АНТИПЛОСКА ЗАДАЧА ДЛЯ ОДНОМІРНОГО П’ЄЗОЕЛЕКТРИЧНОГО КВАЗІКРИСТАЛА З МІЖФАЗНОЮ ТРІЩИНОЮ ПІД ВПЛИВОМ ВНУТРІШНЬОГО ЕЛЕКТРИЧНОГО ЗАРЯДУ

Abstract

The paper considers two coupled one-dimensional quasicrystalline half-spaces and a tunnel crack along their interface. The stress-strain state in the vicinity of the electrically conductive faces of the crack is investigated. It is believed that the polarization of materials is directed in the direction of the crack front and in the same direction the arrangement of atoms is quasi-periodic, and perpendicular to the crack front the arrangement of atoms is periodic. Uniformly distributed antiplane phonon and phason shear loads parallel to the crack faces are applied. The electric charge on the crack faces also takes place. A matrix-vector representations for the derivatives of displacement jumps and stresses are constructed through a vector function that is holomorphic in the whole complex plane, except of the crack region. Satisfying the boundary conditions on the crack faces, using matrix-vector representations, the Riemann-Hilbert linear conjugation problem with corresponding conditions at infinity is formed. An analytical solution of this problem is constructed. Analyzing the solution, we obtain analytical expressions for the phonon and phason stresses, the jumps of displacements along the interface between the materials in the crack region that has an electric charge. Numerical analysis of the solution demonstrated the essential influence of the electric charge of the crack to the phonon and phason stress-strain state in the vicinity of the crack. The analysis was performed for a combination of different quasicrystalline compounds. The main results of the solutions, i.e. phonon and phason stresses along the materials interface and the phonon and phason displacement jumps are presented in the graphic form. Conclusions are made regarding the influence of the electric charge of the crack on the behavior of both the crack itself and the material in its vicinity.