Symplectic Analytical Solutions for the Magnetoelectroelastic Solids Plane Problem in Rectangular Domain

The transversely isotropic magnetoelectroelastic solids plane problem in rectangular domain is derived to Hamiltonian system. In symplectic geometry space with the origin variables— displacements, electric potential, and magnetic potential, as well as their duality variables— lengthways stress, electric displacement, and magnetic induction, on the basis of the obtained eigensolutions of zero-eigenvalue, the eigensolutions of nonzero-eigenvalues are also obtained. The former are the basic solutions of Saint-Venant problem, and the latter are the solutions which have the local effect, decay drastically with respect to distance, and are covered in the Saint-Venant principle. So the complete solution of the problem is given out by the symplectic eigensolutions expansion. Finally, a few examples are selected and their analytical solutions are presented.


Introduction
Magnetoelectroelastic solids are a kind of the emerging functional composite material.Due to possessing mechanical, electric and magnetic field coupling capacity, these materials show better foreground in many high-tech areas see 1 .They not only convert energy from one form to the other among magnetic, electric, and mechanical energies , but also possess some new properties of magnetoelectric effect, which are not found in single-phase piezoelectric or piezomagnetic materials.Over the years, a large amount of studies have been done in mechanics, materials science, and physics fields see 2-6 , and it has become a new cross subject.
Due to multifields coupling, the magnetoelectroelastic solids problem is solved more difficultly than elasticity one.Zhong

Functional Equations and Boundary Conditions
The transversely isotropic magnetoelectroelastic solids are studied here, with the z-axis being the polar direction.If geometry size, load, and so forth, in y direction satisfy the given condition, the problem can be simplified as a plane problem in the xoz plane.The functional equations of the plane problem are as follows see 14 .where u, w are displacement components in the x, z direction, respectively; σ x , σ z , and τ xz are stress components, respectively; D x , D z , and φ are electric displacement components and electric potential, respectively; B x , B z , and ψ are magnetic induction components and magnetic potential, respectively; f x , f z , and ρ are body force components and density of free charges in region V , respectively.c ij , κ ij , and μ ij are elastic modus, dielectric constant, and magnetic constants, respectively; e ij , q ij , and α ij are piezoelectric, piezomagnetic, and electromagnetic constants, respectively.Equation 2.3 can also be rewritten as follows:

2.4
The rectangle domain as showing in Figure 1 is studied in this paper And the boundary conditions are expressed as see 15

2.6
In addition, there are body force and density of free charges in region V .On z 0, l, the boundary conditions are see 15 w w, u u, φ φ, ψ ψ on z 0 or l.

2.7
Or

Duality Equation in Symplectic Geometry Space and Separation of Variables (See [13])
Consider rectangle domain as showing in Figure 1.At first, the coordinate z here is employed to simulate the time variable in the Hamiltonian system, and a symbol " ˙" in the following derivation will be used denoting the differential with respect to z, that is, ˙ ∂/∂z.For the sake of simplicity, the notations σ, τ, D, and B are introduced to represent σ z , τ xz , D z , and B z , respectively.If we omit body forces and density of free charges, a dual equation with the full state function vector is given as v Hv, 3.1 The rectangular domain problems on magnetoelectroelastic solids.
and H is the operator matrix in 13 where μ is the eigenvalue and Y x is the eigenfunction vector, which satisfies the boundary condition 3.3 on x ±h.
The full state vectors v form a symplectic space according to the following definition of symplectic inner product: If only v 1 and v 2 meet the requirement of 3.3 , the following invariant is obtained as Therefore, the operator matrix H is the Hamiltonian operator matrix in the symplectic geometry space.So its eigenvalues have some characteristics, that is, if μ is an eigenvalue, then −μ must also be one, and the eigenfunction vectors satisfy the symplectic adjoint orthogonal relationship.After eigenvalues and eigenfunction vectors of H are given, the origin problem can be solved by the method of eigenfunction expansion see 7, 8 .

Eigenfunction Vectors of Eigenvalue Zero (See [13])
With the free boundary condition at both sides x ±h , the magnetoelectroelastic solids plane problem in rectangular domain exists eigenvalue zero.The eigenvalue zero is a special eigenvalue of the Hamiltonian operator matrix, which eigenfunction vectors are not only the fundamental solution with the special physical significance but also a nondecaying solution.Solve the following eigenequation with conditions 3.3 : After obtaining the fundamental eigensolutions and eigensolutions in Jordan form, the solutions of 3.1 can be given as

6 Journal of Applied Mathematics
The component forms of v 3 0s are

4.3
The constants in 4.2 -4.3 are listed in 13 .The eigensolutions of eigenvalue zero just are the fundamental solution of Saint Venant problem.But the solutions having the local effect are constituted by the eigensolutions of eigenvalue nonzero of 3.1 , which are obtained in the next section.

Eigenfunction Vectors of Eigenvalue Nonzero
The eigenvalue equation 3.5 is a system of ordinary differential equations with respect to x which can be solved by first determining the eigenvalue λ with respect to the x direction.The corresponding equation is det where the constants C 10 ∼ C 25 are listed in the appendix.Expanding the determinant yields the eigenvalue equation Apparently, where c 1 ∼ c 4 constituted by C 10 ∼ C 25 , 5.2 has eight roots

5.3
Only discuss the general case that there are eight different roots λ i .Obviously, − λ i must be an eigenvalue if λ i is an eigenvalue, so the general solution of 5.2 can be expressed as

5.4
It shows that the partial solutions relevant to A are the solutions of symmetric deformation with the z-axis while the partial solutions relevant to D are the solutions of antisymmetric deformation with the z-axis.
Firstly discuss the general solution of symmetric deformation A 8i cosh λ i μx ,
So far, only A 6i i 1, 2, 3, 4 are unknown constants in the general solution.Substituting 5.5 and 5.6 into the boundary conditions 3.3 yields

5.7
For the sake of simplicity, 5.7 is denoted as For nontrivial solution to exist, the determinant of coefficient matrix vanishes.
B ji 0. 5.9 Equation 5.9 can be solved by numerical methods.If μ n n 1, 2, . . . is roots of 5.9 , each μ n in reality has its symplectic adjoint eigenvalue −μ n and their complex conjugate eigenvalues.After obtaining A 6i i, j 1, 2, 3, 4 by substituting μ n into 5.8 , the corresponding eigenvector function is where

5.11
The corresponding solution of 3.1 in symmetric deformation is v n e μ n z Y n .

5.12
Likewise, the corresponding eigenvector function of eigenvalues μ n n 1, 2, . . . in antisymmetric deformation is where

5.14
The corresponding solution of 3.1 in antisymmetric deformation is v n e μ n z Y n .

5.15
Thus, all eigensolutions of nonzero eigenvalues are obtained.These solutions are covered in the Saint-Venant principle and decay with distance depending on the characteristics of eigenvalues see 7, 8 .Together with the eigensolutions of zero eigenvalue, they constitute a complete adjoint symplectic orthonormal basis and the expansion theorem is then applicable.Further discussion concerns solutions of magnetoelectroelastic solids plane problems in rectangular domain.

Solutions of Generalized Plane Problems in Rectangular Domain
The solution of homogeneous 3.1 by the method of separation variables has been discussed in the previous several sections.The analytical expressions of eigensolutions of zero eigenvalue and of nonzero eigenvalues have also been presented.Based on expansion theorems, the general solution of homogeneous equation 3.1 for magnetoelectroelastic solids plane problems in rectangular domain is where a i 0f , a i 0s , a i 0t , a i 0r , a i , and b i are undetermined constants.Determining these constants needs the variational equations corresponding to the two-point boundary conditions obtained by applying the variational principle.
If the end boundary conditions for specified generalized displacements are 2.7 , it can be expressed as q 0 q 0 x w 0 x u 0 x φ 0 x ψ 0 x T at z 0, q l q l x w l x u l x φ l x ψ l x T at z l, where q 0 and q l denote the values of variable q at z 0 and z l, respectively.If the end boundary conditions for specified generalized forces are 2.8 , it can be expressed as where p 0 and p l denote the values of variable p at z 0 and z l, respectively.
The boundary conditions at both ends z 0 or l can also be mixed boundary conditions.
According to approach derived by Zhong et al. in 7, 8 in the Hamiltonian system, the process of obtaining the undetermined constants of the general solution is presented here.For the sake of simplicity, the general solution is given as where n is the number of eigensolutions, v i are all solutions that relate to zero eigenvalue and nonzero eigenvalues, and A i are constants determined by a linear system of equations resulted from the Hamiltonian variational principle.If we specify generalized displacement q 0 at z 0 and generalized force p l at z l, the variational equation is obtained as Equation 6.5 can be expressed as A j p T 0i q 0j q T li p lj − p T 0i q 0 − q T li p l dx δA i 0.

6.6
A i can be determined by the following equations: where 6.8

Numerical Examples
The Saint-Venant principle is applicable to the problem for l h in a rectangular domain.The influence of self-equilibrium forces at both ends z 0 or l is only confined to the vicinity of the region.It is then appropriate to neglect the solutions of nonzero eigenvalues and, therefore, to apply only the solutions of zero eigenvalue in the expansion theorem.The problem is treated as a symmetric deformation one.The solution is formed from 4.2 .
For load a , For load b , For load c , For numerical calculation, the composite materials BaTiO 3 -CoFe 2 O 4 are specified, which material constants are given in 5 .Take σ z 10 N/m 2 , D z 10 −9 C/m 2 , B z 10 −7 N/Am, h 0.03 m, and l 0.1 m.The numerical results at point h, l are given in Table 1, where the values in the parenthesis are the exact ones which are calculated with the formulas presented by Ding and Jiang in 5 .The results show that the numerical solutions by the method have higher accuracy.The eigensolutions of zero eigenvalue correspond to the solutions of Saint-Venant problems.On the other hand, the solutions of the portion covered by the Saint-Venant principle correspond to the eigensolutions with nonzero eigenvalues.Subsequently, a very simple example is presented here.Secondly, obtain nonzero eigenvalues.With reference to the problem, there is only the tension stress σ ∞ for z → ∞ and the deformation is symmetric with respect to z-axis.Solve 5.9 by numerical methods.Table 3 lists the first five eigenvalues in the first quadrant.Each μ n in reality has its symplectic adjoint eigenvalue −μ n and their complex conjugate eigenvalue.
Thirdly, the general solution is then formed from the eigensolution of the zero eigenvalue 4.2 and 4.3 and the eigensolution of the nonzero eigenvalues of the symmetric deformation 5.12 .Consider the deformation is symmetric with respect to z-axis and rigid body translation, constant electric potential or constant magnetic potential has no effect on where n is the number of the eigensolutions of nonzero eigenvalues of the symmetric deformation.For the above general solution, only eigensolutions of α-set in 5.12 are adopted, that is, Re μ i < 0. Thus it consistent with the far-end boundary condition at z → ∞.Finally, Substituting 7.4 into the equation of variation 6.5 and adopting a total of ten eigensolutions of nonzero eigenvalues in the calculation yield σ z /σ ∞ at the fixed end z 0 as illustrated in Figure 2. The figure shows that there is stress singularity at the edge corner.Fluctuation of stress observed is common when truncated finite terms are adopted in the expansion.Such phenomenon has been observed, for instance, when finite terms in a Fourier series are assumed.Figure 3 shows that the normal stress distribution approximates uniformity at z 2h.

Conclusions
In this paper, the transversely isotropic magnetoelectroelastic solids plane problem in rectangular domain is considered from a symplectic approach.The eigensolutions of nonzero-eigenvalues obtained decay drastically with respect to distance can express the end effects and corner stresses.There is no requirement of experience in the symplectic approach for solving the problem since it is a rational, analytical approach to satisfy the boundary conditions in a straightforward manner.

Example 7 . 1 .
Consider a magnetoelectroelastic rectangle domain, under uniform axial tension, electrical displacement, or magnetic induction, respectively.Three load cases are considered and the boundary conditions are given by z l: a σ z P 0 , τ xz D z B z 0, under the uniform tension; b D z D 0 , σ z τ xz B z 0, under the uniform electric displacement; c B z B 0 , σ z τ xz D z 0, under the uniform magnetic induction.z 0: τ xz w φ ψ 0; x ±h: σ x τ xz D x B x 0.

Figure 3 :
Figure 3: Normal stress distribution in the partial region of the left end.

Table 2 :
The eigenvalues with respect to the x-direction.
Example 7.2.Consider a simple tension problem of composite materials BaTiO 3 -CoFe 2 O 4 semi-infinite strip fixed at z 0, and take −h ≤ x ≤ h, h 0.1 m.There is only the tension stress σ ∞ for z → ∞, and τ zx D z B z 0. The stress distribution at the fixed end is determined here.Firstly, by utilizing 5.2 and 5.3 , λ i i 1, 2, . . ., 4 are calculated and listed in Table 2 for the composite materials BaTiO 3 -CoFe 2 O 4 .