1 Introduction

The use of the framework of continuum mechanics for the study of the influence of electromagnetic effects on solids has been largely stimulated by Truesdell and Toupin[14].Their research on the coupling between the mechanical and magnetic responses of magnetoelastic solids was the first of a long list(without any attempt to be exhaustive,see,e.g.[1,4–7,10,12–13]etc.).An important stimulus for the development of these researches was the study of the magnetoelastic buckling problem.Indeed a plate,made of a magnetoelastic material,subject to a transverse magnetic field,buckles when the magnetic field attains a critical value;see also[3,5,9],for a general analysis of the buckling of some magnetoelastic structures.Following a pioneering experimental and theoretical research of Moon and Pao,the first rigorous attempt to analyze this problem is due to Pao and Yeh[12].Maugin and Goudjo[8]considered a plate model with particular attention on the regularity of the boundary.More recently,in the case linear soft magnetoelastic materials,Zhou and Zheng[15–17]have revisited the subject by adapting the usual Kirchhoff-Love and von Kármán models only modifying the equivalent transverse force with the addition of the magnetic effects.

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The paper is organized as follows.In Section 2,we briefly recall the governing equilibrium equations of magnetoelasticity and,then,in Section 3,we state the problem on a variable domain assuming that the magnetic forces are given.In order to apply the Ciarlet’s method(see,e.g.,[2]),we must at first provethat the magnetic forcesgiveriseto a linear and continuous form.This can be achieved under suitable assumptions on the magnetic forces(see lemma 3.1).In Section 4,we introduce the usual scaling on the mechanical quantities and we scale the magnetic quantities in such a way that the Gauss’Law(2.1)is conserved.Using the classical change of variables,we deduce the scaled equations.Then we can use the asymptotic methods,following[2],to obtain the limit problem and the strong convergence result.It is interesting to remark that in the simplest situation of a transversal magnetic field,we recover the Kirchhoff-Love model of Zhou and Zheng,which is,hence,completely justified.

2 Governing Equations of Magnetoelasticity for Linear Soft Ferromagnetic M aterials

In the sequel,Greek indices range in the set{1,2},Latin indices range in the set{1,2,3},and the Einstein’ssummation convention with respect to there peated indicesisadopted.Let us consider a three-dimensional Euclidean space identified by R3 and such that the three vectors e i form an orthonormal basis.We introduce the following notations for the vector product:a∧b=a i e i∧b j e j=a i b jǫij k e k,for all vectors a=(a i)∈R3 and b=(b i)∈R3,whereǫij k denotes the alternator Ricci’s symbol.

When a magnetizable,deformable elastic solidΩis placed in a magnetic field,magnetic moments are induced inside the body.The action of the external magnetic induction B 0 manifests itself in magnetization M(magnetic moment per unit volume).Within the body,the magnetic induction B is not necessarily equal to B 0.The induced magnetization M=(M i)is related to B=(B i)by B=µ0(H+M),where H=(H i)is called magnetic intensity andµ0 is the magnetic permeability of vacuum.Generally,we have H=H(M),but,in the sequel,we restrict our attention to a class of linear isotropic magnetoelastic materials called soft ferromagnetic materials,which are characterized by the fact that their local average magnetization becomes zero when the external field is set to zero.In this particular case,the hysteresis loops are narrow and the influence of induced currents is small in comparison with the effect of magnetization.Therefore,it is possible to use the quasi-static approximation,i.e.,the equations of magneto-statics:

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i.e.,component-wise, and.This choice of the action of the magnetic field is sometimes called the dipole model of microcurrents and has been used in particular by Pao-Yeh for soft ferromagnetic elastic solids(see[12]).Using the Gauss’law and the Ampère’slaw,the magnetic body force f m can also be written as the divergenceof a second order tensorthe so-called Maxwell’s stress tensor

whereχrepresents the magnetic susceptibility andµr:=χ+1 is the relative magnetic permeability.For linear soft ferromagnetic materials,such as steels,iron,cobalt and various alloys,the relative permeability is very large,µr or

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Since the bilinear formis-coercive,in order to apply the Lax-Milgram’s lemma,we have only to prove that the linear formis continuous on V(Ωε).For this when 1the Banach space of v ∈ L p(Ωε)whose first order derivatives(in the distribution sense)also belong to L p(Ωε).The continuity of the linear formis the object of the following lemma.

The magnetic constitutive law takes the following linear form

In this work we use the model proposed by Brown[1]where the action of the magnetic field is given by a magnetic body force(per unit volume)and a magnetic body couple(per unit volume)

or,component-wise, with andAs pointed out by[1],other different choices of the Maxwell’s stress tensor are possible and,indeed,they depend on the choice of the Helmomtz free energy;see in particular[5],for a clear explanation of the influence of the choice of the argumentsin the freeenergy on the Maxwell stress,magnetic body forces and traction boundary conditions.

Considering the expressions above,in the absence of electric field,charge distribution and conduction current,the mechanical governing equations defined in a magnetized bodyΩcan be expressed by

where t ij is the non-symmetric total stress tensor,(n i)represents the unit normal vector to the boundaryΞ⊂∂Ωand M n:=M i n i is the normal surface boundary magnetization.The nonsymmetry of the total stress tensor is due to the presence of a magnetic body couple.In order to simplify the model,we neglect magnetostriction and piezostrictive terms in the constitutive laws and we consider an isotropic linear elastic material.Thus,we obtain that

whereσij denotes the symmetric Cauchy stress tensor,associated with the linearized strain tensorbeing(u i)the displacement field,through the classical Lamé’s constitutive equations.The presence of the termµ0M i H j in the decomposition of the total stress always follows from the form of the Helmoltz free energy(see[1,5]).

In the Pao-Yeh’s case of soft ferromagnetic materials,thanks to(2.2),the magnetic body couple I m= µ0χH∧H=0,and,hence,the stress tensor t ij becomes symmetric,i.e.,t ij=t j i.Moreover,substituting the expression of the Maxwell’sstresstensor into the divergence relation(2.3)and using(2.1),we can find an alternative form of the magnetic body force:

In the sequel,we will focus our attention on the reduced mechanical model arising from the use of the asymptotic methods,assuming that the magnetization M and the magnetic intensity H are a given external magnetic source.

3 Position of the Problem

Letω ⊂ R2 denote a smooth domain in the plane spanned by vectors eα,with boundary γ;γ0⊂ γ is a measurable subset ofγ with strictly positive length measure; is the complement ofγ0 with respect toγ;finally,0< ε<1 is a dimensionless small real parameter which shall tend to zero.For eachε,we define

with hε>0.Hence the boundary∂ΩεofΩεis partitioned into the lateral surfaceΓεand the upper and lower faces and the lateral surface is itself partitioned aswithMoreover,we letthe complement of with respect to ∂Ωε.

We assume thatΩεis constituted by a homogeneous isotropic linear soft ferromagnetic material,whose constitutive law is given in(2.5).We suppose that the Lamé’s coefficients satisfy the classical positivity properties.The plate is clamped on so that uε =0,and,for simplicity,we consider that no mechanical charges are applied to the body.The only source terms are given by and

Letbe the functional space of admissible displacements.The variational formulation of problem(2.4),defined on the variable domainΩε,takes the following form:

where the bilinear form Aε(·,·)and the linear form Lε(·)are,respectively,defined by

with

In order to prove the wellposedness of the problem,by virtue of the Lax-Milgram’s lemma,we rewrite(3.1)in an alternative form:

where

反之，对任意x ∈ X, 令 = γ, 则x ∈ X( 且X( 是X的滤子.如果 则1 X( 这与X( ∅是X的滤子矛盾，于是 ⊇ 令 (x → y) = γ′, 则x ∈ X( → y ∈ X( 且X( 是X的滤子.如果 ∩ (x → y), 则y X( 这与X( 是X的滤子相悖，所以 ⊇ (x → y).同理， ⊇ ∩ y).总之，为X的一个犹豫模糊滤子.

Lemma 3.1 Let us assume

Thenis continuous on V(Ωε).

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Proof(i)Sincethanks to the Sobolev imbedding theorem,(see e.g.[11,Chapter 2,Theorem 3.4])we obtain thatand that;hence,by means of Hölder’s inequality,we can infer that

(ii)Sincethanks to the Sobolev imbedding theorem,we get thatand,thus,It then followsfrom Korn’sinequality

(iii)By virtue of a trace imbedding theorem(see e.g.[11,Chapter 2,Theorem 4.2]),we have thatBesides,being v ∈ V(Ωε),then the same trace imbedding theorem imply thatand so

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Collecting(3.4)–(3.6),we obtain the desired result.

(b)The sequence{u(ε)}ε>0 strongly converges in H 1(Ω;R3)to u0,the solution of the limit problem(5.1).

4 The Asymptotic Expansion

In order to perform an asymptotic analysis,we need to transform problem(3.1),posed on a variable domainΩε,onto a problem posed on a fixed domainΩ (independent ofε).We suppose that the thickness of the plate hε depends linearly on ε,so that hε= εh.Accordingly,we let

经过近现代以来的翻译实践和理论研究，外国人名汉译已经建成一些规范。如陈国华和石春让在《外国人名汉译的原则》一文中提到名从主人、定名不咎、音义兼顾、译音循本、音系对应等四个原则。[9]岳静和付吟璐在《浅析外国人名汉译的规范化问题》一文中强调了注重汉译名的人名区分度、异域辨识度、性别辨识度，以及对源语发音、专名主人的汉语定名、约定俗成原则的尊重。[10]但是由于汉语与大多数国家文字（字母文字）在类型上的差异，导致汉语不能完全音译，进而导致在对应音的选字和字义的考虑两方面的困难。前者基本可以通过译音表加以规范，但后者则涉及广泛，非硬性规定可以奏效。

wheredenotes the restriction ofφ on Γ±.Sincethanks to Lemma 3.1 and Lax-Milgram’s lemma,we can prove that the scaled problem admits one and only one solution.

By using the bijectionπε,one hasandMoreover,we define the following functional spaces:

In order to writ etheexpression of the scaled problem,weneed first to makesomeassumptionson the data which will define their dependences with respect to the small parameterε.In our case,the only external data are represented by the magnetic fieldthe magnetic intensity fieldand the magnetization fieldBy virtue of the constitutive equationsandwe assume that and will share the same dependence onε.The scaling of must reflect the fact that the magnetic field is a solenoidal field,meaning that inΩε.This property must be satisfied also on the fixed domainΩ.Let us suppose that

with B i independent of ε.By applying the change of variables πε,we can write that the scaled divergence vanishes in Ω,so that in Ω.In order to guarantee the consistency of this equation,we ask that p=q+1,finding a relation between the two exponents p and q.In the sequel,we choose q=0 and,hence

With the unknown displacement field uε,we associate the scaled unknown displacement field u(ε)defined by

We likewise associate with any test function vε,the scaled test function v,defined by the scalings:

According to the previous hypothesis,problem(3.1)can be reformulated on a fixed domainΩ independent ofε.Thus we obtain the following scaled variational problem:

where the scaled bilinear form A(·,·)and the scaled linear form L(·)are,respectively,defined by

with

and we define the following change of variables(see[2]):

We are now in position to perform an asymptotic analysis of the scaled problem(4.1).Since the scaled problem(4.1)has a polynomial structure with respect to the small parameterε,we can look for the solution of the problem as a formal series of powers ofε:

Hence,by substituting expressions(4.2)in(4.1)and by identifying the terms with identical power ofε,we can write the following sequence of variational subproblems:

By solving the above variational problems,we can characterize the leading term of the asymptotic expansion u0,the so-called the limit displacement field,and its associated limit problem.

5 The Limit Problem

We define the usual functional space of Kirchhoff-Love admissible displacements:

and

where ν =(να)is the outer unit normal vector to γ.

Theorem 5.1(a)The leading term u0 of the asymptotic expansion(4.2)satisfies the following variational problem:

where

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Thanks to the V(Ωε)-coercivity of the bilinear formand the continuity of the linear formwededuce,using the Lax-Milgram’slemma,that the variational problem(3.1)admits one and only one solution.

Proof The proof is straightforward,following the approach by[2].

Let us focus our attention on the expression of the magnetic force work L(·)of the limit problem.By choosing a test function v ∈ V K L(Ω),namelyandwith and after an integration along x3 and by applying the Gauss-Green’s formula,we get

where the reduced magnetic forces and have the following form:

whereand

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It easy to verify that if the induced magnetic intensity field is normal to the middle plane of the plate,with Hα=0,the form of the limit magnetic force acting on a plate depends just on the jump of the square of magnetic intensities evaluated at the top and bottom faces of the plate.Indeed,sinceone has

beingχvery large for soft ferromagnetic materials.Equation(5.3)is analogue to the one presented in[16]and it can be considered as a mathematical justification of the magnetic force acting on a plate,which is usually employed in magnetic instability problems.

The limit problem(5.1)can be decoupled into a membrane and a bending problem,by virtue of the Kirchhoff-Love limit displacement field.The membrane problem reads as follows:

where

represents the ferromagnetic membrane stress tensor.After an integration by parts,we find that the membrane displacementssolve the following membrane differential problem:

The bending problem takes the following form:

where

represents the ferromagnetic moment stress tensor.After an integration by parts,we find that the transversal displacement solves the following bending differential problem:

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where represents the unit tangent vector to γ.

Considering the case of an induced magnetic intensity field,normal to the middle plane of the plate,with Hα=0,nαβand mαβreduce to the classical elastic membrane stress tensor and moment stress tensor.Besides,sincewe can infer that the membrane problem admits the only zero solution,so that nαβ =0,and thus,in this case,the plate equilibrium problem takes just into account the bending behavior.

6 Concluding Remarks

In this work we derive a model of a soft ferromagnetic isotropic linear plate by means of an asymptotic analysis.In the absence of mechanical loading,thanks to the particular scaling of the magnetic charges,we obtain a complex expression of the reduced magnetic forces acting on the plate.The problem can be decoupled as usual in a membrane problem and in a flexural problem.It is important to notice that in the simple case in which the magnetic charges are normal to the middle plane of the plate,we formally obtain an expression of the magnetic force(5.3),acting on the plate,which is equivalent to the one used in classical literature(see,e.g.,[12,15–16]).Moreover,by virtue of the strong convergence result,we also give a mathematical justification to the limit model.

The present work represents a first step on the asymptotic modeling of soft ferromagnetic plates.Indeed,we do not consider the coupling between the mechanical and magnetic behaviors within the equations of magnetostatics.We assume the magnetic charges as external loads without investigating the asymptotic behavior of the magnetostatic equations for what concerns with the magnetic field,the magnetic intensity and the magnetization.Therefore,the limit problem becomes linear and we cannot see,at first glance,the so-called magnetic buckling phenomenon.

Acknowledgement We thank K.Danas and N.Triantafyllidis for many useful discussions on magnetoelastic materials.

References

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[2]Ciarlet,P.G.,Mathematical Elasticity,II,Theory of Plates,North-Holland,Amsterdam,1997.

[3]Danas,K.and Triantafyllidis,N.,Instability of a magnetoelastic layer resting on a non-magnetic substrate,J.Mech.Phys.Solids,69,2014,67–83.

[4]Dorfmann,A.and Ogden,R.W.,Nonlinear magnetoelastic deformations of elastomers,Acta Mechanica,167,2004,13–28.

[5]Kankanala,S.V.and Triantafyllidis,N.,Magnetoelastic buckling of a rectangular block in plane strain,J.Mech.Phys.Solids,56,2008,1147–1169.

[6]Maugin,G.A.,Continuum Mechanics of Electromagnetic Solids,North-Holland,Amsterdam,1988.

[7]Maugin,G.A.and Eringen,A.C.,Deformable magnetically saturated media I,Field equations,J.Math.Phys.,13,1972,143–155.

[8]Maugin,G.A.and Goudjo,C.,The equations of soft ferro-magnetic elastic plates,Int.J.Solid Struct.,8,1982,889–912.

[9]Miya,K.,Hara,K.and Someya,K.,Experimental and theoretical study on magnetoelastic buckling of a ferromagnetic cantilevered beam-plate,J.Appl.Mech.,45,1978,355–360.

[10]Moon,E.C.,Magneto-Solid Mechanics,Wiley,New York,1984.

[11]Nečas,J.,Direct Methods in the Theory of Elliptic Equations,Springer-Verlag,Berlin,Heidelberg,2012.

[12]Pao,Y.H.and Yeh,C.S.,A linear theory for soft ferromagnetic elastic solids,Int.J.Engng.,11,1973,415–436.

[13]Tiersten,H.F.,Coupled magnetomechanical equations for magnetically saturated insulators,J.Math.Phys.,5,1964,1298–1318.

[14]Truesdell,C.and Toupin,R.,The Classical Field Theories,Handbuch der Physik,III/I,Principles of Classical Mechanics and Field Theories,Flügge,S.(ed.),Springer-Verlag,Berlin,1960,226–790.

[15]Zhou,Y.H.and Zheng,X.J.,A theoretical model of magnetoelastic buckling for soft ferromagnetic thin plates,Acta Mechanica Sinica(English edition),12,1996,213–224.

[16]Zhou,Y.H.and Zheng,X.,General expression of magnetic force for soft ferromagnetic plates in complex magnetic fields,Int.J.Engng.,35,1997,1405–1417.

[17]Zhou,Y.H.,Zheng,X.and Miya K.,Magnetoelastic bending and snapping of ferromagnetic plates in oblique magnetic fields,Fusion Engng.design,30,1995,325–337.