1 Introduction

The development of magnetic and electronic material has been penetrated in every fieldof modern technology. Owing to the trend of device miniaturization,multifunctional materi-als with the properties of electricity and magnetism have attracted more and more interests.Magnetoelectric (ME) structures have the magneto-electro-elastic coupling property,with theability to convert energy from one form (among magnetic,electric,and mechanical energies)to another. Many references^{[1, 2, 3, 4, 5]} show us the recent research of ME materials. Because ofthe coupling effects,ME materials have significant application prospects in sensor technology,memory devices,and smart structures^{[6, 7, 8, 9]}.

For ME materials,the ME effect is one of the most important research topics. Recently,it is confirmed that the ME effect can be enhanced in the functionally graded multiferroicmaterials^{[10, 11]}. The functionally graded material (FGM) is a kind of material with mate-rial composition and properties varying continuously along certain directions. The intelligentdevices made of FGMs have no discernible internal boundaries and do not produce stress con-centration while they are loaded. The concept of FGMs was first proposed in 1984 by mate-rial scientists as a means of preparing thermal barrier materials^[12]. Since then,FGMs havebeen applied to various fields such as electronic,chemical,optical,acoustic,and biomedicaltechnologies^{[13, 14, 15, 16]}. For practical problems in multifunctional materials,the direct solution^[17]and various effective numerical methods (such as the finite element method^[18] and the bound-ary element method^{[19, 20]}) are generally adopted to study the transformation between force,electricity,and magnetism. However,in order to verify the correctness of numerical calculationmethod and check the precision of numerical results,it is still necessary and meaningful to seekanalytical solutions.

Shi^[21] and Shi and Chen^[22] studied the response of functionally graded piezoelectric beamsand presented the analytical solutions. However,in their analysis,only one or two materialparameters were assumed to vary in the form of finite power series along the thickness directionwith other parameters being kept constant. Zhong and Yu^[23] provided the analytical solutionfor the problem of a cantilever functionally graded elastic beam subjected to different loadsby assuming that all the elastic moduli of the material have the same variations along thebeam-thickness direction. Pan and Han^[24] presented an exact solution for functionally gradedlayered magneto-electro-elastic plates with all material constants varying in the same expo-nential way along the thickness direction by the pseudo-Stroh formalism and the propagatormatrix method. Ding et al.^[25] obtained elasticity solutions for plane anisotropic functionallygraded beams by considering the elastic compliance parameters being arbitrary functions of thethickness coordinate. Huang et al.^[26] investigated the bending problem of anisotropic function-ally graded magneto-electro-elastic (FGMEE) plane beams subjected to polynomial loads ontheir upper/lower surface with various boundary conditions being applied at the two ends ofthe beams. Li et al.^[27] presented a three-dimensional analytical solution for FGMEE circularplates subjected to a uniform load. By introducing stress,electric displacement,and magneticinduction functions,Huang et al.^[28] obtained an analytical solution and a semi-analytical solu-tion for the anisotropic FGMEE beam subjected to distributed sinusoidal loadings. However,their analytical solution is only suitable for FGMEE beams with exponential variation or powerfunction variation along the thickness. To the authors’ knowledge,no general solution has beenfound to be adaptive to arbitrary material property variation for FGMEE beams until now.

In this paper,we present a general solution of a cantilever FGMEE with arbitrary gradedvariations of material property distribution based on the two-dimensional theory of multiferroicmaterial. In Section 2,the basic equations and boundary conditions are described. In Section 3,the stress,the electric displacement,and the magnetic induction are expressed,respectively,ina particular form involving certain unknown functions of the thickness coordinate. Then,an an-alytical solution of a plane problem is formulated. In Section 4,the numerical results and dis-cussion are given for specific examples. Finally,some conclusions are summarized in Section 5. 2 Basic equations

As shown in Fig. 1,a magneto-electro-elastic cantilever with a rectangular cross-section ofthe height h and the width b is subjected to a uniform load q on the upper surface and aconcentrated force P and moment M at the end of the beam. When the width b is muchsmaller than the height and the length,this problem can be treated as a plane stress problem.

The constitutive equations for this plane stress problem can be written as

where σ_x,σ_z,and τ_zx denote the stress components,"x,"z,and zx are the strain components,and E_k,D_k,H_k,and B_k (k = x,y) are,respectively,the components of electric field,electricdisplacement,magnetic field,and magnetic flux. s_ij ,d_ij ,q_ij ,λ_ij ,μ_ij ,and μ_ij denote thecoefficients of elastic compliance,piezoelectricity,piezomagnetism,dielectric impermeability,magnetic permeability,and magneto-electricity,respectively.

Assume an FGM with the function variation in the z-direction. The material coefficients in Eq. (1) can be described as

where s_ij⁰ ,d_ij⁰ ,q_ij⁰ ,μ_ij⁰ ,μ_ij⁰ ,and α_ij⁰ are the material coefficients for z = z0 and F (z0) = 1.

The equilibrium equations without the body force and the electric charge can be written as

The geometrical equations are

The equation of strain compatibility for the plane stress problem is expressed as

For the cantilever beam,the boundary conditions are given as follows:

(i) The boundary conditions for the external force are

At x = 0 and z = While at x = 0 and z = −

(ii) The boundary conditions for electric and magnetic fields are

at the ends of the beam (i.e.,x = 0 and x = l),and at the upper surface and under the surface (i.e.,z = ± ).

(iii) The displacement boundary conditions at the clamped end (z = 0 and x = l) are

3 Stress,electric displacement,and magnetic induction functions and analyticalsolution of FGMEE plane problem

Firstly,we introduce the Airy stress function U(x,z) which satisfies the following equations:

Substituting Eqs. (5),(6),and (14) into Eq. (1) and based on the compatibility equation (7) and the last two equations of equilibrium equations (3),we obtain

Then,we assume By substituting the assumptions into Eq. (15),we can get the following relations: where

The exact expressions of f(z) and f₁ (z) ,f₂(z),· · · ,f₈(z) can be obtained by solving Eqs. (17)-(19).

Firstly,we introduce some new symbols

Since the expressions of f(z) and f₁ (z) ,f₂(z),· · · ,f₈(z) are obtained,the Airy stress func-tion U(x,z),the electric potential φ,and the magnetic potential all can be determined. Therefore,the stress,electric field,and the magnetic field can be expressed asφ

Substituting Eqs. (20)-(26) into Eq. (1),we can obtain the expressions of the strains,the electric displacements,and the magnetic fluxes as follows:

where

Integrating Eq. (27) with respect to x, the displacement in the x-direction can be derived as

Similarly,the displacement in the z-direction can be obtained by integrating Eq. (28) withrespect to z as follows:

Notice that there are two unknown functions g₁(z) and g₂ (x) in the expressions of displace-ments u and w. Using the shear strain-displacement relations in Eqs. (4),(31),(34),and (35),the forms of g₁(z) and g₂ (x) are expressed as

The determinations of all the unknown coefficients included in Eqs. (20)-(37) are given in Appendix A. 4 Examples and discussion

In the following numerical example,the dimensions ofME cantilever beam shown in Fig. 1 areset as l = 0.3m,h = 0.02m,and b = 0.02m. The loads are as follows: q = 1N/m²,P = 10N,and M = 1N·m. Two kinds of gradient function of material parameters are considered: theexponential function F(z) = e^k (k = 0,1,2) and the linear function F(z) = 1 + k (k = 0,1,1.9),where k is the gradient factor characterizing the degree of the material gradient inthe z-direction. It is obvious that k = 0 corresponds to the homogeneous material case. Thephysical parameters of the magneto-electro-elastic material at z = 0 are recorded in Table 1.

In Table 1,the units are the elastic stiffness s_ij ,10⁻¹² m²/N,the piezoelectric constant d_ij ,10⁻¹³ C/(N·m),the piezomagnetic constant q_ij ,10⁻⁹ N/(A·m),the dielectric constant λ_ij ,10⁻⁸ F/m,the magnetic permeability m_ij ,10⁻⁶ N·s²/C²,and the electromagnetic contant α_ij ,10⁻⁸ N·s/(V·C).

Figures 2 and 3 show the z-dependent variation of all physical quantities (stresses,electricdisplacements,magnetic fluxes,and displacements in the x- and z-directions) at the locationof x = l/6. In all the figures,we compare five kinds of ME materials: (i) the homogeneousmaterial (i.e.,k = 0); (ii) the exponential gradient function with k = 1; (iii) the linear gradientfunction with k = 1; (iv) the exponential gradient function with k = 2; (v) the linear gradientfunction with k = 1.9.

Observing Figs. 2(a)−2(g),it can be seen that when k = 1,the curves of physical quantities of material parameters withexponential variation and linear variation are essentially coincident,which indicates that the two kinds of gradient function have the same effect.

Figure 2(a) shows the z-dependent variation of σ_x. It gives the information that for theFGM,the stress σ_x is a nonlinear distribution with respect to z,and the location of centralaxis is upward with the increase of gradient factor k. In particular,the absolute value of thestress σ_x on the upper surface of the beam is larger than that on the lower surface. For the caseof homogeneous material,σ_x changes linearly with respect to z and is also antisymmetric withrespect to the central axis. These coincide with the results of classical solution of elasticity.

Figure 2(b) shows the z-dependent variation of stress σ_z. It is a nonlinear distribution andincreasing along the z-direction. For the material with the same gradient distribution,theabsolute value of σz decreases with the increase of k.

Figure 2(c) gives the distribution of shear stress τ_zx along the z-direction. It is obvious thatthe distribution of τ_zx is parabolic. For the homogeneous material (i.e.,k = 0),τ_zx takes themaximum value in the neutral axis which is also consistent with the classical solution. However,for the FGM,although τ_zx also has the maximum value in the neutral axis,it is no longer asymmetric distribution on the cross section. It is noteworthy that the maximum absolute valueof τ_zx on the cross section increases with k when the beam is subjected to the same externalforce and has the same type of gradient distribution.

Figure 2(d) shows that the electric displacement D_x is also a parabolic distribution alongthe z-direction. For the homogeneous material,the electric displacement has a symmetricdistribution with respect to z = 0. The values of D_x on the lower and upper surfaces are equalto each other,and they are all nonzero. It indicates that the beam achieves electro-mechanicalconversion under the effect of external forces. For the FGM,D_x on the lower surface is larger than that on the upper surface. At the same time,the differential of electric displacement on the upper and lower surfaces enlarges when k increases.

Figure 2(e) shows that D_z has a nonlinear distribution with respect to z,and it reaches thepeak value at the location of h/4 and 3h/4,respectively. For a given gradient distribution,thelarger the gradient factor k is,the more D_z increases.

The distributions of magnetic fluxes B_x and B_z in Figs. 2(f) and 2(g) have the similar trendsas those of D_x and D_z. Regardless of the gradient variation and the value of k,B_x (or B_z) is twomagnitudes larger than D_x (or D_z). This phenomenon tells us that the magneto-mechanicalconversion is more remarkable than the electro-mechanical conversion.

Figures 3(a)-3(b) show the x-dependency of displacements u and w at the section centroid.Figure 3(a) expresses that with the increase of k in the gradient function,the displacement uincreases. However,in Fig. 3(b),for the FGM with exponential variation,the deflection w isnot affected by the gradient factor k. For the FGM with linear variation,w decreases with theincrease of k.

5 Conclusions

We derive an analytic solution of the two-dimensional plane problem for a functionallygraded ME cantilever beam by the semi-inverse method. This solution can be used to solvethe problems of the beam with any gradient distribution under different boundary conditions(such as the uniform load on the surface,the concentrated force,and the moment at the freeend of the beam). This solution can also be used as a validation of numerical solutions of thefunctionally graded ME composite. Our main conclusions are as follows:

(i) If the gradient factor k is 1,the exponential and linear gradients have the same effect onall the physical quantities.

(ii) The central axis moves up with the increase of the gradient factor k. The shear stressreaches its maximum at the central axis,and its absolute value increases with the increase ofk.

(iii) The components of electric displacement and magnetic flux in the same direction havethe similar distribution,but the value of the magnetic flux is two magnitudes larger than thatof the electric displacement.

(iv) If the gradient factor k increases,the difference of electric displacements D_x (or themagnetic flux B_x) on the upper and lower surfaces enlarges,and D_x (or B_z) increases too.

(v) No matter the material parameters are exponential or linear variation,the displacementu increases with the increase of k,while the deflection w has very different results. For the FGMwith exponential variation,the changes of gradient factor have no effect on the deflection w. For the FGM with linear variation,the deflection w decreases with the increase of the gradient factor k. Appendix A

The expressions of undetermined coefficients contained in Eqs. (20)-(37) are given as follows. By virtue of Eq. (9),Eq. (10),and the first expression of Eq. (8),we can obtain

where

Using the boundary conditions (11) and (12),we can determine the following parameters:

By the force boundary conditions in Eq. (8),C13 and C14 can be determined.

where

According to Eq. (15),we have