Analytic solutions for a finite crack rupturing with type-II at one tip with exciting and decaying processes

Under the action of Rayleigh damping, when the shear stress exerts at the boundary of the crack and causes one tip of the crack to rupture with varying velocity, by the singular perturbation method^[1], we reduce the governing nonlinear partial differential equations to a system of linear ones and solve them by using generalized Fourier series.