s s + s s , s s s , and to find these from the bicubic (5.29), similarlyto the analysis conducted in the purely elastic case (see Destrade et al. [34]).However, the resulting algebraic expressions rapidly become too cumbersomefor this approach to be pursued.Instead, we propose either(a) to turn directly to a numerical treatment once Ω has been determinedby curve fitting from experimental data for a given magnetoelastic solid,or(b) to use a simple form for Ω that allows some progress to be made.Regarding approach (a), we remark that, as emphasized by Dorfmannand Ogden [12, 22, 23, 24], there is a shortage of, and a pressing need for,suitable experimental data and for the derivation of functions Ω from suchdata. In the next section we focus primarily on the analytical approach (b). As a prototype for the energy function Ω, we proposeΩ = µ (0)[(1 + γ )( I −
3) + (1 − γ )( I − µ − ( αI + βI ) , (5.37)where µ (0) is the shear modulus of the material in the absence of magneticfields and α , β , γ are dimensionless material constants, α and β being mag-netoelastic coupling parameters. For α = β = 0, (5.37) reduces to the strainenergy of the elastic Mooney-Rivlin material, a model often used for elas-tomers.In respect of (5.37) the stress τ in (2.17) reduces to τ = − p I + µ (1 + γ ) b + µ (1 − γ )( I b − b ) + 2 µ − β B ⊗ B , (5.38)while H in (2.19) becomes H = 2 µ − ( α b − B + β B ) . (5.39)Clearly, equation (5.38) shows that the parameter α does not affect the stress.By contrast β , if positive, stiffens the material in the direction of the magneticfield, i.e. a larger normal stress in this direction is required to achieve a givenextension in this direction than would be the case without the magneticfield. On the other hand, by reference to (5.39), we see that α provides a20easure of how the magnetic properties of the material are influenced by thedeformation (through b ). If β = 0 the stress is unaffected by the magneticfield. On the other hand, if α = 0 then the magnetic constitutive equation(5.39) is unaffected by the deformation. Thus, a two-way coupling requiresinclusion of both constants.The quantities defined in (5.2) and (5.3) now reduce to a = µ (0) λ , b = µ (0)( λ + λ − + βλ − ¯ I ) , c = µ (0)( λ − + βλ − ¯ I ) ,d = q µ − µ (0) βλ − ¯ B l , e = 2 q µ − µ (0) βλ − ¯ B l ,f = µ − ( αλ − + β ) , g = µ − ( αλ + β ) , (5.40)where ¯ B l , a dimensionless measure of the magnetic induction vector ampli-tude, and ¯ I are defined by¯ B l = B l / p µ µ (0) , ¯ I = ¯ B l . (5.41)Note the connections 2 b = a + c, e = 2 d. (5.42)Now we find that the bicubic (5.29) factorizes in the form( s − s − λ )[ αλ + βλ − ( α + βλ + αβ ¯ I ) s ] = 0 , (5.43)and it follows that the relevant roots are s = 1 , s = λ , s = λ s αλ + βα + βλ + αβ ¯ I . (5.44)Note that for s to be real for all λ > B l , the inequalities α ≥ , β > α > , β ≥ α = β = 0.) It is assumed here that these inequalities are satisfied, so that s is indeed a qualifying root satisfying (5.27).The equation (5.32) becomes s j ( s j − βλ − ¯ B l ˆ A j − [( αλ − + β ) s j − αλ − β ] ˆ D j = 0 , j = 1 , , , (5.46)where ˆ A j = q µ − µ (0) A j , ˆ D j = µ − D j , (5.47)and the s j are given by (5.44). 21ext, consider the four remaining boundary conditions (5.33)–(5.36). Inorder to keep the number of parameters to a minimum (so far, we have λ , ¯ B l , α , β ), and to make a simple connection with known results for the surfacestability of an elastic Mooney-Rivlin material, we assume that there is noapplied mechanical traction on the boundary x = 0, and hence τ = τ ⋆ = µ − λ − I . (5.48)The boundary conditions (5.21)–(5.25) now read[1 + ( β −
1) ¯ I ]( ˆ A + ˆ A + ˆ A ) + (1 + β ¯ I )( s ˆ A + s ˆ A + s ˆ A ) − βλ ¯ B l ( s ˆ D + s ˆ D + s ˆ D ) − λ ¯ B l C ⋆ = 0 , ( λ + 2 + 2 β ¯ I )( s ˆ A + s ˆ A + s ˆ A ) − (1 + β ¯ I )( s ˆ A + s ˆ A + s ˆ A )+ βλ ¯ B l [( s −
2) ˆ D + ( s −
2) ˆ D + ( s −
2) ˆ D ] + λ ¯ B l C ⋆ = 0 , ¯ B l ( s ˆ A + s ˆ A + s ˆ A ) − λ ( ˆ D + ˆ D + ˆ D ) + λC ⋆ = 0 ,λ ¯ B l ( β − A + ˆ A + ˆ A ) + λβ ¯ B l ( s ˆ A + s ˆ A + s ˆ A ) − ( α + βλ )( s ˆ D + s ˆ D + s ˆ D ) − λ C ⋆ = 0 . (5.49)From the seven equations (5.46) and (5.49), we have derived a bifurcationcriterion (vanishing of the determinant of coefficients) using a computer alge-bra package, but it is too long to reproduce here. It is a complicated rationalfunction of the four parameters λ , ¯ B l , α , β . However, it is easy to solvenumerically, and for the numerical examples we fix the material parameters α and β and find the critical stretch λ cr in compression as a function of ¯ B l .For ¯ B l = 0, we recovered the well-known critical compression stretch for sur-face instability of the elastic Mooney-Rivlin material in plane strain, namely λ cr = 0 . α = 0 . α = 2 .
0) and curves for β = 0 . , . , . , . , . λ cr is an even function of ¯ B l and we therefore restricted attention to positive¯ B l (within the range 0 ≤ ¯ B l ≤ B l becomes largerand larger (not shown here) indicates that the half-space becomes more andmore unstable in compression. Moreover, it can even become unstable intension ( λ cr > α , β , and ¯ B l the critical stretch ratio is smaller than that for the purelyelastic case ( λ cr < . µ (0), α , β , γ , two of which, µ (0)22nd β , may be determined from shear tests. Indeed Dorfmann and Ogden[24] show that in general the shear modulus for isotropic nonlinear magne-toelasticity is 2[Ω + Ω + I Ω + I Ω (3 + 2 κ )], where κ is the amount ofshear in a simple shear test. Here the modulus is independent of κ and isgiven by µ ( B l ) = µ (0) + 2 µ − βI . (5.50)This highlights the role of β in increasing the mechanical stiffness of thematerial — through the shear modulus. Jolly et al. [9] conducted doublelap shear tests on magneto-sensitive elastomers containing 10, 20, and 30%by volume of iron particles. From their Figure 7, we see that in the range0 ≤ B l ≤ . µ ( B l ) resemble those of a parabolicprofile such as the one suggested by (5.50). For the 10% iron by volumeelastomer specimen, Table 1 in Jolly et al. [9] gives µ (0) = 0 .
26 MPa, andat B l = 0 . µ (0 . − µ (0) ≃ .
07 MPa,indicating that β ≃ .
18. Similarly, for the 20% and the 30% iron by volumeelastomer specimens we find β ≃ .
53 and β ≃ .
72, respectively.Figure 2a (Figure 2b) illustrates the variation of the critical compressionstretch with the amplitude of the dimensional magnetic induction vector,from 0 to 0.5 Tesla, for the 20% (30%) iron by volume elastomer, and forseveral values of α . We remark than the presence of the magnetic fieldmakes the two specimens slightly more stable than in the purely elastic casebecause all the critical compression stretch values are smaller than 0.5437. Itis also clear that increasing the value of α makes the half-space more stable.However, it is worth noting that the 30% iron by volume specimen is slightlyless stable than the 20% iron by volume specimen for the same values of α . Figure 2 A Derivatives of the invariants with respectto F and B l We derive the expressions for the first derivatives of the six invariants withrespect to F , ∂I ∂F iα = 2 F iα , ∂I ∂F iα = 2( c γγ F iα − c αγ F iγ ) ,∂I ∂F iα = 2 I F − αi , ∂I ∂F iα = 0 , ∂I ∂F iα = 2 B lα ( F iγ B lγ ) ,∂I ∂F iα = 2( F iγ B lγ c αβ B lβ + F iγ c γβ B lβ B lα ) , (A.1)23nd with respect to B l , ∂I ∂B lα = 0 , ∂I ∂B lα = 0 , ∂I ∂B lα = 0 ,∂I ∂B lα = 2 B lα , ∂I ∂B lα = 2 c αβ B lβ , ∂I ∂B lα = 2 c αγ c γβ B lβ . (A.2)The second derivatives of the invariants are computed as follows: first,the second derivatives with respect to F , ∂ I ∂F iα ∂F jβ = 2 δ ij δ αβ ,∂ I ∂F iα ∂F jβ = 2(2 F iα F jβ − F iβ F jα + c γγ δ ij δ αβ − b ij δ αβ − c αβ δ ij ) ,∂ I ∂F iα ∂F jβ = 4 I F − αi F − βj − I F − αj F − βi ,∂ I ∂F iα ∂F jβ = 0 ,∂ I ∂F iα ∂F jβ = 2 δ ij B lα B lβ ,∂ I ∂F iα ∂F jβ = 2[ δ ij ( c αγ B lγ B lβ + c βγ B lγ B lα ) + δ αβ F iγ B lγ F jδ B lδ + F iγ B lγ F jα B lβ + F jγ B lγ F iβ B lα + b ij B lα B lβ ];(A.3)next, the mixed derivatives with respect to F and B l , ∂ I ∂F iα ∂B l β = 0 , ∂ I ∂F iα ∂B l β = 0 , ∂ I ∂F iα ∂B l β = 0 , ∂ I ∂F iα ∂B l β = 0 ,∂ I ∂F iα ∂B l β = 2 δ αβ F iγ B l γ + 2 B l α F iβ ,∂ I ∂F iα ∂B l β = 2 F iβ c αγ B l γ + 2 F iγ B l γ c αβ + 2 F iγ c γβ B l α + 2 δ αβ F iγ c γδ B lδ ; (A.4)finally, the second derivatives with respect to B l , ∂ I ∂B lα ∂B lβ = 0 , ∂ I ∂B lα ∂B lβ = 0 , ∂ I ∂B lα ∂B lβ = 0 ,∂ I ∂B lα ∂B lβ = 2 δ αβ , ∂ I ∂B lα ∂B lβ = 2 c αβ , ∂ I ∂B lα ∂B lβ = 2 c αγ c γβ . 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Dependence of the critical stretch λ cr < B l of the magnetic field, for several values of the mag-netoelastic coupling parameters α and β .28 igures b = 1.0b = 1.5b = 0.5b = 2.0b = 0.0l cr B a = 0.5 (a) α = 0 . b = 1.0b = 1.5b = 0.5b = 2.0b = 0.0a = 2.0l cr B (b) α = 2 . Figure 1: Dependence of the critical stretch λ cr < B l of the magnetic field for several values of themagnetoelastic coupling parameters α and β .29 .460.480.50.520.54 0 0.1 0.2 0.3 0.4 0.5 l a = 0.1a = 0.5a = 2.0a = 7.0 cr B (T) (a) β = 0 . l a = 0.1a = 0.5a = 2.0a = 7.0 cr B (T) (b) β = 0 . Figure 2: Dependence of the critical stretch λ cr < B l of the magnetic field, for several values of the mag-netoelastic coupling parameters α and ββ