Symplectic-isogeometric Analysis Coupling Method for Fracture of Multi-material Piezoelectric Quasicrystals


Student thesis: Doctoral Thesis

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Awarding Institution
  • Xinsheng Xu (External person) (External Supervisor)
  • C W LIM (Supervisor)
  • Jian LU (Co-supervisor)
Award date8 May 2023


Quasicrystals (QCs) are special solids between crystals and amorphous materials. Due to their unique microstructures, these materials exhibit excellent wear resistance, corrosion resistance and thermal insulation. Therefore, they are very promising in aviation, aerospace, new energy applications and the likes. In recent years, with further studies in microstructures, researchers believe that QCs also have piezoelectric effects. Compared with traditional piezoelectric materials (crystal materials), piezoelectric quasicrystals (PQCs) can significantly improve the service life and environmental adaptability of microelectromechanical systems (MEMS) devices, which enable MEMS devices to work in more rigorous environments for a long-term service. However, QC materials are brittle and therefore cracks and defects easily occur in QCs, which bring great harm to the equipment. Therefore, it is of great practical significance to investigate the fracture mechanics of QCs and PQCs. Such analyses provide a theoretical framework for the anti-fracture design and safety assessment of equipment made of such materials.

In this work, the anti-plane fracture problems of QCs, PQCs and multiple material junctions are studied. In the analytical research, a Hamiltonian-based fracture model of the QCs/PQCs is built. The analytical solutions of finite sized QCs/PQCs under different crack face conditions are derived. In the numerical research, the analytical solutions obtained is combined with isogeometric analysis (IGA). A symplectic-IGA coupling method for fracture analysis of QCs/PQCs with arbitrary boundary is developed. The fracture parameters and the explicit expressions of the singular physical fields near the crack tip are directly obtained. The main works in this thesis are as follows:

(1) Analytical models for the anti-plane fracture problem of QCs/PQCs are constructed under the Hamiltonian framework. Analytical solutions of the physical fields and explicit expressions of the corresponding fracture parameters are obtained. Firstly, phonon displacement, phason displacement and electric potential in QCs/PQCs are defined as the original variables. The dual vector formed by phonon stress, phason stress and electric displacement is constructed by means of Legendre transformation. Secondly, the Hamiltonian equations (governing equations) are derived with the help of the Hamiltonian variational principle. The higher order partial differential governing equations of the original problem in the traditional system are transformed to a set of lower ordinary differential equations in the Hamiltonian system. Thirdly, the symplectic eigenvalue and eigensolutions can be directly solved through the variable separation method. All physical variables can be analytically expressed by linearly combining symplectic eigensolutions. Lastly, the adjoint symplectic orthogonality relationship and the boundary conditions (displacement condition, stress condition and mixed condition) are applied to solve the undetermined coefficients of the symplectic eigensolutions. Analytical expressions of the physical variables in QCs/PQCs with circular boundary are derived. Explicit expressions of the fracture parameters are obtained simultaneously.

(2) A novel symplectic-IGA coupling method is developed to solve the anti-plane fracture problem of QCs/PQCs by combining symplectic method and IGA. The explicit expressions around the crack tip and the stress intensity factors are obtained directly. Firstly, the IGA equations of the anti-plane problem of QCs/PQCs are formulated by model discretization. Secondly, the entire model is divided into two parts. One is the singular region which contains the crack tip, while the other is the non-singular region far from the crack tip. Thirdly, the symplectic transformation matrix is obtained by substituting coordinates of the large amount of control points in the singular region into the symplectic eigensolutions. Then the symplectic-IGA coupling equations can be derived by combining the symplectic transformation matrix and the IGA equations. The unknowns of the symplectic-IGA coupling equations are the undetermined coefficients of a small amount of the symplectic eigensolutions and the displacement/electric potential in the non-singular region. Finally, the undetermined coefficients of the eigensolutions are derived by solving the symplectic-IGA coupling equations. Explicit expressions of the physical fields in the singular region and the fracture parameters are obtained.

(3) Based on the aforementioned study, a unified analytical model for the anti-plane interfacial fracture problem of the PQC-QC-crystal junction is built. The symplectic-IGA coupling method for this problem is developed. Analytical solutions to all the physical fields near the crack tip and the explicit expression of the fracture parameters are obtained. Firstly, a unified description of the basic equations of the traditional crystals, QCs and PQCs is formulated. The generalized displacement can be described by the displacement in crystals or the phonon/phason displacement in QCs or the phonon/phason displacement and the electric potential in PQCs. The dual variable which is described by stress in crystals or phonon/phason stress in QCs or phonon/phason stress and electric displacement in PQCs are obtained by Legendre transformation. Secondly, the governing equations of the anti-plane problem of the three-material junction are derived by means of the Hamiltonian variation principle. Analytical solutions of the three-material junction under different interface conditions are obtained. Thirdly, the symplectic-IGA coupling equations for anti-plane problem of the three-material junctions are derived by means of the obtained analytical solutions. Finally, the intensity factors of each physical field in three-material junctions with cracks at different interfaces are analyzed. The effects of material constants, geometric configuration, interface conditions and other factors on fracture parameters are reported.

    Research areas

  • Hamiltonian system, Interface crack, Isogeometric analysis, Quasicrystals/piezoelectric quasicrystals, Symplectic-isogeometric analysis coupling method