Fracture Analysis On Magnetoelectroelastic Composite And Its Symplectic Numerical Method

With the development of material science and electronic technology, functional composites including piezoelectric and magnetoelectroelastic materials have been widely applied for manufacturing intelligent devices such as transducers, magnetic field probes, medical ultrasonic imaging and actuators, due to their excellent mechanical-electric, mechanical-magnetic and magneto-electric coupling effects. Because of the inherent brittleness and mismatched property of the composite material, various cracks or defects which happened in manufacturing and consumption alter the integrity of the structure and cause physical field stress concentration near the defects, and inevitably lead to fracture failure of structure. Therefore, it is of great theoretical and practical significance to study the fracture problem of intelligent composite structures in structural design and evaluation.In this dissertation, a symplectic dual approach based on the Hamiltonian system is proposed to analyze the fracture problem of piezoelectric/magnetoelectroelastic composites. The analytical expressions of the physical field of the crack tip and intensity factor of the reaction force, electric field and magnetic field singularity are derived explicitly in the series of symplectic eigensolutions. In addition, based on the eigensolutions and the traditional finite element method, a finite element discretized symplectic method (FEDSM) which can overcome the mesh and path sensitivity of the grid is constructed. The main works of the thesis are listed as follows: (1) A Hamiltonian solution system for the interfacial fracture problem of finite size piezoelectric/magnetoelectroelastic composites is established.In symplectic space, the displacements and stresses, electric potentials and electric displacements, magnetic potentials and magnetic induction functions are conjugating (dual) variables, respectively. By introducing the basic unknowns consisting of the original variables and dual variables, a unified form of Hamiltonian canonical equations of piezoelectric/magnetoelectroelastic composites are constructed and the corresponding eigenvalues are obtained by using the method of separation of variables. Analytic expressions of physical fields can be expressed by the linear combination of symplectic eigensolutions with coefficients to be determined by the boundary conditions and symplectic adjoint orthogonality among the eigenfunctions. The intensity factors and energy release rate for four crack surface electromagnetic assumptions are obtained by using the fracture mechanics formula. This method overcome the limitations of the traditional semi-inverse methods, and it is directly rational solving method. The results showed that the generalized stress variables at the crack tip show the square root singularities; The generalized displacement intensity factors are independent of the material constants and directly relate to the undetermined coefficients of eigenfunctions associated with ?=1/2; the generalized stress intensity factors could be represented by the combination of generalized displacement intensity factors and material constant. Numerical examples for various boundary conditions, including mixed boundary conditions, are presented to show the characteristic of mechanical, electric and magnetic field at the crack tip and the effects of material parameters, geometry size and external load on the fracture parameters are revealed.(2) A novel finite element discretized symplectic method has been developed for evaluating the fracture problem of piezoelectric/magnetoelectroelastic composites.By using the analytic symplectic eigensolutions and the traditional finite element method, a finite element discretized symplectic method for fracture analysis of piezoelectric/magnetoelectroelastic composite is presented. The overall cracked body is mesh by conventional finite elements and divided into a finite size singular region near the crack tip (near field) and a regular region far away from the crack tip (far field). In the near field, with the aid of the employment of the exact analytical symplectic eigenfunctions as global functions, a large number of displacement unknowns of conventional elements are condensed into a small set of coefficients of symplectic eigenfunctions. The node unknowns in far field remain as usual. Finally, the explicit expressions of the physical field in the near field and the fracture parameters of the crack are obtained directly by the coefficients of the symplectic series. Compared with other numerical methods, FEDSM has three advantages:(?) No special finite elements and no mesh refinement are needed near the crack tip, the mesh sensitivity is eliminated; (?) The number of unknown variables in the near region is reduced to a very low level. This results in reducing the dimension of the stiffness matrix, and the computational efficiency and accuracy are greatly improved; (?) The fracture parameters are obtained by the coefficients of the symplectic series and no post-processing is needed, the path sensitivity is eliminated. Comparison with published results shows excellent accuracy and agreement of the new FEDSM. Comprehensive numerical examples for parallel cracks, branched cracks and interfacial cracks emanating from an elliptical hole are also presented and the calculation results provide direct theoretical guidance and technical support for the research and development, design and manufacture of composite materials.