Analytical solution for mode III dynamic rupture of standard linear viscoelastic solid with nonlinear damping

Introducing the nonlinear Rayleigh damping into the governing equation of the Mode III dynamic rupture for standard viscoelastic solid, this equation is a partial differential and integral equation. First, eliminating the integral term, a PDE of third-order is obtained. Then, applying the small parameter expansion method, linearized asymptotic governing equation for each order of the small parameter is obtained. Dividing the third-order PDE into an elastic part with known solution, the rest part pertains to viscous effect which is neither a Mathieu equation nor a Hill arte. The WKBJ method is still adopted to solve it analytically.