Variational principles of nonlinear magnetoelastostatics and their correspondences
VVariational principles of nonlinearmagnetoelastostatics and their correspondences
Basant Lal Sharma , Prashant Saxena ∗ Department of Mechanical Engineering, Indian Institute of Technology KanpurKanpur, Uttar Pradesh 208016, India Glagsow Computational Engineering Centre, James Watt School of EngineeringUniversity of Glasgow, Glasgow G12 8LT, UK
Abstract
We derive the equations of nonlinear magnetoelastostatics using severalvariational formulations involving the mechanical deformation and an inde-pendent field representing the magnetic component. An equivalence is alsodiscussed, modulo certain boundary integrals or constant integrals, betweenthese formulations using the Legendre transform and properties of Maxwell’sequations. The second variation based bifurcation equations are stated forthe incremental fields as well for all five variational principles. When the to-tal potential energy is defined over the infinite space surrounding the body,we find that the inclusion of certain term in the energy principle, associatedwith the externally applied magnetic field, leads to slight changes in theMaxwell stress tensor and associated boundary conditions. On the otherhand, when the energy contained in the magnetic field is restricted to fi-nite volumes, we find that there is a correspondence between the discussedformulations and associated expressions of physical entities. In view of adiverse set of boundary data and nature of externally applied controls inthe problems studied in the literature, along with a equally diverse list ofvariational principles employed in modeling, our analysis emphasizes carein the choice of variational principle and unknown fields so that consistencywith other choices is also satisfied. ∗ Corresponding author email: [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] S e p onlinear magnetoelastostatics Introduction
Magnetoelastostatics concerns the analysis of suitable phenomenological modelsfor a physical description of the equilibrium in a certain type of deformable solidsassociated with multi-functional processes involving magnetic and elastic effects.The main property characterizing these solids is the coupling between elastic de-formation and magnetisation that they experience in the presence of externallyapplied mechanical as well as magnetic force fields (Jolly et al., 1996; Lokanderand Stenberg, 2003; Boczkowska et al., 2010; Danas et al., 2012). The so calledmagnetoelastic coupling is known to occur due to a phenomenon involving re-configurations of small magnetic domains while a continuum vector field is borneout of an averaging of microscopic and distributed sub-fields (Chatzigeorgiou et al.,2014; Kovetz, 1990). Thus an imposition of the magnetic field also induces a de-formation of the material specimen in addition to the magnetic effects caused bythe traditional mechanical forces (Brown, 1966).With a history of more than five decades (Tiersten, 1964, 1965; Maugin andEringen, 1972; Maugin, 1988; DeSimone and James, 2002; Kankanala and Tri-antafyllidis, 2004; Dorfmann and Ogden, 2004), the mathematical modeling ofmagnetoelasticity continues to be a vibrant area of research. The presence of strongmagnetoelastic coupling in some manufactured materials such as magnetorheolog-ical elastomers (MREs) (Jolly et al., 1996) allows the subject to be relevant for alarge number of potential engineering and technological applications. MREs arecomposites made of ferromagnetic particles embedded in a polymer matrix. Mag-netization of the ferromagnetic domains in the presence of an external magneticfield and the resulting interactions leads to a change in macroscopically observablemechanical properties. As a result they find applications in micro-roboticss (Huet al., 2018; Ren et al., 2019), sensors and actuators (B¨ose et al., 2012; Psarraet al., 2017), active vibration control (Ginder et al., 2000), and waveguides (Sax-ena, 2018; Karami Mohammadi et al., 2019). Constitutive modelling of MREs hasbeen undertaken by appropriately considering the micromechanics and derivationof coupled field equations using homogenization (Ponte Casta˜neda and Galipeau,2011; Chatzigeorgiou et al., 2014); consideration of energy dissipation due to vis-coelasticity of underlying matrix (Saxena et al., 2013, 2014; Ethiraj and Miehe,2016; Haldar et al., 2016) ; and consideration of anisotropy due to ferromagneticparticle alignment (Bustamante, 2010; Danas et al., 2012; Saxena et al., 2015).Derivation of a consistent set of partial differential equations (PDEs) andboundary conditions that describe equilibrium, analysis of the stability of equi-librium, and solution of the relevant PDEs via numerical techniques such as thefinite element method requires development of appropriate variational principles.In this paper, we shall be concerned with the variational principles that have beenpostulated for the materials under the magnetoelastostatics assumptions and ig-2onlinear magnetoelastostaticsnore any dynamic or dissipative effects. Current variational principles of mag-netoelastostatics typically fall into two classes: principles based on the magneticfield or the magnetic induction as independent variable (Bustamante and Ogden,2012; Vogel et al., 2013) and principles based on a variant of the magnetizationas an independent variable (Kankanala and Triantafyllidis, 2004; Liu, 2014). Thetypical starting point, definition of the total potential energy, is different in allthese cases while it results in certain correspondence between the Euler–Lagrangeequations derived.The twofold motivation of this paper is the study of equations for the staticsproblem as well as the counterparts of bifurcation equations within the several vari-ational formulations. Within the magnetization based principles we discuss threedifferent formulations that utilize magnetization field per unit volume, magnetiza-tion per unit mass, and another adaption of magnetization field as an independententity. In fact, one of these variational principles analyzed in this paper was pos-tulated originally, very early, by (Brown, 1965, Eq. 8), while another one has beenutilized in the work of Kankanala and Triantafyllidis (2004) for a specific instabil-ity problem. In addition to these three magnetization based principles, two moreformulations are presented which are analogues of electroelastostatics as derivedin (Saxena and Sharma, 2020). For each of these variational principles, we derivethe equation of equilibrium as well as the equation for the description of a stateat bifurcation point. As part of the first variation based analysis, we find thatthe expression for the Maxwell stress is susceptible to inclusion of certain integralterms that define suitable magnetic energy over an infinite space; the peculiarsituation is however completely different from those formulations in which energyis defined over a finite domain of space. Moreover, we present certain argumentsbased on Legendre transform as well as application of divergence theorem (us-ing the properties of Maxwell fields) that suggest a direct equivalence betweenseemingly different formulations.
Outline
This paper is organised as follows. After briefly introducing the mathematicalpreliminaries, we introduce the system under study and present the basic equa-tions of nonlinear magnetoelastostatics in Section 1. In Sections 2–4, we presentthe first variation of the potential energy functional corresponding to three dif-ferent magnetization vectors M , M and K , respectively, and then derive or statethe equations for critical point by linearising the equilibrium equations. Someauxiliary details are presented in the first appendix. In Appendix sections B andC, we present the derivations of first and second variations of the potential energyfunctionals corresponding to the magnetic induction B and the magnetic field H ,respectively. 3onlinear magnetoelastostaticsTable 1: Notation x Position vector(spatial) X Position vector(referential)grad Gradient (spatial) Grad Gradient (referential)div Divergence (spatial) Div Divergence(referential)curl Curl (spatial) Curl Curl (referential) n Unit outward normal(spatial) n Unit outward normal(referential) h Magnetic field vector(spatial) H Magnetic field vector(referential) b Magnetic inductionvector (spatial) B Magnetic inductionvector (referential) m Magnetisation vectorper unit volume(spatial) M Magnetisation vectorper unit volume(referential) m Magnetisation vectorper unit mass(spatial) M Magnetisation vectorper unit mass(referential) ρ mass density (spatial) ρ mass density(referential) φ Magnetic scalarpotential (spatial) Φ Magnetic scalarpotential (referential) a Magnetic vectorpotential (spatial) A Magnetic vectorpotential (referential) σ Cauchy stress tensor P First Piola–Kirchhoffstress tensor F Grad χ P m Maxwell stress tensor J det F K J F −(cid:62) M (cid:74) {·} (cid:75) Jump of a quantity {·} across a boundary (cid:74) {·} (cid:75) = {·} + − {·} − {·} , G Partial derivativewith respect to G Notation
We use the direct notation of tensor algebra and tensor calculus throughout thepaper. The scalar product of two vectors a and b is denoted as a · b = [ a ] i [ b ] i where a repeated index implies summation according to Einstein’s summationconvention. The vector (cross) product of two vectors a and b is denoted as a ∧ b with [ a ∧ b ] i = ε ijk [ a ] j [ b ] k , ε ijk being the permutation symbol. The tensor productof two vectors a and b is a second order tensor H = a ⊗ b with [ H ] ij = [ a ] i [ b ] j .Operation of a second order tensor H on a vector a is given by [ H a ] i = [ H ] ij [ a ] j .Scalar product of two tensors H and G is denoted as H · G = [ H ] ij [ G ] ij . (cid:107)·(cid:107) represents the usual (Euclidean) norm for the mentioned vector entity. A list ofkey variables employed throughout this manuscript is presented in Table 1.For tensor calculus and variational method, we refer to (Knowles, 1997; Itskov,2018) and (Gelfand and Fomin, 2003), respectively, whereas the notation anddefinitions of physical entities in continuum mechanics typically follow (Gurtin,1981). Consider a deformable body, in which its boundary or interior does not possess anydistributed dipoles, occupying a three dimensional region B lying inside anotherregion V as schematically depicted in Figure 1. We denote the region exterior tothe body, relative to V , by B (cid:48) so that B (cid:48) = V \ ( B ∪ ∂ B ) . We assume that thebody occupies a region B in its reference configuration while V is the referentialregion corresponding to V , as explained below. The points in regions B and B corresponding to the same material point of the body are naturally mapped intoeach other by the deformation function χ : B → B . (1.1)In order to make sense of the referential (Lagrangian) description of fields in currentregion V , but exterior to the body, in a meaningful manner, we also define anextension of the deformation function χ to the part of region exterior to the bodysuch that sufficient continuity requirements are maintained; the latter region isdenoted by B (cid:48) = V \ ( B ∪ ∂ B ) . Thus, by an abuse of notation, we assume an extension of mapping χ on a largerregion, also denoted by χ , i.e., χ : V → V . (1.2)5onlinear magnetoelastostaticsFigure 1: A representation of the problem depicting the body in its reference andcurrent configurations embedded in a volume V .In typical situations in practice, it is assumed that ∂ V and ∂ V coincide (forinstance this is the scenario depicted in Fig. 1).Following the standard notation in continuum mechanics, we define the defor-mation gradient for points in the reference configuration B and on its exteriorrelative to V as F : = Grad χ. The extension of the natural definition of deformation and its gradient associatedwith χ on B to V permits us later to perform some useful manipulations on thereference configuration as well as on the exterior of the body B (cid:48) in the referenceconfiguration.The magnetic field vector, magnetic induction vector, and the magnetisationvector are denoted in the reference configuration as ( H , B , M ), respectively, and inthe current configuration as ( h , b , m ), respectively. These three vector fields arerelated by the well known constitutive relation b = µ h + m . (1.3)Further, the vector fields ( h , b , m ) must satisfy the Maxwell’s equationsdiv b = 0 and curl h = in B ∪ B (cid:48) . (1.4)The divergence-free and curl-free conditions (1.4) for b and h , respectively, lead tothe existence of magnetic potential (vector) field a and magnetic potential (scalar)field φ on B ∪ B (cid:48) ; the respective expressions of b and h are given by b = curl a , h = − grad φ. (1.5)6onlinear magnetoelastostaticsFollowing tradition in continuum mechanics (Gurtin, 1981), let J denote the de-terminant of the deformation gradient, i.e., J = det F (note that J > B aswell as B (cid:48) ). The referential (Lagrangian) counterparts of b and h , defined by B = J F − b , H = F (cid:62) h , (1.6)naturally satisfy the Maxwell’s equations (1.4) in the reference configuration asDiv B = 0 and Curl H = in B ∪ B (cid:48) . (1.7)Suitable referential (Lagrangian) counterparts of the magnetic vector potential andmagnetic scalar potential (1.5) on B ∪ B (cid:48) , based on the referential equations (1.7),are given by B = Curl A , H = − Grad Φ . (1.8)Concerning notational issues, a typical point in B (as well as B (cid:48) ) is denoted by X ,which is related (after deformation) to the point in B (resp. B (cid:48) ) by the deformationfunction, assumed to be a sufficiently smooth mapping, χ such that x = χ ( X )and X = χ − ( x ) (Gurtin, 1981), i.e., X (cid:55)→ x , x (cid:55)→ X . (1.9)It can be shown using tensor algebra and calculus that A ( X ) = F (cid:62) ( X ) a ( x ) , Φ( X ) = φ ( x ) , (1.10)for all X ∈ B ∪ B (cid:48) . Upon substituting the transformations (1.6) into the consti-tutive relation (1.3), we obtain the relation J − C B = µ H + M , (1.11)where M denotes the referential (Lagrangian) magnetisation (per unit volume)vector field. Clearly, M is related to the current (spatial, Eulerian) magnetisation(per unit volume) vector field m by the definition (recall (1.9)) M ( X ) : = F (cid:62) ( X ) m ( x ) , (1.12)for all X ∈ B ∪ B (cid:48) (as m is zero in B (cid:48) , we also get vanishing M in B (cid:48) ). From thepoint of view of practical applications motivated by physics oriented models, it isalso useful to define the magnetisation (per unit mass) m : B → R . It is easy tosee that the defining relation is m ( x ) : = ρ ( x ) − m ( x ) , x ∈ B , (1.13)where ρ stands for the mass density, i.e., a scalar field on B . The referential(Lagrangian) counterpart of the spatial field m is denoted by M , which is definedby M ( X ) : = J − ( X ) F (cid:62) ( X ) m ( x ) , X ∈ B . (1.14)7onlinear magnetoelastostatics Remark 1.1.
When the density ρ in the reference configuration is a constant,in particular for a homogeneous body, it is easy to see that M and M are simplyproportional (i.e., M = ρ M as ρ = ρJ ). Holding the viewpoint of several practical applications where magnetoelasticmaterials are involved, in certain situations it is quite convenient to distinguishthe externally applied fields and the fields generated due to presence of the magne-toelastic body. In such a typical scenario, an external magnetic field h e is appliedthat results in the generation of a magnetic flux density b e with the relation b e = µ h e , (1.15)where µ is the (constant) magnetic permeability of vacuum. The presence of themagnetoelastic body creates a perturbation (sometimes described as the self-field )in the magnetic field that is denoted by h s and a corresponding self-field for themagnetic flux vector denoted by b s (Brown, 1966, Ch. 5). Remark 1.2.
In general, in this paper the decoration with superscript ‘ s ’ de-notes the self-field or stray field while the superscript ‘ e ’ denotes the externallyapplied entity. Thus, the total magnetic field and induction vector field is given by the sum h = h e + h s , b = b e + b s . (1.16)The relationship between the three magnetic vector fields b s , h s , and the magneti-sation per unit volume m , is naturally given by b s = µ h s + m , (1.17)which holds on account of the relations (1.15) and (1.16). Remark 1.3.
Concerning the units of magnetisation vector m , we note thatdefinition of the magnetisation vector is not standardised in literature anddepending on the choice of units either of m and µ m have been used. Thus,the constitutive equation relating the three magnetic variables is also sometimeswritten as b s = µ [ h s + m ] for a different set of units. A detailed discussionon this topic can be found in (Maugin, 1988). Consider the body B in its reference configuration (lying inside a containing space V ). Noting that H = − Grad Φ by (1.8) , it is assumed that the total potentialenergy of the system is a functional of the deformation χ (1.1) (and (1.2)) andthe referential magnetisation M (1.12) with the explicit expression given by (Liu,2014) E I [ χ, M ] : = (cid:90) B Ω( F , M ) dv + µ (cid:90) V J (cid:107) F −(cid:62) Grad Φ (cid:107) dv − (cid:90) B (cid:101) f e · χdv − (cid:90) ∂ B (cid:101) t e · χds + (cid:90) ∂ V φ e n · B ds , (2.1)where Ω is the (magnetoelastic) stored energy density per unit volume that de-pends on the deformation gradient F and the referential magnetisation vector M .Integrals in equation (2.1) are defined on the reference configuration and the spa-tial fields are mapped to the reference configuration by using the mapping χ asplacement. In this expression of the potential energy functional, it is assumedthat φ e stands for the externally applied magnetic potential on the boundary ofthe containing region V . Note that (cid:101) f e is the body force (vector) field per unitvolume while (cid:101) t e is the applied traction (vector) field due to dead loads at theboundary of the body in its current configuration; here also recall the notationdescribed in Remark 1.2. In order to describe the state of magnetoelastic equilibrium, the particular defor-mation χ and magnetisation M at such a equilibrium corresponds to a extremumpoint of E , that is, when the first variation of the potential energy functionalvanishes. In other words, it is assumed that χ and M satisfy δ E I ≡ δ E I [ χ, M ; ( δχ, δ M )] = 0 , (2.2)for arbitrary but admissible variations δχ and δ M . The variation of the potentialenergy functional E I up to the first order is given by δ E I = E I [ χ + δχ, M + δ M ] − E I [ χ, M ]= (cid:90) B [Ω , F · δ F + Ω , M · δ M ] dv − (cid:90) B (cid:101) f e · δχdv − (cid:90) ∂ B (cid:101) t e · δχds + (cid:90) V [ − (cid:98) P m · δ F − J µ [ C − H ] · Grad δ Φ] dv + (cid:90) ∂ V φ e n · δ B ds . (2.3)9onlinear magnetoelastostaticswhere (cid:98) P m is a tensor field defined by (cid:98) P m = µ J (cid:20) − (cid:2) F −(cid:62) H (cid:3) · (cid:2) F −(cid:62) H (cid:3) I + (cid:2) F −(cid:62) H (cid:3) ⊗ (cid:2) F −(cid:62) H (cid:3)(cid:21) F −(cid:62) , (2.4)where I is the identity tensor. We are able to understand the physical nature of (cid:98) P m by noticing that, in the region B (cid:48) exterior to the body, the magnetisation M = ;this results in (cid:98) P m = P m , where P m denotes the well known Maxwell stress tensordefined by P m : = 1 µ J (cid:20) [ F B ] ⊗ [ F B ] −
12 [ F B ] · [ F B ] I (cid:21) F −(cid:62) . (2.5)In order to further simplify the first variation expression (2.3), we apply thedivergence theorem on the last term and use the condition from a variation ofequation (1.7) that Div( δ B ) = 0 to get (cid:90) ∂ V n · φδ B ds = (cid:90) V Div ( φδ B ) dv = (cid:90) V Grad( φ ) · δ B dv = − (cid:90) V H · δ B dv . (2.6)At this point we recall several identities for variations of C , J , etc, from AppendixA. Using the constitutive relation (1.11), an increment of magnetic induction B up to first order can be written as δ B = [[ F −(cid:62) · δ F ] I − C − [ δ F ] (cid:62) F − F − [ δ F ]] B − µ J C − Grad δ Φ + J C − δ M . (2.7)Upon substituting (2.6) and (2.7) in the last term of equation (2.3), we thus obtain δ E I = (cid:90) B [Ω , F · δ F + Ω , M · δ M − (cid:101) f e · δχ ] dv − (cid:90) ∂ B (cid:101) t e · δχds + (cid:90) V [[˚ P m − (cid:98) P m ] · δ F − J C − H · δ M ] dv , (2.8)where we have defined the tensor˚ P m : =[ − [ B · H ] I + [ F B ] ⊗ [ F −(cid:62) H ] + [ F −(cid:62) H ] ⊗ [ F B ]] F −(cid:62) = 2 (cid:98) P m + J [ − [ C − M · H ] I + [ F −(cid:62) M ] ⊗ [ F −(cid:62) H ]+ [ F −(cid:62) H ] ⊗ [ F −(cid:62) M ]] F −(cid:62) . (2.9)10onlinear magnetoelastostaticsAs observed above, M = in the region B (cid:48) , which leads to ˚ P m = 2 P m .Upon splitting the (third term) integral over V in (2.8) to a sum of the integralson disjoint regions B and B (cid:48) , we obtain δ E I = (cid:90) B [[Ω F + ˚ P m − (cid:98) P m ] · δ F − (cid:101) f e · δχ + [Ω , M − J C − H ] · δ M ] dv − (cid:90) ∂ B (cid:101) t e · δχds + (cid:90) B (cid:48) P m · δ F dv . This is rewritten with the use of divergence theorem as δ E I = (cid:90) B [ − [Div(Ω , F + ˚ P m − (cid:98) P m ) + (cid:101) f e ] · δχ + [Ω , M − J C − H ] · δ M ] dv + (cid:90) ∂ B [[[Ω , F + ˚ P m − (cid:98) P m ] | − − P m | + ] n − (cid:101) t e ] · δχds − (cid:90) B (cid:48) Div P m · δχdv + (cid:90) ∂ V P m n · δχds . Following the traditional definition, at this point, by virtue of inspection of theabove form of the first variation of the potential energy functional, we define thefirst Piola–Kirchhoff stress in the body as P : = Ω , F + ˚ P m − (cid:98) P m , in B , (2.10)while we have the natural stress tensor, i.e. Maxwell stress, P = P m defined byequation (2.5) exterior to the body, i.e., in B (cid:48) . Remark 2.1.
The Cauchy stress σ in the body is related to the first Piola–Kirchhoff stress P by the Piola transform as σ cof( F ) = P ; also sometimesreferred as the Nanson’s relation. Upon using the relation (1.6) and the tensorfield stated as (2.5) , the counterpart σ m of the Cauchy stress σ in B (cid:48) (vacuum)is given by the expression σ = σ m = µ (cid:2) b ⊗ b − [ b · b ] I (cid:3) in B (cid:48) . Upon applying the condition (2.2) to the first variation calculated above, thecoefficients appearing with the arbitrary variations δχ and δ M also should vanishfor the requirement that δ E I must be zero at equilibrium (i.e., χ, M correspondingto a extremum point of E ). Vanishing of the coefficients of δ M results in thefollowing constitutive relation between H and MH = J − C Ω , M in B . (2.11)11onlinear magnetoelastostaticsUpon substituting the above expression for H in equations (2.4), (2.9), and (2.10)the total first Piola–Kirchhoff stress can be rewritten in terms of the independentquantities F and M as P = Ω , F + ˚ P m − (cid:98) P m (2.12)= Ω , F + µ J − [ −
12 Ω , M · (cid:2) C Ω , M (cid:3) I + F Ω , M ⊗ (cid:2) F Ω , M (cid:3) ] F −(cid:62) + [ − (cid:2) M · Ω , M (cid:3) I + F −(cid:62) M ⊗ F Ω , M + F Ω , M ⊗ F −(cid:62) M ] F −(cid:62) . (2.13)Also P = J ( J − Ω , F F (cid:62) + h ⊗ b − µ ( h · h ) I + { m ⊗ h − ( m · h ) I } ) F −(cid:62) , which differsfrom that given by (Kankanala and Triantafyllidis, 2004) (see their eq. (2.26)) dueto the presence of terms in the curly brackets. Vanishing of the coefficients of δχ results in the following equationsDiv P + (cid:101) f e = in B , (2.14a)Div P = in B (cid:48) , (2.14b) (cid:74) P (cid:75) n + (cid:101) t e = on ∂ B , (2.14c) P n = on ∂ V . (2.14d)Here (cid:74) {·} (cid:75) = {·} + − {·} − with plus sign representing that side of the boundary(surface) which is reached along the unit outward normal vector. Remark 2.2.
We note that in this formulation based on the magnetisationvector, we have to apriori use both the Maxwell’s equations (1.7) to imposeconditions on B and H unlike the two formulations based on B and H presentedin Appendix sections B and C. in which one condition was imposed and theother was derived. Also unlike those two formulations, stress does not have asimple expression of being a derivative of the stored energy density with respectto the deformation gradient tensor. The procedure implies the constitutiverelation (2.11) between H and M . In terms of the variations (cid:52) χ and (cid:52) M , we find the perturbation in the first Piola–Kirchhoff stress using equation (2.13) as (cid:52) P = Ω , FF (cid:52) F + 12 [Ω , F M + Ω ∗ F M ] (cid:52) M − µ J − [ F −(cid:62) · (cid:52) F ][ −
12 Ω , M · (cid:2) C Ω , M (cid:3) I + Ω , M ⊗ (cid:2) C Ω , M (cid:3) ] F −(cid:62) − µ J − [ −
12 Ω , M · (cid:2) C Ω , M (cid:3) I + Ω , M ⊗ (cid:2) C Ω , M (cid:3) ] F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) µ J − [ − (cid:20) F Ω , M · [ (cid:52) F Ω , M + F (cid:2) Ω , MM (cid:52) M + 12 Ω , M F (cid:52) F + 12 Ω ∗ M F (cid:52) F (cid:3) ] (cid:21) I + (cid:2) Ω , MM (cid:52) M + 12 Ω , M F (cid:52) F + 12 Ω ∗ M F (cid:52) F (cid:3) ⊗ (cid:2) C Ω , M (cid:3) + Ω , M ⊗ (cid:20) C (cid:2) Ω , MM (cid:52) M + 12 Ω , M F (cid:52) F + 12 Ω ∗ M F (cid:52) F (cid:3) + [[ (cid:52) F ] (cid:62) F + F (cid:62) (cid:52) F ]Ω , M (cid:21) ] F −(cid:62) − (cid:20) − (cid:2) M · Ω , M (cid:3) I + M ⊗ Ω , M + Ω , M ⊗ M (cid:21) F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) + (cid:18) − (cid:20) (cid:52) M · Ω , M + M · [Ω , MM (cid:52) M + 12 Ω , M F (cid:52) F + 12 Ω ∗ M F (cid:52) F ] (cid:21) I + (cid:52) M ⊗ Ω , M + M ⊗ [Ω , MM (cid:52) M + 12 Ω , M F (cid:52) F + 12 Ω ∗ M F (cid:52) F ]+ [Ω , MM (cid:52) M + 12 Ω , M F (cid:52) F + 12 Ω ∗ M F (cid:52) F ] ⊗ M + Ω , M ⊗ M (cid:19) F −(cid:62) , (2.15)where we have defined two third order tensors Ω ∗ F M and Ω ∗ M F which have thefollowing property[Ω ∗ F M u ] · U = [Ω , M F U ] · u , [Ω ∗ M F U ] · u = [Ω , F M u ] · U , (2.16) u being an arbitrary vector and U being an arbitrary second order tensor. Forthe bifurcation analysis of critical point ( χ, M ), using (2.14), the perturbations (cid:52) χ and (cid:52) M in the equilibrium state need to satisfy the following following partialdifferential equations and boundary conditions:Div (cid:52) P = in B , (2.17a)Div (cid:52) P = in B (cid:48) , (2.17b) (cid:74) (cid:52) P (cid:75) n = on ∂ B , (2.17c) (cid:52) P n = on ∂ V . (2.17d)The set of equations (2.17) need to be solved for the non-trivial unknown functions( (cid:52) χ, (cid:52) M ) describing the onset of bifurcation. Remark 2.3.
Perturbation in the Maxwell stress (cid:52) P m in B (cid:48) in terms of (cid:52) F and (cid:52) H is given by Equation (C.22) . The boundary condition (2.17c) connects (cid:52) P (2.15) and (cid:52) P m (C.22) through the constitutive relation (2.11) for H . Remark 2.4.
In the context of the first variation as well as the critical pointperturbation, above expressions and equations are similar to those obtained intwo other formulations based on B and H . These are summarized in Appendix B and C, where the derivations provided in (Saxena and Sharma, 2020) forthe case of electroelastic materials are closely followed.
Suppose that the physical space exterior to B is the entire space outside; in otherwords, we assume that V = R . (3.1)We consider that scenario when the potential energy functional depends on themagnetic energy stored in the entire space, due to the so called stray field h s , andalso includes a contribution of the work done by an external magnetic field h e on the magnetisation induced in the body. As a consequence of this, unlike theformulation presented in Section 2 and in Appendix B and C, we do not have anycontribution due to those terms that involve an integral on the boundary of theregion exterior to the body, i.e., on ∂ V . In particular, the total (magnetoelastic)stored energy E in the considered system is the sum of the energy stored in the bodyand the stray magnetic field energy of the entire space. The explicit mathematicalexpression of the energy, as a functional of the deformation χ (1.1) (and (1.2)) andthe spatial magnetisation m (1.13), (1.14) (per unit mass), is given by E ( χ, m ) : = (cid:90) B ρ (cid:98) Ω( F , m ) dv + (cid:90) R µ h s · h s dv, (3.2)where we have defined (cid:98) Ω as the Helmholtz energy (per unit mass). Followingthe physical nature of the stray fields, also by convention, it is assumed that thestray magnetic field h s decays (in a suitable manner) far away from the body, thatis (cid:107) h s (cid:107) → (cid:107) x (cid:107) → ∞ (recall that x denotes the position vector in currentconfiguration).The work done on the magnetoelastic body (same as the negative of the po-tential energy of the applied dead loading) by externally applied mechanical andmagnetic forces is given by (Kankanala and Triantafyllidis, 2004) (cid:90) B ρ h e · m dv + (cid:90) B ρ f e · χdv + (cid:90) ∂ B t e · χds, (3.3)where f e denotes the body force (per unit mass) and t e denotes the mechanicaltraction (per unit area of the current configuration) while and the first term isidentified as the Zeeman energy (Feynmann et al., 1965). It is emphasized that f e , t e and h e are external dead loads . 14onlinear magnetoelastostaticsUsing (3.2) and (3.3), the potential energy E II of the system comprising of thebody and the surrounding space is then given by E minus magnetoelastic workdone, i.e., E II ( χ, m ) : = (cid:90) B (cid:2) ρ (cid:98) Ω( F , m ) − h e · ( ρ m ) − ρ f e · χ (cid:3) dv − (cid:90) ∂ B t e · χds + 12 µ (cid:90) B h s · h s dv + 12 µ (cid:90) B (cid:48) h s · h s dv. (3.4) Remark 3.1.
We emphasize that even though the Eulerian expression of thepotential energy E II is the same as that provided by Kankanala and Triantafyl-lidis (2004), our formulation is markedly different from theirs; since we con-sider the mechanical deformation χ and the magnetisation in the body m asthe only two unknown fields of the problem. Moreover, our referential formu-lation is quite different from that of Kankanala and Triantafyllidis (2004) asdiscussed below. In terms of χ and m , the magnetic vector field b s can befound by employing the Maxwell’s equations stated in Section 1. As b s = µ h s + ρ m (3.5) and h s = − grad φ s by (1.17) , (1.13) , while φ s is found from the condition (1.4) , i.e. div b s = 0 , that b s satisfies. It is preferable to write the potential energy in equation (3.4) in the reference(Lagrangian) configuration, i.e. all field variables are functions of the referenceposition vector X instead of the current position vector x (= χ ( X )).The Helmholtz energy function (cid:98) Ω( F , m ) in (3.4) is mapped to (cid:98) Ω( F , M ) (recall(1.9)), i.e., (cid:98) Ω( F ( χ − ( x )) , m ( x )) = (cid:98) Ω( F ( X ) , M ( X )) , X ∈ B . It is emphasized that the field h e depends directly on the spatial location x (unlike f e ) and is therefore explicitly mentioned as such.Recall equation (1.13) and Remark 1.1, and in particular the relations m = ρ m = ρJ F −(cid:62) M = ρ F −(cid:62) M = F −(cid:62) M , and M = ρ M . Using the transformations stated above, we can redefine the expression of thepotential energy functional (3.4) in a referential description as E II ( χ, M ) : = (cid:90) B ρ (cid:2)(cid:98) Ω( F , M ) − J h e ( χ ( X )) · F −(cid:62) M (cid:3) dv − (cid:90) B (cid:101) f e · χdv − (cid:90) ∂ B (cid:101) t e · χds + 12 µ (cid:90) B J F −(cid:62) H s · F −(cid:62) H s dv + 12 µ (cid:90) B (cid:48) h s · h s dv, (3.6)where (cid:101) t e is the force per unit area of the current configuration placed on thereference configuration, i.e., (cid:101) t e ( X ) ds = t e ( x ) ds, X ∈ ∂ B . Also the body forceper unit volume (cid:101) f e on the reference configuration is related to f e by (cid:101) f e ( X ) dv = ρ f e ( x ) dv, X ∈ B . In the first term of (3.6), we have highlighted the dependenceon X for additional clarity.Recall the extension of χ to B (cid:48) is also denoted by χ and the mapping χ issufficiently smooth and it maps ∂ B to ∂ B such that it identifies with χ in thatregion and its gradient F identifies with the deformation gradient F of χ on thecommon boundary ∂ B . In vacuum far from ∂ B the deformation gradient F canvery well be assumed to be identity for convenience. We can rewrite the last termof the potential energy in equation (3.6) so that the entire expressions becomes E II ( χ, M ) = (cid:90) B ρ (cid:2)(cid:98) Ω( F , M ) − J h e ( χ ( X )) · F −(cid:62) M (cid:3) dv − (cid:90) B (cid:101) f e · χdv − (cid:90) ∂ B (cid:101) t e · χds + 12 µ (cid:90) B J C − H s · H s dv + 12 µ (cid:90) B (cid:48) J C − H s · H s dv . (3.7) Upon using the expressions for increments, E II ( χ + δχ, M + δ M ) − E II ( χ, M ) = δ E II ( χ, M )[ δχ, δ M ]+ 12 δ E II ( χ, M )[ δχ, δ M ] + o [ δχ, δ M ] , (3.8)where o [ δχ, δ M ] are the terms of order higher than two in δχ and δ M ; δ E II and δ E II are the first and the second variations of E II , respectively.The first variation δ E II [ δχ, δ M ], written simply as δ E II , is given by δ E II = (cid:90) B ρ (cid:104)(cid:98) Ω , F · δ F + (cid:98) Ω , M · δ M − J (grad (cid:62) h e ) F −(cid:62) M · δχ − J h e · F −(cid:62) δ M − J h e · δ F −(cid:62) M − δJ h e · F −(cid:62) M (cid:3) dv − (cid:90) B (cid:101) f e · δχdv − (cid:90) ∂ B (cid:101) t e · δχds µ (cid:90) B J C − H s · δ H s dv + 12 µ (cid:90) B (cid:2) J δ C − H s · H s + δJ C − H s · H s (cid:3) dv + µ (cid:90) B (cid:48) J C − H s · δ H s dv + 12 µ (cid:90) B (cid:48) (cid:2) J δ C − H s · H s + δJ C − H s · H s (cid:3) dv . (3.9)Using the identities for variations of C , J from Appendix A,12 µ (cid:90) B (cid:2) J δ C − H s · H s + δJ C − H s · H s (cid:3) dv = (cid:90) B [ − ˇ P m · δ F ] dv , (3.10)where ˇ P m : = µ J (cid:20) h s ⊗ h s −
12 [ h s · h s ] I (cid:21) F −(cid:62) . (3.11) Remark 3.2. ˇ P m resembles the tensor P m as defined in (2.5) in the regionexterior to the body B . Indeed, µ (cid:90) B (cid:48) (cid:2) J δ C − H s · H s + δJ C − H s · H s (cid:3) dv = (cid:90) B (cid:48) (cid:2) − ˇ P m · δ F (cid:3) dv , which leads to ˇ P m = µ J ( h s ⊗ h s − ( h s · h s ) I ) F −(cid:62) , that can be compared withthe Maxwell stress tensor P m = µ J ( h ⊗ h − ( h · h ) I ) F −(cid:62) from equation (2.5) ,exterior to the body B . Thus, it is not same as that obtained by the other threeformulations; in particular, ˇ P m decays as (cid:107) X (cid:107) → ∞ . This anomaly is due tothe presence of an applied external field in infinite space which corresponds toa non-vanishing ‘external’ Maxwell stress. We write a first order variation of the magnetic induction vector using theconstitutive relation (1.11) (with M = ρ M , δ H s = − Grad δ Φ s ) as δ B s = δ ( J C − )( µ H s + ρ M ) + J C − δ ( µ H s + ρ M )= (cid:104) [ F −(cid:62) · δ F ] I − C − δ F (cid:62) F − F − δ F (cid:105) B s − µ J C − Grad δ Φ s + ρ J C − δ M . (3.12)We use the divergence theorem and use the condition from a variation of equation(1.7) that Div( δ B s ) = 0 to get − (cid:90) B H s · δ B s dv = (cid:90) ∂ B n · Φ s δ B s ds , (3.13) − (cid:90) B (cid:48) H s · δ B s dv = (cid:90) ∂ B (cid:48) n · Φ s δ B s ds = − (cid:90) ∂ B n · Φ s δ B s ds . (3.14)17onlinear magnetoelastostaticsAlso, due to (3.12), − µ (cid:90) B J C − H s · δ H s dv = (cid:90) B − ˚ P m · δ F dv + (cid:90) B H s · ( ρ J C − δ M − δ B s ) dv , (3.15)where ˚ P m is defined by˚ P m : = 2 ˇ P m + ρ J (cid:0) − ( C − M · H s ) I + ( F −(cid:62) M ) ⊗ ( F −(cid:62) H s )+ ( F −(cid:62) H s ) ⊗ ( F −(cid:62) M ) (cid:1) F −(cid:62) . (3.16)Similarly, − µ (cid:90) B (cid:48) J C − H s · δ H s dv = (cid:90) B (cid:48) − P m · δ F dv + (cid:90) ∂ B (cid:48) n · Φ s δ B s ds , (3.17)where (cid:82) ∂ B (cid:48) n · Φ s δ B s ds = − (cid:82) ∂ B n · Φ s δ B s ds . Upon changing the derivatives from current to reference configuration, we getgrad h e = (cid:2) Grad h e (cid:3) F − . (3.18)Using these expressions, the first variation δ E II can, therefore, be rewritten as δ E II = (cid:90) B ρ (cid:0)(cid:98) Ω , F · δ F + (cid:98) Ω , M · δ M − J F −(cid:62) (Grad (cid:62) h e ) F −(cid:62) M · δχ − J h e · F −(cid:62) δ M − J h e · δ F −(cid:62) M − δJ h e · F −(cid:62) M (cid:3) dv − (cid:90) B (cid:101) f e · δχdv − (cid:90) ∂ B (cid:101) t e · δχds + (cid:90) B ( − ˇ P m · δ F ) dv + (cid:90) B (cid:48) ( − ˇ P m · δ F ) dv + (cid:90) B ˚ P m · δ F dv − (cid:90) B H s · ( ρ J C − δ M ) dv − (cid:90) ∂ B n · Φ s δ B s ds + (cid:90) B (cid:48) ˚ P m · δ F dv + (cid:90) ∂ B n · Φ s δ B s ds . (3.19)Assuming the continuity of Φ s , i.e., Φ s | + − Φ s | − on the boundary ∂ B , the twoterms involving Φ s and δ B s cancel; the latter is obtained by using the variation ofthe condition (cid:74) B s (cid:75) · n = 0 . (3.20)18onlinear magnetoelastostaticsApply the divergence theorem on the terms containing gradients of δχ to get δ E II = (cid:90) B (cid:18) − (cid:0) Div( ρ (cid:98) Ω , F + P m + ˚ P m − ˇ P m ) + (cid:101) f e + ρ J F −(cid:62) (Grad (cid:62) h e ) F −(cid:62) M (cid:1) · δχ + ρ ( (cid:98) Ω , M − J F − h e − J F − h s ) · δ M (cid:19) dv + (cid:90) ∂ B (cid:0) ( ρ (cid:98) Ω , F + P m + ˚ P m − ˇ P m ) − n − (˚ P m − ˇ P m ) + n − (cid:101) t e (cid:1) · δχds − (cid:90) B (cid:48) Div(˚ P m − ˇ P m ) · δχdv , (3.21)where P m is defined by P m : = ρ ( J F −(cid:62) M ⊗ h e − ( J F −(cid:62) M · h e ) I ) F −(cid:62) = ρ ( J F −(cid:62) M ⊗ F −(cid:62) F (cid:62) h e − ( J F − F −(cid:62) M · F (cid:62) h e ) I ) F −(cid:62) , (3.22)and we have used the assumptions that χ and δχ are continuous across ∂ B ; and h s → as (cid:107) X (cid:107) → ∞ . Note that˚ P m + P m = 2 ˇ P m + ρ J (cid:0) − ( C − M · F (cid:62) ( h s + h e )) I + ( F −(cid:62) M ) ⊗ F −(cid:62) F (cid:62) ( h s + h e )+ ( F −(cid:62) H s ) ⊗ ( F −(cid:62) M ) (cid:1) F −(cid:62) . (3.23)In vacuum M = which leads to˚ P m + P m = ˚ P m = 2 ˇ P m . With the defining expression P = ρ (cid:98) Ω , F + P m + ˚ P m − ˇ P m , (3.24)the tensor P can be identified as the total first Piola–Kirchhoff stress tensor and P (cid:63) = P = ˇ P m can be identified as the Maxwell stress tensor in vacuum.Corresponding to the equilibrium condition of vanishing of the first variation δ E II of the potential energy E II , using the classical methods in the calculus ofvariations (Gelfand and Fomin, 2003), i.e., δ E II ( χ, M )[ δχ, δ M ] = 0 , (3.25)since the increment δ M is arbitrary, we arrive at the constitutive relation (cid:98) Ω , M = J F − [ h s + h e ] = J C − [ H s + H e ] , (3.26)which is, remarkably, same as (2.11). 19onlinear magnetoelastostatics Remark 3.3.
In particular, inside the body P is given by (as h = J − F (cid:98) Ω , M ) P = ρ (cid:98) Ω , F + ρ J (cid:0) − ( C − M · F (cid:62) h ) I + ( F −(cid:62) M ) ⊗ F −(cid:62) F (cid:62) h + h s ⊗ ( F −(cid:62) M ) (cid:1) F −(cid:62) + (cid:20) µ J h s ⊗ h s − µ J h s · h s ] I (cid:21) F −(cid:62) = ρ (cid:98) Ω , F + µ J − (cid:20) − J h s · h s ] I + J h s ⊗ h s (cid:21) F −(cid:62) + ρ [ − ( M · (cid:98) Ω , M ) I + ( F −(cid:62) M ) ⊗ F (cid:98) Ω , M + J h s ⊗ ( F −(cid:62) M )] F −(cid:62) , (3.27) which differs from the expression (2.13) by the following term: P N = µ J (cid:20) −
12 [ h e · h e ] I + h e ⊗ h e (cid:21) F −(cid:62) + µ J (cid:20) − [ h s · h e ] I + h s ⊗ h e + h e ⊗ h s (cid:21) F −(cid:62) + ρ J h e ⊗ ( F −(cid:62) M ) F −(cid:62) . (3.28) With ψ ( a ) = − ( a · a ) I + a ⊗ a , the first and second line in P N can be writtenas ψ ( h e + h s ) − ψ ( h s ) . The difference between the two definitions of the stresstensor is not surprising. It is known that these could be different expressions,yet physically equivalent, as they depend on the formulation, see for example(Hutter and van de Ven, 1978) who presented this aspect of the Maxwell stresstensor while analyzing several formulations of electromagnetism in the theoryof deformable media. Since the increment δχ is arbitrary, we arrive at the following equation ofequilibrium in magnetoelastostatics (for a system of magnetoelastic body and itssurrounding vacuum)Div P + (cid:101) f e + ρ J F −(cid:62) [Grad (cid:62) h e ] F −(cid:62) M = in B , (3.29a)Div ˇ P m = in B (cid:48) (3.29b)[ P − ˇ P m ] n = (cid:101) t e on ∂ B . (3.29c) For the analysis of the critical point ( χ, M ), the perturbations (cid:52) χ and δ M inthe equilibrium state need to satisfy certain incremental equations and boundary20onlinear magnetoelastostaticsconditions. They are derived by a perturbation of (3.29) and are stated below.Recalling from equation (3.26) that h s = J − F (cid:98) Ω , M − h e , perturbation in the firstPiola–Kirchhoff stress can be written using the equation (3.27) as (cid:52) P = ρ (cid:20)(cid:98) Ω , FF (cid:52) F + 12 (cid:2)(cid:98) Ω , F M + (cid:98) Ω ∗ F M (cid:3) (cid:52) M (cid:21) − µ J − [ F −(cid:62) · (cid:52) F ] (cid:20) − J h s · h s ] I + J h s ⊗ h s (cid:21) F −(cid:62) + µ J − (cid:34) − J (cid:20) [ F −(cid:62) · (cid:52) F ][ h s · h s ] + h s · (cid:52) h s (cid:21) I + 2 J [ F −(cid:62) · (cid:52) F ] h s ⊗ h s + J (cid:2) (cid:52) h s ⊗ h s + h s ⊗ (cid:52) h s (cid:3)(cid:35) F −(cid:62) − µ J − (cid:20) − J h s · h s ] I + J h s ⊗ h s (cid:21) F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) + ρ (cid:34) − (cid:104) (cid:52) M · (cid:98) Ω , M + M · (cid:20) (cid:2)(cid:98) Ω , M F + (cid:98) Ω ∗ M F (cid:3) (cid:52) F + (cid:98) Ω , MM (cid:52) M (cid:21)(cid:105) I + [ − F −(cid:62) (cid:52) FF −(cid:62) M + F −(cid:62) (cid:52) M ] ⊗ F (cid:98) Ω , M + F −(cid:62) M ⊗ (cid:20) (cid:52) F (cid:98) Ω , M + 12 F (cid:2)(cid:98) Ω , M F + (cid:98) Ω ∗ M F (cid:3) (cid:52) F + F (cid:98) Ω , MM (cid:52) M (cid:21) + (cid:2) J [ F −(cid:62) · (cid:52) F ] h s + J (cid:52) h s (cid:3) ⊗ [ F −(cid:62) M ] + J h s ⊗ (cid:2) F −(cid:62) (cid:52) M − F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) (cid:3)(cid:35) F −(cid:62) − ρ (cid:20) − ( M · (cid:98) Ω , M ) I + ( F −(cid:62) M ) ⊗ F (cid:98) Ω , M + J h s ⊗ ( F −(cid:62) M ) (cid:21) F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) , (3.30)where we can obtain the expression for (cid:52) h s from equation (3.26) as (cid:52) h s = J − F (cid:2)(cid:98) Ω , M F (cid:52) F + (cid:98) Ω , MM (cid:52) M (cid:3) − J − [ F −(cid:62) · (cid:52) F ] F (cid:98) Ω , M + J − (cid:52) F (cid:98) Ω , M . (3.31)We have also introduced two second order tensors (cid:98) Ω ∗ M F and (cid:98) Ω ∗ F M with the property (cid:104)(cid:98) Ω ∗ M F U (cid:105) u = (cid:104)(cid:98) Ω , F M u (cid:105) · U , (cid:104)(cid:98) Ω ∗ F M u (cid:105) · U = (cid:104)(cid:98) Ω , M F U (cid:105) · u . (3.32)for arbitrary vector u and arbitrary second order tensor U . The expression for (cid:52) ˇ P m is obtained from equation (3.11) as (cid:52) ˇ P m = µ J (cid:2) F −(cid:62) · (cid:52) F (cid:3)(cid:20) h s ⊗ h s −
12 [ h s · h s ] I (cid:21) F −(cid:62) µ J (cid:20) h s ⊗ (cid:52) h s + (cid:52) h s ⊗ h s − (cid:2) h s · (cid:52) h s (cid:3) I (cid:21) F −(cid:62) − µ J (cid:20) h s ⊗ h s −
12 [ h s · h s ] I (cid:21) F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) . (3.33)Finally above leads to the following partial differential equations and boundaryconditions Div (cid:52) P + ρ J F −(cid:62) (cid:2) Grad (cid:62) h e (cid:3) F −(cid:62) (cid:52) M + ρ J (cid:2) F −(cid:62) · (cid:52) F (cid:3) F −(cid:62) (cid:2) Grad (cid:62) h e (cid:3) F −(cid:62) M − ρ J F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) (cid:2) Grad (cid:62) h e (cid:3) F −(cid:62) M − ρ J F −(cid:62) (cid:2) Grad (cid:62) h e (cid:3) F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) M = 0 in B , (3.34a)Div (cid:52) ˇ P m = 0 in B (cid:48) (3.34b) (cid:2) (cid:52) P − (cid:52) ˇ P m (cid:3) n = 0 on ∂ B . (3.34c) In the back drop of the two formulations provided thus far based on the magneti-sation, we investigate in this section the expressions provided by Kankanala andTriantafyllidis (2004) which also assume that the stored energy density dependson the magnetisation as the additional field besides the deformation gradient. Fol-lowing Kankanala and Triantafyllidis (2004), in this case, the magnetisation perunit mass pulled back to the reference configuration (recall (1.9)), i.e., K ( X ) : = m ( x ) = J ( X ) F −(cid:62) ( X ) M ( X ) , X ∈ B . (4.1)is itself treated as a material field. In particular, note that the direction of thereferential vector field K on B is same as that of the spatial vector field m on B , while it differs from the choice of the referential field M due to the presence ofcofactor map for F (Nanson’s relation). The total potential energy of the systemis written as E III ( χ, K ) : = (cid:90) B ρ (cid:2)(cid:101) Ω( F , K ) − h e ( χ ( X )) · K (cid:3) dv − (cid:90) B (cid:101) f e · χdv − (cid:90) ∂ B (cid:101) t e · χds + 12 µ (cid:90) B J C − H s · H s dv + 12 µ (cid:90) B (cid:48) J C − H s · H s dv . (4.2)Here (cid:102) f e represents the body force (per unit volume) and (cid:101) t e denotes the mechanicaltraction. In contrast to (3.3), the term corresponding to the Zeeman energy is22onlinear magnetoelastostaticswritten differently. Note from equation (1.17) that J − F B s = µ F −(cid:62) H s + ρ K , i.e., B s = µ J C − H s + ρ F − K , so that δ B s = δ ( J C − )[ µ H s ] + ρ δ ( F − K ) , = (cid:104) [ F −(cid:62) · δ F ] I − C − δ F (cid:62) F − F − δ F (cid:105) µ J C − H s − ρ F − δ FF − K − µ J C − Grad δ Φ s + ρ F − δ K , = (cid:104) [ F −(cid:62) · δ F ] I − C − δ F (cid:62) F (cid:105) µ J C − H s − F − δ F B s − µ J C − Grad δ Φ s + ρ F − δ K . (4.3)Using this relation, we can rewrite the following integral that occurs in the firstvariation of potential energy as − µ (cid:90) B J C − H s · δ H s dv = (cid:90) B − (cid:101) P m · δ F dv + (cid:90) B H s · ( ρ F − δ K − δ B s ) dv , (4.4)where the integrand of the first term on the right hand side, i.e., − (cid:101) P m · δ F , can beexpanded as − (cid:101) P m · δ F = µ J ( F −(cid:62) · δ F ) C − H s · H s − µ J C − δ F (cid:62) FC − H s · H s − F −(cid:62) H s ⊗ B s · δ F = ( − P m − ρ h s ⊗ K F −(cid:62) ) · δ F . (4.5)Thus, (cid:101) P m − ˇ P m = ˇ P m + ρ h s ⊗ K F −(cid:62) = ˇ P m + J h s ⊗ m F −(cid:62) . (4.6)From (3.10) we already know a part of the expression of the first variation ofstray field energy term. Therefore, we write the first variation of the potentialenergy (4.2) as δ E III ( χ, K ) = (cid:90) B ρ (cid:2)(cid:101) Ω , F · δ F + (cid:101) Ω , K · δ K − F −(cid:62) (Grad (cid:62) h e ) K · δχ − h e · δ K (cid:3) dv − (cid:90) B (cid:101) f e · δχdv − (cid:90) ∂ B (cid:101) t e · δχds + (cid:90) B ( − ˇ P m · δ F ) dv + (cid:90) B (cid:48) ( − ˇ P m · δ F ) dv + (cid:90) B (cid:101) P m · δ F dv − (cid:90) B H s · ( ρ F − δ K ) dv − (cid:90) ∂ B n · Φ s δ B s ds (cid:90) B (cid:48) (cid:101) P m · δ F dv + (cid:90) ∂ B n · Φ s δ B s ds . (4.7)On applying the divergence theorem on the terms containing gradient of δχ , weget δ E III = (cid:90) B (cid:18) − (cid:2) Div( ρ (cid:101) Ω , F + (cid:101) P m − ˇ P m ) + (cid:101) f e + ρ F −(cid:62) (Grad (cid:62) h e ) K (cid:3) · δχ + ρ (cid:2)(cid:101) Ω , K − h e − h s (cid:3) · δ K (cid:19) dv + (cid:90) ∂ B (cid:0) ( ρ (cid:101) Ω , F + (cid:101) P m − ˇ P m ) − n − ( (cid:101) P m − ˇ P m ) + n − (cid:101) t e (cid:1) · δχds − (cid:90) B (cid:48) Div( (cid:101) P m − ˇ P m ) · δχdv . (4.8)Since the increments δχ and δ K are arbitrary, we arrive at the following Euler–Lagrange equations for this variational problemDiv P + (cid:101) f e + ρ F −(cid:62) (Grad (cid:62) h e ) K = in B , (4.9a) (cid:74) P (cid:75) n + (cid:101) t e = on ∂ B , (4.9b)Div P = in B (cid:48) (4.9c) h = (cid:101) Ω , K in B , (4.9d)where we have recognised the total first Piola–Kirchhoff stress tensor in the bodyand in vacuum as P = ρ (cid:101) Ω , F + (cid:101) P m − ˇ P m in B , (4.10a) P = (cid:101) P m − ˇ P m in B (cid:48) . (4.10b) Remark 4.1.
From (3.11) and (4.6) (recall Remark 2.1), we can write thetotal Cauchy stress on B as σ = J − PF (cid:62) = ρ (cid:101) Ω , F F (cid:62) + h s ⊗ b s − µ ( h s · h s ) I . (4.11) Thus far, we have presented three different magnetisation based formulations wherethe difference between these variational principles occurs due the choice of partic-ular magnetisation field. In addition to these, we have also presented two other24onlinear magnetoelastostaticsformulations in Appendix B and C where in place of the magnetisation field thestored energy density depends on B and H , respectively. Since the mechanicalwork terms involving the body force and the surface traction are same in all theseformulations (in the referential description), i.e., W M : = − (cid:90) B (cid:101) f e · χdv − (cid:90) ∂ B (cid:101) t e · χds , (which also equals its spatial description − (cid:82) B ρ f e · χdv − (cid:82) ∂ B t e · χds ) so we some-times compare only the remaining terms. Using the constitutive relation (1.11)and the fact that M vanishes outside the body B , we get12 µ (cid:90) V J (cid:107) F −(cid:62) H (cid:107) dv − (cid:90) V H · B dv = − µ (cid:90) V J (cid:107) F −(cid:62) H (cid:107) dv − (cid:90) B J C − M · H dv . (5.1)In a similar manner, we find that12 µ (cid:90) B (cid:48) J (cid:107) F −(cid:62) H (cid:107) dv − (cid:90) B (cid:48) H · B dv = − µ (cid:90) B (cid:48) J (cid:107) F −(cid:62) H (cid:107) dv . (5.2)At this point, it is useful to recall Remark 1.2. Using (1.7) and (1.8) , − (cid:90) ∂ V Φ B e · n ds = − (cid:90) V Div(Φ B e ) dv = − (cid:90) V [Φ Div B e + Grad Φ · B e ] dv = (cid:90) V H · B e dv . (5.3)In general, we have (cid:90) V H · B dv = − (cid:90) ∂ V n · Φ B ds , (cid:90) V H · B e dv = − (cid:90) ∂ V n · Φ B e ds , (cid:90) V H e · B dv = − (cid:90) ∂ V n · Φ e B ds . (5.4)Also, these relations can be re-written further, for example, (cid:82) V H · B e dv = − (cid:82) ∂ V n · Φ B e ds = − µ (cid:82) ∂ V n · Φ J C − H e ds . M , B , and H The variational formulations based on B and H can be related by applying aLegendre-type transform on the energy functions ˚Ω and ˇΩ as ˚Ω( F , B ) = ˇΩ( F , H ) + B · H (Dorfmann and Ogden, 2004). Moreover, we note that the three variationalformulations based on M , B , and H can be mutually related by a set of Legendre-type transform on the stored energy density functions Ω , ˚Ω , and ˇΩ, respectively,so thatΩ( F , M ) = ˚Ω( F , B ) − µ J C − H · H = ˚Ω( F , B ) + 1 µ M · B − µ J C − M · M − µ J − C B · B , (5.5)Ω( F , M ) = ˇΩ( F , H ) + B · H − µ J C − H · H = ˇΩ( F , H ) + J C − H · M + 12 µ J C − H · H , (5.6)˚Ω( F , B ) = ˇΩ( F , H ) + B · H . (5.7)By a direct calculation, it can be verified that the above relations result in themagnetic constitutive relations (B.8), (C.3) , and (2.11); in particular,˚Ω , B = H , ˇΩ , H = − B , Ω , M = J C − H in B . As such, these relations lead to different convexity properties for ˚Ω( F , B ), ˇΩ( F , H ),and Ω( F , M ) in general.As a consequence of above, it is natural to establish the relationship betweenthe three variational formulations based on B , H and M . Recall that the totalpotential energy (2.1) is a functional of the deformation χ (1.1) (and (1.2)) andthe referential magnetisation M (1.12). Indeed, the variational formulation (2.1)can be expressed as E I [ χ, M ] + W M = (cid:90) B Ω( F , M ) dv + 12 µ (cid:90) V J (cid:107) F −(cid:62) Grad Φ (cid:107) dv + (cid:90) ∂ V Φ e n · B ds = (cid:90) B Ω( F , M ) dv + 12 µ (cid:90) V J (cid:107) F −(cid:62) H (cid:107) dv − (cid:90) V H e · B dv = (cid:20)(cid:90) B Ω( F , M ) dv + 12 µ (cid:90) B J (cid:107) F −(cid:62) H (cid:107) dv (cid:21) + 12 µ (cid:90) B (cid:48) J (cid:107) F −(cid:62) H (cid:107) dv − (cid:90) V H e · B dv , (5.8)26onlinear magnetoelastostaticswhich can be written as E I [ χ, M ] + W M = E IV [ χ, A ] + W M . (5.9)Above is the exact relationship between the variational principles analyzed in Sec-tion 2 and Appendix B. Recall that the total potential energy (B.1) is a functionalof the deformation χ (1.1) (and (1.2)) and the referential counterpart B of b (viathe referential magnetic vector potential A (1.8) ). Also,12 µ (cid:90) V J (cid:107) F −(cid:62) H (cid:107) dv − (cid:90) V H · B dv = − µ (cid:90) V J (cid:107) F −(cid:62) H (cid:107) dv − (cid:90) B J C − M · H dv , (5.10)so that E I [ χ, M ] + W M = (cid:90) B Ω( F , M ) dv − µ (cid:90) V J (cid:107) F −(cid:62) H (cid:107) dv + (cid:90) V ( H − H e ) · B dv − (cid:90) B J C − M · H dv = (cid:90) B (Ω( F , M ) − J C − H · M − µ J (cid:107) F −(cid:62) H (cid:107) ) dv − µ (cid:90) B (cid:48) J (cid:107) F −(cid:62) H (cid:107) dv + (cid:90) V ( H − H e ) · B dv . (5.11)Hence, (5.8) can be written as E I [ χ, M ] + W M = E V [ χ, Φ] + W M − (cid:90) V H · B e dv + (cid:90) V ( H − H e ) · B dv , (5.12)which is the relationship between the variational principles analyzed in Section 2and Appendix C. Here we recall that the total potential energy (C.1) is a functionalof the deformation χ (1.1) (and (1.2)) and the referential magnetic field vector H (via the referential magnetic scalar potential Φ (1.8) ). Remark 5.1.
Upon using equations (B.4) , (1.7) and (1.8) , we can write (cid:90) ∂ V [ H e ∧ A ] · n ds = (cid:90) V Div [ H e ∧ A ] dv = (cid:90) V [ Curl H e · A − [ Curl A ] · H e ] dv = − (cid:90) V B · H e dv . (5.13)27onlinear magnetoelastostatics Hence, the total potential energy functional (B.1) can be re-written asE IV [ χ, A ] + W M = (cid:90) B ˚Ω( F , B ) dv + 12 µ (cid:90) B (cid:48) J − (cid:107) F B (cid:107) dv − (cid:90) V H e · B dv = (cid:90) B ˚Ω( F , B ) dv + 12 µ (cid:90) B (cid:48) J − (cid:107) F B (cid:107) dv + (cid:90) ∂ V n · Φ e B ds . (5.14) In the variational formulation for the total potential energy functional (C.1) we haveE V [ χ, Φ] + W M = (cid:90) B ˇΩ( F , H ) dv − µ (cid:90) B (cid:48) J (cid:107) F −(cid:62) H (cid:107) dv + (cid:90) V H · B e dv = (cid:90) B ˇΩ( F , H ) dv − µ (cid:90) B (cid:48) J (cid:107) F −(cid:62) H (cid:107) dv − (cid:90) ∂ V n · Φ B e ds . (5.15) Hence,E V [ χ, Φ] + W M = (cid:90) B ˇΩ( F , H ) dv + 12 µ (cid:90) B (cid:48) J (cid:107) F −(cid:62) H (cid:107) dv − (cid:90) B (cid:48) H · B dv + (cid:90) V H · B e dv = (cid:90) B ( ˇΩ( F , H ) + B · H ) dv + 12 µ (cid:90) B (cid:48) J − (cid:107) F B (cid:107) dv − (cid:90) V H · ( B − B e ) dv , (5.16) which can be written asE V [ χ, Φ] + W M = E IV [ χ, A ] + W M + (cid:90) V H e · B dv − (cid:90) V H · ( B − B e ) dv = E IV [ χ, A ] + W M + (cid:90) V H · B e dv − (cid:90) V ( H − H e ) · B dv . (5.17) Also (cid:90) V H · B e dv = µ (cid:90) V H · J C − H e dv = (cid:90) V ( J − C B − M1 B ) · J C − H e dv (cid:90) V ( B − J C − M1 B ) · H e dv = (cid:90) V H e · B dv − (cid:90) B J F −(cid:62) M · F −(cid:62) H e dv . (5.18) Hence,E V [ χ, Φ] + W M = (cid:90) B ( ˇΩ( F , H ) + B · H ) dv + 12 µ (cid:90) B (cid:48) J − (cid:107) F B (cid:107) dv − (cid:90) V ( H − H e ) · B dv − (cid:90) B F −(cid:62) H e · J F −(cid:62) M dv . (5.19) M , M , and K Following the arguments in Section 3, we assumed that V = V = R , (5.20)for the formulation presented in Section 2. This needed some changes in theexpression (2.1). Clearly, the only term that needs to be re-written is the lastterm in (2.1). Using the nature of magnetic field in vacuum we have by (5.4) , (cid:82) ∂ V φ e n · B ds = − (cid:82) V H e · B dv . Hence, based on the definitions (2.1) and (3.7),we get E I − E II = E I [ χ, M ] − E II [ χ, M ]= (cid:90) B (cid:2) Ω( F , M ) − ρ (cid:98) Ω( F , M ) + ρ J h e ( χ ( X )) · F −(cid:62) M (cid:3) dv ++ µ (cid:90) V J C − H · H dv − (cid:90) V H e · B dv − µ (cid:90) B J C − H s · H s dv − µ (cid:90) B (cid:48) J C − H s · H s dv . (5.21)The first term in the second line can be rewritten in view of (1.16) as µ (cid:90) V J C − H · H dv = µ (cid:90) V J C − (cid:20) H e · H e + 2 H e · H s + H s · H s (cid:21) dv . (5.22)Upon substituting the above back to (5.21), we get E I − E II = (cid:90) B (cid:2) Ω( F , M ) − ρ (cid:98) Ω( F , M ) + ρ J h e ( χ ( X )) · F −(cid:62) M (cid:3) dv µ (cid:90) V J C − H e · H s dv + µ (cid:90) V J C − H e · H e dv − (cid:90) V H e · B s dv (5.23) − µ (cid:90) V J C − H e · H e dv = (cid:90) B (cid:2) Ω( F , M ) − ρ (cid:98) Ω( F , M ) + ρ J h e ( χ ( X )) · F −(cid:62) M (cid:3) dv − (cid:90) B J C − H e · M dv − µ (cid:90) V J C − H e · H e dv . (5.24)Since, M = ρ M and H e = F (cid:62) h e , the above can be rewritten as E I − E II = (cid:90) B (cid:2) Ω( F , M ) − ρ (cid:98) Ω( F , M ) (cid:3) dv − µ (cid:90) V J C − H e · H e dv . (5.25)Thus, the two potential energies differ not only by the definition of the respectivestored energy density functions but also an extra term; the latter term, clearly, isa constant term though it could be infinite for H e (cid:54) = 0 while the former can bemade zero by naturally identifying the stored energy density functions.From equations (3.7) and (4.2) (using (4.1)), we get E II [ χ, M ] − E III [ χ, K ] = (cid:90) B ρ (cid:2)(cid:98) Ω( F , M ) − (cid:101) Ω( F , K ) (cid:3) dv . (5.26)These two potential energies differ only by the definition of the respective storedenergy density functions which can be naturally identified to achieve an equiva-lence. Since h e is the gradient of a potential, and in view of (1.15), by a direct calculationwe have (cid:82) R h e · b s dv = 0, as a result of which we get (cid:90) B ρ h e · K dv = (cid:90) B h e · m dv = (cid:90) R h e · ( b s − µ h s ) dv = − µ (cid:90) R h e · h s dv. (5.27)Note that µ (cid:82) R h e · h s dv is non-zero, indeed, with h s = − grad φ s , µ h e = b e and B r ⊂ R as a ball of radius r , we find it to be equal to − (cid:90) R b e · grad φ s dv = lim r →∞ (cid:20) (cid:90) B r div b e φ s dv − (cid:90) ∂B r φ s b e · n ds (cid:21) , b e = 0 in the first term but φ s may not necessarily go to zero in thesecond term as r = (cid:107) x (cid:107) → ∞ . Thus, an equivalent potential energy functional is E III ( χ, K ) + (cid:90) B ρ h e · K dv + µ (cid:90) R h e · h s dv, in addition to which by including the constant term µ (cid:82) R h e · h e dv too, we get(recall (3.2)) E III ( χ, m ) = (cid:90) B ρ (cid:98) Ω( F , m ) dv + W M + 12 µ (cid:90) B h · h dv + 12 µ (cid:90) B (cid:48) h · h dv, (5.28)with its referential form (to be compared with (4.2)) as (cid:99) E III ( χ, K ) = (cid:90) B ρ (cid:101) Ω( F , K ) dv + W M + 12 µ (cid:90) B J C − H · H dv + 12 µ (cid:90) B (cid:48) J C − H · H dv . (5.29)Above expression coincides with the potential energy functional of (2.1) exceptfor the last term (which is absent in the present scenario as V = R ) and moreimportantly a different measure of magnetisation; note that K ( X ) = m ( x ) = ρ − ( x ) F −(cid:62) ( X ) M ( X )by (1.12) and (1.13). Similar to (4.3), we have δ B = (cid:104) [ F −(cid:62) · δ F ] I − C − δ F (cid:62) F (cid:105) µ J C − H − F − δ F B − µ J C − Grad δ Φ + ρ F − δ K , (5.30)and similar to (3.14) , we have (with B R ⊂ R as a ball of radius R ) − (cid:90) B (cid:48) H · δ B dv = (cid:90) B (cid:48) Grad Φ · δ B dv = lim R →∞ ( (cid:90) ∂B R Φ n · δ B ds − (cid:90) B R Φdiv δ B dv ) − (cid:90) ∂ B n · Φ δ B ds = − (cid:90) ∂ B n · Φ δ B ds , (5.31)31onlinear magnetoelastostaticsassuming that δ B · n vanishes as R = (cid:107) X (cid:107) → ∞ in a suitable manner. Bycarrying out the first variation analysis similar to that presented above in thissection we get δ (cid:99) E III ( χ, K ) = (cid:90) B ρ (cid:2)(cid:101) Ω , F · δ F + (cid:101) Ω , K · δ K (cid:3) dv − (cid:90) B (cid:101) f e · δχdv − (cid:90) ∂ B (cid:101) t e · δχds + (cid:90) B ( − (cid:98) P m · δ F ) dv + (cid:90) B (cid:48) ( − (cid:98) P m · δ F ) dv + (cid:90) B ˚ P m · δ F dv − (cid:90) B H s · ( ρ F − δ K ) dv − (cid:90) ∂ B n · Φ s δ B s ds + (cid:90) B (cid:48) ˚ P m · δ F dv + (cid:90) ∂ B n · Φ s δ B s ds , (5.32)where (cid:98) P m is defined by (2.4) and ˚ P m is defined by (2.9). The Euler–Lagrangeequations by setting δ (cid:99) E III = 0 are derived asDiv ( P ) + (cid:101) f e = in B , (5.33a) (cid:74) P (cid:75) n + (cid:101) t e = on ∂ B , (5.33b)Div ( P ) = in B (cid:48) , (5.33c) F H s = (cid:101) Ω , K in B , (5.33d)by recognising that for this potential energy functional, the first Piola–Kirchhoffstress is given by P = ρ (cid:101) Ω , F + ˚ P m − (cid:98) P m in B , (5.34a) P = ˚ P m − (cid:98) P m in B (cid:48) . (5.34b)Upon a direct comparison of the above with (4.9) and (4.10), we note that dueto the inclusion of extra terms with h e , the expressions for first Piola–Kirchhoffstress and the Maxwell stress in vacuum are different. This leads to the vanishingof the equivalent of electromagnetic body force term in (5.33)a and a modifiedconstitutive equation (5.33)d. In this paper, we have presented five variational formulations of nonlinear mag-netoelastostatics that differ from each other with respect to the independent fieldvariable for the magnetic effect. The formulations based on the magnetic field H ,the magnetic induction B , and referential magnetization vector per unit volume M M was originally postulated by Brown (1965) and that basedon a pull-back of the magnetization per unit mass to reference configuration K wasgiven by Kankanala and Triantafyllidis (2004). A direct equivalence between allfive principles by means of Legendre transform and properties of Maxwell equationsis the highlight of Section 5 of this paper.The principles can broadly be divided into two categories. For the first kindbased on H , B , and M , the total energy is defined over a bounded domain V with the external magnetic loading being specified by means of potential on theboundary ∂ V . For the second kind based on M and K , integral is defined overan infinite space and the notion of an external field becomes necessary to supplyexternal loading. The choice to include this (constant) external field in the totalenergy can lead to a different definition of the Maxwell stress, and result in changesin the body force and traction terms. Our analysis suggests caution with the choiceof variational principle appropriate to the physical problem and control variables.The analysis presented in this paper can be easily extended to the specialcase of incompressibility. For this purpose, see Remark 4 in the recent expositionand formulation for the electroelastic counterpart (Saxena and Sharma, 2020).Further extension of the present analysis to include mixed boundary conditions anddiscontinuities in the magnetoelastic body or free space can shed further light onthe issues around correspondences between the five principles. Inclusion of kineticenergy and the effect of time-dependent boundaries is another possible interestingarea for extension of the analysis presented here. We have restricted our analysisto nonlinear deformation and coupling. A linearised analysis to study deformationclose to the reference configuration may lead to simplifications and influence theequivalence analysis presented in Section 5. These avenues are currently underinvestigation and shall appear in suitable forum elsewhere. Acknowledgements
Basant Lal Sharma acknowledges the support of SERB MATRICS grant MTR/2017/000013. Prashant Saxena acknowledges the support of startup funds from theJames Watt School of Engineering at the University of Glasgow.33onlinear magnetoelastostatics
A Variation of some relevant kinematic quanti-ties
We list the first and second variations of key kinematic variables (see, for example,(Saxena and Sharma, 2020) for detailed derivations). Upon a perturbation χ → χ + δχ , we get F ( χ + δχ ) = Grad χ + Grad( δχ ) ⇒ δ F = Grad( δχ ) , δ F = . Theright Cauchy–Green deformation tensor changes as C ( χ + δχ ) = C + δ C + δ C . . . , with δ C = F (cid:62) δ F + [ δ F ] (cid:62) F , δ C = [ δ F ] (cid:62) δ F . (A.1)For the determinant J = det F , we get J ( χ + δχ ) = J + δJ + δ J + . . . with δJ = J F −(cid:62) · δ F , δ J = F · cof( δ F ) . (A.2)As δ F = Grad( δχ ), the second of the above expressions, δ J , is written in compo-nent form as δ J = ε imn ε jpq F ij [ δχ m,p ][ δχ n,q ] . Here ε ijk is the third order permuta-tion tensor. It can also be shown that δ J = 12 J [ (cid:2) F −(cid:62) · δ F (cid:3)(cid:2) F −(cid:62) · δ F (cid:3) − F −(cid:62) (cid:2) δ F ] (cid:62) F −(cid:62) · δ F ] . (A.3)Taylor’s expansion for the inverse of determinant J − is J − ( χ + δχ ) = J + J + J + . . . where J = J − , J = − J − F −(cid:62) · δ F , J = − J − F · cof( δ F ) + J − (cid:2) F −(cid:62) · δ F (cid:3) . Using the expression (A.3), we rewrite J as J = (2 J ) − [[ F −(cid:62) · δ F ] + F −(cid:62) [ δ F ] (cid:62) F −(cid:62) · δ F ] . For the inverse tensors, [ F ( χ + δχ )] − = F − + D F − + D F − + . . . , with D F − = − F − [ δ F ] F − , D F − = F − [ δ F ] F − [ δ F ] F − . (A.4)and [ C ( χ + δχ )] − = C − + D C − + D C − + . . . with D C − = − C − [ δ F ] (cid:62) F −(cid:62) − F − [ δ F ] C − , (A.5) D C − = C − [ δ F ] (cid:62) F −(cid:62) [ δ F ] (cid:62) F −(cid:62) + F − [ δ F ] C − [ δ F ] (cid:62) F −(cid:62) + F − [ δ F ] F − [ δ F ] C − . (A.6) B Variational formulation based on magnetic in-duction
Using the fact that B is found in terms of A by (1.8) , i.e., B = Curl A , the totalpotential energy of the system, i.e., the body B and its exterior B (cid:48) , is written34onlinear magnetoelastostaticsas a functional depending on the deformation χ and A as (Dorfmann and Ogden,2014) E IV [ χ, A ] : = (cid:90) B ˚Ω( F , B ) dv + 12 µ (cid:90) B (cid:48) J − [ F B ] · [ F B ] dv + (cid:90) ∂ V [ h e ∧ a ] · n ds − (cid:90) B (cid:101) f e · χdv − (cid:90) ∂ B (cid:101) t e · χds , (B.1)where ˚Ω is the (scalar) total (magnetoelastic) stored energy density per unit vol-ume, h e is the externally applied magnetic (vector) field whose tangential compo-nent is prescribed on ∂ V . The integral terms in equation (B.1) involve the referenceconfiguration as the spatial fields are mapped to the reference configuration, withthe exception of the third term which is written in terms of the current region V . It assumed that the boundary (typically, infinitally far away) is fixed (i.e., itdoes not change in space between reference and spatial description), so that thethird term in equation (B.1) is also rewritten in the reference configuration simplyas (cid:82) ∂ V [ H e ∧ A ] · n ds . Notice that n and n are used to denote the respectiveoutward unit normals for the region V and V (as well as B and B ). B.1 Equilibrium: first variation
In order to describe the deformation χ and the referential magnetic vector po-tential A when the body is in a state of equilibrium, the first variation of thepotential energy functional should vanish, that is, using the functional (B.1), δ E IV ≡ δ E IV [ χ, A ; ( δχ, δ A )] = 0 . An expansion of the functional E IV up to thefirst order, owing to a variation of its arguments χ and A , is given by E IV [ χ + δχ, A + δ A ] = (cid:90) B ˚Ω( F + δ F , B + δ B ) dv + 12 µ (cid:90) B (cid:48) [ J + δJ ] − [[ F + δ F ] [ B + δ B ]] · [[ F + δ F ] [ B + δ B ]] dv + (cid:90) ∂ V [ H e ∧ [ A + δ A ]] · n ds − (cid:90) B (cid:101) f e · [ χ + δχ ] dv − (cid:90) ∂ B (cid:101) t e · [ χ + δχ ] ds . (B.2)Taking advantage of the referential description, noting that δ D = Curl δ A , whileusing expressions for first order variations as derived in (Saxena and Sharma, 2020),we simplify further the expression of E IV [ χ + δχ, A + δ A ] stated above. Thus, it isfound that the first variation (2.2) of E IV is given by δ E IV = E IV [ χ + δχ, A + δ A ] − E IV [ χ, A ]35onlinear magnetoelastostatics= (cid:90) B (cid:104) ˚Ω , F · δ F + ˚Ω , B · Curl δ A (cid:105) dv + 12 µ (cid:90) B (cid:48) (cid:20) − J − (cid:2) F −(cid:62) · δ F (cid:3) [ F B ] · [ F B ] + 2 J − [[ F B ] ⊗ B ] · δ F + 2[ C B ] · Curl δ A (cid:21) dv + (cid:90) ∂ V [ n ∧ H e ] · δ A ds − (cid:90) B (cid:101) f e · δχdv − (cid:90) ∂ B (cid:101) t e · δχds . (B.3)Using an elementary identity for vector fields u and v , namely, v · Curl u = Div[ u ∧ v ] + [Curl v ] · u , (B.4)we expand the above expression for δ E IV as δ E IV = (cid:90) B (cid:104) ˚Ω , F · δ F + [Curl˚Ω , B ] · δ A (cid:105) dv + (cid:90) ∂ B − n · (cid:104) ˚Ω , B ∧ δ A (cid:105) ds − µ (cid:90) ∂ B +0 n · [ C B ∧ δ A ] ds + 12 µ (cid:90) B (cid:48) (cid:20) − J − (cid:2) F −(cid:62) · δ F (cid:3) [ F B ] · [ F B ] + 2 J − [[ F B ] ⊗ B ] · δ F + [Curl( C B )] · δ A (cid:21) dv + (cid:90) ∂ V [ n ∧ [ H e − µ C B ]] · δ A ds − (cid:90) B (cid:101) f e · δχdv − (cid:90) ∂ B (cid:101) t e · δχds . (B.5)Inspection of above leads to consideration of the definition of a tensor field givenby (2.5). Using the definition (2.5), we rewrite the first variation δ E IV of the totalpotential as δ E IV = (cid:90) B (cid:104) − [Div (cid:16) ˚Ω , F (cid:17) + (cid:101) f e ] · δχ + [Curl˚Ω , B ] · δ A (cid:105) dv + (cid:90) ∂ B [[ (cid:104) ˚Ω , F | − − P m | + (cid:105) n − (cid:101) t e ] · δχ + [ n ∧ [˚Ω , B | − − µ C B | + ]] · δ A ] ds + (cid:90) B (cid:48) [ − Div P m · δχ + 12 µ [Curl C B ] · δ A ] dv + (cid:90) ∂ V [ P m n · δχ + [ n ∧ [ H e − µ C B ]] · δ A ] ds . (B.6)The total (first Piola–Kirchhoff) stress P in the body is P = ˚Ω , F , in B , and the(Maxwell) stress exterior to the body is given by (2.5), i.e., P = P m , in B (cid:48) . δχ and δ A should vanish for δ E IV to vanish.As a consequence of the vanishing of the coefficients of δχ results in the followingequations Div P + (cid:101) f e = in B , (B.7a)Div P = in B (cid:48) , (B.7b) (cid:74) P (cid:75) n + (cid:101) t e = on ∂ B , (B.7c) P n = on ∂ V . (B.7d)We thus obtain the magnetic field H in the body as H = ˚Ω , B = 1 µ (cid:2) J − C B − M (cid:3) in B , (B.8)and exterior to the body as H = 1 µ J − C B in B (cid:48) , (B.9)because the magnetisation M vanishes in B (cid:48) and use has been made of the con-stitutive relation (1.11). Since the body B and normal to the boundary n canbe chosen arbitrarily, we get the following relations from the vanishing of thecoefficients of δ A Curl( H ) = in B ∪ B (cid:48) , (B.10a) n ∧ (cid:74) H (cid:75) = on ∂ B , (B.10b) n ∧ [ H e − H ] = on ∂ V . (B.10c) Remark B.1.
We note that in this formulation based on the magnetic in-duction vector, we have apriori assumed that the equation (1.7) is satisfiedby B and have recovered the equation (1.7) for the magnetic field H as theEuler-Lagrange equation for the variational (potential energy minimisation)problem. This procedure implies the constitutive assumption H = ˚Ω , B . B.2 Critical point: second variation
For the analysis of critical point ( χ, A ), we need to find the functions (cid:52) χ and (cid:52) A such that the bilinear functional defined below vanishes at the critical point, thatis δ E IV ≡ δ E IV [ χ, A ; ( δχ, δ A ) , ( (cid:52) χ, (cid:52) A )] = 0 . Upon using the expressions derived37onlinear magnetoelastostaticsin (Saxena and Sharma, 2020), the bilinear functional associated with the secondvariation of E IV is expanded into the form δ E IV = (cid:90) B [[˚Ω , FF (cid:52) F + 12 ˚Ω , F B (cid:52) B + 12 ˚Ω ∗ F B (cid:52) B ] · δ F + [˚Ω , BB (cid:52) B + 12 ˚Ω , B F (cid:52) F + 12 ˚Ω ∗ B F (cid:52) F ] · δ B ] dv + 12 µ (cid:90) B (cid:48) J − [[ F B ] · [ F B ][[ F −(cid:62) · (cid:52) F ][ F −(cid:62) · δ F ] + F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) · δ F − (cid:52) F B ] · [ F B ] + [ F (cid:52) B ] · [ F B ]] F −(cid:62) · δ F − δ F B ] · [ F B ] + [ F δ B ] · [ F B ]] F −(cid:62) · (cid:52) F + 2[ δ F (cid:52) B + (cid:52) F δ B ] · [ F B ] + 2 δ F B · F (cid:52) B + 2 (cid:52) F B · F δ B + 2[ (cid:52) F B ] · [ δ F B ] + 2[ F (cid:52) B ] · [ F δ B ]] dv . (B.11)In the expression stated above we have defined the third order tensors ˚Ω ∗ F B and˚Ω ∗ B F according to the following property[˚Ω ∗ F B u ] · U = [˚Ω , B F U ] · u , [˚Ω ∗ B F U ] · u = (cid:104) ˚Ω , F B u (cid:105) · U , (B.12)which holds for arbitrary u and U , while u is a vector and U is a second ordertensor. Using the expression (B.11) of δ E IV , in the region B (cid:48) the terms containing δ B can be written in the form v · δ B , where the vector field v is defined by v : = 1 µ J [ − [ F −(cid:62) · (cid:52) F ] F (cid:62) F B + [ (cid:52) F ] (cid:62) F B + F (cid:62) (cid:52) F B + F (cid:62) F (cid:52) B ] . (B.13)Since equation (1.11) gives H = J − µ − C B in B (cid:48) , it is easy to see that v = (cid:52) H . Also, in the expression (B.11) of δ E IV , in the region B (cid:48) the terms containing δ F can be written in the form T · δ F where the second order tensor T is defined by T : = 12 µ J (cid:34) [ F B ] · [ F B ] (cid:20)(cid:2) F −(cid:62) · (cid:52) F (cid:3) F −(cid:62) + F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) (cid:21) − (cid:20)(cid:104) (cid:52) F B (cid:105) · (cid:2) F B (cid:3) + (cid:104) F (cid:52) B (cid:105) · (cid:2) F B (cid:3)(cid:21) F −(cid:62) − (cid:2) F −(cid:62) · (cid:52) F (cid:3) [ F B ] ⊗ B + 2[ F B ] ⊗ (cid:52) B + 2[ F (cid:52) B ] ⊗ B + 2[ (cid:52) F B ] ⊗ B (cid:35) . (B.14)By expanding the expression stated in equation (2.5), to first order perturbation,it is seen that T = (cid:52) P m . Based on a repeated application of the triple product38onlinear magnetoelastostaticsidentity involving the curl operator (B.4) and the divergence theorem, while ob-serving that the variations δχ and δ A are arbitrary, the equation δ E IV = 0 (B.11)finally leads to the following partial differential equationsDiv(˚Ω , FF (cid:52) F + 12 [˚Ω , F B + ˚Ω ∗ F B ] (cid:52) B ) = 0 in B , (B.15)Curl(˚Ω , BB (cid:52) B + 12 [˚Ω , B F + ˚Ω ∗ B F ] (cid:52) F ) = 0 in B , (B.16)[[˚Ω , FF (cid:52) F + 12 [˚Ω , F B + ˚Ω ∗ F B ] (cid:52) B ] | − − T | + ] n = 0 on ∂ B , (B.17)[˚Ω , BB (cid:52) B + 12 [˚Ω , B F + ˚Ω ∗ B F ] (cid:52) F | − − v | + ] ∧ n = 0 on ∂ B , (B.18)Div T = 0 in B (cid:48) , (B.19)Curl v = 0 in B (cid:48) , (B.20) T n = 0 on ∂ V , (B.21) v ∧ n = 0 on ∂ V . (B.22) Remark B.2.
Note that since we have proved T = (cid:52) P m and v = (cid:52) H , italso follows that the above set of equations for the variations (cid:52) B and (cid:52) F in B (cid:48) can be alternatively obtained by perturbing the corresponding equations of equi-librium (B.7a) – (B.10c) . However, perturbation of the equilibrium equations in B do not result in the above equations due to presence of the [˚Ω , F B + ˚Ω ∗ F B ] and [˚Ω , B F + ˚Ω ∗ B F ] terms. This general argument can be relaxed in cases whenthe energy density function ˚Ω is assumed to be sufficiently continuous as hasbeen considered, for example, by Bustamante and Ogden (2012). C Variational formulation based on magnetic field
Noting that H = − Grad Φ, the total potential energy of the system is written as(Dorfmann and Ogden, 2014) E V [ χ, Φ] : = (cid:90) B ˇΩ( F , H ) dv − µ (cid:90) B (cid:48) J (cid:2) F −(cid:62) H (cid:3) · (cid:2) F −(cid:62) H (cid:3) dv − (cid:90) ∂ V φ b e · n ds − (cid:90) B (cid:101) f e · χdv − (cid:90) ∂ B (cid:101) t e · χds , (C.1)where ˇΩ is the stored energy density per unit volume that depends on the deforma-tion gradient F and the referential magnetic field vector H . The third term in equa-tion (C.1) is in the current configuration but the same argument as that following(B.1) allows it to be rewritten in the reference configuration as − (cid:82) ∂ V Φ B e · n ds . C.1 Equilibrium: first variation
At an state of equilibrium, χ and Φ are such that the first variation of the potentialenergy functional vanishes satisfying an analogue of equation (2.2), i.e., δ E V ≡ δ E V [ χ, Φ; ( δχ, δ
Φ)] = 0 . The variation of the functional E V up to the first order in( δχ, δ Φ) is given by δ E V = E V [ χ + δχ, Φ + δ Φ] − E V [ χ, Φ] = (cid:90) B (cid:2) ˇΩ , F · δ F − ˇΩ , H · Grad δ Φ (cid:3) dv − µ (cid:90) B (cid:48) (cid:20) J F −(cid:62) · δ F [ F −(cid:62) H ] · [ F −(cid:62) H ] − J (cid:2) F −(cid:62) [ δ F ] (cid:62) F −(cid:62) H (cid:3) · [ F −(cid:62) H ]+ 2 J [ F −(cid:62) H ] · [ F −(cid:62) δ H ] (cid:21) dv − (cid:90) ∂ V δ Φ B e · n ds − (cid:90) B (cid:101) f e · δχdv − (cid:90) ∂ B (cid:101) t e · δχds . (C.2)We define the first Piola–Kirchhoff stress P and magnetic induction B in the bodyas P = ˇΩ , F , B = − ˇΩ , H in B , (C.3)the (Maxwell) stress P m exterior to the body as stated earlier in equation (2.5)and recall the relation J − F B = µ F −(cid:62) H in vacuum from equation (1.11). Usingthe above relations (C.3), we rewrite the first variation (C.2) as δ E V = (cid:90) B (cid:2) Div (cid:0) P (cid:62) δχ (cid:1) − [Div P + (cid:101) f e ] · δχ + Div ( δ Φ B ) − δ Φ Div B (cid:3) dv + (cid:90) B (cid:48) (cid:2) Div (cid:0) P (cid:62) m δχ (cid:1) − [Div P m ] · δχ + Div ( δ Φ B ) − δ Φ Div B (cid:3) dv − (cid:90) ∂ V δ Φ B e · n ds − (cid:90) ∂ B (cid:101) t e · δχds . (C.4)After an application of divergence theorem to (C.4), we get δ E V = (cid:90) B [ − [Div( P ) + (cid:101) f e ] · δχ − δ Φ Div B ] dv + (cid:90) ∂ B [[[ P | − − P m | + ] n − (cid:101) t e ] · δχ + δ Φ[ B | − − B | + ] · n ] ds + (cid:90) B (cid:48) [ − [Div P m ] · δχ − δ Φ Div B ] dv + (cid:90) ∂ V [ P m n · δχ + δ Φ[ B − B e ] · n ] dv . (C.5)40onlinear magnetoelastostaticsSince the two variations δχ and δ Φ are arbitrary, their coefficients in each of theintegrals must vanish. Accordingly, using the coefficient of δχ in (C.5), we get theequations Div P + (cid:101) f e = in B , (C.6a)Div P = in B (cid:48) , (C.6b) (cid:74) P (cid:75) n + (cid:101) t e = on ∂ B , (C.6c) P n = on ∂ V , (C.6d)while the coefficient of δ Φ in (C.5) leads to the equationsDiv B = 0 in B , (C.7a)Div B = 0 in B (cid:48) , (C.7b) (cid:74) B (cid:75) · n = 0 on ∂ B , (C.7c) (cid:74) B (cid:75) · n = 0 on ∂ V . (C.7d) Remark C.1.
Parallel to the remark B.1 at the end of Section B.1, we notethat in this formulation based on the magnetic field (equivalently, the magneticscalar potential), we have apriori assumed the equation (1.7) that H shouldsatisfy and have recovered the equation (1.7) for the magnetic induction B asan Euler-Lagrange equation of this minimisation problem. This procedure tooimplies the constitutive assumption B = − ˇΩ , H while it has been also indepen-dently derived earlier (Dorfmann and Ogden, 2004). C.2 Critical point: second variation
For the analysis of critical point ( χ, Φ), we need to find (cid:52) χ and (cid:52) Φ such thatcertain bilinear functional based on the second variation vanishes at the criticalpoint, that is δ E V ≡ δ E V [ χ, Φ; ( δχ, δ Φ) , ( (cid:52) χ, (cid:52) Φ)] = 0 . The second variation ofthe functional in (C.1) based on the magnetic field H is given by δ E V = (cid:90) B (cid:20) Div (cid:18)(cid:2) ˇΩ , FF (cid:52) F + 12 ˇΩ , F H (cid:52) H + 12 ˇΩ ∗ F H (cid:52) H (cid:3) (cid:62) δχ (cid:19) − Div (cid:18) ˇΩ , FF (cid:52) F + 12 ˇΩ , F H (cid:52) H + 12 ˇΩ ∗ F H (cid:52) H (cid:19) · δχ − Div (cid:18)(cid:2)
12 ˇΩ ∗ H F (cid:52) F + 12 ˇΩ , H F (cid:52) F + ˇΩ , HH (cid:52) H (cid:3) δ Φ (cid:19) + Div (cid:18)
12 ˇΩ ∗ H F (cid:52) F + 12 ˇΩ , H F (cid:52) F + ˇΩ , HH (cid:52) H (cid:19) δ Φ (cid:21) dv (cid:90) B (cid:48) (cid:2) Div( (cid:101) T (cid:62) δχ ) − Div (cid:101) T · δχ + Div ( (cid:101) v δ Φ) − Div (cid:101) v δ Φ (cid:3) dv , (C.8)where we have introduced the tensor (cid:101) T and the vector (cid:101) v as (cid:101) T : = J µ (cid:34) F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) H ⊗ F − F −(cid:62) H + F −(cid:62) H ⊗ F − (cid:52) FF − F −(cid:62) H − F −(cid:62) (cid:52) H ⊗ F − F −(cid:62) H − F −(cid:62) H ⊗ F − F −(cid:62) (cid:52) H + F −(cid:62) H ⊗ F − F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) H − (cid:2) F −(cid:62) · (cid:52) F (cid:3) F −(cid:62) H ⊗ F − F −(cid:62) H + (cid:20) − (cid:2) F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) H (cid:3) · (cid:2) F −(cid:62) H (cid:3) + (cid:2) F −(cid:62) H (cid:3) · F −(cid:62) (cid:2) (cid:52) H (cid:3)(cid:21) F −(cid:62) − (cid:2) F −(cid:62) H (cid:3) · (cid:2) F −(cid:62) H (cid:3)(cid:20)(cid:2) F −(cid:62) · (cid:52) F (cid:3) F −(cid:62) − F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) (cid:21) , (C.9) (cid:101) v : = J µ (cid:20) F − (cid:52) FF − F −(cid:62) + F − F −(cid:62) (cid:2) (cid:52) F (cid:3) (cid:62) F −(cid:62) − (cid:2) F −(cid:62) · (cid:52) F (cid:3) F − F −(cid:62) (cid:21) H − J µ F − F −(cid:62) (cid:52) H , (C.10)while we have also utilized the definitions of two third order tensors ˇΩ ∗ F H and ˇΩ ∗ H F ,according to the relations[ ˇΩ ∗ F H u ] · U = [ ˇΩ , H F U ] · u , [ ˇΩ ∗ H F U ] · u = [ ˇΩ , F H u ] · U , (C.11)where u and U are arbitrary vector and arbitrary second order tensor, respectively.An application of divergence theorem to (C.8) gives δ E V = (cid:90) B (cid:34) − Div( ˇΩ , FF (cid:52) F + 12 ˇΩ , F H (cid:52) H + 12 ˇΩ ∗ F H (cid:52) H ) · δχ + Div( 12 ˇΩ ∗ H F (cid:52) F + 12 ˇΩ , H F (cid:52) F + ˇΩ , HH (cid:52) H ) δ Φ (cid:35) dv + (cid:90) ∂ B (cid:34)(cid:20) ˇΩ , FF (cid:52) F + 12 ˇΩ , F H (cid:52) H + 12 ˇΩ ∗ F H (cid:52) H (cid:21) | − − (cid:101) T | + (cid:35) n · δχds − (cid:90) ∂ B (cid:34)(cid:20)
12 ˇΩ ∗ H F (cid:52) F + 12 ˇΩ , H F (cid:52) F + ˇΩ , HH (cid:52) H (cid:21) | − − (cid:101) v | + (cid:35) · n δ Φ ds + (cid:90) B (cid:48) (cid:20) − Div (cid:101) T · δχ − Div (cid:101) v δ Φ (cid:21) dv + (cid:90) ∂ V (cid:20) (cid:101) T n · δχ + (cid:101) v · n δ Φ (cid:21) ds . (C.12)42onlinear magnetoelastostaticsSince the variations δχ and δ Φ are arbitrary, we arrive at the following equationsfor the unknown functions ( (cid:52) χ, (cid:52) Φ)Div( ˇΩ , FF (cid:52) F + 12 ˇΩ , F H (cid:52) H + 12 ˇΩ ∗ F H (cid:52) H ) = in B , (C.13)Div( 12 ˇΩ ∗ H F (cid:52) F + 12 ˇΩ , H F (cid:52) F + ˇΩ , HH (cid:52) H ) = 0 in B , (C.14)[ (cid:2) ˇΩ , FF (cid:52) F + 12 ˇΩ , F H (cid:52) H + 12 ˇΩ ∗ F H (cid:52) H (cid:3) | − − (cid:101) T | + ] n = on ∂ B , (C.15)[[ 12 ˇΩ ∗ H F (cid:52) F + 12 ˇΩ , H F (cid:52) F + ˇΩ , HH (cid:52) H ] | − − (cid:101) v | + ] · n = 0 on ∂ B , (C.16)Div (cid:101) T = in B (cid:48) , (C.17)Div (cid:101) v = 0 in B (cid:48) , (C.18) (cid:101) T n = on ∂ V , (C.19) (cid:101) v · n = 0 on ∂ V , (C.20)describing the onset of bifurcation. Remark C.2.
Note that a variation of the relation B = J µ C − H from equa-tion (1.11) gives (cid:52) B = (cid:101) v , since (cid:52) B = J µ (cid:2) F − F −(cid:62) (cid:52) H − F − (cid:52) FF − F −(cid:62) H − F − F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) H + (cid:2) F −(cid:62) · (cid:52) F (cid:3) F − F −(cid:62) (cid:3) . (C.21) A variation of the Maxwell stress (2.5) (after writing it in terms of H usingthe relation (1.11) ) gives (cid:52) P m = (cid:101) T , since (cid:52) P m = J µ (cid:34) F −(cid:62) (cid:52) H ⊗ F − F −(cid:62) H + F −(cid:62) H ⊗ F − F −(cid:62) (cid:52) H + (cid:2) F −(cid:62) · (cid:52) F (cid:3) F −(cid:62) H ⊗ F − F −(cid:62) H − F −(cid:62) [ (cid:52) F ] −(cid:62) H ⊗ F − F −(cid:62) H − F −(cid:62) H ⊗ F − F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) H − F −(cid:62) H ⊗ F − [ (cid:52) F ] F − F −(cid:62) H + 12 [ F −(cid:62) H ] · [ F −(cid:62) H ][ F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) − [ F −(cid:62) · (cid:52) F ] F −(cid:62) ]+ (cid:2) − [ F −(cid:62) [ (cid:52) F ] (cid:62) F −(cid:62) H ] · [ F −(cid:62) H ] + [ F −(cid:62) H ] · F −(cid:62) [ (cid:52) H ] (cid:3) F −(cid:62) (cid:35) . (C.22) Alternative to the statements (cid:101) v = (cid:52) H and (cid:101) T = (cid:52) P m , it can be also shown that the above set of equations for the perturbations (cid:52) H and (cid:52) F can be ob-tained by linearising the equations of equilibrium (C.6a) – (C.7d) . References
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