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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.

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 [

Due to multifields coupling, the magnetoelectroelastic solids problem is solved more difficultly than elasticity one. Zhong et al. (see [

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 [

The transversely isotropic magnetoelectroelastic solids are studied here, with the

Governing equations:

Gradient equations:

Constitutive equations:

The rectangle domain as showing in Figure

The rectangular domain problems on magnetoelectroelastic solids.

And the boundary conditions are expressed as (see [

On

Consider rectangle domain as showing in Figure

Consider the following boundary conditions on two sides in [

The full state vectors

If only

With the free boundary condition at both sides(

The eigenvalue equation (

Only discuss the general case that there are eight different roots

It shows that the partial solutions relevant to

Firstly discuss the general solution of symmetric deformation

So far, only

The corresponding solution of (

Likewise, the corresponding eigenvector function of eigenvalues

The corresponding solution of (

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 [

The solution of homogeneous (

If the end boundary conditions for specified generalized displacements are (

If the end boundary conditions for specified generalized forces are (

The boundary conditions at both ends (

According to approach derived by Zhong et al. in [

The Saint-Venant principle is applicable to the problem for

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

The problem is treated as a symmetric deformation one. The solution is formed from (

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 [

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

Case | ^{-13} m) | ^{-12} m) | ^{-3} V) | ^{-4} A) | ^{2}) | ^{-9} C/m^{2}) | ^{-7} N/Am) |
---|---|---|---|---|---|---|---|

a | −9.4998 | 5.6834 | 9.4955 | 2.1391 | 10.0000 | 0.0000 | 0.0000 |

(−9.4998) | (5.6834) | (9.4955) | (2.1391) | (10.0000) | (0.0000) | (0.0000) | |

b | −2.1079 | 0.94955 | −6.2894 | 0.25669 | 0.0000 | 1.0000 | 0.0000 |

(−2.1079) | (0.94955) | (−6.2894) | (0.25669) | (0.0000) | (1.0000) | (0.0000) | |

c | 5.0767 | 2.1391 | 2.5669 | −7.5213 | 0.0000 | 0.0000 | 1.0000 |

(5.0767) | (2.1391) | (2.5669) | (−7.5213) | (0.0000) | (0.0000) | (1.0000) |

Consider a simple tension problem of composite materials BaTiO_{3}-CoFe_{2}O_{4} semi-infinite strip fixed at

Firstly, by utilizing (_{3}-CoFe_{2}O_{4}.

Secondly, obtain nonzero eigenvalues. With reference to the problem, there is only the tension stress

Thirdly, the general solution is then formed from the eigensolution of the zero eigenvalue (

Finally, Substituting (

The eigenvalues with respect to the

1 | 2 | 3 | 4 | |
---|---|---|---|---|

0.58880 | 1.0711 | 0.60197 + 1.1734 | 0.60197 − 1.1734 |

Nonzero eigenvalues for symmetric deformation.

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

1.8209 + 1.2193 | 2.8349 | 3.8972 | 3.9331 + 2.1502 | 5.8470 |

Normal stress analysis with fixed end.

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

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.

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