13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- For the sake of convenience, the time factor will be suppressed and only the magnitude w(x,y) will be considered. The harmonic displacement field must satisfy the Helmholtz equation [13], (2) where cT=( µ/ ρ) 1/2 is the shear wave speed of the medium. The non-vanishing shear stress components are (3) where µ is the shear modulus of the material. Figure 1. Interacting defects subjected to an incident wave Instead of solving the original interaction problem, single defect problems will be considered. For any individual defect the interaction with other ones will be treated as an unknown wave, pseudo-incident wave. This wave represents the scattered waves from all other defects and will be determined by considering the consistency condition between defects. 2.1 Single crack problem Consider the single defect problem first. For a single crack subjected to a dynamic antiplane loading, the boundary conditions along the crack surfaces are, (4) with τ1 being the shear stress caused by the incident wave. x is the axis along the crack surface and c is the half length of the crack. By making use of Fourier transform, the general solution of the displacement and stress fields in the transformation domain can be expressed as (5) where y is an axis starts from the centre of the crack and is perpendicular to the crack surfaces, s is the Fourier transform parameter of x, f(x) represents the deformation of the crack surfaces, defined by (6) and The solution of the problem can be obtained by using Chebyshev polynomial expansion of f(x) as, (7)
RkJQdWJsaXNoZXIy MjM0NDE=