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

Abstract

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.

1. 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 et al. (see [7, 8]) introduced a symplectic dual method based on the conservative Hamiltonian system to solve the elastic problem which is different from the traditional semi-inverse solution method. The complete solutions space can be obtained and a satisfactory solution can be obtained under the boundary conditions (see [9–11]). The symplectic dual method has been developed for studying piezoelectric effects (see [12]) and magnetoelectroelastic problems (see [13]).

With the symplectic approach, the plane problem of magnetoelectroelastic solids in rectangular domain is derived into the Hamiltonian system by means of the generalized variable principle of the magnetoelectroelastic solids. 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, symplectic dual equations are employed. Yao and Li have obtained all the eigensolutions of zero-eigenvalue, which have their specific physical interpretation and are the basic solutions of plane Saint-Venant problem in [13]. This paper gets the eigensolutions of nonzero-eigenvalues, which are the solutions that 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.

2. 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]).(1)Governing equations: 𝜕𝜎_𝑥+𝜕𝑥𝜕𝜏_𝑥𝑧𝜕𝑧+𝑓_𝑥=0,𝜕𝜏_𝑥𝑧+𝜕𝑥𝜕𝜎_𝑧𝜕𝑧+𝑓_𝑧=0,𝜕𝐷_𝑥+𝜕𝑥𝜕𝐷_𝑧𝜕𝑧=𝜌,𝜕𝐵_𝑥+𝜕𝑥𝜕𝐵_𝑧𝜕𝑧=0. (2.1)(2)Gradient equations: 𝜀_𝑥=𝜕𝑢𝜕𝑥,𝜀_𝑧=𝜕𝑤𝜕𝑧,𝛾_𝑥𝑧=𝜕𝑤+𝜕𝑥𝜕𝑢,𝐸𝜕𝑧_𝑥=−𝜕𝜙𝜕𝑥,𝐸_𝑧=−𝜕𝜙𝜕𝑧,𝐻_𝑥=−𝜕𝜓𝜕𝑥,𝐻_𝑧=−𝜕𝜓.𝜕𝑧 (2.2)(3)Constitutive equations: 𝜎_𝑥=𝑐₁₁𝜀_𝑥+𝑐₁₃𝜀_𝑧−𝑒₃₁𝐸_𝑧−𝑞₃₁𝐻_𝑧,𝜎_𝑧=𝑐₁₃𝜀_𝑥+𝑐₃₃𝜀_𝑧−𝑒₃₃𝐸_𝑧−𝑞₃₃𝐻_𝑧,𝜏_𝑥𝑧=𝑐₄₄𝛾_𝑥𝑧−𝑒₁₅𝐸_𝑥−𝑞₁₅𝐻_𝑥,𝐷_𝑥=𝑒₁₅𝛾_𝑥𝑧+𝜅₁₁𝐸_𝑥+𝛼₁₁𝐻_𝑥,𝐷_𝑧=𝑒₃₁𝜀_𝑥+𝑒₃₃𝜀_𝑧+𝜅₃₃𝐸_𝑧+𝛼₃₃𝐻_𝑧,𝐵_𝑥=𝑞₁₅𝛾_𝑥𝑧+𝛼₁₁𝐸_𝑥+𝜇₁₁𝐻_𝑥,𝐵_𝑧=𝑞₃₁𝜀_𝑥+𝑞₃₃𝜀_𝑧+𝛼₃₃𝐸_𝑧+𝜇₃₃𝐻_𝑧, (2.3)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: 𝜀_𝑥=𝑠₁₁𝜎_𝑥+𝑠₁₃𝜎_𝑧+𝑑₃₁𝐷_𝑧+𝑏₃₁𝐵_𝑧,𝜀_𝑧=𝑠₁₃𝜎_𝑥+𝑠₃₃𝜎_𝑧+𝑑₃₃𝐷_𝑧+𝑏₃₃𝐵_𝑧,𝛾_𝑥𝑧=𝑠₄₄𝜏_𝑥𝑧+𝑑₁₅𝐷_𝑥+𝑏₁₅𝐵_𝑥,𝐸_𝑥=−𝑑₁₅𝜏_𝑥𝑧+𝜆₁₁𝐷_𝑥+𝛽₁₁𝐵_𝑥,𝐸_𝑧=−𝑑₃₁𝜎_𝑥−𝑑₃₃𝜎_𝑧+𝜆₃₃𝐷_𝑧+𝛽₃₃𝐵_𝑧,𝐻_𝑥=−𝑏₁₅𝜏_𝑥𝑧+𝛽₁₁𝐷_𝑥+𝜈₁₁𝐵_𝑥,𝐻_𝑧=−𝑏₃₁𝜎_𝑥−𝑏₃₃𝜎_𝑧+𝛽₃₃𝐷_𝑧+𝜈₃₃𝐵_𝑧. (2.4)

The rectangle domain as showing in Figure 1 is studied in this paper 𝑉∶0≤𝑧≤𝑙,−ℎ≤𝑥≤ℎ. (2.5)

Figure 1 

The rectangular domain problems on magnetoelectroelastic solids.

And the boundary conditions are expressed as (see [15]) 𝜎_𝑥=𝐹_𝑥1(𝑧),𝜏_𝑥𝑧=𝐹_𝑧1(𝑧),𝐷_𝑥=𝐷_𝑥1(𝑧),𝐵_𝑥=𝐵_𝑥1(𝑧)𝜎on𝑥=−ℎ,_𝑥=𝐹_𝑥2(𝑧),𝜏_𝑥𝑧=𝐹_𝑧2(𝑧),𝐷_𝑥=𝐷_𝑥2(𝑧),𝐵_𝑥=𝐵_𝑥2(𝑧)on𝑥=ℎ. (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=𝜓on𝑧=0or𝑙. (2.7) Or 𝜎_𝑧=𝜎_𝑧,𝜏_𝑥𝑧=𝜏_𝑥𝑧,𝐷_𝑧=𝐷_𝑧,𝐵_𝑧=𝐵_𝑧on𝑧=0or𝑙. (2.8)

3. 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 𝐯̇=𝐇𝐯, (3.1) and 𝐇 is the operator matrix in [13] 𝐇=0𝐶₁₀𝜕𝜕𝑥00𝐶₁₁0𝐶₁₄𝐶₁₅−𝜕𝜕𝑥0−𝐶₂₁𝜕𝜕𝑥−𝐶₂₂𝜕𝜕𝑥0𝐶₁₇000𝐶₁₈𝜕𝜕𝑥00𝐶₁₄0−𝐶₁₂−𝐶₁₆0𝐶₁₉𝜕𝜕𝑥00𝐶₁₅0−𝐶₁₆−𝐶₁₃𝜕00000−𝜕𝑥000−𝐶₂₀𝜕²𝜕𝑥²00𝐶₁₀𝜕𝜕𝑥0𝐶₁₈𝜕𝐶𝜕𝑥¹⁹𝜕𝜕𝑥00𝐶₂₃𝜕²𝜕𝑥²−𝐶₂₄𝜕²𝜕𝑥²0−𝐶₂₁𝜕𝜕𝑥0000−𝐶₂₄𝜕²𝜕𝑥²𝐶₂₅𝜕²𝜕𝑥²0−𝐶₂₂𝜕𝜕𝑥00, (3.2) where the constants 𝐶₁₀∼𝐶₂₅ are listed in the appendix.

Consider the following boundary conditions on two sides in [13]: 1𝑠₁₁𝜕𝑢𝜕𝑥−𝑠₁₃𝜎−𝑑₃₁𝐷−𝑏₃₁𝐵1=0,𝜏=0,Δ−𝜈₁₁𝜕𝜙𝜕𝑥+𝛽₁₁𝜕𝜓𝜕𝑥+𝐶₁𝜏1=0,Δ𝛽₁₁𝜕𝜙𝜕𝑥−𝜆₁₁𝜕𝜓𝜕𝑥+𝐶₂𝜏=0,on𝑥=±ℎ, (3.3) where constant Δ,𝐶₁, and 𝐶₂ are listed in the appendix. Equation (3.1) can usually be solved by using the method of partition of variables. Let 𝐯(𝑥,𝑧)=𝜉(𝑧)𝐘(𝑥), (3.4) and substitute (3.4) into (3.1); the following equations can be obtained: 𝐇𝐘(𝑥)=𝜇𝐘(𝑥),𝜉(𝑧)=𝑒^𝜇𝑧, (3.5) 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: 𝐯⁽¹⁾,𝐯=^(2)defℎ_−ℎ𝐯^𝑇(1)𝐉𝐯⁽²⁾𝑑𝑥,𝐉=𝟎𝐈₄−𝐈₄𝟎. (3.6)

If only 𝐯⁽¹⁾ and 𝐯⁽²⁾ meet the requirement of (3.3), the following invariant is obtained as 𝐯⁽¹⁾,𝐇𝐯⁽²⁾≡𝐯⁽²⁾,𝐇𝐯⁽¹⁾. (3.7) 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]).

4. 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): 𝐇𝐘(𝑥)=0. (4.1) After obtaining the fundamental eigensolutions and eigensolutions in Jordan form, the solutions of (3.1) can be given as 𝐯⁽⁰⁾_0𝑓=10000000^𝑇,𝐯⁽⁰⁾_0𝑠=01000000^𝑇,𝐯⁽⁰⁾_0𝑡=00100000^𝑇,𝐯⁽⁰⁾_0𝑟=00010000^𝑇,𝐯⁽¹⁾_0𝑓=𝑧𝑏₁𝑥00𝑎₁0𝑎₂𝑎₃^𝑇,𝐯⁽¹⁾_0𝑠=−𝑥𝑧000000^𝑇,𝐯⁽¹⁾_0𝑡=0𝑏₂𝑥𝑧0𝑎₂0𝑎₄𝑎₅^𝑇,𝐯⁽¹⁾_0𝑟=0𝑏₃𝑥0𝑧𝑎₃0𝑎₅𝑎₆^𝑇,𝐯⁽²⁾_0𝑠=−𝑥𝑧0.5𝑧²−𝑏₁𝑥²00−𝑎₁𝑥0−𝑎₂𝑥−𝑎₃𝑥^𝑇. (4.2) The component forms of 𝐯⁽³⁾_0𝑠 are 𝑤=𝑎₈𝑥³+𝑏₅1𝑥−2𝑥𝑧²1,𝑢=−2𝑏₁𝑥²1𝑧+6𝑧³,𝜙=𝑎₉𝑥³−3ℎ²𝑥,𝜓=𝑎₁₀𝑥³−3ℎ²𝑥,𝜎=−𝑎₁1𝑥𝑧,𝜏=2𝑎₁𝑥²−ℎ²,𝐷=−𝑎₂𝑥𝑧,𝐵=−𝑎₃𝑥𝑧. (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.

5. 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−𝜇𝐶₁₀𝜆00𝐶₁₁0𝐶₁₄𝐶₁₅−𝜆−𝜇−𝐶₂₁𝜆−𝐶₂₂𝜆0𝐶₁₇000𝐶₁₈𝜆−𝜇0𝐶₁₄0−𝐶₁₂−𝐶₁₆0𝐶₁₉𝜆0−𝜇𝐶₁₅0−𝐶₁₆−𝐶₁₃0000−𝜇−𝜆000−𝐶₂₀𝜆²00𝐶₁₀𝜆−𝜇𝐶₁₈𝜆𝐶₁₉𝜆00𝐶₂₃𝜆²−𝐶₂₄𝜆²0−𝐶₂₁𝜆−𝜇000−𝐶₂₄𝜆²𝐶₂₅𝜆²0−𝐶₂₂𝜆0−𝜇=0, (5.1) where the constants 𝐶₁₀∼𝐶₂₅ are listed in the appendix. Expanding the determinant yields the eigenvalue equation 𝑐₁𝜆⁸+𝑐₂𝜆⁶𝜇²+𝑐₃𝜆⁴𝜇⁴+𝑐₄𝜆²𝜇⁶+𝜇⁸=0. (5.2) Apparently, where 𝑐₁∼𝑐₄ constituted by 𝐶₁₀∼𝐶₂₅, (5.2) has eight roots 𝜆_𝑖=𝜆_𝑖𝜇,𝜆_4+𝑖=−𝜆_𝑖𝜇(𝑖=1,2,3,4). (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 𝑤=⁴_𝑖=1𝐴_1𝑖cosh𝜆_𝑖𝜇𝑥+⁴_𝑖=1𝐷_1𝑖sinh𝜆_𝑖𝜇𝑥,𝑢=⁴_𝑖=1𝐴_2𝑖sinh𝜆_𝑖𝜇𝑥+⁴_𝑖=1𝐷_2𝑖cosh𝜆_𝑖𝜇𝑥,𝜙=⁴_𝑖=1𝐴_3𝑖cosh𝜆_𝑖𝜇𝑥+⁴_𝑖=1𝐷_3𝑖sinh𝜆_𝑖𝜇𝑥,𝜓=⁴_𝑖=1𝐴_4𝑖cosh𝜆_𝑖𝜇𝑥+⁴_𝑖=1𝐷_4𝑖sinh𝜆_𝑖𝜇𝑥,𝜎=⁴_𝑖=1𝐴_5𝑖cosh𝜆_𝑖𝜇𝑥+⁴_𝑖=1𝐷_5𝑖sinh𝜆_𝑖𝜇𝑥,𝜏=⁴_𝑖=1𝐴_6𝑖sinh𝜆_𝑖𝜇𝑥+⁴_𝑖=1𝐷_6𝑖cosh𝜆_𝑖𝜇𝑥,𝐷=⁴_𝑖=1𝐴_7𝑖cosh𝜆_𝑖𝜇𝑥+⁴_𝑖=1𝐷_7𝑖sinh𝜆_𝑖𝜇𝑥,𝐵=⁴_𝑖=1𝐴_8𝑖cosh𝜆_𝑖𝜇𝑥+⁴_𝑖=1𝐷_8𝑖sinh𝜆_𝑖𝜇𝑥. (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 𝑤=⁴_𝑖=1𝐴_1𝑖cosh𝜆_𝑖𝜇𝑥,𝑢=⁴_𝑖=1𝐴_2𝑖sinh𝜆_𝑖𝜇𝑥,𝜙=⁴_𝑖=1𝐴_3𝑖cosh𝜆_𝑖𝜇𝑥,𝜓=⁴_𝑖=1𝐴_4𝑖cosh𝜆_𝑖𝜇𝑥,𝜎=⁴_𝑖=1𝐴_5𝑖cosh𝜆_𝑖𝜇𝑥,𝜏=⁴_𝑖=1𝐴_6𝑖sinh𝜆_𝑖𝜇𝑥,𝐷=⁴_𝑖=1𝐴_7𝑖cosh𝜆_𝑖𝜇𝑥,𝐵=⁴_𝑖=1𝐴_8𝑖cosh𝜆_𝑖𝜇𝑥, (5.5) where constants 𝐴_𝑗𝑖(𝑗=1,2,…,8;𝑖=1,2,3,4) are not all independent. Substitute (5.5) into (3.5); 𝐴_𝑗𝑖 can be expressed by the independent constants 𝐴_6𝑖(𝑖=1,2,3,4)𝐴_𝑗𝑖=𝑓_𝑁𝑗𝜆_𝑖𝑓_𝐷𝑗𝜆_𝑖𝜇𝐴_6𝑖𝑗=1,2,3,4,𝐴_5𝑖=−𝜆_𝑖𝐴_6𝑖,𝐴_𝑗𝑖=𝑓_𝑁𝑗𝜆_𝑖𝑓_𝐷𝑗𝜆_𝑖𝐴_6𝑖𝑗=7,8, (5.6) where 𝑓_𝑁𝑗(𝜆_𝑖) and 𝑓_𝐷𝑗(𝜆_𝑖) are expressions of 𝜆_𝑖.

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 ⁴_𝑖=1𝜆_𝑖𝑓_𝑁2𝜆_𝑖𝑓_𝐷2𝜆_𝑖+𝑠₁₃𝜆_𝑖−𝑑₃₁𝑓_𝑁7𝜆_𝑖𝑓_𝐷7𝜆_𝑖−𝑏₃₁𝑓_𝑁8𝜆_𝑖𝑓_𝐷8𝜆_𝑖cosh𝜆_𝑖𝜇ℎ𝐴_6𝑖=0,_4𝑖=1sinh𝜆_𝑖𝜇ℎ𝐴_6𝑖=0,_4𝑖=1𝐶₁−𝜈₁₁𝜆_𝑖𝑓_𝑁3𝜆_𝑖𝑓_𝐷3𝜆_𝑖+𝛽₁₁𝜆_𝑖𝑓_𝑁4𝜆_𝑖𝑓_𝐷4𝜆_𝑖sinh𝜆_𝑖𝜇ℎ𝐴_6𝑖=0,_4𝑖=1𝐶₂+𝛽₁₁𝜆_𝑖𝑓_𝑁3𝜆_𝑖𝑓_𝐷3𝜆_𝑖−𝜆₁₁𝜆_𝑖𝑓_𝑁4𝜆_𝑖𝑓_𝐷4𝜆_𝑖sinh𝜆_𝑖𝜇ℎ𝐴_6𝑖=0. (5.7) For the sake of simplicity, (5.7) is denoted as 𝐵_𝑗𝑖𝐴_6𝑖={0}𝑖,𝑗=1,2,3,4. (5.8) 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 𝐘_𝑛=𝑤_𝑛𝑢_𝑛𝜙_𝑛𝜓_𝑛𝜎_𝑛𝜏_𝑛𝐷_𝑛𝐵_𝑛^𝑇, (5.10) where 𝑤_𝑛=⁴_𝑖=1𝑓_𝑁1𝜆_𝑖𝑓_𝐷1𝜆_𝑖𝜇𝐴_6𝑖cosh𝜆_𝑖𝜇_𝑛𝑥,𝑢_𝑛=⁴_𝑖=1𝑓_𝑁2𝜆_𝑖𝑓_𝐷2𝜆_𝑖𝜇𝐴_6𝑖sinh𝜆_𝑖𝜇_𝑛𝑥,𝜙_𝑛=⁴_𝑖=1𝑓_𝑁3𝜆_𝑖𝑓_𝐷3𝜆_𝑖𝜇𝐴_6𝑖cosh𝜆_𝑖𝜇_𝑛𝑥,𝜓_𝑛=⁴_𝑖=1𝑓_𝑁4𝜆_𝑖𝑓_𝐷4𝜆_𝑖𝜇𝐴_6𝑖cosh𝜆_𝑖𝜇_𝑛𝑥,𝜎_𝑛=⁴_𝑖=1−𝜆_𝑖𝐴_6𝑖cosh𝜆_𝑖𝜇_𝑛𝑥,𝜏_𝑛=⁴_𝑖=1𝐴_6𝑖sinh𝜆_𝑖𝜇_𝑛𝑥,𝐷_𝑛=⁴_𝑖=1𝑓_𝑁7𝜆_𝑖𝑓_𝐷7𝜆_𝑖𝐴_6𝑖cosh𝜆_𝑖𝜇_𝑛𝑥,𝐵_𝑛=⁴_𝑖=1𝑓_𝑁8𝜆_𝑖𝑓_𝐷8𝜆_𝑖𝐴_6𝑖cosh𝜆_𝑖𝜇_𝑛𝑥. (5.11)

The corresponding solution of (3.1) in symmetric deformation is 𝐯_𝑛=𝑒^𝜇𝑛𝑧𝐘_𝑛. (5.12)

Likewise, the corresponding eigenvector function of eigenvalues 𝜇^′_𝑛(𝑛=1,2,…) in antisymmetric deformation is 𝐘^′_𝑛=𝑤^′_𝑛𝑢^′_𝑛𝜙^′_𝑛𝜓^′_𝑛𝜎^′_𝑛𝜏^′_𝑛𝐷^′_𝑛𝐵^′_𝑛^𝑇, (5.13) where 𝑤^′_𝑛=⁴_𝑖=1𝐷_1𝑖sinh𝜆_𝑖𝜇^′_𝑛𝑥,𝑢^′_𝑛=⁴_𝑖=1𝐷_2𝑖cosh𝜆_𝑖𝜇^′_𝑛𝑥,𝜙^′_𝑛=⁴_𝑖=1𝐷_3𝑖sinh𝜆_𝑖𝜇^′_𝑛𝑥,𝜓^′_𝑛=⁴_𝑖=1𝐷_4𝑖sinh𝜆_𝑖𝜇^′_𝑛𝑥,𝜎^′_𝑛=⁴_𝑖=1𝐷_5𝑖sinh𝜆_𝑖𝜇^′_𝑛𝑥,𝜏^′_𝑛=⁴_𝑖=1𝐷_6𝑖cosh𝜆_𝑖𝜇^′_𝑛𝑥,𝐷^′_𝑛=⁴_𝑖=1𝐷_7𝑖sinh𝜆_𝑖𝜇^′_𝑛𝑥,𝐵^′_𝑛=⁴_𝑖=1𝐷_8𝑖sinh𝜆_𝑖𝜇^′_𝑛𝑥. (5.14)

The corresponding solution of (3.1) in antisymmetric deformation is 𝐯^′_𝑛=𝑒^{𝜇′𝑛𝑧}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.

6. 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 𝐯=¹_𝑖=0𝑎^(𝑖)_0𝑓𝐯^(𝑖)_0𝑓+𝑎^(𝑖)_0𝑡𝐯^(𝑖)_0𝑡+𝑎^(𝑖)_0𝑟𝐯^(𝑖)_0𝑟+³_𝑖=0𝑎^(𝑖)_0𝑠𝐯^(𝑖)_0𝑠+^∞_𝑖=1𝑎_𝑖𝐯_𝑖+^∞_𝑖=1𝑏̃_𝑖𝐯^′_𝑖, (6.1) where 𝑎^(𝑖)_0𝑓, 𝑎^(𝑖)_0𝑠, 𝑎^(𝑖)_0𝑡, 𝑎^(𝑖)_0𝑟, 𝑎_𝑖, and 𝑏̃_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 𝐪₀=𝐪₀(𝑥)=𝑤₀(𝑥)𝑢₀(𝑥)𝜙₀(𝑥)𝜓₀(𝑥)^𝑇𝐪at𝑧=0,_𝑙=𝐪_𝑙(𝑥)=𝑤_𝑙(𝑥)𝑢_𝑙(𝑥)𝜙_𝑙(𝑥)𝜓_𝑙(𝑥)^𝑇at𝑧=𝑙, (6.2) where 𝐪₀ 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 𝐩₀=𝐩₀(𝑥)=𝜎₀(𝑥)𝜏₀(𝑥)𝐷₀(𝑥)𝐵₀(𝑥)^𝑇𝐩𝑧=0,_𝑙=𝐩_𝑙(𝑥)=𝜎_𝑙(𝑥)𝜏_𝑙(𝑥)𝐷_𝑙(𝑥)𝐵_𝑙(𝑥)^𝑇𝑧=𝑙, (6.3) where 𝐩₀ and 𝐩_𝑙 denote the values of variable 𝐩 at 𝑧=0 and 𝑧=𝑙, respectively.

The boundary conditions at both ends (𝑧=0or𝑙) 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 𝐯=^𝑛_𝑖=1𝐴_𝑖𝐯_𝑖, (6.4) 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 𝐪₀ at 𝑧=0 and generalized force 𝐩_𝑙 at 𝑧=𝑙, the variational equation is obtained as ^ℎ_−ℎ^𝑛_𝑖=1𝛿𝐴_𝑖𝐩^𝑇_0𝑖^𝑛_𝑗=1𝐴_𝑗𝐪_0𝑗−𝐪₀d𝑥+^ℎ_−ℎ^𝑛_𝑖=1𝛿𝐴_𝑖𝐪^𝑇_𝑙𝑖^𝑛_𝑗=1𝐴_𝑗𝐩_𝑙𝑗−𝐩_𝑙d𝑥=0. (6.5) Equation (6.5) can be expressed as ^𝑛_𝑖=1^𝑛_𝑗=1^ℎ_−ℎ𝐴_𝑗𝐩^𝑇_0𝑖𝐪_0𝑗+𝐪^𝑇_𝑙𝑖𝐩_𝑙𝑗−𝐩^𝑇_0𝑖𝐪₀−𝐪^𝑇_𝑙𝑖𝐩_𝑙d𝑥𝛿𝐴_𝑖=0. (6.6)𝐴_𝑖 can be determined by the following equations: 𝑐̃₁₁𝑐̃₁₂⋯𝑐̃_1𝑛𝑐̃₂₁𝑐̃₂₂⋯𝑐̃_2𝑛𝑐̃⋮⋮⋱⋮_𝑛1𝑐̃_𝑛2⋯𝑐̃_𝑛𝑛𝐴₁𝐴₂⋮𝐴_𝑛=𝑑₁𝑑₂⋮𝑑_𝑛, (6.7) where 𝑐̃_𝑖𝑗=^ℎ_−ℎ𝐩^𝑇_0𝑖𝐪_0𝑗+𝐪^𝑇_𝑙𝑖𝐩_𝑙𝑗d𝑥,𝑑_𝑖=^ℎ_−ℎ𝐩^𝑇_0𝑖𝐪₀+𝐪^𝑇_𝑙𝑖𝐩_𝑙d𝑥𝑖,𝑗=1,2,…,𝑛. (6.8)

7. 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=0or𝑙) 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.

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 by𝑧=𝑙: (a)𝜎_𝑧=𝑃₀, 𝜏_𝑥𝑧=𝐷_𝑧=𝐵_𝑧=0, under the uniform tension;(b)𝐷_𝑧=𝐷₀, 𝜎_𝑧=𝜏_𝑥𝑧=𝐵_𝑧=0, under the uniform electric displacement;(c)𝐵_𝑧=𝐵₀, 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.
The problem is treated as a symmetric deformation one. The solution is formed from (4.2). For load (a), 𝐯₁=𝑠₃₃𝐯⁽¹⁾_0𝑓+𝑑₃₃𝐯⁽¹⁾_0𝑡+𝑏₃₃𝐯⁽¹⁾_0𝑟𝑃₀. (7.1)For load (b), 𝐯₂=𝑑₃₃𝐯⁽¹⁾_0𝑓−𝜆₃₃𝐯⁽¹⁾_0𝑡−𝛽₃₃𝐯⁽¹⁾_0𝑟𝐷₀. (7.2)For load (c), 𝐯₃=𝑏₃₃𝐯⁽¹⁾_0𝑓−𝛽₃₃𝐯⁽¹⁾_0𝑟−𝜈₃₃𝐯⁽¹⁾_0𝑟𝐵₀. (7.3)
For numerical calculation, the composite materials BaTiO₃-CoFe₂O₄ are specified, which material constants are given in [5]. Take 𝜎_𝑧=10N/m², 𝐷_𝑧=10⁻⁹C/m², 𝐵_𝑧=10⁻⁷N/Am, ℎ=0.03m, and 𝑙=0.1m. 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.

Table 1 

The numerical results in the rectangular domain problem under three-load case.

Example 7.2. Consider a simple tension problem of composite materials BaTiO₃-CoFe₂O₄ semi-infinite strip fixed at 𝑧=0, and take −ℎ≤𝑥≤ℎ, ℎ=0.1m. 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 BaTiO₃-CoFe₂O₄.
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 stress, only 𝐯⁽¹⁾_0𝑓, 𝐯⁽¹⁾_0𝑡, and 𝐯⁽¹⁾_0𝑟 are selected in (4.2). Their coefficients are determined by the boundary conditions at 𝑧→∞. So the general solution can be expressed as 𝐯=𝜎_∞𝑠₃₃𝐯⁽¹⁾_0𝑓+𝑑₃₃𝐯⁽¹⁾_0𝑡+𝑏₃₃𝐯⁽¹⁾_0𝑟+^𝑛_𝑖=1𝐴_𝑖𝐯_𝑖, (7.4) 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, Re𝜇_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.

Table 2 

The eigenvalues with respect to the 𝑥-direction.

Table 3 

Nonzero eigenvalues for symmetric deformation.

Figure 2 

Normal stress analysis with fixed end.

Figure 3 

Normal stress distribution in the partial region of the left end.

8. 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.

Appendix

Δ=𝜆₁₁𝜈₁₁−𝛽²₁₁,𝐶₁=𝜈₁₁𝑑₁₅−𝛽₁₁𝑏₁₅,𝐶₂=𝜆₁₁𝑏₁₅−𝛽₁₁𝑑₁₅,𝐶₁₀=𝑠₁₃𝑠₁₁,𝐶₁₁=𝑠₁₁𝑠₃₃−𝑠²₁₃𝑠₁₁,𝐶₁₂=𝑠₁₁𝜆₃₃+𝑑²₃₁𝑠₁₁,𝐶₁₃=𝑠₁₁𝜈₃₃+𝑏²₃₁𝑠₁₁,𝐶₁₄=𝑠₁₁𝑑₃₃−𝑠₁₃𝑑₃₁𝑠₁₁,𝐶₁₅=𝑠₁₁𝑏₃₃−𝑠₁₃𝑏₃₁𝑠₁₁,𝐶₁₆=𝑠₁₁𝛽₃₃+𝑏₃₁𝑑₃₁𝑠₁₁,𝐶₁₇=𝑠₄₄+𝑑₁₅𝐶₁+𝑏₁₅𝐶₂Δ,𝐶₁₈=𝑑₃₁𝑠₁₁,𝐶₁₉=𝑏₃₁𝑠₁₁,𝐶₂₀=1𝑠₁₁,𝐶₂₁=𝐶₁Δ,𝐶₂₂=𝐶₂Δ,𝐶₂₃=𝜈₁₁Δ,𝐶₂₄=𝛽₁₁Δ,𝐶₂₅=𝜆₁₁Δ. (A.1)

Acknowledgment

The work was supported by the National Natural Science Foundation of China (no. 10902072).