An analytical approach for the mixed-mode crack in linear viscoelastic media

• We propose an analytical approach for the cracked linear viscoelastic media.
• Analytical time-dependent solutions of stresses and displacements are obtained.
• Analytical expressions of SIFs and J-integral are obtained simultaneously.
• The independent variables are uncoupled using Hamiltonian formulation.
• Solutions to Hamiltonian equations are symplectic orthogonal guaranteeing convergence.

In this paper, an analytical approach is developed for the fracture analysis of linear viscoelastic media. By the Laplace transform, the governing equations for the time domain (t-domain) are changed into frequency domain (s-domain). Then, a Hamiltonian system is established by introducing the dual variables of displacements and energy variational principle. In the framework of symplectic mathematics, the unknown vector consisting of displacements and stresses is expanded in terms of symplectic eigensolutions whose coefficients can be determined from the outer boundary conditions. Then t-domain solution is finally obtained by inverse Laplace transform and exact forms of fracture parameters including stress intensity factor (SIF) and J-integral are derived simultaneously. Numerical examples are provided to verify the validity of the present method. A parametric study of viscoelastic parameters and outer boundary conditions is carried out also.